Finite Group Databasebeta

← #17976337

Subgroups — #17976337

Order 4352.

840 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763860C211yesmaximal, characteristic, complemented
2C2×#11763812C211yesmaximal, characteristic
3#14544958C211yesmaximal, characteristic, complemented
4#14544886C211yesmaximal, characteristic, complemented
5#14544886C211yesmaximal, characteristic, complemented
6#14544881C211yesmaximal, characteristic, complemented
7#14544841C211yesmaximal, characteristic, complemented
8#14544837C211yesmaximal, characteristic, complemented
9#14544731C211yesmaximal, characteristic
10#14544731C211yesmaximal, characteristic
11#14544584C211yesmaximal, characteristic, complemented
12#14544526C211yesmaximal, characteristic
13#14544526C211yesmaximal, characteristic
14#14544510C211yesmaximal, characteristic
15C22⋊Q8×Dic17C211yesmaximal, characteristic
16C22×#63247C2211yescharacteristic
17C22×#63247C2211yescharacteristic
18#11763860—24no—
19#11763812—24no—
20C2×#10587544C2211yescharacteristic
21C2×#10587544C2211yescharacteristic
22C2×#10587544C2211yescharacteristic
23C2×#10587544C2211yescharacteristic
24C2×C4⋊#3624C2211yescharacteristic
25C2×C4⋊#3624C2211yescharacteristic
26C2×C4⋊#3624C2211yescharacteristic
27C2×C4⋊#3624C2211yescharacteristic
28C2×C22⋊#3624C2211yescharacteristic
29C2×C22⋊#3624C2211yescharacteristic
30C2×#10587542C2211yescharacteristic
31C2×#10587542C2211yescharacteristic
32C2×#10587542C2211yescharacteristic
33C2×#10587542C2211yescharacteristic
34C2×C4×#3624C2211yescharacteristic
35C2×C4×#3624C2211yescharacteristic
36#11763675C2211yescharacteristic
37#11763675C2211yescharacteristic
38#11763675C2211yescharacteristic
39#11763675C2211yescharacteristic
40#11763675C2211yescharacteristic
41#11763675C2211yescharacteristic
42#11763675C2211yescharacteristic
43#11763675C2211yescharacteristic
44#11763675C2211yescharacteristic
45#11763675C2211yescharacteristic
46C2×Q8×Dic17C2211yescharacteristic
47C22×C4×Dic17C2211yescharacteristic
48#11762716C411yescharacteristic
49#11762708C411yescharacteristic
50#11762695C411yescharacteristic
51#11762695C411yescharacteristic
52#11762684C411yescharacteristic
53#11762684C411yescharacteristic
54#11762669C411yescharacteristic
55C4⋊C4×Dic17C2211yescharacteristic
56C4⋊C4×Dic17C2211yescharacteristic
57C4⋊C4×Dic17C2211yescharacteristic
58C22⋊C4×Dic17C2211yescharacteristic
59C22⋊C4×Dic17C2211yescharacteristic
60C2×C17×C22⋊Q8C411yescharacteristic
61C22×Q8×C17C2×C411yescharacteristic
62C23×C4×C17C2×C411yescharacteristic
63C23×Dic17C2×C411yescharacteristic
64C23×Dic17C2311yescharacteristic
65C2×#63247—22no—
66C2×#63247—22no—
67C2×#63247—22no—
68C2×#63247—22no—
69C2×#63247—41no—
70C2×#63247—41no—
71C22×#3624C2311yescharacteristic
72C22×#3624C2311yescharacteristic
73C22×#3624C2311yescharacteristic
74C22×#3624C2311yescharacteristic
75C22×#3624C2311yescharacteristic
76C22×#3624C2311yescharacteristic
77C22×#3624C2311yescharacteristic
78C22×#3624C2311yescharacteristic
79C17×C22⋊Q8—24no—
80C2×C4⋊C4×C17C2×C411yescharacteristic
81C2×C4⋊C4×C17C2×C411yescharacteristic
82C2×C4⋊C4×C17C2×C411yescharacteristic
83C2×C22⋊C4×C17C2×C411yescharacteristic
84C2×C22⋊C4×C17C2×C411yescharacteristic
85C2×C22⋊Dic17C2×C411yescharacteristic
86C2×C22⋊Dic17C2×C411yescharacteristic
87Q8×Dic17—22no—
88Q8×Dic17—22no—
89C2×C4⋊Dic17C2×C411yescharacteristic
