Finite Group Databasebeta

← #17980227

Subgroups — #17980227

Order 4352.

1748 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545693C214yesmaximal, complemented
2#14545688C214yesmaximal, complemented
3#14545684C214yesmaximal, complemented
4#14545682C211yesmaximal, characteristic, complemented
5#14545680C211yesmaximal, characteristic, complemented
6#14545679C211yesmaximal, characteristic, complemented
7#14545677C211yesmaximal, characteristic, complemented
8#14545579C214yesmaximal, complemented
9#14545572C214yesmaximal, complemented
10#14545473C214yesmaximal
11#14545460C211yesmaximal, characteristic
12C2×#11763825C211yesmaximal, characteristic
13C2×#11763361C211yesmaximal, characteristic
14#11763861C2218yescomplemented
15#11763861C2218yescomplemented
16#11763860C2214yescomplemented
17#11763860C2214yescomplemented
18#11763859C2214yescomplemented
19#11763859C2214yescomplemented
20#11763859C2218yescomplemented
21#11763858C2214yescomplemented
22#11763858C2214yescomplemented
23#11763858C2218yescomplemented
24Q8×#3624C2214yescomplemented
25Q8×#3624C2214yescomplemented
26#11763833C2218yes—
27#11763833C2218yes—
28#11763830C2214yes—
29#11763830C2212yes—
30#11763829C2214yes—
31#11763829C2212yes—
32#11763826C2214yes—
33#11763826C2212yes—
34#11763825C2211yescharacteristic
35#11763825C2211yescharacteristic
36#11763825C2214yes—
37#11763825C2211yescharacteristic
38#11763825C2211yescharacteristic
39#11763825C2212yes—
40#11763825C2212yes—
41#11763825C2212yes—
42#11763815C2218yes—
43#11763812C2212yes—
44C2×C4⋊#3624C2212yes—
45#11763779C2212yes—
46C2×C22⋊#3624C2212yes—
47#11763773C2218yes—
48C4×#63247C2212yes—
49C2×C4×#3624C2212yes—
50#11763361C412yes—
51#11763361C412yes—
52#11763361C412yes—
53#11763361C412yes—
54#11763361C2212yes—
55#11763361C2212yes—
56#11763361C2212yes—
57#11763361C2212yes—
58C2×#10587574C411yescharacteristic
59C2×#10587630C411yescharacteristic
60C2×#10587575C412yes—
61C2×#10587610C411yescharacteristic
62C2×#10587564C412yes—
63C2×Q8×Dic17C2211yescharacteristic
64C2×#10587573C2211yescharacteristic
65C2×#10587573C2212yes—
66C2×#10587563C2212yes—
67C22×C4×Dic17C2211yescharacteristic
68C2×C17×C22⋊Q8C411yescharacteristic
69C22×Q8×C17C2×C411yescharacteristic
70C23×C4×C17C2×C411yescharacteristic
71C23×Dic17C2×C411yescharacteristic
72C23×Dic17C2311yescharacteristic
73C22×Dic34C2×C411yescharacteristic
74C22×Dic34C2×C412yes—
75C2×#63247C2314yes—
76C22×#3624C2314yes—
77C17×C22⋊Q8C2×C412yes—
78C17×C22⋊Q8C2×C412yes—
79C17×C22⋊Q8C2×C412yes—
80C17×C22⋊Q8C2×C412yes—
81C2×C4⋊C4×C17C2×C412yes—
82C2×C4⋊C4×C17C2×C411yescharacteristic
83C2×C22⋊C4×C17C2×C412yes—
84C2×C22⋊Dic17C2×C412yes—
85Q8×Dic17C2×C412yes—
