Finite Group Databasebeta

← #34031909

Subgroups — #34031909

Order 18688.

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#22626842C213yesmaximal, complemented
2#22626840C213yesmaximal, complemented
3#22626836C213yesmaximal, complemented
4#22626835C213yesmaximal, complemented
5#22626834C211yesmaximal, characteristic, complemented
6#22626833C211yesmaximal, characteristic, complemented
7#22626825C211yesmaximal, characteristic, complemented
8#22626824C211yesmaximal, characteristic, complemented
9Q8×#11924037C211yesmaximal, characteristic, complemented
10#22626726C213yesmaximal, complemented
11#22626719C213yesmaximal, complemented
12C2×Q8×#10596833C211yesmaximal, characteristic, complemented
13#22626626C213yesmaximal
14#22626614C213yesmaximal
15Q8×#11924027C211yesmaximal, characteristic
16#17989639C2216yes—
17#17989639C2216yes—
18#17989638C2213yes—
19#17989638C2213yes—
20#17989638C2213yes—
21#17989638C2213yes—
22#17989637C2213yes—
23#17989637C2213yes—
24#17989637C2216yes—
25#17989636C2213yes—
26#17989636C2213yes—
27#17989636C2216yes—
28Q8×#10596833C2211yescharacteristic
29Q8×#10596833C2211yescharacteristic
30Q8×#10596833C2213yes—
31Q8×#10596833C2213yes—
32Q8×#10596833C2211yescharacteristic
33Q8×#10596833C2211yescharacteristic
34Q8×#10596833C2213yes—
35Q8×#10596833C2213yes—
36#17989612C2213yes—
37#17989612C2216yes—
38#17989612C2213yes—
39#17989611C2213yes—
40#17989611C2216yes—
41#17989611C2213yes—
42#17989609C2213yes—
43#17989608C2213yes—
44#17989606C2213yes—
45#17989606C2213yes—
46#17989605C2213yes—
47#17989604C2213yes—
48#17989603C2213yes—
49#17989603C2213yes—
50#17989593C2213yes—
51#17989593C2213yes—
52#17989590C2213yes—
53C2×C4⋊#10596833C2213yes—
54#17989560C2211yescharacteristic
55#17989560C2213yes—
56#17989551C2216yes—
57C4×#11924037C2213yes—
58C2×C4×#10596833C2213yes—
59Q8×Dic146C411yescharacteristic
60#17989312C413yes—
61#17989295C413yes—
62#17989295C413yes—
63#17989294C413yes—
64Q8×C22⋊D73C411yescharacteristic
65Q8×C22⋊D73C411yescharacteristic
66#17989142C2213yes—
67#17989139C2213yes—
68#17989139C2213yes—
69C4×#11924027C2213yes—
70C2×Q8×Dic73C2211yescharacteristic
71C2×Q8×Dic73C2211yescharacteristic
72C4×Q8×D73C2211yescharacteristic
73D4×Q8×C73C411yescharacteristic
74C2×#11924037C2311yescharacteristic
75C2×#11924037C2313yes—
76C22×#10596833C2311yescharacteristic
77C22×#10596833C2313yes—
78#14556658C2313yes—
79#14556658C2313yes—
80#14556658C2313yes—
81#14556658C2316yes—
82#14556658C2313yes—
83C4⋊#10596833C2313yes—
84C4⋊#10596833C2313yes—
85C4⋊#10596833C2313yes—
86C4⋊#10596833C2313yes—
87C4⋊#10596833C2316yes—
88C4⋊#10596833C2313yes—
89C4⋊#10596833C2313yes—
90C4⋊#10596833C2313yes—
91C4⋊#10596833C2313yes—
92#14556648C2311yescharacteristic
93#14556648C2313yes—
94#14556648C2311yescharacteristic
95#14556648C2313yes—