90C2×C4⋊Dic17C2×C411yescharacteristic
91C2×C4⋊Dic17C2×C411yescharacteristic
92C2×C4⋊Dic17C2×C411yescharacteristic
93C2×C4⋊Dic17C2×C411yescharacteristic
94C2×C4⋊Dic17C2×C411yescharacteristic
95C2×C4×Dic17—22no—
96C2×C4×Dic17—22no—
97C2×C4×Dic17C2×C411yescharacteristic
98C2×C4×Dic17C2×C411yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2311yescharacteristic
104C2×C4×Dic17C2311yescharacteristic
105C22⋊#3624—22no—
106C22⋊#3624Q814yescomplemented
107#10587544—22no—
108#10587544—22no—
109#10587544—41no—
110#10587544D414yes—
111#10587542—22no—
112#10587542—22no—
113#10587542—22no—
114#10587542—41no—
115C4⋊#3624—22no—
116C4⋊#3624—22no—
117C4⋊#3624—22no—
118C4⋊#3624—41no—
119C4×#3624—22no—
120C4×#3624—41no—
121#10587502C2×C411yescharacteristic
122#10587502C2×C411yescharacteristic
123#10587502C2×C411yescharacteristic
124#10587502C2×C411yescharacteristic
125#10587502C2×C411yescharacteristic
126#10587502C2×C411yescharacteristic
127#10587502C2×C411yescharacteristic
128#10587502C2×C411yescharacteristic
129#10587502C2×C411yescharacteristic
130#10587502C2×C411yescharacteristic
131C24×C17C22×C411yescharacteristic
132C2×Q8×C17—22no—
133C2×Q8×C17—22no—
134C2×Q8×C17—41no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—41no—
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17C22×C411yescharacteristic
142C22×C4×C17C22×C411yescharacteristic
143C22×C4×C17C22×C411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—41no—
150C22×Dic17—41no—
151C22×Dic17C22×C411yescharacteristic
152C22×Dic17C22×C411yescharacteristic
153C22×Dic17C22×C411yescharacteristic
154C22×Dic17C22×C411yescharacteristic
155C22×Dic17C22×C411yescharacteristic
156C22×Dic17C22×C411yescharacteristic
157C22×Dic17C22×C411yescharacteristic
158C22×Dic17C2×Q812yes—
159C22×Dic17#4112yes—
160C22×Dic17C2411yescharacteristic
161#63247—44no—
162#63247—44no—
163C2×#3624—21no—
164C2×#3624—21no—
165C2×#3624—21no—
166C2×#3624—21no—
167C2×#3624—21no—
168C2×#3624—21no—
169C2×#3624—21no—
170C2×#3624—21no—
171C2×#3624—21no—
172C2×#3624—21no—
173C2×#3624—21no—
174C2×#3624—21no—
175C2×#3624—21no—
176C2×#3624—21no—
177C2×#3624—21no—
178C2×#3624—21no—
179C2×#3624—21no—
180C2×#3624—21no—
181C2×#3624—21no—
182C2×#3624—21no—
183C2×#3624C2×D412yes—
184C2×#3624C2×D412yes—
185C2×#3624C2×Q812yes—
186C2×#3624C2×Q812yes—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—22no—
189C4⋊C4×C17M1614yescomplemented
190C22⋊C4×C17—22no—
191C22⋊C4×C17—22no—
192C22⋊Dic17—41no—
193C22⋊Dic17—41no—
194C4⋊Dic17—22no—
195C4⋊Dic17—22no—
196C4⋊Dic17—22no—
197C4⋊Dic17—41no—
198C4⋊Dic17—41no—
199C4⋊Dic17—41no—
200C4×Dic17—22no—
201C4×Dic17—22no—
202C4×Dic17—21no—
203C4×Dic17—21no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17C2×D412yes—
213C4×Dic17#4112yes—
214#22955—171nomaximal, Sylow
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17C4×Q812yes—
222C23×C17C23×C411yescharacteristic
223Q8×C17—44no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—44no—
239C2×C4×C17C4×D412yes—
240C2×C4×C17C2×M1612yes—
241C2×C4×C17C2×M1612yes—
242C2×C4×C17C2×M1612yes—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—44no—
267C2×Dic17C4×D412yes—
268C2×Dic17C4×Q812yes—
269C2×Dic17C22×D411yescharacteristic
270C2×Dic17C22×Q811yescharacteristic
271C2×Dic17C2×#4111yescharacteristic
272C2×Dic17C2×#4111yescharacteristic
273C2×Dic17H(5,𝔽2)11yescharacteristic
