Finite Group Databasebeta

← #33951007

Subgroups — #33951007

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
1Q8×#11924028C211yesmaximal, characteristic
2#22619032C213yesmaximal, complemented
3#22619029C213yesmaximal, complemented
4#22619028C213yesmaximal, complemented
5#22619027C213yesmaximal, complemented
6#22619019C211yesmaximal, characteristic, complemented
7#22619012C211yesmaximal, characteristic, complemented
8#22619010C211yesmaximal, characteristic, complemented
9Q8×#11924011C211yesmaximal, characteristic, complemented
10#22618879C213yesmaximal, complemented
11#22618862C213yesmaximal, complemented
12C8×Q8×D73C211yesmaximal, characteristic, complemented
13#22618598C213yesmaximal
14#22618570C213yesmaximal
15C73×#2984C211yesmaximal, characteristic, complemented
16Q8×Dic146C411yescharacteristic
17#17989315C413yes—
18#17989297C413yes—
19#17989292C413yes—
20#17989292C413yes—
21Q8×D292C411yescharacteristic
22Q8×C22⋊D73C411yescharacteristic
23Q8×C22⋊D73C411yescharacteristic
24#17989144C2213yes—
25#17989141C2213yes—
26#17989141C2213yes—
27C4×#11924028C2213yes—
28C4×Q8×D73C2211yescharacteristic
29C4×Q8×D73C2211yescharacteristic
30#17988717C2213yes—
31#17988717C2213yes—
32#17988715C2216yes—
33#17988715C2216yes—
34#17988714C2213yes—
35#17988714C2213yes—
36#17988714C2216yes—
37Q8×#10596831C2211yescharacteristic
38Q8×#10596831C2211yescharacteristic
39Q8×#10596831C2213yes—
40Q8×#10596831C2213yes—
41C8×Dic146C2213yes—
42C8×Dic146C2213yes—
43#17988711C2213yes—
44#17988711C2213yes—
45#17988711C2216yes—
46#17988649C2216yes—
47#17988647C2213yes—
48#17988647C2213yes—
49#17988645C2213yes—
50#17988645C2216yes—
51#17988645C2213yes—
52#17988643C2213yes—
53#17988640C2213yes—
54#17988640C2213yes—
55#17988639C2213yes—
56#17988638C2213yes—
57#17988638C2213yes—
58#17988612C2213yes—
59#17988612C2213yes—
60#17988609C2213yes—
61C4⋊C8×D73C2213yes—
62#17988552C2211yescharacteristic
63#17988552C2213yes—
64#17988544C2216yes—
65C4×#11924011C2213yes—
66C4×C8×D73C2213yes—
67C2×C4×Q8×C73C2211yescharacteristic
68#446×C73C2213yes—
69#446×C73C2213yes—
70C8×Q8×C73C2211yescharacteristic
71C8×Q8×C73C2211yescharacteristic
72#432×C73C2213yes—
73#431×C73C2213yes—
74C2×#11924028C2311yescharacteristic
75C2×Q8×D73C2×C411yescharacteristic
76C2×Q8×D73C2×C411yescharacteristic
77C2×#11924011C2313yes—
78C2×#11924011C2311yescharacteristic
79C2×C8×D73C2311yescharacteristic
80C2×C8×D73C2313yes—
81#14556591C2×C413yes—
82#14556591C2×C413yes—
83#14556590C2×C413yes—
84#14556590C2×C413yes—
85#14556589C2×C413yes—
86#14556589C2×C413yes—
87#14556583C2×C416yes—
88#14556581C2×C416yes—
89#14556581C2×C416yes—
90#14556571C2×C413yes—
91#14556571C2×C413yes—
92#14556570C2×C413yes—
