Finite Group Databasebeta

← #17980249

Subgroups — #17980249

Order 4352.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545695C213yesmaximal, complemented
2#14545693C213yesmaximal, complemented
3#14545689C213yesmaximal, complemented
4#14545688C213yesmaximal, complemented
5#14545687C211yesmaximal, characteristic, complemented
6#14545686C211yesmaximal, characteristic, complemented
7#14545678C211yesmaximal, characteristic, complemented
8#14545677C211yesmaximal, characteristic, complemented
9Q8×#63247C211yesmaximal, characteristic, complemented
10#14545579C213yesmaximal, complemented
11#14545572C213yesmaximal, complemented
12C2×Q8×#3624C211yesmaximal, characteristic, complemented
13#14545479C213yesmaximal
14#14545467C213yesmaximal
15Q8×#63254C211yesmaximal, characteristic
16#11763861C2216yes—
17#11763861C2216yes—
18#11763860C2213yes—
19#11763860C2213yes—
20#11763860C2213yes—
21#11763860C2213yes—
22#11763859C2213yes—
23#11763859C2213yes—
24#11763859C2216yes—
25#11763858C2213yes—
26#11763858C2213yes—
27#11763858C2216yes—
28Q8×#3624C2211yescharacteristic
29Q8×#3624C2211yescharacteristic
30Q8×#3624C2213yes—
31Q8×#3624C2213yes—
32Q8×#3624C2211yescharacteristic
33Q8×#3624C2211yescharacteristic
34Q8×#3624C2213yes—
35Q8×#3624C2213yes—
36#11763834C2213yes—
37#11763834C2216yes—
38#11763834C2213yes—
39#11763833C2213yes—
40#11763833C2216yes—
41#11763833C2213yes—
42#11763831C2213yes—
43#11763830C2213yes—
44#11763828C2213yes—
45#11763828C2213yes—
46#11763827C2213yes—
47#11763826C2213yes—
48#11763825C2213yes—
49#11763825C2213yes—
50#11763815C2213yes—
51#11763815C2213yes—
52#11763812C2213yes—
53C2×C4⋊#3624C2213yes—
54#11763782C2211yescharacteristic
55#11763782C2213yes—
56#11763773C2216yes—
57C4×#63247C2213yes—
58C2×C4×#3624C2213yes—
59Q8×Dic34C411yescharacteristic
60#11763534C413yes—
61#11763517C413yes—
62#11763517C413yes—
63#11763516C413yes—
64Q8×C22⋊D17C411yescharacteristic
65Q8×C22⋊D17C411yescharacteristic
66#11763364C2213yes—
67#11763361C2213yes—
68#11763361C2213yes—
69C4×#63254C2213yes—
70C2×Q8×Dic17C2211yescharacteristic
71C2×Q8×Dic17C2211yescharacteristic
72C4×Q8×D17C2211yescharacteristic
73D4×Q8×C17C411yescharacteristic
74C22×Q8×C17C2×C411yescharacteristic
75C22×Q8×C17C2×C411yescharacteristic
76C2×Q8×D17C2×C411yescharacteristic
77C2×#63254C2311yescharacteristic
78C2×#63247C2311yescharacteristic
79C2×#63247C2313yes—
80C22×#3624C2311yescharacteristic
81C22×#3624C2313yes—
82C17×C4⋊Q8C2×C413yes—
83C17×C22⋊Q8C2×C413yes—
84C17×C22⋊Q8C2×C413yes—
85C4×Q8×C17C2×C411yescharacteristic
86C4×D4×C17C2×C413yes—
87#10587632C2×C413yes—
88#10587632C2×C413yes—
89Q8×Dic17C2×C411yescharacteristic
90Q8×Dic17C2×C411yescharacteristic
91Q8×Dic17C2×C413yes—
92Q8×Dic17C2×C413yes—