96C22⋊#10596833C2313yes—
97C22⋊#10596833C2311yescharacteristic
98C22⋊#10596833C2311yescharacteristic
99C22⋊#10596833C2313yes—
100#14556646C2313yes—
101#14556646C2316yes—
102#14556646C2316yes—
103#14556646C2313yes—
104C4×#10596833C2313yes—
105C4×#10596833C2313yes—
106C4×#10596833C2316yes—
107C4×#10596833C2313yes—
108C4×#10596833C2313yes—
109C4×#10596833C2313yes—
110C4×#10596833C2313yes—
111C4×#10596833C2316yes—
112C2×#11924027C2311yescharacteristic
113C2×Q8×D73C2×C411yescharacteristic
114#14556591C2×C413yes—
115#14556590C2×C413yes—
116#14556590C2×C413yes—
117#14556589C2×C413yes—
118#14556589C2×C413yes—
119#14556582C2×C416yes—
120#14556581C2×C416yes—
121#14556581C2×C416yes—
122#14556571C2×C416yes—
123#14556570C2×C413yes—
124#14556570C2×C413yes—
125#14556568C2×C416yes—
126#14556568C2×C416yes—
127#14556567C2×C413yes—
128#14556567C2×C413yes—
129Q8×Dic73C2×C411yescharacteristic
130Q8×Dic73C2×C411yescharacteristic
131Q8×Dic73C2×C413yes—
132Q8×Dic73C2×C413yes—
133Q8×Dic73C2311yescharacteristic
134Q8×Dic73C2313yes—
135Q8×Dic73C2313yes—
136C4×Dic146C2×C413yes—
137C4×Dic146C2×C413yes—
138#14556556C2×C413yes—
139#14556556C2313yes—
140#14556556C2316yes—
141D4×Dic73C2×C413yes—
142C4×C22⋊D73C2×C413yes—
143C4×C22⋊D73C2×C413yes—
144#14556551C2×C413yes—
145#14556551C2×C413yes—
146#14556549C2313yes—
147#14556547C2313yes—
148#14556547C2313yes—
149C2×C4⋊Dic73C2×C413yes—
150C2×C4⋊Dic73C2×C413yes—
151C4⋊C4×D73C2×C413yes—
152C2×C4×Dic73C2313yes—
153C2×C4×Dic73C2313yes—
154C42×D73C2313yes—
155C22×Q8×C73C2×C411yescharacteristic
156C22×Q8×C73C2×C411yescharacteristic
157C4⋊Q8×C73C2×C413yes—
158C22⋊Q8×C73C2×C413yes—
159C22⋊Q8×C73C2×C413yes—
160C4×Q8×C73C2×C411yescharacteristic
161C4×D4×C73C2×C413yes—
162#11924037—22no—
163#11924037—26no—
164C2×#10596833—26no—
165C2×#10596833—22no—
166C2×#10596833C2413yes—
167C2×#10596833C2411yescharacteristic
168C2×#10596833C2411yescharacteristic
169C2×#10596833C2413yes—
170C2×#10596833C2411yescharacteristic
171C2×#10596833C2413yes—
172C2×#10596833C2413yes—
173C2×#10596833C2411yescharacteristic
174#11924027—22no—
175#11924027—22no—
176C2×Dic146C22×C411yescharacteristic
177C2×Dic146C22×C413yes—
178C2×Dic146C22×C416yes—
179Q8×D73—24no—
180C2×C22⋊D73C22×C411yescharacteristic
181C2×C22⋊D73C22×C411yescharacteristic
182C22×Dic73C22×C413yes—
183C22×Dic73C22×C413yes—
184C22×Dic73C2411yescharacteristic
185C22×Dic73C2411yescharacteristic
186C2×C4×D73C22×C413yes—
187C2×C4×D73C22×C413yes—
188C2×C4×D73C2411yescharacteristic
189C4⋊Dic73C22×C413yes—
190C4⋊Dic73C22×C413yes—
191C4⋊Dic73C22×C413yes—
192C4⋊Dic73C22×C416yes—
193C4⋊Dic73C22×C416yes—
194C4⋊Dic73C22×C413yes—
195#11924007C22×C416yes—
196#11924007C22×C413yes—
197#11924007C22×C413yes—
198#11924007C22×C413yes—
199#11924007C22×C413yes—
200#11924007C22×C416yes—
201#11924007C22×C416yes—