93#14556570C2×C413yes—
94#14556570C2×C416yes—
95#14556569C2×C416yes—
96#14556569C2×C416yes—
97#14556567C2×C413yes—
98#14556567C2×C413yes—
99Q8×Dic73C2×C411yescharacteristic
100Q8×Dic73C2×C411yescharacteristic
101C4×Dic146C2×C413yes—
102C4×Dic146C2×C413yes—
103C4×Dic146C2316yes—
104C4×Dic146C2313yes—
105C4×Dic146C2313yes—
106#14556556C2×C413yes—
107#14556556C2×C413yes—
108C4×D292C2×C413yes—
109C4×D292C2×C413yes—
110C4×C22⋊D73C2×C413yes—
111C4×C22⋊D73C2×C413yes—
112C4×C22⋊D73C2×C413yes—
113C4×C22⋊D73C2×C413yes—
114#14556550C2313yes—
115#14556548C2313yes—
116#14556548C2313yes—
117C4⋊C4×D73C2×C413yes—
118C4⋊C4×D73C2×C413yes—
119C42×D73C2313yes—
120C42×D73C2313yes—
121C4⋊#10596831C2313yes—
122C4⋊#10596831C2313yes—
123C4⋊#10596831C2313yes—
124C4⋊#10596831C2313yes—
125#14556520C2316yes—
126#14556520C2313yes—
127#14556520C2313yes—
128#14556520C2313yes—
129#14556520C2313yes—
130#14556520C2316yes—
131C22⋊#10596831C2311yescharacteristic
132C22⋊#10596831C2313yes—
133#14556503C2311yescharacteristic
134#14556503C2311yescharacteristic
135#14556503C2313yes—
136#14556503C2313yes—
137#14556501C2316yes—
138#14556501C2313yes—
139#14556501C2313yes—
140C4×#10596831C2313yes—
141C4×#10596831C2313yes—
142C4×#10596831C2316yes—
143C8×Dic73C2316yes—
144C8×Dic73C2313yes—
145C8×Dic73C2313yes—
146C22×Q8×C73C2×C411yescharacteristic
147C4×Q8×C73C2×C411yescharacteristic
148C4×Q8×C73C2×C413yes—
149C4×Q8×C73C2313yes—
150C4×Q8×C73C2311yescharacteristic
151C2×C4⋊C4×C73C2×C413yes—
152C2×C42×C73C2313yes—
153C4⋊C8×C73C2313yes—
154C4⋊C8×C73C2313yes—
155C4⋊C8×C73C2313yes—
156C4⋊C8×C73C2313yes—
157C22⋊C8×C73C2311yescharacteristic
158C22⋊C8×C73C2313yes—
159#97×C73C2316yes—
160C4×C8×C73C2313yes—
161C4×C8×C73C2313yes—
162#11924028—22no—
163#11924028—22no—
164C2×Dic146C22×C413yes—
165C2×Dic146C22×C413yes—
166C2×Dic146C22×C411yescharacteristic
167C2×Dic146C22×C416yes—
168Q8×D73—24no—
169Q8×D73—24no—
170C2×D292C22×C411yescharacteristic
171C2×C22⋊D73C22×C411yescharacteristic
172C2×C22⋊D73C22×C411yescharacteristic
173C2×C4×D73C22×C413yes—
174C2×C4×D73C22×C413yes—
175C2×C4×D73C22×C413yes—
176C2×C4×D73C22×C413yes—
177C2×C4×D73C2411yescharacteristic
178C2×C4×D73C2411yescharacteristic
179#11924011—22no—
180#11924011—26no—
181C2×#10596831C2413yes—
182C2×#10596831C2411yescharacteristic
183C2×#10596831C2411yescharacteristic
184C2×#10596831C2413yes—
185C8×D73—26no—
186C8×D73—22no—
187C4⋊Dic73C22×C413yes—
188C4⋊Dic73C22×C413yes—
189C4⋊Dic73C22×C413yes—
190C4⋊Dic73C22×C413yes—
191C4⋊Dic73C22×C416yes—
192#11924007C22×C416yes—
193#11924007C22×C416yes—
194#11924007C22×C413yes—
195#11924007C22×C413yes—
196#11924007C22×C413yes—
197#11924007C22×C413yes—
198#11924007C22×C416yes—
