Finite Group Databasebeta

← #16956306

Subgroups — #16956306

Order 4352.

2510 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D17C211yesmaximal, characteristic
2C2×#11762680C214yesmaximal
3C2×#11762678C211yesmaximal, characteristic, complemented
4C2×#11762666C211yesmaximal, characteristic
5C2×C4×#63227C211yesmaximal, characteristic
6#14527047C214yesmaximal, complemented
7#14527014C211yesmaximal, characteristic, complemented
8#14527013C211yesmaximal, characteristic, complemented
9C2×C17×#384C211yesmaximal, characteristic, complemented
10C23×C4×D17C2212yes—
11C23×C4×D17C2211yescharacteristic
12C22×C4⋊Dic17C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×#63225C2212yes—
15C22×#63225C2214yes—
16C2×C4⋊C4×D17—24no—
17C2×C4⋊C4×D17—24no—
18C2×C4⋊C4×D17—24no—
19C22×#63227C2212yes—
20C22×#63227C2212yes—
21C22×C4×Dic17C2211yescharacteristic
22#11762680—216no—
23#11762678—24no—
24#11762666—24no—
25C4×#63227C418yescomplemented
26C2×#10587502C2214yes—
27C2×#10587502C2214yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C22×C4⋊C4×C17C2211yescharacteristic
31C22×C4⋊C4×C17C2214yes—
32C22×C4⋊C4×C17C2211yescharacteristic
33C22×C42×C17C2211yescharacteristic
34C17×#384—24no—
35C24×D17C2311yescharacteristic
36C23×C4×C17C2312yes—
37C23×C4×C17C2312yes—
38C23×C4×C17C2312yes—
39C23×C4×C17C2311yescharacteristic
40C23×Dic17C2312yes—
41C23×Dic17C2312yes—
42C23×Dic17C2312yes—
43C23×Dic17C2311yescharacteristic
44C22×C4×D17—22no—
45C22×C4×D17—22no—
46C22×C4×D17—22no—
47C22×C4×D17—22no—
48C22×C4×D17—24no—
49C22×C4×D17—24no—
50C22×C4×D17—24no—
51C22×C4×D17—28no—
52C22×C4×D17—44no—
53C22×C4×D17C2×C414yes—
54C22×C4×D17D414yes—
55C22×C4×D17Q814yes—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—28no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17—22no—
61C2×C4⋊C4×C17—22no—
62C2×C4⋊C4×C17—28no—
63C2×C4⋊C4×C17—44no—
64C2×C4⋊C4×C17D414yes—
65C2×C42×C17—22no—
66C2×C42×C17—22no—
67C2×C42×C17C2×C414yescomplemented
68C2×#63227—24no—
69C2×#63227—28no—
70C2×#63227C2×C418yes—
71C2×#63227D418yes—
72C2×#63227Q818yes—
73C2×C4⋊Dic17—22no—
74C2×C4⋊Dic17—22no—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17D414yes—
79C2×#63225—24no—
80C2×#63225—24no—
81C2×#63225—24no—
82C2×#63225—28no—
83C2×#63225—28no—
84C2×#63225—44no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17C2×C414yescomplemented
88C4⋊C4×D17—232no—
89#10587502—28no—
90#10587502—24no—
91#10587502—28no—
92#10587502C2×C418yescomplemented
93C24×C17C2411yescharacteristic
94C23×D17—21no—
95C23×D17—21no—
96C23×D17—21no—
97C23×D17—21no—
98C23×D17—22no—
99C23×D17—22no—
100C23×D17C22×C412yes—
101C23×D17C2×D412yes—
102C23×D17C2×D414yes—
103C23×D17C2×Q812yes—
104C23×D17C2×Q814yes—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—24no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—24no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—24no—
116C22×C4×C17—22no—
117C22×C4×C17—24no—
118C22×C4×C17C22×C414yes—
119C22×C4×C17C22×C412yes—
120C22×C4×C17C2×D414yes—
121C22×C4×C17C2×D412yes—
122C22×C4×C17C2×D412yes—
123C22×C4×C17C2×D414yes—
124C22×C4×C17C2×D412yes—
125C22×C4×C17C2×Q814yes—
126C22×C4×C17C2×Q812yes—
127C22×C4×C17#4112yes—
128C22×C4×C17#4112yes—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—22no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—24no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—24no—