202C22⋊Dic73C22×C413yes—
203C22⋊Dic73C22×C413yes—
204#11924005C22×C413yes—
205#11924005C22×C413yes—
206C4×Dic73C22×C413yes—
207C4×Dic73C22×C413yes—
208C4×Dic73C22×C413yes—
209C4×Dic73C22×C413yes—
210C4×Dic73C22×C413yes—
211C4×Dic73C2413yes—
212C4×Dic73C2413yes—
213C4×Dic73C2413yes—
214C4×Dic73C2413yes—
215C2×Q8×C73—24no—
216C2×Q8×C73—24no—
217C2×Q8×C73C22×C413yes—
218C2×Q8×C73C22×C413yes—
219C2×Q8×C73C22×C411yescharacteristic
220C2×D4×C73C22×C411yescharacteristic
221C22×C4×C73C22×C413yes—
222C22×C4×C73C22×C413yes—
223C4⋊C4×C73C22×C413yes—
224C4⋊C4×C73C22×C413yes—
225C4⋊C4×C73C22×C416yes—
226C22⋊C4×C73C22×C413yes—
227C22⋊C4×C73C22×C413yes—
228C42×C73C22×C413yes—
229C23×C73C23×C411yescharacteristic
230C23×C73C23×C411yescharacteristic
231C22×D73C23×C411yescharacteristic
232Q8×C73—26no—
233Q8×C73—26no—
234Q8×C73#13114yes—
235D4×C73—22no—
236C2×C4×C73—26no—
237C2×C4×C73—26no—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C23×C413yes—
240C2×C4×C73C23×C413yes—
241C2×C4×C73C23×C413yes—
242C2×C4×C73C23×C413yes—
243C22⋊D73—22no—
244C22⋊D73—22no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—43no—
248C2×Dic73—43no—
249C2×Dic73C23×C411yescharacteristic
250C2×Dic73C23×C413yes—
251C2×Dic73C23×C413yes—
252C2×Dic73C23×C413yes—
253C2×Dic73C23×C413yes—
254C2×Dic73C23×C411yescharacteristic
255C2×Dic732−1+412yes—
256C2×Dic732−1+412yes—
257C2×Dic73C2511yescharacteristic
258C4×D73—26no—
259C4×D73—21no—
260C4×D73—21no—
261C4×D73—43no—
262Dic146—26no—
263Dic146—22no—
264Dic146—46no—
265#10596833—23no—
266#10596833—21no—
267#10596833—21no—
268#10596833—23no—
269#10596833—21no—
270#10596833—23no—
271#10596833—23no—
272#10596833—21no—
273C22×C73—21no—
274C22×C73—21no—
275C22×C73—21no—
276C22×C73—21no—
277C22×C73—21no—
278C22×C73#51912yes—
279C22×C73C24×C411yescharacteristic
280C2×D73—21no—
281C2×D73—21no—
282C2×D73—21no—
283C4×C73—21no—
284C4×C73—23no—
285C4×C73—23no—
286C4×C73—23no—
287C4×C73C2×#13116yes—
288Dic73—21no—
289Dic73—23no—
290Dic73—23no—
291Dic73—23no—
292Dic73—23no—
293Dic73#51912yes—
294Dic73C2×2−1+411yescharacteristic
295Dic73C2×2−1+411yescharacteristic
296#60253—731nomaximal, Sylow
297C2×C73—22no—
298C2×C73—41no—
299C2×C73C2×#51911yescharacteristic
300C2×C73C22×#13111yescharacteristic
301C2×C73#357311yescharacteristic
302D73—41no—
303Q8×#41—731no—
304#2986—733no—
305#2986—733no—
306#2985—733no—
307#2985—733no—
308#2984—731no—
309#2984—731no—
310#2982—731no—
311#2982—731no—
312Q8×M16—731no—
313#2962—733no—
314#2960—733no—
315C2×C8×Q8—731no—
316#2925—733no—
317#2922—733no—
318C73#6025311yescharacteristic, complemented, Sylow
319Q82—731no—
320#557—733no—
321#554—733no—
322#554—733no—
323#554—733no—
324D4×Q8—731no—
325D4×Q8—731no—
326D4×Q8#6440811yescharacteristic
327#533—733no—
328#533—733no—
329#533—733no—
