Finite Group Databasebeta

← C2×#22623552 #33993073

Subgroups — C2×#22623552 #33993073

Order 18688.

7706 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623552C2132yesmaximal, direct product factor
2C2×#17989316C211yesmaximal, characteristic, complemented
3C2×#17989311C213yesmaximal, complemented
4C2×#17989279C216yesmaximal, complemented
5C2×#17989248C212yesmaximal, complemented
6C2×#17989241C213yesmaximal, complemented
7C2×#17989142C213yesmaximal, complemented
8C2×#17989123C216yesmaximal, complemented
9C2×C4×#11924029C211yesmaximal, characteristic, complemented
10C2×C4×#11924027C213yesmaximal, complemented
11C22×Q8×Dic73C212yesmaximal, complemented
12C2×D4×Q8×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2211yescharacteristic
14C22×#11924027C2213yes—
15#17989316C22116yescomplemented
16#17989311C22148yescomplemented
17#17989279C22196yescomplemented
18#17989248C22132yescomplemented
19#17989241C22148yescomplemented
20#17989142C22148yescomplemented
21C2×#14556591C2213yes—
22#17989123C22196yescomplemented
23C2×#14556586C22112yes—
24C2×#14556582C2216yes—
25C2×#14556582C2216yes—
26C2×#14556580C2216yes—
27C2×#14556575C22112yes—
28C2×#14556571C2216yes—
29C2×#14556567C2216yes—
30C2×#14556565C2216yes—
31C4×#11924029C22116yescomplemented
32C4×#11924027C22148yescomplemented
33C2×Q8×Dic73C2212yes—
34C2×Q8×Dic73C2211yescharacteristic
35C2×Q8×Dic73C2216yes—
36C2×Q8×Dic73C2216yes—
37C2×Q8×Dic73C22132yescomplemented
38C2×C4×Dic146C2213yes—
39C2×#14556556C2213yes—
40C2×C4×D292C2213yes—
41C2×D4×Dic73C2211yescharacteristic
42C2×#14556553C2213yes—
43C2×C4×C22⋊D73C2216yes—
44C2×#14556551C2216yes—
45C2×#14556549C2213yes—
46C2×#14556549C2213yes—
47C2×#14556549C2216yes—
48C2×#14556547C2216yes—
49C22×C4⋊Dic73C2216yes—
50C22×C4×Dic73C2216yes—
51C2×C42×D73C2213yes—
52C23×Q8×C73C2212yes—
53D4×Q8×C73C22116yescomplemented
54C2×C4⋊Q8×C73C2213yes—
55C2×C22⋊Q8×C73C2216yes—
56C2×C4×Q8×C73C2211yescharacteristic
57C2×C4×D4×C73C2213yes—
58C2×#11924029—28no—
59C2×#11924029C2318yes—
60C2×#11924027—224no—
61C2×#11924027C23124yes—
62C22×Dic146C2313yes—
63C22×D292C2313yes—
64C22×C22⋊D73C2316yes—
65C23×Dic73C2316yes—
66C23×Dic73C2312yes—
67C22×C4×D73C2313yes—
68C22×C4×D73C2313yes—
69C22×C4×D73C2313yes—
70#14556591C23124yes—
71#14556586C23196yes—
72#14556582C23148yes—
73#14556582C23148yes—
74#14556580C23148yes—
75#14556575C23196yes—
76#14556571C23148yes—
77#14556567C23148yes—
78#14556565C23148yes—
79Q8×Dic73C23148yes—
80Q8×Dic73C23116yes—
81Q8×Dic73C23148yes—
82Q8×Dic73C2318yes—
83C4×Dic146C23124yes—
84#14556556C23124yes—
85C4×D292C23124yes—
86D4×Dic73C2318yes—