141C22×Dic17—24no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—24no—
146C22×Dic17C22×C412yes—
147C22×Dic17C22×C414yes—
148C22×Dic17C2×D414yes—
149C22×Dic17C2×D412yes—
150C22×Dic17C2×D414yes—
151C22×Dic17C2×Q812yes—
152C22×Dic17C2×Q814yes—
153C2×C4×D17—28no—
154C2×C4×D17—28no—
155C2×C4×D17—28no—
156C2×C4×D17—28no—
157C2×C4×D17—28no—
158C2×C4×D17—28no—
159C2×C4×D17—216no—
160C2×C4×D17—416no—
161C2×C4×D17—416no—
162C2×C4×D17—48no—
163C2×C4×D17—48no—
164C2×C4×D17C4⋊C4116yescomplemented
165C4⋊C4×C17—28no—
166C4⋊C4×C17—416no—
167C4⋊C4×C17—44no—
168C42×C17—28no—
169#63227—48no—
170#63227C4⋊C4132yescomplemented
171C4⋊Dic17—28no—
172C4⋊Dic17—44no—
173#63225—216no—
174#63225—416no—
175C4×Dic17—28no—
176#13880—171nomaximal, Sylow
177C23×C17C23×C411yescharacteristic
178C23×C17C22×D411yescharacteristic
179C23×C17C22×D412yes—
180C23×C17C22×D411yescharacteristic
181C23×C17C22×D411yescharacteristic
182C23×C17C22×Q811yescharacteristic
183C23×C17C22×Q812yes—
184C23×C17C2×#4111yescharacteristic
185C23×C17C2×#4111yescharacteristic
186C23×C17H(5,𝔽2)12yes—
187C23×C172−1+412yes—
188C22×D17—22no—
189C22×D17—22no—
190C22×D17—24no—
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—24no—
205C22×D17—24no—
206C22×D17C2×C4⋊C414yes—
207C22×D17C2×C4⋊C418yes—
208C22×D17#12714yes—
209C22×D17C4⋊Q814yes—
210C22×D17C4⋊Q818yes—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—28no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—44no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—44no—
226C2×C4×C17—44no—
227C2×C4×C17C2×C4⋊C414yes—
228C2×C4×C17C2×C4⋊C418yescomplemented
229C2×C4×C17C4×D414yes—
230C2×C4×C17C4×D414yes—
231C2×C4×C17#12014yes—
232C2×C4×C17C22⋊Q814yes—
233C2×C4×C17C22⋊Q814yes—
234C2×C4×C17#12314yes—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—28no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—44no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—44no—
256C2×Dic17—44no—
257C2×Dic17C2×C4⋊C414yes—
258C2×Dic17C2×C4⋊C418yescomplemented
259C4×D17—232no—
260C4×D17—432no—
261C23×C4⋊C4—171no—
262C2×#385—174no—
263C2×#384—171no—
264C2×#384D1711yescharacteristic, complemented
265C2×#380—171no—
266C2×C4×C22⋊C4—171no—
267#1444—174no—
268C4⋊#95—171no—
269C4⋊#95—171no—
270C22×C17C22×C4⋊C411yescharacteristic
271C22×C17C22×C4⋊C412yes—
272C22×C17C2×C4×D411yescharacteristic
273C22×C17C2×C4×D411yescharacteristic
274C22×C17#52012yes—
275C22×C17C2×#12011yescharacteristic
276C22×C17C2×C22⋊Q811yescharacteristic
277C22×C17C2×C22⋊Q811yescharacteristic
278C22×C17C2×#12311yescharacteristic
279C22×C17C2×#12711yescharacteristic
280C22×C17C2×C4⋊Q811yescharacteristic
281C22×C17C2×C4⋊Q812yes—
282C22×C17#53511yescharacteristic
283C22×C17#53611yescharacteristic
284C22×C17#54412yes—
285C22×C17D4212yes—
286C22×C17D4212yes—
287C22×C17#54712yes—
288C22×C17#54712yes—
289C22×C17D4×Q812yes—
290C22×C17D4×Q812yes—
291C22×C17#55412yes—
292C22×C17#55412yes—
293C2×D17—24no—
294C2×D17—28no—
295C2×D17—28no—
296C2×D17—24no—
297C2×D17—24no—
298C2×D17—28no—
299C2×D17—28no—
300C2×D17—24no—
301C2×D17—24no—
302C2×D17—24no—
303C2×D17#38418yescomplemented
304C4×C17—28no—
305C4×C17—44no—
306C4×C17—82no—
307C4×C17#38518yescomplemented
308Dic17—28no—
309Dic17—24no—
310Dic17—44no—
311Dic17—82no—
312C24×C4—171no—
313C24×C4—172no—
314C22×C4⋊C4—171no—
315C22×C4⋊C4—172no—
316C22×C4⋊C4—171no—
317C22×C4⋊C4—174no—