93Q8×Dic17C2311yescharacteristic
94Q8×Dic17C2313yes—
95Q8×Dic17C2313yes—
96#10587630C2×C413yes—
97#10587630C2×C413yes—
98D4×Dic17C2×C413yes—
99C4×C22⋊D17C2×C413yes—
100C4×C22⋊D17C2×C413yes—
101C2×C4⋊Dic17C2×C413yes—
102C2×C4⋊Dic17C2×C413yes—
103#10587610C2×C413yes—
104#10587610C2×C413yes—
105C2×C4×Dic17C2313yes—
106C2×C4×Dic17C2313yes—
107#10587583C2×C416yes—
108#10587578C2313yes—
109C4⋊C4×D17C2×C413yes—
110#10587576C2×C416yes—
111#10587575C2×C416yes—
112#10587575C2×C416yes—
113#10587574C2×C413yes—
114#10587574C2×C413yes—
115#10587573C2×C413yes—
116#10587573C2313yes—
117#10587573C2316yes—
118#10587567C2×C413yes—
119#10587567C2×C413yes—
120#10587564C2×C416yes—
121#10587564C2×C416yes—
122#10587563C2313yes—
123#10587563C2313yes—
124C42×D17C2313yes—
125#10587555C2×C413yes—
126C4×Dic34C2×C413yes—
127C4×Dic34C2×C413yes—
128C22⋊#3624C2313yes—
129C22⋊#3624C2311yescharacteristic
130C22⋊#3624C2311yescharacteristic
131C22⋊#3624C2313yes—
132#10587544C2316yes—
133#10587544C2313yes—
134#10587544C2313yes—
135#10587544C2313yes—
136#10587544C2313yes—
137#10587543C2311yescharacteristic
138#10587543C2313yes—
139#10587543C2311yescharacteristic
140#10587543C2313yes—
141#10587542C2313yes—
142#10587542C2313yes—
143#10587542C2316yes—
144#10587542C2316yes—
145C4⋊#3624C2313yes—
146C4⋊#3624C2313yes—
147C4⋊#3624C2316yes—
148C4⋊#3624C2313yes—
149C4⋊#3624C2313yes—
150C4⋊#3624C2313yes—
151C4⋊#3624C2313yes—
152C4⋊#3624C2313yes—
153C4⋊#3624C2313yes—
154C4×#3624C2313yes—
155C4×#3624C2313yes—
156C4×#3624C2313yes—
157C4×#3624C2313yes—
158C4×#3624C2313yes—
159C4×#3624C2313yes—
160C4×#3624C2316yes—
161C4×#3624C2316yes—
162C2×Q8×C17—24no—
163C2×Q8×C17—24no—
164C2×Q8×C17C22×C413yes—
165C2×Q8×C17C22×C413yes—
166C2×Q8×C17C22×C411yescharacteristic
167C2×D4×C17C22×C411yescharacteristic
168C22×C4×C17C22×C413yes—
169C22×C4×C17C22×C413yes—
170C2×C22⋊D17C22×C411yescharacteristic
171C2×C22⋊D17C22×C411yescharacteristic
172C22×Dic17C22×C413yes—
173C22×Dic17C22×C413yes—
174C22×Dic17C2411yescharacteristic
175C22×Dic17C2411yescharacteristic
176Q8×D17—24no—
177#63254—22no—
178#63254—22no—
179C2×C4×D17C22×C413yes—
180C2×C4×D17C22×C413yes—
181C2×C4×D17C2411yescharacteristic
182C2×Dic34C22×C411yescharacteristic
183C2×Dic34C22×C413yes—
184C2×Dic34C22×C416yes—
185#63247—22no—
186#63247—26no—
187C2×#3624—26no—
188C2×#3624—22no—
189C2×#3624C2413yes—
190C2×#3624C2411yescharacteristic
191C2×#3624C2411yescharacteristic
192C2×#3624C2413yes—
193C2×#3624C2411yescharacteristic
194C2×#3624C2413yes—
195C2×#3624C2413yes—
196C2×#3624C2411yescharacteristic
197C4⋊C4×C17C22×C416yes—
198C4⋊C4×C17C22×C413yes—
199C4⋊C4×C17C22×C413yes—
200C22⋊C4×C17C22×C413yes—
201C22⋊C4×C17C22×C413yes—
202C42×C17C22×C413yes—