199#11924007C22×C416yes—
200C22⋊Dic73C22×C413yes—
201C22⋊Dic73C22×C413yes—
202#11924005C22×C413yes—
203#11924005C22×C413yes—
204#11924005C22×C413yes—
205#11924005C22×C413yes—
206C4×Dic73C22×C413yes—
207C4×Dic73C22×C413yes—
208C4×Dic73C22×C413yes—
209C4×Dic73C22×C413yes—
210C4×Dic73C2413yes—
211C4×Dic73C2413yes—
212C2×Q8×C73—24no—
213C2×Q8×C73C22×C413yes—
214C2×Q8×C73C22×C411yescharacteristic
215C22×C4×C73C22×C413yes—
216C22×C4×C73C22×C413yes—
217C22×C4×C73C2411yescharacteristic
218C2×C8×C73C2411yescharacteristic
219C2×C8×C73C2411yescharacteristic
220C2×C8×C73C2413yes—
221C2×C8×C73C2413yes—
222C4⋊C4×C73C22×C413yes—
223C4⋊C4×C73C22×C416yes—
224C4⋊C4×C73C22×C413yes—
225C42×C73C22×C413yes—
226C42×C73C22×C413yes—
227C42×C73C2413yes—
228C42×C73C2413yes—
229C23×C73C23×C411yescharacteristic
230C22×D73C23×C411yescharacteristic
231C22×D73C23×C411yescharacteristic
232Q8×C73—26no—
233Q8×C73#13114yes—
234C2×C4×C73—26no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—43no—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C23×C413yes—
240C2×C4×C73C23×C413yes—
241C2×C4×C73C23×C413yes—
242C2×C4×C73C23×C413yes—
243C2×C4×C73C2511yescharacteristic
244C22⋊D73—22no—
245C22⋊D73—22no—
246C2×Dic73C23×C411yescharacteristic
247C2×Dic73C23×C413yes—
248C2×Dic73C23×C413yes—
249C2×Dic73C23×C413yes—
250C2×Dic73C23×C413yes—
251C2×Dic73C23×C411yescharacteristic
252D292—22no—
253C4×D73—26no—
254C4×D73—26no—
255C4×D73—21no—
256C4×D73—21no—
257C4×D73—43no—
258C4×D73—43no—
259C4×D732−1+412yes—
260C4×D732−1+412yes—
261Dic146—26no—
262Dic146—26no—
263Dic146—22no—
264Dic146—46no—
265C8×C73—21no—
266C8×C73—23no—
267C8×C73—21no—
268C8×C73—23no—
269#10596831—23no—
270#10596831—21no—
271#10596831—21no—
272#10596831—23no—
273C22×C73—21no—
274C22×C73—21no—
275C22×C73—21no—
276C22×C73C24×C411yescharacteristic
277C2×D73—21no—
278C2×D73—21no—
279C2×D73—21no—
280C2×D73—21no—
281C2×D73—21no—
282C2×D73#51912yes—
283C4×C73—21no—
284C4×C73—23no—
285C4×C73—23no—
286C4×C73—23no—
287C4×C73C2×#13116yes—
288C4×C73C2×2−1+411yescharacteristic
289C4×C73C2×2−1+411yescharacteristic
290Dic73—21no—
291Dic73—23no—
292Dic73—23no—
293Dic73—23no—
294Dic73—23no—
295Dic73#51912yes—
296#60253—731nomaximal, Sylow
297C2×C73—41no—
298C2×C73C2×#51911yescharacteristic
299C2×C73C22×#13111yescharacteristic
300C2×C73#357311yescharacteristic
301D73—22no—
302D73—41no—
303Q8×#41—731no—
304#2986—733no—
305#2986—733no—
306#2985—733no—
307#2985—733no—
308#2984—731no—
309#2984D7311yescharacteristic, complemented
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—731no—
327#533—733no—
328#533—733no—
329#533—733no—
330C4×#41—733no—
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8C2×D7311yescharacteristic