86Q8×Dic17C2×C412yes—
87Q8×Dic17C2312yes—
88Q8×Dic17C2312yes—
89#10587630C2×C412yes—
90#10587630C2×C412yes—
91#10587630C2×C412yes—
92#10587630C2×C412yes—
93C2×C4⋊Dic17C2×C412yes—
94C2×C4⋊Dic17C2×C411yescharacteristic
95#10587610C2×C412yes—
96#10587610C2×C412yes—
97#10587610C2×C412yes—
98#10587610C2×C412yes—
99C2×#63225C2×C412yes—
100C2×#63225C2×C412yes—
101C2×#63225C2×C412yes—
102C2×#63225C2×C412yes—
103C2×#63225C2×C412yes—
104C2×C4×Dic17C2×C411yescharacteristic
105C2×C4×Dic17C2×C411yescharacteristic
106C2×C4×Dic17C2×C411yescharacteristic
107C2×C4×Dic17C2×C411yescharacteristic
108C2×C4×Dic17C2×C411yescharacteristic
109C2×C4×Dic17C2×C411yescharacteristic
110C2×C4×Dic17C2311yescharacteristic
111C2×C4×Dic17C2311yescharacteristic
112C2×C4×Dic17C2311yescharacteristic
113C2×C4×Dic17C2311yescharacteristic
114C2×C4×Dic17C2311yescharacteristic
115C2×C4×Dic17C2311yescharacteristic
116C2×C4×Dic17C2312yes—
117C2×C4×Dic17C2312yes—
118#10587575C2×C414yes—
119#10587575C2×C414yes—
120#10587575C2×C414yes—
121#10587575C2×C414yes—
122#10587574C2×C412yes—
123#10587574C2×C412yes—
124#10587574C2×C412yes—
125#10587574C2×C412yes—
126#10587573C2×C412yes—
127#10587573C2×C412yes—
128#10587573C2×C414yes—
129#10587573C2×C414yes—
130#10587573C2312yes—
131#10587573C2312yes—
132#10587573C2314yes—
133#10587573C2314yes—
134#10587564C2×C414yes—
135#10587564C2×C414yes—
136#10587564C2×C414yes—
137#10587564C2×C414yes—
138#10587563C2×C414yes—
139#10587563C2×C414yes—
140#10587563C2314yes—
141#10587563C2314yes—
142C22⋊#3624C2312yes—
143C22⋊#3624C2312yes—
144C22⋊#3624C2314yes—
145C22⋊#3624C2312yes—
146C22⋊#3624C2312yes—
147C22⋊#3624C2314yes—
148#10587544C2314yes—
149#10587544C2318yes—
150#10587544C2318yes—
151#10587542C2314yes—
152#10587542C2318yes—
153#10587542C2318yes—
154C4⋊#3624C2312yes—
155C4⋊#3624C2312yes—
156C4⋊#3624C2314yes—
157C4⋊#3624C2312yes—
158C4⋊#3624C2312yes—
159C4⋊#3624C2318yes—
160C4⋊#3624C2318yes—
161C4×#3624C2312yes—
162C4×#3624C2314yes—
163C4×#3624C2312yes—
164C4×#3624C2312yes—
165C4×#3624C2312yes—
166C4×#3624C2318yes—
167C4×#3624C2318yes—
168C24×C17C22×C411yescharacteristic
169C2×Q8×C17—24no—
170C2×Q8×C17C22×C412yes—
171C2×Q8×C17C22×C412yes—
172C22×C4×C17—24no—
173C22×C4×C17C22×C411yescharacteristic
174C22×C4×C17C22×C411yescharacteristic
175C22×C4×C17C22×C411yescharacteristic
176C22×C4×C17C22×C411yescharacteristic
177C22×C4×C17C22×C411yescharacteristic
178C22×C4×C17C22×C412yes—
179C22×C4×C17C22×C412yes—
180C22×C4×C17C22×C411yescharacteristic
181C22×Dic17—24no—
182C22×Dic17—42no—
183C22×Dic17C22×C411yescharacteristic
184C22×Dic17C22×C412yes—
185C22×Dic17C22×C411yescharacteristic
186C22×Dic17C22×C411yescharacteristic