203C22⋊Dic17C22×C413yes—
204C22⋊Dic17C22×C413yes—
205#63227C22×C413yes—
206#63227C22×C413yes—
207C4⋊Dic17C22×C413yes—
208C4⋊Dic17C22×C413yes—
209C4⋊Dic17C22×C413yes—
210C4⋊Dic17C22×C413yes—
211C4⋊Dic17C22×C416yes—
212C4⋊Dic17C22×C416yes—
213#63225C22×C413yes—
214#63225C22×C413yes—
215#63225C22×C416yes—
216#63225C22×C413yes—
217#63225C22×C413yes—
218#63225C22×C416yes—
219#63225C22×C416yes—
220C4×Dic17C22×C413yes—
221C4×Dic17C22×C413yes—
222C4×Dic17C22×C413yes—
223C4×Dic17C22×C413yes—
224C4×Dic17C22×C413yes—
225C4×Dic17C2413yes—
226C4×Dic17C2413yes—
227C4×Dic17C2413yes—
228C4×Dic17C2413yes—
229#60253—171nomaximal, Sylow
230C23×C17C23×C411yescharacteristic
231C23×C17C23×C411yescharacteristic
232C22×D17C23×C411yescharacteristic
233Q8×C17—26no—
234Q8×C17—26no—
235Q8×C17#13114yes—
236D4×C17—22no—
237C2×C4×C17—26no—
238C2×C4×C17—26no—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C23×C413yes—
241C2×C4×C17C23×C413yes—
242C2×C4×C17C23×C413yes—
243C2×C4×C17C23×C413yes—
244C22⋊D17—22no—
245C22⋊D17—22no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—43no—
249C2×Dic17—43no—
250C2×Dic17C23×C413yes—
251C2×Dic17C23×C413yes—
252C2×Dic17C23×C413yes—
253C2×Dic17C23×C411yescharacteristic
254C2×Dic17C23×C413yes—
255C2×Dic17C23×C411yescharacteristic
256C2×Dic172−1+412yes—
257C2×Dic172−1+412yes—
258C2×Dic17C2511yescharacteristic
259C4×D17—26no—
260C4×D17—21no—
261C4×D17—21no—
262C4×D17—43no—
263Dic34—26no—
264Dic34—22no—
265Dic34—46no—
266#3624—23no—
267#3624—21no—
268#3624—21no—
269#3624—23no—
270#3624—21no—
271#3624—23no—
272#3624—23no—
273#3624—21no—
274Q8×#41—171no—
275#2986—173no—
276#2986—173no—
277#2985—173no—
278#2985—173no—
279#2984—171no—
280#2984—171no—
281#2982—171no—
282#2982—171no—
283Q8×M16—171no—
284#2962—173no—
285#2960—173no—
286C2×C8×Q8—171no—
287#2925—173no—
288#2922—173no—
289C22×C17—21no—
290C22×C17—21no—
291C22×C17—21no—
292C22×C17—21no—
293C22×C17—21no—
294C22×C17#51912yes—
295C22×C17C24×C411yescharacteristic
296C2×D17—21no—
297C2×D17—21no—
298C2×D17—21no—
299C4×C17—21no—
300C4×C17—23no—
301C4×C17—23no—
302C4×C17—23no—
303C4×C17C2×#13116yes—
304Dic17—21no—
305Dic17—23no—
306Dic17—23no—
307Dic17—23no—
308Dic17—23no—
309Dic17#51912yes—
310Dic17C2×2−1+411yescharacteristic
311Dic17C2×2−1+411yescharacteristic
312Q82—171no—
313#557—173no—
314#554—173no—
315#554—173no—
316#554—173no—
317D4×Q8—171no—
318D4×Q8—171no—
319D4×Q8#59511yescharacteristic
320#533—173no—
321#533—173no—
322#533—173no—
323C4×#41—173no—
324C2×C4×Q8—171no—
325C2×C4×Q8—171no—
326C2×C4×Q8—171no—
327#446—173no—
328#446—173no—
329#446—176no—
330#446—173no—
331#446—176no—
332#446—173no—
333#446—176no—
334#446—173no—
335#446—173no—
336C8×Q8—171no—
337C8×Q8—171no—