87#14556553C23124yes—
88C4×C22⋊D73C23148yes—
89#14556551C23148yes—
90#14556549C23148yes—
91#14556549C23124yes—
92#14556549C23124yes—
93#14556547C23148yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2316yes—
96C2×C4⋊Dic73C23148yes—
97C2×C4⋊Dic73C2313yes—
98C2×C4⋊Dic73C23112yes—
99C2×C4⋊Dic73C2311yescharacteristic
100C2×#11924007C2316yes—
101C2×#11924007C2316yes—
102C2×C22⋊Dic73C2312yes—
103C2×#11924005C2316yes—
104C2×#11924005C2316yes—
105C2×#11924005C23112yes—
106C2×C4×Dic73C23148yes—
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2313yes—
109C2×C4×Dic73C2313yes—
110C2×C4×Dic73C2316yes—
111C2×C4×Dic73C2316yes—
112C2×C4×Dic73C2316yes—
113C42×D73C23124yes—
114C22×Q8×C73—216no—
115C22×Q8×C73C23116yes—
116C22×Q8×C73C2316yes—
117C22×Q8×C73C2311yescharacteristic
118C22×D4×C73C2311yescharacteristic
119C23×C4×C73C2316yes—
120C4⋊Q8×C73C23124yes—
121C22⋊Q8×C73C23148yes—
122C4×Q8×C73C2318yes—
123C4×D4×C73C23124yes—
124C2×C4⋊C4×C73C2313yes—
125C2×C4⋊C4×C73C2313yes—
126C2×C4⋊C4×C73C2316yes—
127C2×C22⋊C4×C73C2316yes—
128C2×C42×C73C2313yes—
129C23×D73C2413yes—
130#11924029—232no—
131#11924027—296no—
132C2×Dic146—212no—
133C2×Dic146C24112yes—
134C2×D292—212no—
135C2×D292C24112yes—
136C2×C22⋊D73—224no—
137C2×C22⋊D73C24124yes—
138C22×Dic73—224no—
139C22×Dic73—28no—
140C22×Dic73C2412yes—
141C22×Dic73C2418yes—
142C22×Dic73C2413yes—
143C22×Dic73C2411yescharacteristic
144C22×Dic73C2416yes—
145C22×Dic73C24124yes—
146C22×Dic73C2411yescharacteristic
147C2×C4×D73—212no—
148C2×C4×D73—212no—
149C2×C4×D73—212no—
150C2×C4×D73C24112yes—
151C2×C4×D73C24112yes—
152C2×C4×D73C24112yes—
153C4⋊Dic73C24112yes—
154C4⋊Dic73C24124yes—
155C4⋊Dic73C24148yes—
156C4⋊Dic73C2414yes—
157C4⋊Dic73C24124yes—
158#11924007C24124yes—
159#11924007C24124yes—
160C22⋊Dic73C2418yes—
161#11924005C24124yes—
162#11924005C24124yes—
163#11924005C24148yes—
164C4×Dic73C2414yes—
165C4×Dic73C24112yes—
166C4×Dic73C24112yes—
167C4×Dic73C24124yes—
168C4×Dic73C24124yes—
169C4×Dic73C24124yes—
170C24×C73C2412yes—
171C2×Q8×C73—264no—
172C2×Q8×C73—224no—
173C2×Q8×C73#4118yescomplemented
174C2×Q8×C73C24124yes—
175C2×Q8×C73C2414yes—
176C2×D4×C73#4118yescomplemented
177C2×D4×C73C2414yes—
178C22×C4×C73—224no—
179C22×C4×C73C2413yes—
180C22×C4×C73C2416yes—
181C22×C4×C73C24124yes—
182C22×C4×C73C2411yescharacteristic
183C22×C4×C73C2413yes—
184C4⋊C4×C73C24112yes—
185C4⋊C4×C73C24112yes—
186C4⋊C4×C73C24124yes—
187C22⋊C4×C73C24124yes—
188C42×C73C24112yes—
189C23×C73—24no—
190C23×C73—24no—
191C23×C73C2×#4118yes—
192C23×C73C2514yes—
193C23×C73C2511yescharacteristic
194C22×D73—26no—
195C22×D73—26no—