187C22×Dic17C22×C411yescharacteristic
188C22×Dic17C22×C411yescharacteristic
189C22×Dic17C22×C411yescharacteristic
190C22×Dic17C22×C412yes—
191C22×Dic17C22×C411yescharacteristic
192C22×Dic17C22×C411yescharacteristic
193C22×Dic17C22×C411yescharacteristic
194C22×Dic17C2411yescharacteristic
195C22×Dic17C2411yescharacteristic
196C22×Dic17C2411yescharacteristic
197C2×Dic34—24no—
198C2×Dic34—44no—
199C2×Dic34C22×C414yes—
200C2×Dic34C22×C412yes—
201C2×Dic34C22×C412yes—
202C2×Dic34C22×C414yes—
203#63247—28no—
204C2×#3624—28no—
205C2×#3624C2414yes—
206C2×#3624C2414yes—
207C2×#3624C2414yes—
208C2×#3624C2414yes—
209C4⋊C4×C17C22×C412yes—
210C4⋊C4×C17C22×C412yes—
211C4⋊C4×C17C22×C414yes—
212C4⋊C4×C17C22×C414yes—
213C22⋊C4×C17C22×C414yes—
214C22⋊C4×C17C22×C414yes—
215C22⋊Dic17C22×C414yes—
216C22⋊Dic17C22×C414yes—
217C4⋊Dic17C22×C414yes—
218C4⋊Dic17C22×C414yes—
219C4⋊Dic17C22×C412yes—
220C4⋊Dic17C22×C412yes—
221#63225C22×C414yes—
222#63225C22×C414yes—
223#63225C22×C414yes—
224#63225C22×C414yes—
225#63225C22×C414yes—
226#63225C22×C414yes—
227#63225C22×C414yes—
228#63225C22×C414yes—
229#63225C22×C414yes—
230#63225C22×C414yes—
231C4×Dic17C22×C414yes—
232C4×Dic17C22×C414yes—
233C4×Dic17C22×C411yescharacteristic
234C4×Dic17C22×C411yescharacteristic
235C4×Dic17C22×C411yescharacteristic
236C4×Dic17C22×C411yescharacteristic
237C4×Dic17C22×C411yescharacteristic
238C4×Dic17C22×C411yescharacteristic
239C4×Dic17C22×C411yescharacteristic
240C4×Dic17C22×C411yescharacteristic
241C4×Dic17C22×C411yescharacteristic
242C4×Dic17C22×C411yescharacteristic
243C4×Dic17C22×C411yescharacteristic
244C4×Dic17C22×C411yescharacteristic
245C4×Dic17C2411yescharacteristic
246C4×Dic17C2411yescharacteristic
247C4×Dic17C2411yescharacteristic
248C4×Dic17C2414yes—
249C4×Dic17C2414yes—
250C4×Dic17C2411yescharacteristic
251#60250—171nomaximal, Sylow
252C23×C17—22no—
253C23×C17—22no—
254C23×C17—41no—
255C23×C17C23×C411yescharacteristic
256C23×C17C23×C411yescharacteristic
257C23×C17C23×C411yescharacteristic
258Q8×C17—24no—
259Q8×C17—24no—
260C2×C4×C17—22no—
261C2×C4×C17—24no—
262C2×C4×C17—22no—
263C2×C4×C17—22no—
264C2×C4×C17—22no—
265C2×C4×C17—22no—
266C2×C4×C17—42no—
267C2×C4×C17#13114yes—
268C2×C4×C17C23×C414yes—
269C2×C4×C17C23×C414yes—
270C2×C4×C17C23×C411yescharacteristic
271C2×C4×C17C23×C411yescharacteristic
272C2×C4×C17C23×C411yescharacteristic
273C2×C4×C17C23×C411yescharacteristic
274C2×Dic17—22no—
275C2×Dic17—24no—
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—22no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—42no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17C23×C414yes—
288C2×Dic17C23×C414yes—