338C8×Q8—173no—
339C8×Q8—173no—
340C8×Q8—173no—
341C8×Q8—173no—
342C8×Q8—176no—
343C8×Q8—173no—
344C8×Q8—171no—
345C8×Q8—171no—
346C8×Q8—173no—
347#433—173no—
348#433—173no—
349#433—173no—
350#433—173no—
351#433—176no—
352#433—176no—
353#432—173no—
354#432—173no—
355#431—173no—
356#431—173no—
357#431—173no—
358#431—173no—
359#431—173no—
360#431—173no—
361#424—173no—
362#424—173no—
363#423—173no—
364C2×C4⋊C8—173no—
365#408—171no—
366#408—173no—
367#405—176no—
368C4×M16—173no—
369C2×C4×C8—173no—
370C2×C17—22no—
371C2×C17—41no—
372C2×C17C2×#51911yescharacteristic
373C2×C17C22×#13111yescharacteristic
374C2×C17#357311yescharacteristic
375D17—41no—
376C2×#41—171no—
377C22×Q8—171no—
378C22×Q8C2×#59511yescharacteristic
379C22×Q8C2×#59511yescharacteristic
380C2×M16—171no—
381C2×M16—173no—
382C22×C8—171no—
383C22×C8—173no—
384C4⋊Q8—173no—
385C4⋊Q8—173no—
386C4⋊Q8—173no—
387C4⋊Q8—173no—
388C4⋊Q8—173no—
389C4⋊Q8C2×#59513yes—
390#125—176no—
391#125—176no—
392#125—176no—
393C22⋊Q8—173no—
394C22⋊Q8—173no—
395C22⋊Q8—173no—
396C22⋊Q8—173no—
397C22⋊Q8—176no—
398C22⋊Q8—176no—
399C22⋊Q8—176no—
400C22⋊Q8C2×#59513yes—
401C22⋊Q8C2×#59513yes—
402C4×Q8—171no—
403C4×Q8—173no—
404C4×Q8—173no—
405C4×Q8—171no—
406C4×Q8—171no—
407C4×Q8—173no—
408C4×Q8—173no—
409C4×Q8—173no—
410C4×Q8—173no—
411C4×Q8—176no—
412C4×Q8—173no—
413C4×Q8—173no—
414C4×Q8C2×#59511yescharacteristic
415C4×D4—173no—
416C4×D4—173no—
417C4×D4—173no—
418C4×D4—173no—
419C4×D4—173no—
420C4×D4C2×#59513yes—
421#117—173no—
422#117—173no—
423#117—173no—
424C2×C4⋊C4—173no—
425C2×C4⋊C4—173no—
426C2×C4⋊C4—173no—
427C2×C42—173no—
428C2×C42—173no—
429C2×C42—173no—
430C4⋊C8—173no—
431C4⋊C8—173no—
432C4⋊C8—176no—
433C4⋊C8—173no—
434C4⋊C8—173no—
435C4⋊C8—173no—
436C4⋊C8—173no—
437C4⋊C8—173no—
438C4⋊C8—173no—
439C4⋊C8—176no—
440C4⋊C8—173no—
441C4⋊C8—173no—
442C4⋊C8—173no—
443C4⋊C8—173no—
444C22⋊C8—171no—
445C22⋊C8—173no—
446C22⋊C8—171no—
447C22⋊C8—173no—
448C22⋊C8—171no—
449C22⋊C8—171no—
450C22⋊C8—173no—
451C22⋊C8—173no—
452#97—173no—
453#97—173no—
454#97—176no—
455#97—176no—
456C4×C8—173no—
457C4×C8—173no—
458C4×C8—173no—
459C4×C8—173no—
460C4×C8—173no—
461C4×C8—173no—
462C4×C8—176no—
463C4×C8—176no—
464C17#6025311yescharacteristic, complemented, Sylow
465#41—342no—
466#41—342no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—171no—
470C2×Q8—173no—
471C2×Q8—176no—
472C2×Q8—344no—
473C2×Q8C22×#59513yes—
474C2×Q8C22×#59513yes—
475C2×Q8C22×#59511yescharacteristic
476C2×D4—171no—
477C2×D4—171no—
478C2×D4C22×#59511yescharacteristic
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—173no—
482C22×C4—173no—
483C22×C4—173no—
484C22×C4—171no—
485C22×C4—173no—
486C22×C4C22×#59513yes—
487C22×C4C22×#59513yes—
488M16—342no—