318C22×C4⋊C4—344no—
319C22×C4⋊C4—344no—
320C22×C4⋊C4—344no—
321C22×C4⋊C4C2×D1711yescharacteristic
322C22×C4⋊C4C2×D1714yes—
323C22×C4⋊C4C2×D1711yescharacteristic
324C22×C22⋊C4—172no—
325C22×C22⋊C4—172no—
326C22×C42—171no—
327C22×C42C2×D1711yescharacteristic
328#385—3416no—
329#384—24no—
330#384—344no—
331#380—344no—
332C4×C22⋊C4—178no—
333C2×#95—172no—
334C2×#95—174no—
335C2×#95—174no—
336C2×#95—172no—
337C2×C17C2×#38411yescharacteristic
338C2×C17C2×#38511yescharacteristic
339C2×C17#231711yescharacteristic
340C2×C17D4×C4⋊C412yes—
341C2×C17D4×C4⋊C412yes—
342C2×C17#243311yescharacteristic
343C2×C17#243411yescharacteristic
344C2×C17#244012yes—
345C2×C17#244012yes—
346C2×C17#250111yescharacteristic
347C2×C17#250211yescharacteristic
348D17—28no—
349C25—171no—
350C23×C4—172no—
351C23×C4—171no—
352C23×C4—174no—
353C23×C4—174no—
354C23×C4—174no—
355C23×C4—172no—
356C23×C4—172no—
357C23×C4—344no—
358C23×C4—344no—
359C23×C4—348no—
360C23×C4—342no—
361C23×C4—342no—
362C23×C4—342no—
363C23×C4—342no—
364C23×C4—344no—
365C23×C4—684no—
366C23×C4C22×D1712yes—
367C23×C4C22×D1712yes—
368C23×C4C22×D1712yes—
369C23×C4C22×D1711yescharacteristic
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—44no—
378C2×C4⋊C4—174no—
379C2×C4⋊C4—3432no—
380C2×C4⋊C4—344no—
381C2×C4⋊C4—344no—
382C2×C4⋊C4—342no—
383C2×C4⋊C4—344no—
384C2×C4⋊C4—342no—
385C2×C4⋊C4—348no—
386C2×C4⋊C4—348no—
387C2×C4⋊C4—342no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—684no—
391C2×C4⋊C4C22⋊D1714yes—
392C2×C22⋊C4—178no—
393C2×C22⋊C4—178no—
394C2×C22⋊C4—178no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—348no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—174no—
400C2×C42—342no—
401C2×C42—342no—
402C2×C42C4×D1714yescomplemented
403#95—178no—
404#95—344no—
405#95—348no—
406#95—348no—
407C17#1388011yescharacteristic, complemented, Sylow
408C24—172no—
409C24—174no—
410C24—172no—
411C24—174no—
412C24—172no—
413C24—341no—
414C24—341no—
415C24—342no—
416C24—341no—
417C24—342no—
418C24—341no—
419C24C23×D1711yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—172no—
434C22×C4—174no—
435C22×C4—174no—
436C22×C4—172no—
437C22×C4—174no—
438C22×C4—172no—
439C22×C4—1716no—
440C22×C4—174no—
441C22×C4—3416no—
442C22×C4—341no—
443C22×C4—342no—
444C22×C4—341no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—341no—
448C22×C4—342no—
449C22×C4—341no—
450C22×C4—348no—
451C22×C4—348no—
452C22×C4—348no—
453C22×C4—348no—
454C22×C4—348no—
455C22×C4—348no—
456C22×C4—344no—
457C22×C4—342no—
458C22×C4—344no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—344no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—344no—
465C22×C4—6816no—
466C22×C4—6816no—
467C22×C4—688no—
468C22×C4—688no—
469C22×C4C2×C4×D1714yes—
470C22×C4C2×C4×D1712yes—
471C22×C4#6325212yes—
472C22×C4D4×D1714yes—
473C22×C4D4×D1712yes—
474C22×C4D4×D1712yes—
475C22×C4Q8×D1714yes—
476C22×C4Q8×D1712yes—
477C22×C4#6325612yes—
478C22×C4C2×C22⋊D1714yes—
479C22×C4C2×C22⋊D1712yes—
480C4⋊C4—28no—
481C4⋊C4—416no—
482C4⋊C4—44no—
483C4⋊C4—3416no—
484C4⋊C4—348no—
485C4⋊C4—6816no—
486C4⋊C4—684no—
487C22⋊C4—1732no—
488C22⋊C4—688no—
489C42—28no—
490C42—348no—
491C23—178no—
492C23—174no—
493C23—174no—
494C23—174no—
495C23—178no—
496C23—344no—
497C23—344no—
498C23—344no—
499C23—344no—
500C23—344no—
501C23—344no—
502C23—342no—
503C23—342no—
504C23—344no—
505C23—342no—