289C2×Dic17C23×C411yescharacteristic
290C2×Dic17C23×C411yescharacteristic
291C2×Dic17C23×C411yescharacteristic
292C2×Dic17C23×C411yescharacteristic
293C2×Dic17C23×C411yescharacteristic
294C2×Dic17C23×C411yescharacteristic
295C2×Dic17C23×C411yescharacteristic
296C2×Dic172−1+414yes—
297C2×Dic17C2511yescharacteristic
298Dic34—24no—
299Dic34—24no—
300Dic34—28no—
301Dic34—44no—
302#3624—24no—
303#3624—24no—
304#3624—24no—
305#3624—24no—
306C2×#533—171no—
307#2986—174no—
308#2985—174no—
309#2984—174no—
310#2983—171no—
311#2983—171no—
312#2982—171no—
313#2982—171no—
314#2962—174no—
315#2960—174no—
316#2924—174no—
317#2920—171no—
318C2×#431—171no—
319C22×C17—21no—
320C22×C17—21no—
321C22×C17—21no—
322C22×C17—21no—
323C22×C17—21no—
324C22×C17—21no—
325C22×C17—21no—
326C22×C17—44no—
327C22×C17#51912yes—
328C22×C17C2×#13112yes—
329C22×C17C24×C411yescharacteristic
330C4×C17—21no—
331C4×C17—21no—
332C4×C17—24no—
333C4×C17—24no—
334C4×C17C2×#13112yes—
335C4×C17C2×#13112yes—
336Dic17—21no—
337Dic17—21no—
338Dic17—24no—
339Dic17—24no—
340Dic17—21no—
341Dic17—21no—
342Dic17—21no—
343Dic17#51912yes—
344Dic17#56812yes—
345Dic17C2×2−1+412yes—
346#533—172no—
347#533—172no—
348#533—172no—
349#533—172no—
350#533—172no—
351#533—172no—
352#533—172no—
353#533—172no—
354C2×C4⋊Q8—171no—
355C2×C4⋊Q8—171no—
356C2×#125—172no—
357C2×C22⋊Q8—171no—
358C2×C22⋊Q8—172no—
359C2×C22⋊Q8#59511yescharacteristic
360C2×C4×Q8—171no—
361C2×C4×Q8—171no—
362C2×C4×Q8—172no—
363C2×#117—172no—
364C22×C42—171no—
365#446—174no—
366#446—174no—
367#446—178no—
368#446—174no—
369#446—178no—
370#446—174no—
371#446—178no—
372C8×Q8—174no—
373C8×Q8—174no—
374C8×Q8—178no—
375C8×Q8—174no—
376C8×Q8—174no—
377#433—178no—
378#433—178no—
379#432—172no—
380#432—172no—
381#432—174no—
382#432—174no—
383#431—171no—
384#431—171no—
385#431—172no—
386#431—172no—
387#431—172no—
388#431—171no—
389#431—171no—
390#431—172no—
391#431—174no—
392#431—174no—
393#424—178no—
394#423—172no—
395C2×C4⋊C8—172no—
396#407—172no—
397C2×C22⋊C8—172no—
398#405—178no—
399C4×M16—172no—
400C2×C4×C8—172no—
401C2×C17—22no—
402C2×C17—22no—
403C2×C17—41no—
404C2×C17C2×#51911yescharacteristic
405C2×C17C22×#13111yescharacteristic
406C2×C17C2×#56811yescharacteristic
407C22×Q8—171no—
408C22×Q8—172no—
409C22×Q8C2×#59511yescharacteristic
410C23×C4—171no—
411C23×C4—171no—
412C23×C4C2×#59511yescharacteristic
413C2×M16—174no—
414C22×C8—174no—
415C4⋊Q8—172no—
416C4⋊Q8—172no—
417C4⋊Q8—172no—
418C4⋊Q8—172no—
419C4⋊Q8—172no—
420C4⋊Q8—172no—
421C4⋊Q8—172no—
422C4⋊Q8—172no—
423#125—174no—
424#125—174no—
425#125—174no—
426#125—174no—
427C22⋊Q8—172no—
428C22⋊Q8—172no—
429C22⋊Q8—172no—
430C22⋊Q8—172no—