489M16—346no—
490C2×C8—173no—
491C2×C8—171no—
492C2×C8—171no—
493C2×C8—171no—
494C2×C8—173no—
495C2×C8—173no—
496C2×C8—171no—
497C2×C8—173no—
498C2×C8—346no—
499C2×C8—342no—
500C4⋊C4—173no—
501C4⋊C4—176no—
502C4⋊C4—176no—
503C4⋊C4—176no—
504C4⋊C4—173no—
505C4⋊C4—173no—
506C4⋊C4—173no—
507C4⋊C4—173no—
508C4⋊C4—173no—
509C4⋊C4—173no—
510C4⋊C4—173no—
511C4⋊C4—176no—
512C4⋊C4—176no—
513C4⋊C4C22×#59513yes—
514C4⋊C4C22×#59513yes—
515C4⋊C4C22×#59516yes—
516C22⋊C4—173no—
517C22⋊C4—173no—
518C22⋊C4—173no—
519C22⋊C4—173no—
520C22⋊C4C22×#59513yes—
521C22⋊C4C22×#59513yes—
522C42—173no—
523C42—173no—
524C42—173no—
525C42—173no—
526C42—173no—
527C42—173no—
528C42—173no—
529C42—173no—
530C42—173no—
531C42C22×#59513yes—
532C23—171no—
533C23C23×#59511yescharacteristic
534C23C23×#59511yescharacteristic
535Q8—26no—
536Q8—26no—
537Q8—342no—
538Q8—346no—
539Q8—686no—
540Q8#1058767814yes—
541D4—22no—
542D4—342no—
543D4—342no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—172no—
547C2×C4—172no—
548C2×C4—171no—
549C2×C4—171no—
550C2×C4—171no—
551C2×C4—173no—
552C2×C4—173no—
553C2×C4—173no—
554C2×C4—173no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—346no—
560C2×C4—683no—
561C2×C4—683no—
562C2×C4—683no—
563C2×C4C23×#59511yescharacteristic
564C2×C4C23×#59513yes—
565C2×C4C23×#59513yes—
566C2×C4C23×#59513yes—
567C2×C4C23×#59513yes—
568C8—343no—
569C8—341no—
570C8—341no—
571C8—343no—
572C8—341no—
573C8—341no—
574C8—343no—
575C8—343no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—341no—
582C22—341no—
583C22—341no—
584C22#1176387712yes—
585C22C24×#59511yescharacteristic
586C4—21no—
587C4—23no—
588C4—23no—
589C4—23no—
590C4—171no—
591C4—171no—
592C4—172no—
593C4—341no—
594C4—343no—
595C4—343no—
596C4—343no—
597C4—343no—
598C4C2×#1058767816yes—
599C2—22no—
600C2—41no—
601C2—681no—
602C2C2×#1176387711yescharacteristic
603C2C22×#1058767811yescharacteristic
604C2#1454578011yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingD4×Q8×C17C4
prime radicalC2×#63254C23
cosocleC2×Dic17C25
hypercenterD4×Q8#595
hypocenterC17#60253
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #14545695C2semidirect
2. #14545693C2semidirect
3. #14545689C2semidirect
4. #14545688C2semidirect
5. #14545687C2semidirect
6. #14545686C2semidirect
7. #14545678C2semidirect
8. #14545677C2semidirect
9. Q8×#63247C2semidirect
10. #14545579C2semidirect
11. #14545572C2semidirect
12. C2×Q8×#3624C2semidirect
464. C17#60253semidirect

Maximal subgroups

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

subgroupindex
#145456952
#145456932
#145456892
#145456882
#145456872
#145456862
#145456782
#145456772
Q8×#632472
#145455792
#145455722
C2×Q8×#36242
#145454792
#145454672
Q8×#632542
#6025317