330C4×#41—733no—
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8—731no—
334#446—733no—
335#446—733no—
336#446—736no—
337#446—733no—
338#446—733no—
339#446—733no—
340#446—733no—
341#446—736no—
342#446—736no—
343C8×Q8—731no—
344C8×Q8—731no—
345C8×Q8—733no—
346C8×Q8—733no—
347C8×Q8—733no—
348C8×Q8—733no—
349C8×Q8—731no—
350C8×Q8—731no—
351C8×Q8—733no—
352C8×Q8—736no—
353C8×Q8—733no—
354#433—733no—
355#433—733no—
356#433—733no—
357#433—733no—
358#433—736no—
359#433—736no—
360#432—733no—
361#432—733no—
362#431—733no—
363#431—733no—
364#431—733no—
365#431—733no—
366#431—733no—
367#431—733no—
368#424—733no—
369#424—733no—
370#423—733no—
371C2×C4⋊C8—733no—
372#408—731no—
373#408—733no—
374#405—736no—
375C4×M16—733no—
376C2×C4×C8—733no—
377C2×#41—731no—
378C22×Q8—731no—
379C22×Q8C2×#6440811yescharacteristic
380C22×Q8C2×#6440811yescharacteristic
381C2×M16—731no—
382C2×M16—733no—
383C22×C8—731no—
384C22×C8—733no—
385C4⋊Q8—733no—
386C4⋊Q8—733no—
387C4⋊Q8—733no—
388C4⋊Q8—733no—
389C4⋊Q8—733no—
390C4⋊Q8C2×#6440813yes—
391#125—736no—
392#125—736no—
393#125—736no—
394C22⋊Q8—733no—
395C22⋊Q8—733no—
396C22⋊Q8—733no—
397C22⋊Q8—733no—
398C22⋊Q8—736no—
399C22⋊Q8—736no—
400C22⋊Q8—736no—
401C22⋊Q8C2×#6440813yes—
402C22⋊Q8C2×#6440813yes—
403C4×Q8—731no—
404C4×Q8—733no—
405C4×Q8—733no—
406C4×Q8—731no—
407C4×Q8—731no—
408C4×Q8—733no—
409C4×Q8—733no—
410C4×Q8—733no—
411C4×Q8—733no—
412C4×Q8—736no—
413C4×Q8—733no—
414C4×Q8—733no—
415C4×Q8C2×#6440811yescharacteristic
416C4×D4—733no—
417C4×D4—733no—
418C4×D4—733no—
419C4×D4—733no—
420C4×D4—733no—
421C4×D4C2×#6440813yes—
422#117—733no—
423#117—733no—
424#117—733no—
425C2×C4⋊C4—733no—
426C2×C4⋊C4—733no—
427C2×C4⋊C4—733no—
428C2×C42—733no—
429C2×C42—733no—
430C2×C42—733no—
431C4⋊C8—733no—
432C4⋊C8—733no—
433C4⋊C8—736no—
434C4⋊C8—733no—
435C4⋊C8—733no—
436C4⋊C8—736no—
437C4⋊C8—733no—
438C4⋊C8—733no—
439C4⋊C8—733no—
440C4⋊C8—733no—
441C4⋊C8—733no—
442C4⋊C8—733no—
443C4⋊C8—733no—
444C4⋊C8—733no—
445C22⋊C8—731no—
446C22⋊C8—731no—
447C22⋊C8—733no—
448C22⋊C8—731no—
449C22⋊C8—731no—
450C22⋊C8—733no—
451C22⋊C8—733no—
452C22⋊C8—733no—
453#97—733no—
454#97—733no—
455#97—736no—
456#97—736no—
457C4×C8—733no—
458C4×C8—733no—
459C4×C8—733no—
460C4×C8—733no—
461C4×C8—733no—
462C4×C8—733no—
463C4×C8—736no—
464C4×C8—736no—
465#41—1462no—
466#41—1462no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—736no—
470C2×Q8—731no—
471C2×Q8—733no—
472C2×Q8—1464no—
473C2×Q8C22×#6440813yes—
474C2×Q8C22×#6440813yes—
475C2×Q8C22×#6440811yescharacteristic
476C2×D4—731no—
477C2×D4—731no—
478C2×D4C22×#6440811yescharacteristic
479C22×C4—733no—
480C22×C4—733no—
481C22×C4—733no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—733no—
485C22×C4—731no—
486C22×C4C22×#6440813yes—
487C22×C4C22×#6440813yes—
488M16—1462no—
489M16—1466no—