431C22⋊Q8—174no—
432C22⋊Q8—174no—
433C22⋊Q8—174no—
434C22⋊Q8—174no—
435C22⋊Q8C2×#59512yes—
436C22⋊Q8C2×#59512yes—
437C22⋊Q8C2×#59512yes—
438C22⋊Q8C2×#59512yes—
439C4×Q8—172no—
440C4×Q8—172no—
441C4×Q8—172no—
442C4×Q8—172no—
443C4×Q8—172no—
444C4×Q8—172no—
445C4×Q8—174no—
446C4×Q8—174no—
447C4×Q8—172no—
448C4×Q8—172no—
449C4×Q8—174no—
450C4×Q8—174no—
451#117—174no—
452#117—174no—
453#117—174no—
454#117—174no—
455C2×C4⋊C4—172no—
456C2×C4⋊C4—172no—
457C2×C4⋊C4—172no—
458C2×C4⋊C4—172no—
459C2×C4⋊C4—172no—
460C2×C4⋊C4—172no—
461C2×C4⋊C4—171no—
462C2×C4⋊C4C2×#59512yes—
463C2×C4⋊C4C2×#59511yescharacteristic
464C2×C22⋊C4—172no—
465C2×C22⋊C4C2×#59512yes—
466C2×C42—172no—
467C2×C42—172no—
468C2×C42—171no—
469C2×C42—171no—
470C2×C42—171no—
471C2×C42—171no—
472C2×C42—171no—
473C2×C42—171no—
474C2×C42—171no—
475C2×C42—171no—
476C2×C42—171no—
477C2×C42—171no—
478C2×C42—171no—
479C2×C42—171no—
480C4⋊C8—172no—
481C4⋊C8—172no—
482C4⋊C8—174no—
483C4⋊C8—172no—
484C4⋊C8—172no—
485C4⋊C8—178no—
486C4⋊C8—178no—
487C4⋊C8—174no—
488C4⋊C8—178no—
489C4⋊C8—178no—
490C22⋊C8—172no—
491C22⋊C8—172no—
492C22⋊C8—174no—
493C22⋊C8—172no—
494C22⋊C8—172no—
495C22⋊C8—174no—
496#97—174no—
497#97—178no—
498#97—178no—
499C4×C8—172no—
500C4×C8—174no—
501C4×C8—172no—
502C4×C8—172no—
503C4×C8—172no—
504C4×C8—178no—
505C4×C8—178no—
506C17#6025011yescharacteristic, complemented, Sylow
507C24C22×#59511yescharacteristic
508C2×Q8—24no—
509C2×Q8—174no—
510C2×Q8—172no—
511C2×Q8—172no—
512C2×Q8—174no—
513C2×Q8—344no—
514C2×Q8—684no—
515C2×Q8C22×#59512yes—
516C2×Q8C22×#59512yes—
517C22×C4—24no—
518C22×C4—171no—
519C22×C4—171no—
520C22×C4—171no—
521C22×C4—171no—
522C22×C4—171no—
523C22×C4—171no—
524C22×C4—171no—
525C22×C4—172no—
526C22×C4—171no—
527C22×C4—171no—
528C22×C4—171no—
529C22×C4—171no—
530C22×C4—171no—
531C22×C4—172no—
532C22×C4—344no—
533C22×C4—682no—
534C22×C4C22×#59511yescharacteristic
535C22×C4C22×#59511yescharacteristic
536C22×C4C22×#59511yescharacteristic
537C22×C4C22×#59511yescharacteristic
538C22×C4C22×#59511yescharacteristic
539C22×C4C22×#59512yes—
540C22×C4C22×#59512yes—
541C22×C4C22×#59511yescharacteristic
542M16—348no—
543C2×C8—174no—
544C2×C8—174no—
545C2×C8—174no—
546C2×C8—174no—
547C2×C8—348no—
548C4⋊C4—174no—
549C4⋊C4—174no—
550C4⋊C4—174no—
551C4⋊C4—174no—
552C4⋊C4—174no—
553C4⋊C4—174no—
554C4⋊C4—174no—
555C4⋊C4—174no—
556C4⋊C4—174no—
557C4⋊C4—174no—
558C4⋊C4—174no—
559C4⋊C4—174no—
560C4⋊C4—172no—
561C4⋊C4—172no—
562C4⋊C4C22×#59514yes—
563C4⋊C4C22×#59514yes—
564C4⋊C4C22×#59512yes—
565C4⋊C4C22×#59512yes—
566C22⋊C4—174no—
567C22⋊C4—174no—
568C22⋊C4C22×#59514yes—
569C22⋊C4C22×#59514yes—
570C42—171no—
571C42—171no—