274C2×Dic172−1+411yescharacteristic
275#3624—22no—
276#3624—22no—
277#3624—22no—
278#3624—22no—
279#3624—41no—
280#3624—41no—
281#3624—41no—
282#3624—41no—
283C2×#446—171no—
284C2×#423—171no—
285C4×C22⋊Q8—171no—
286#1985—171no—
287#1940—171no—
288#1940—171no—
289#1939—171no—
290#1917—171no—
291#1916—171no—
292#1851—171no—
293#1851—171no—
294#1771—171no—
295#1753—171no—
296#1753—171no—
297#1749—171no—
298C22×C17—22no—
299C22×C17—22no—
300C22×C17—22no—
301C22×C17—41no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17C2×C4×D411yescharacteristic
306C22×C17C2×C4×Q811yescharacteristic
307C22×C17#52011yescharacteristic
308C22×C17C22×M1611yescharacteristic
309C22×C17#56811yescharacteristic
310C22×C17#56811yescharacteristic
311C22×C17#56811yescharacteristic
312C4×C17—22no—
313C4×C17—22no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—41no—
318Dic17—22no—
319Dic17—22no—
320Dic17—41no—
321Dic17—41no—
322Dic17—41no—
323Dic17—41no—
324Dic17—41no—
325Dic17#54611yescharacteristic
326Dic17D4×Q811yescharacteristic
327Dic17#55211yescharacteristic
328Dic17#55411yescharacteristic
329C22×M16—171no—
330C22×M16—171no—
331C2×C22⋊Q8#59511yescharacteristic
332C2×C4×Q8—171no—
333C22×C42—171no—
334#446—344no—
335#423—344no—
336C2×C4⋊C8—171no—
337C2×C4⋊C8—171no—
338C2×C4⋊C8—171no—
339C2×C4⋊C8—171no—
340C2×C4⋊C8—171no—
341C2×C4⋊C8—171no—
342C2×C4⋊C8—171no—
343C2×C4⋊C8—171no—
344C2×C22⋊C8—171no—
345C2×C22⋊C8—171no—
346C2×#97—171no—
347C2×#97—171no—
348C2×#97—171no—
349C2×#97—171no—
350C2×C4×C8—171no—
351C2×C4×C8—171no—
352#391—171no—
353#389—171no—
354#387—171no—
355#387—171no—
356#385—171no—
357#385—171no—
358#380—171no—
359C4×C4⋊C4—171no—
360C4×C4⋊C4—171no—
361C4×C4⋊C4—171no—
362C4×C22⋊C4—171no—
363C4×C22⋊C4—171no—
364#336—171no—
365#336—171no—
366#336—171no—
367#336—171no—
368#336—171no—
369#336—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374C2×C17—42no—
375C2×C17#236011yescharacteristic
376C2×C17#293011yescharacteristic
377C2×C17D4×M1611yescharacteristic
378C2×C17#296111yescharacteristic
379C2×C17Q8×M1611yescharacteristic
380C2×C17#298911yescharacteristic
381C2×C17#299511yescharacteristic
382C22×Q8C2×#59511yescharacteristic
383C23×C4—171no—
384C23×C4—171no—
385C23×C4C2×#59511yescharacteristic
386C2×M16—342no—
387C2×M16—342no—
388C2×M16—342no—
389C2×M16—342no—
390C2×M16—681no—
391C2×M16—681no—
392C22×C8—171no—
393C22×C8—171no—
394C22×C8—171no—
395C22×C8—171no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C22⋊Q8—24no—
401C4×Q8—342no—
402C4×Q8—342no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4C2×#59511yescharacteristic
410C2×C4⋊C4C2×#59511yescharacteristic
411C2×C4⋊C4C2×#59511yescharacteristic
412C2×C22⋊C4—171no—
413C2×C22⋊C4—171no—
414C2×C22⋊C4C2×#59511yescharacteristic
415C2×C22⋊C4C2×#59511yescharacteristic
416C2×C42—171no—
417C2×C42—171no—
418C2×C42—171no—
419C2×C42—171no—
420C2×C42—171no—
421C2×C42—171no—
422C2×C42—171no—
423C2×C42—171no—
424C2×C42—342no—
425C2×C42—342no—
426C4⋊C8—174no—
427C4⋊C8—342no—
428C4⋊C8—342no—
429C4⋊C8—342no—
430C4⋊C8—342no—
431C4⋊C8—342no—
432C4⋊C8—681no—
433C4⋊C8—681no—
434C22⋊C8—174no—
435C22⋊C8—342no—
436#97—342no—
437#97—342no—
438#97—342no—
439#97—681no—
440C4×C8—342no—
441C4×C8—681no—
442#95—171no—
443#95—171no—
444#95—171no—
445#95—171no—
446#95—171no—
447#95—171no—
448#95—171no—
449#95—171no—
450#95—171no—
451#95—171no—