490C2×C8—733no—
491C2×C8—731no—
492C2×C8—731no—
493C2×C8—731no—
494C2×C8—733no—
495C2×C8—733no—
496C2×C8—731no—
497C2×C8—733no—
498C2×C8—1466no—
499C2×C8—1462no—
500C4⋊C4—736no—
501C4⋊C4—733no—
502C4⋊C4—733no—
503C4⋊C4—733no—
504C4⋊C4—733no—
505C4⋊C4—736no—
506C4⋊C4—736no—
507C4⋊C4—733no—
508C4⋊C4—736no—
509C4⋊C4—736no—
510C4⋊C4—733no—
511C4⋊C4—733no—
512C4⋊C4—733no—
513C4⋊C4C22×#6440813yes—
514C4⋊C4C22×#6440813yes—
515C4⋊C4C22×#6440816yes—
516C22⋊C4—733no—
517C22⋊C4—733no—
518C22⋊C4—733no—
519C22⋊C4—733no—
520C22⋊C4C22×#6440813yes—
521C22⋊C4C22×#6440813yes—
522C42—733no—
523C42—733no—
524C42—733no—
525C42—733no—
526C42—733no—
527C42—733no—
528C42—733no—
529C42—733no—
530C42—733no—
531C42C22×#6440813yes—
532C23—731no—
533C23C23×#6440811yescharacteristic
534C23C23×#6440811yescharacteristic
535Q8—26no—
536Q8—26no—
537Q8—2926no—
538Q8—1462no—
539Q8—1466no—
540Q8#1455667814yes—
541D4—22no—
542D4—1462no—
543D4—1462no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—2923no—
547C2×C4—2923no—
548C2×C4—2923no—
549C2×C4—732no—
550C2×C4—732no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—731no—
554C2×C4—733no—
555C2×C4—733no—
556C2×C4—733no—
557C2×C4—733no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1466no—
563C2×C4C23×#6440811yescharacteristic
564C2×C4C23×#6440813yes—
565C2×C4C23×#6440813yes—
566C2×C4C23×#6440813yes—
567C2×C4C23×#6440813yes—
568C8—1463no—
569C8—1461no—
570C8—1461no—
571C8—1463no—
572C8—1461no—
573C8—1461no—
574C8—1463no—
575C8—1463no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—1461no—
582C22—1461no—
583C22—1461no—
584C22#1798965512yes—
585C22C24×#6440811yescharacteristic
586C4—21no—
587C4—23no—
588C4—23no—
589C4—23no—
590C4—731no—
591C4—731no—
592C4—732no—
593C4—1461no—
594C4—1463no—
595C4—1463no—
596C4—1463no—
597C4—1463no—
598C4C2×#1455667816yes—
599C2—22no—
600C2—41no—
601C2—2921no—
602C2C2×#1798965511yescharacteristic
603C2C22×#1455667811yescharacteristic
604C2#2262692711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
FittingD4×Q8×C73C4
prime radicalC2×#11924027C23
cosocleC2×Dic73C25
hypercenterD4×Q8#64408
hypocenterC73#60253
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #22626842C2semidirect
2. #22626840C2semidirect
3. #22626836C2semidirect
4. #22626835C2semidirect
5. #22626834C2semidirect
6. #22626833C2semidirect
7. #22626825C2semidirect
8. #22626824C2semidirect
9. Q8×#11924037C2semidirect
10. #22626726C2semidirect
11. #22626719C2semidirect
12. C2×Q8×#10596833C2semidirect
318. C73#60253semidirect

Maximal subgroups

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

subgroupindex
#226268422
#226268402
#226268362
#226268352
#226268342
#226268332
#226268252
#226268242
Q8×#119240372
#226267262
#226267192
C2×Q8×#105968332
#226266262
#226266142
Q8×#119240272
#6025373