196C22×D73—26no—
197C22×D73C2516yes—
198Q8×C73—248no—
199Q8×C73C2×#41116yescomplemented
200D4×C73C2×#41116yescomplemented
201C2×C4×C73—212no—
202C2×C4×C73—26no—
203C2×C4×C73—248no—
204C2×C4×C73C2×#41112yes—
205C2×C4×C73C2×#4114yes—
206C2×C4×C73C2516yes—
207C2×C4×C73C25112yes—
208C2×C4×C73C2512yes—
209C2×C4×C73C2516yes—
210C22⋊D73—248no—
211C2×Dic73—22no—
212C2×Dic73—26no—
213C2×Dic73—212no—
214C2×Dic73—248no—
215C2×Dic73—24no—
216C2×Dic73—216no—
217C2×Dic732−1+414yes—
218C2×Dic73C2514yes—
219C2×Dic73C2516yes—
220C2×Dic73C2512yes—
221C2×Dic73C25112yes—
222C2×Dic73C2512yes—
223D292—224no—
224C4×D73—224no—
225C4×D73—224no—
226C4×D73—224no—
227Dic146—224no—
228C22×C73—24no—
229C22×C73—24no—
230C22×C73—216no—
231C22×C73C22×#4112yes—
232C22×C73C22×#4118yes—
233C22×C73C22×#4112yes—
234C22×C73C2×2−1+412yes—
235C22×C73C2611yescharacteristic
236C2×D73—26no—
237C2×D73—26no—
238C2×D73—26no—
239C2×D73—412no—
240C4×C73—212no—
241C4×C73—26no—
242C4×C73C22×#41112yes—
243C4×C73C22×#4114yes—
244Dic73—212no—
245Dic73—22no—
246Dic73—26no—
247Dic73—24no—
248Dic73C2×2−1+414yes—
249C2×#3482—731nomaximal, Sylow
250C2×C73—28no—
251C2×C73#348214yescomplemented
252C2×C73C23×#4111yescharacteristic
253C2×C73C23×#4111yescharacteristic
254C2×C73C22×2−1+411yescharacteristic
255D73—46no—
256#3482—7332no—
257C2×#557—731no—
258C2×#556—733no—
259C2×#552—736no—
260C2×D4×Q8D7311yescharacteristic, complemented
261C2×#548—732no—
262C2×#547—733no—
263C2×#533—733no—
264C2×#529—736no—
265C2×C4×#41—731no—
266C2×C4×#41—733no—
267C22×C4×Q8—732no—
268C73C2×#348211yescharacteristic, complemented, Sylow
269C22×#41—731no—
270C22×#41—733no—
271C23×Q8C2×D7312yes—
272#557—7316no—
273#556—7348no—
274#552—7396no—
275D4×Q8C2×D73116yescomplemented
276#548—7332no—
277#547—7348no—
278#533—7348no—
279C2×C4⋊Q8—733no—
280C2×C4⋊Q8C2×D7313yes—
281#529—7396no—
282C2×#126—7312no—
283C2×#125—736no—
284C2×#125—736no—
285C2×#124—736no—
286C2×#123—7312no—
287C2×C22⋊Q8—736no—
288C2×C22⋊Q8—736no—
289C2×C22⋊Q8C2×D7316yes—
290C2×#121—736no—
291C4×#41—7316no—
292C4×#41—7348no—
293C2×C4×Q8—731no—
294C2×C4×Q8—736no—
295C2×C4×Q8—732no—
296C2×C4×Q8—736no—
297C2×C4×Q8—7332no—
298C2×C4×Q8—733no—
299C2×C4×Q8—733no—
300C2×C4×Q8C2×D7311yescharacteristic
301C2×C4×D4—731no—
302C2×C4×D4—736no—
303C2×C4×D4—733no—
304C2×C4×D4—736no—
305C2×C4×D4—733no—
306C2×C4×D4C2×D7313yes—
307C2×#117—733no—
308C2×#117—733no—
309C2×#117—736no—
310C2×#117—736no—
311C22×C4⋊C4—736no—
312C22×C42—736no—
313C22×C42—733no—
314C2×#41—738no—
315C2×#41—7324no—
316C2×#41—1468no—
317C2×#41—14624no—
318C22×Q8—216no—
319C22×Q8—733no—
320C22×Q8C22×D7316yes—