334#446—733no—
335#446—736no—
336#446—733no—
337#446—736no—
338#446—736no—
339#446—733no—
340#446—733no—
341#446C2×D7313yes—
342#446C2×D7313yes—
343C8×Q8—731no—
344C8×Q8—733no—
345C8×Q8—733no—
346C8×Q8—731no—
347C8×Q8—733no—
348C8×Q8—733no—
349C8×Q8—736no—
350C8×Q8—733no—
351C8×Q8—733no—
352C8×Q8C2×D7311yescharacteristic
353C8×Q8C2×D7311yescharacteristic
354#433—733no—
355#433—733no—
356#433—733no—
357#433—733no—
358#433—736no—
359#433—736no—
360#432—733no—
361#432C2×D7313yes—
362#431—733no—
363#431—733no—
364#431—733no—
365#431—733no—
366#431—733no—
367#431C2×D7313yes—
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×Q8—731no—
380C22×Q8C4×D7311yescharacteristic
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⋊Q8—733no—
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—733no—
400C22⋊Q8—733no—
401C22⋊Q8—736no—
402C22⋊Q8—736no—
403C4×Q8—731no—
404C4×Q8—731no—
405C4×Q8—733no—
406C4×Q8—733no—
407C4×Q8—733no—
408C4×Q8—733no—
409C4×Q8—733no—
410C4×Q8—733no—
411C4×Q8—736no—
412C4×Q8C4×D7311yescharacteristic
413C4×Q8C4×D7313yes—
414C4×Q8C22×D7311yescharacteristic
415C4×Q8C22×D7313yes—
416C4×D4—733no—
417C4×D4—733no—
418C4×D4—733no—
419C4×D4—733no—
420C4×D4—733no—
421C4×D4—733no—
422#117—733no—
423#117—733no—
424#117—733no—
425C2×C4⋊C4—733no—
426C2×C4⋊C4—733no—
427C2×C4⋊C4C4×D7313yes—
428C2×C42—733no—
429C2×C42—733no—
430C2×C42C22×D7313yes—
431C4⋊C8—736no—
432C4⋊C8—736no—
433C4⋊C8—733no—
434C4⋊C8—733no—
435C4⋊C8—733no—
436C4⋊C8—733no—
437C4⋊C8—733no—
438C4⋊C8—733no—
439C4⋊C8—733no—
440C4⋊C8—733no—
441C4⋊C8C22×D7313yes—
442C4⋊C8C22×D7313yes—
443C4⋊C8C22×D7313yes—
444C4⋊C8C22×D7313yes—
445C22⋊C8—731no—
446C22⋊C8—733no—
447C22⋊C8—733no—
448C22⋊C8—731no—
449C22⋊C8—731no—
450C22⋊C8—733no—
451C22⋊C8C22×D7311yescharacteristic
452C22⋊C8C22×D7313yes—
453#97—733no—
454#97—733no—
455#97—736no—
456#97C22×D7316yes—
457C4×C8—733no—
458C4×C8—733no—
459C4×C8—736no—
460C4×C8—733no—
461C4×C8—733no—
462C4×C8—736no—
463C4×C8C22×D7313yes—
464C4×C8C22×D7313yes—
465#41—1462no—
466#41—1462no—
467C2×Q8—24no—
468C2×Q8—736no—
469C2×Q8—731no—
470C2×Q8—733no—
471C2×Q8—733no—
472C2×Q8—1464no—
473C2×Q8—1464no—
474C2×Q8C2×C4×D7313yes—
475C2×Q8C2×C4×D7311yescharacteristic
476C2×D4—731no—
477C2×D4—731no—
478C2×D4—731no—
479C22×C4—733no—
480C22×C4—733no—
481C22×C4—731no—
482C22×C4—733no—
483C22×C4—733no—
484C22×C4—731no—
485C22×C4C2×C4×D7313yes—
486C22×C4C2×C4×D7313yes—
487C22×C4C23×D7311yescharacteristic
488M16—1462no—
489M16—1466no—
490C2×C8—733no—
491C2×C8—731no—
492C2×C8—731no—
493C2×C8—733no—
494C2×C8—1466no—
495C2×C8—1462no—
496C2×C8C23×D7311yescharacteristic
497C2×C8C23×D7313yes—
498C2×C8C23×D7313yes—