506C23—342no—
507C23—342no—
508C23—342no—
509C23—342no—
510C23—342no—
511C23—344no—
512C23—344no—
513C23—344no—
514C23C22×C4×D1711yescharacteristic
515C23C2×#6325211yescharacteristic
516C23C2×D4×D1712yes—
517C23C2×D4×D1711yescharacteristic
518C23C2×D4×D1711yescharacteristic
519C23#1058769112yes—
520C23C2×Q8×D1712yes—
521C23C2×Q8×D1711yescharacteristic
522C23C2×#6325611yescharacteristic
523C23#1058769412yes—
524C23C22×C22⋊D1711yescharacteristic
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—178no—
542C2×C4—174no—
543C2×C4—344no—
544C2×C4—344no—
545C2×C4—344no—
546C2×C4—342no—
547C2×C4—344no—
548C2×C4—342no—
549C2×C4—344no—
550C2×C4—342no—
551C2×C4—342no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—344no—
555C2×C4—3432no—
556C2×C4—344no—
557C2×C4—348no—
558C2×C4—6832no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—684no—
562C2×C4—684no—
563C2×C4—684no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4—682no—
567C2×C4C4⋊C4×D1714yes—
568C2×C4C4⋊C4×D1718yescomplemented
569C2×C4#1058757914yes—
570C2×C4#1058758014yes—
571C2×C4#1058758214yes—
572C2×C4C4×C22⋊D1714yes—
573C2×C4#1058762314yes—
574C2×C4#1058763214yes—
575C22—178no—
576C22—344no—
577C22—344no—
578C22—344no—
579C22—348no—
580C22—348no—
581C22—344no—
582C22—348no—
583C22—344no—
584C22—348no—
585C22—344no—
586C22C2×C4⋊C4×D1711yescharacteristic
587C22C2×C4⋊C4×D1712yes—
588C22C2×C4×C22⋊D1711yescharacteristic
589C22C2×#1058757911yescharacteristic
590C22#1176329012yes—
591C22C2×#1058762311yescharacteristic
592C22C2×#1058763211yescharacteristic
593C22C2×#1058758211yescharacteristic
594C22C2×#1058758011yescharacteristic
595C22#127×D1711yescharacteristic
596C22C4⋊Q8×D1711yescharacteristic
597C22C4⋊Q8×D1712yes—
598C22#1176337211yescharacteristic
599C22#1176338411yescharacteristic
600C22#1176343612yes—
601C22D4×C22⋊D1712yes—
602C22#1176344412yes—
603C22#1176346012yes—
604C22#1176346212yes—
605C22Q8×C22⋊D1712yes—
606C22#1176347412yes—
607C22#1176351412yes—
608C22#1176352012yes—
609C4—28no—
610C4—44no—
611C4—82no—
612C4—348no—
613C4—344no—
614C4—684no—
615C4—1362no—
616C4#1176268018yescomplemented
617C2—348no—
618C2D17×#38411yescharacteristic
619C2C2×#1176268011yescharacteristic
620C2#1453140511yescharacteristic
621C2C4⋊C4×C22⋊D1712yes—
622C2#1453168412yes—
623C2#1453241811yescharacteristic
624C2#1453242511yescharacteristic
625C2#1453248212yes—
626C2#1453248912yes—
627C2#1453325211yescharacteristic
628C2#1453325911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#384C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC2×#384D17
hypocenterC17#13880
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
3. C2×#11762678C2semidirect
6. #14527047C2semidirect
7. #14527014C2semidirect
8. #14527013C2semidirect
9. C2×C17×#384C2semidirect
25. C4×#63227C4semidirect
67. C2×C42×C17C2×C4semidirect
87. C2×C4×Dic17C2×C4semidirect
92. #10587502C2×C4semidirect
164. C2×C4×D17C4⋊C4semidirect
170. #63227C4⋊C4semidirect
228. C2×C4×C17C2×C4⋊C4semidirect
258. C2×Dic17C2×C4⋊C4semidirect
264. C2×#384D17semidirect
303. C2×D17#384semidirect
307. C4×C17#385semidirect
402. C2×C42C4×D17semidirect
407. C17#13880semidirect
568. C2×C4C4⋊C4×D17semidirect
616. C4#11762680semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D172
C2×#117626802
C2×#117626782
C2×#117626662
C2×C4×#632272
#145270472
#145270142
#145270132
C2×C17×#3842
#1388017