321C22×Q8C22×D73116yes—
322C22×Q8C22×D7311yescharacteristic
323C22×D4—736no—
324C22×D4—733no—
325C22×D4C22×D7311yescharacteristic
326C23×C4—736no—
327C23×C4—732no—
328C23×C4—733no—
329C23×C4—733no—
330C23×C4—733no—
331C23×C4C22×D7316yes—
332C4⋊Q8—7324no—
333C4⋊Q8C22×D73124yes—
334#126—7396no—
335#125—7348no—
336#125—7348no—
337#124—7348no—
338#123—7396no—
339C22⋊Q8—7348no—
340C22⋊Q8—7348no—
341C22⋊Q8C22×D73148yes—
342#121—7348no—
343C4×Q8—738no—
344C4×Q8—7348no—
345C4×Q8—7316no—
346C4×Q8—7348no—
347C4×Q8—7324no—
348C4×Q8—7324no—
349C4×Q8C22×D7318yes—
350C4×D4—738no—
351C4×D4—7348no—
352C4×D4—7324no—
353C4×D4—7348no—
354C4×D4—7324no—
355C4×D4C22×D73124yes—
356#117—7324no—
357#117—7324no—
358#117—7348no—
359#117—7348no—
360C2×C4⋊C4—733no—
361C2×C4⋊C4—736no—
362C2×C4⋊C4—7312no—
363C2×C4⋊C4—7348no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—736no—
366C2×C4⋊C4—736no—
367C2×C4⋊C4—736no—
368C2×C4⋊C4C22×D7313yes—
369C2×C4⋊C4C22×D7313yes—
370C2×C4⋊C4C22×D7316yes—
371C2×C22⋊C4—732no—
372C2×C22⋊C4—736no—
373C2×C22⋊C4—736no—
374C2×C22⋊C4—7312no—
375C2×C22⋊C4C22×D7316yes—
376C2×C42—733no—
377C2×C42—736no—
378C2×C42—736no—
379C2×C42—736no—
380C2×C42—733no—
381C2×C42—731no—
382C2×C42—7348no—
383C2×C42—7324no—
384C2×C42C22×D7313yes—
385C24—733no—
386C24C23×D7312yes—
387#41—14632no—
388#41—14696no—
389C2×Q8—264no—
390C2×Q8—224no—
391C2×Q8—7312no—
392C2×Q8—14612no—
393C2×Q8#1192402718yescomplemented
394C2×Q8C23×D73124yes—
395C2×Q8C23×D7314yes—
396C2×D4—7324no—
397C2×D4—7312no—
398C2×D4—14624no—
399C2×D4—14612no—
400C2×D4#1192402918yescomplemented
401C2×D4C23×D7314yes—
402C22×C4—224no—
403C22×C4—731no—
404C22×C4—738no—
405C22×C4—7324no—
406C22×C4—732no—
407C22×C4—731no—
408C22×C4—733no—
409C22×C4—736no—
410C22×C4—7312no—
411C22×C4—7312no—
412C22×C4—7312no—
413C22×C4—14612no—
414C22×C4—1468no—
415C22×C4—14624no—
416C22×C4—14612no—
417C22×C4—14612no—
418C22×C4C23×D73124yes—
419C22×C4C23×D7316yes—
420C22×C4C23×D7313yes—
421C22×C4C23×D7313yes—
422C22×C4C23×D7311yescharacteristic
423C4⋊C4—7324no—
424C4⋊C4—7324no—
425C4⋊C4—7312no—
426C4⋊C4—7324no—
427C4⋊C4—7348no—
428C4⋊C4—734no—
429C4⋊C4—7324no—
430C4⋊C4C23×D73112yes—
431C4⋊C4C23×D73112yes—
432C4⋊C4C23×D73124yes—
433C22⋊C4—738no—
434C22⋊C4—7324no—
435C22⋊C4—7324no—
436C22⋊C4—7348no—
437C22⋊C4C23×D73124yes—
438C42—7312no—
439C42—734no—
440C42—7312no—
441C42—7324no—
442C42—7324no—
443C42—7324no—
444C42C23×D73112yes—
445C23—24no—
446C23—24no—
447C23—736no—
448C23—1466no—
449C23—1466no—
450C23—1466no—
451C23C2×#1192402918yes—
452C23C24×D7314yes—
453C23C24×D7311yescharacteristic
454Q8—248no—