499C2×C8C23×D7311yescharacteristic
500C4⋊C4—736no—
501C4⋊C4—736no—
502C4⋊C4—733no—
503C4⋊C4—733no—
504C4⋊C4—733no—
505C4⋊C4—733no—
506C4⋊C4—736no—
507C4⋊C4—736no—
508C4⋊C4—733no—
509C4⋊C4—733no—
510C4⋊C4—733no—
511C4⋊C4—733no—
512C4⋊C4—736no—
513C4⋊C4C2×C4×D7313yes—
514C4⋊C4C2×C4×D7316yes—
515C4⋊C4C2×C4×D7313yes—
516C22⋊C4—733no—
517C22⋊C4—733no—
518C22⋊C4—733no—
519C22⋊C4—733no—
520C22⋊C4—733no—
521C22⋊C4—733no—
522C42—733no—
523C42—733no—
524C42—733no—
525C42—733no—
526C42—733no—
527C42—733no—
528C42C2×C4×D7313yes—
529C42C2×C4×D7313yes—
530C42C23×D7313yes—
531C42C23×D7313yes—
532C23—731no—
533C23—731no—
534C23C22×C4×D7311yescharacteristic
535Q8—26no—
536Q8—2926no—
537Q8—1462no—
538Q8—1466no—
539Q8—1466no—
540Q8#1455659914yes—
541D4—1462no—
542D4—1462no—
543D4—1462no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—26no—
547C2×C4—43no—
548C2×C4—2923no—
549C2×C4—2923no—
550C2×C4—731no—
551C2×C4—732no—
552C2×C4—732no—
553C2×C4—731no—
554C2×C4—733no—
555C2×C4—733no—
556C2×C4—733no—
557C2×C4—733no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1466no—
561C2×C4—1466no—
562C2×C4C22×C4×D7313yes—
563C2×C4C22×C4×D7311yescharacteristic
564C2×C4C22×C4×D7313yes—
565C2×C4C22×C4×D7313yes—
566C2×C4C22×C4×D7313yes—
567C2×C4C24×D7311yescharacteristic
568C8—21no—
569C8—23no—
570C8—23no—
571C8—21no—
572C8—1463no—
573C8—1461no—
574C8—1461no—
575C8—1463no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—732no—
580C22—1461no—
581C22—1461no—
582C22—1461no—
583C22—1461no—
584C22—1461no—
585C22C23×C4×D7311yescharacteristic
586C4—23no—
587C4—23no—
588C4—23no—
589C4—21no—
590C4—732no—
591C4—1463no—
592C4—1463no—
593C4—1463no—
594C4—1463no—
595C4—1461no—
596C4C2×#1455659916yes—
597C42−1+4×D7311yescharacteristic
598C42−1+4×D7311yescharacteristic
599C2—41no—
600C2—2921no—
601C2—1462no—
602C2#519×D7311yescharacteristic
603C2C22×#1455659911yescharacteristic
604C2#2262464211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
FittingC73×#2984C2
prime radicalC2×#11924028C23
cosocleC2×C4×C73C25
hypercenter#2984D73
hypocenterC73#60253
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #22619032C2semidirect
3. #22619029C2semidirect
4. #22619028C2semidirect
5. #22619027C2semidirect
6. #22619019C2semidirect
7. #22619012C2semidirect
8. #22619010C2semidirect
9. Q8×#11924011C2semidirect
10. #22618879C2semidirect
11. #22618862C2semidirect
12. C8×Q8×D73C2semidirect
15. C73×#2984C2semidirect
309. #2984D73semidirect
318. C73#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#119240282
#226190322
#226190292
#226190282
#226190272
#226190192
#226190122
#226190102
Q8×#119240112
#226188792
#226188622
C8×Q8×D732
#226185982
#226185702
C73×#29842
#6025373