452C17#2295511yescharacteristic, complemented, Sylow
453C24C22×#59511yescharacteristic
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—41no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—171no—
461C22×C4—172no—
462C22×C4—172no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—171no—
468C22×C4—171no—
469C22×C4—171no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—342no—
475C22×C4—681no—
476C22×C4—681no—
477C22×C4C22×#59511yescharacteristic
478C22×C4C22×#59511yescharacteristic
479C22×C4C22×#59511yescharacteristic
480C22×C4C22×#59511yescharacteristic
481C22×C4C22×#59511yescharacteristic
482C22×C4C22×#59511yescharacteristic
483M16—684no—
484M16—684no—
485C2×C8—172no—
486C2×C8—172no—
487C2×C8—172no—
488C2×C8—172no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—342no—
512C4⋊C4—342no—
513C4⋊C4—342no—
514C4⋊C4—681no—
515C4⋊C4—681no—
516C4⋊C4—681no—
517C4⋊C4#6324714yescomplemented
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—681no—
521C22⋊C4—681no—
522C42—172no—
523C42—172no—
524C42—341no—
525C42—341no—
526C42—341no—
527C42—341no—
528C42—341no—
529C42—341no—
530C42—342no—
531C42—341no—
532C42—341no—
533C42—341no—
534C42—341no—
535C42—342no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23Q8×#59512yes—
543C23C23×#59511yescharacteristic
544Q8—44no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—44no—
560C2×C4—171no—
561C2×C4—171no—
562C2×C4—171no—
563C2×C4—171no—
564C2×C4—171no—
565C2×C4—171no—
566C2×C4—172no—
567C2×C4—172no—
568C2×C4—342no—
569C2×C4—342no—
570C2×C4—342no—
571C2×C4—342no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—341no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—684no—
592C2×C4D4×#59512yes—
593C2×C4C2×#6324712yes—
594C2×C4C2×#6324712yes—
595C2×C4C2×#6324712yes—
596C8—342no—
597C8—342no—
598C8—342no—
599C8—342no—
600C8—681no—
601C8—681no—
602C8—681no—
603C8—681no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×D4×#59511yescharacteristic
612C22C2×Q8×#59511yescharacteristic
613C22#1176387811yescharacteristic
614C22C22×#6324711yescharacteristic
615C22#1176388711yescharacteristic
616C22#1176388811yescharacteristic
617C22#1176388911yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—171no—
625C4—171no—
626C4—171no—
627C4—171no—
628C4—342no—
629C4—342no—
630C4—681no—
631C4—681no—
632C4—681no—
633C4—681no—
634C4—681no—
635C2—42no—
636C2#1454530811yescharacteristic
637C2#1454550411yescharacteristic
638C2D4×#6324711yescharacteristic
639C2#1454557511yescharacteristic
640C2Q8×#6324711yescharacteristic
641C2#1454570311yescharacteristic
642C2#1454574111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC2×C17×C22⋊Q8C4
prime radicalC24×C17C22×C4
cosocleC22×Dic17C24
hypercenterC2×C22⋊Q8#595
hypocenterC17#22955
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11763860C2semidirect
3. #14544958C2semidirect
4. #14544886C2semidirect
5. #14544886C2semidirect
6. #14544881C2semidirect
7. #14544841C2semidirect
8. #14544837C2semidirect
11. #14544584C2semidirect
106. C22⋊#3624Q8semidirect
189. C4⋊C4×C17M16semidirect
452. C17#22955semidirect
517. C4⋊C4#63247semidirect

Maximal subgroups

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

subgroupindex
C2×#117638602
C2×#117638122
#145449582
#145448862
#145448812
#145448412
#145448372
#145447312
#145445842
#145445262
#145445102
C22⋊Q8×Dic172
#2295517