572C42—171no—
573C42—174no—
574C42—174no—
575C42—171no—
576C42—171no—
577C42—171no—
578C42—171no—
579C42—171no—
580C42—171no—
581C42—171no—
582C42—171no—
583C42—171no—
584C42—171no—
585C42—171no—
586C42—171no—
587C42—171no—
588C42—174no—
589C42—174no—
590C23—22no—
591C23—22no—
592C23—41no—
593C23C23×#59511yescharacteristic
594C23C23×#59511yescharacteristic
595C23C23×#59511yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—348no—
599Q8—344no—
600Q8—344no—
601Q8—684no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—174no—
610C2×C4—171no—
611C2×C4—171no—
612C2×C4—171no—
613C2×C4—171no—
614C2×C4—171no—
615C2×C4—171no—
616C2×C4—171no—
617C2×C4—171no—
618C2×C4—174no—
619C2×C4—174no—
620C2×C4—342no—
621C2×C4—342no—
622C2×C4—342no—
623C2×C4—342no—
624C2×C4—342no—
625C2×C4—342no—
626C2×C4—342no—
627C2×C4—342no—
628C2×C4—344no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—682no—
633C2×C4#1058767814yes—
634C2×C4C23×#59511yescharacteristic
635C2×C4C23×#59511yescharacteristic
636C2×C4C23×#59514yes—
637C2×C4C23×#59514yes—
638C2×C4C23×#59511yescharacteristic
639C2×C4C23×#59511yescharacteristic
640C8—344no—
641C8—344no—
642C8—344no—
643C8—344no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—44no—
652C22#1176387712yes—
653C22C2×#1058767812yes—
654C22C24×#59511yescharacteristic
655C4—21no—
656C4—21no—
657C4—24no—
658C4—24no—
659C4—172no—
660C4—172no—
661C4—172no—
662C4—341no—
663C4—341no—
664C4—344no—
665C4—341no—
666C4—341no—
667C4—341no—
668C4—344no—
669C4C2×#1058767812yes—
670C4C2×#1058767812yes—
671C2—22no—
672C2—22no—
673C2—41no—
674C2C2×#1176387711yescharacteristic
675C2C22×#1058767811yescharacteristic
676C2D17⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC2×C17×C22⋊Q8C4
prime radicalC24×C17C22×C4
cosocleC2×Dic17C25
hypercenterC2×C22⋊Q8#595
hypocenterC17#60250
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #14545693C2semidirect
2. #14545688C2semidirect
3. #14545684C2semidirect
4. #14545682C2semidirect
5. #14545680C2semidirect
6. #14545679C2semidirect
7. #14545677C2semidirect
8. #14545579C2semidirect
9. #14545572C2semidirect
14. #11763861C22semidirect
15. #11763861C22semidirect
16. #11763860C22semidirect
17. #11763860C22semidirect
18. #11763859C22semidirect
19. #11763859C22semidirect
20. #11763859C22semidirect
21. #11763858C22semidirect
22. #11763858C22semidirect
23. #11763858C22semidirect
24. Q8×#3624C22semidirect
25. Q8×#3624C22semidirect
506. C17#60250semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145456932
#145456882
#145456842
#145456822
#145456802
#145456792
#145456772
#145455792
#145455722
#145454732
#145454602
C2×#117638252
C2×#117633612
#6025017