455Q8—14624no—
456Q8C2×#11924027116yescomplemented
457D4—14648no—
458D4—14624no—
459D4C2×#11924029116yescomplemented
460C2×C4—212no—
461C2×C4—26no—
462C2×C4—248no—
463C2×C4—734no—
464C2×C4—732no—
465C2×C4—734no—
466C2×C4—732no—
467C2×C4—736no—
468C2×C4—7312no—
469C2×C4—14624no—
470C2×C4—14616no—
471C2×C4—1462no—
472C2×C4—1466no—
473C2×C4—14612no—
474C2×C4—14648no—
475C2×C4—1464no—
476C2×C4—14624no—
477C2×C4—14624no—
478C2×C4C2×#11924027112yes—
479C2×C4C2×#1192402914yes—
480C2×C4C24×D73112yes—
481C2×C4C24×D7316yes—
482C2×C4C24×D7316yes—
483C2×C4C24×D7312yes—
484C22—216no—
485C22—24no—
486C22—24no—
487C22—29212no—
488C22—1466no—
489C22—1466no—
490C22—1466no—
491C22C22×#1192402712yes—
492C22C22×#1192402918yes—
493C22C22×#1192402912yes—
494C222−1+4×D7312yes—
495C22C25×D7311yescharacteristic
496C4—212no—
497C4—26no—
498C4—734no—
499C4—1462no—
500C4—1466no—
501C4—14612no—
502C4—1464no—
503C4C22×#11924027112yes—
504C4C22×#1192402914yes—
505C2—28no—
506C2—2926no—
507C2#2262355214yesdirect product factor
508C2C23×#1192402711yescharacteristic
509C2C23×#1192402911yescharacteristic
510C2C2×2−1+4×D7311yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×D4×Q8×C73C2
socleC23×C73C25
hypercenterC2×D4×Q8D73
hypocenterC73C2×#3482
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #22623552C2direct (N × H)
2. C2×#17989316C2semidirect
3. C2×#17989311C2semidirect
4. C2×#17989279C2semidirect
5. C2×#17989248C2semidirect
6. C2×#17989241C2semidirect
7. C2×#17989142C2semidirect
8. C2×#17989123C2semidirect
9. C2×C4×#11924029C2semidirect
10. C2×C4×#11924027C2semidirect
11. C22×Q8×Dic73C2semidirect
12. C2×D4×Q8×C73C2semidirect
15. #17989316C22semidirect
16. #17989311C22semidirect
17. #17989279C22semidirect
18. #17989248C22semidirect
19. #17989241C22semidirect
20. #17989142C22semidirect
22. #17989123C22semidirect
31. C4×#11924029C22semidirect
32. C4×#11924027C22semidirect
37. C2×Q8×Dic73C22semidirect
53. D4×Q8×C73C22semidirect
173. C2×Q8×C73#41semidirect
176. C2×D4×C73#41semidirect
199. Q8×C73C2×#41semidirect
200. D4×C73C2×#41semidirect
251. C2×C73#3482semidirect
260. C2×D4×Q8D73semidirect
268. C73C2×#3482semidirect
275. D4×Q8C2×D73semidirect
393. C2×Q8#11924027semidirect
400. C2×D4#11924029semidirect
456. Q8C2×#11924027semidirect
459. D4C2×#11924029semidirect
507. C2#22623552direct (N × H)

Maximal subgroups

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

subgroupindex
#226235522
C2×#179893162
C2×#179893112
C2×#179892792
C2×#179892482
C2×#179892412
C2×#179891422
C2×#179891232
C2×C4×#119240292
C2×C4×#119240272
C22×Q8×Dic732
C2×D4×Q8×C732
C2×#348273