Finite Group Databasebeta

← C2×#22602346 #32905710

Subgroups — C2×#22602346 #32905710

Order 18176.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22602346C2132yesmaximal, direct product factor
2C2×#17986728C211yesmaximal, characteristic, complemented
3C2×#17986693C216yesmaximal, complemented
4C2×#17986671C211yesmaximal, characteristic, complemented
5C2×#17986670C213yesmaximal, complemented
6C2×#17986668C212yesmaximal, complemented
7C2×#17986654C213yesmaximal, complemented
8C2×#17986600C216yesmaximal, complemented
9C2×#17986549C213yesmaximal, complemented
10C2×#17986533C213yesmaximal, complemented
11C2×#17986481C212yesmaximal, complemented
12C2×C4×#41×C71C211yesmaximal, characteristic, complemented
13C22×#11765866C2212yes—
14C22×#11765865C2216yes—
15#17986728C22116yescomplemented
16#17986693C22196yescomplemented
17#17986671C22116yescomplemented
18#17986670C22148yescomplemented
19#17986668C22132yescomplemented
20#17986654C22148yescomplemented
21#17986600C22196yescomplemented
22#17986549C22148yescomplemented
23C2×#14556128C2213yes—
24C2×#14556125C2213yes—
25C2×#14556124C2216yes—
26#17986533C22148yescomplemented
27C2×#14556122C2216yes—
28C2×#14556120C2211yescharacteristic
29C2×#14556119C2216yes—
30C2×#14556116C2213yes—
31C2×#14556111C2216yes—
32C2×#14556110C22112yes—
33C2×#14556107C2213yes—
34C2×#14556103C2216yes—
35C2×#14556102C2216yes—
36C2×#14556102C2213yes—
37C2×#14556102C2216yes—
38C2×#14556101C22112yes—
39C2×#14556099C2216yes—
40#17986481C22132yescomplemented
41C2×Q8×Dic71C2212yes—
42C2×C4×Dic142C2213yes—
43C2×C4×D284C2216yes—
44C2×C4×D284C2213yes—
45C2×C4×D284C2216yes—
46C2×#14556090C2216yes—
47C2×C4×C22⋊D71C2216yes—
48C2×#14556085C2216yes—
49C2×C4⋊C4×D71C2216yes—
50C22×#41×C71C2211yescharacteristic
51C4×#41×C71C22116yescomplemented
52C2×C4×Q8×C71C2211yescharacteristic
53C2×C4×Q8×C71C2211yescharacteristic
54C2×C4×D4×C71C2213yes—
55C2×C4×D4×C71C2213yes—
56C2×#117×C71C2213yes—
57C22×C42×C71C2213yes—
58C2×#11765866—216no—
59C2×#11765866C23116yes—
60C2×#11765865—424no—
61C2×#11765865C23148yes—
62C22×Dic142C2313yes—
63C22×D284C2313yes—
64C22×D284C2316yes—
65C22×D284C2316yes—
66C22×C22⋊D71C23112yes—
67C22×C22⋊D71C2316yes—
68C22×C4×D71C2316yes—
69C22×C4×D71C23112yes—
70#14556128C23124yes—
71#14556125C23124yes—
72#14556124C23148yes—
73#14556122C23148yes—
74#14556120C2318yes—
75#14556119C23148yes—
76#14556116C23124yes—
77#14556111C23148yes—
78#14556110C23196yes—
79#14556107C23124yes—
80#14556103C23148yes—
81#14556102C23148yes—
82#14556102C23148yes—
83#14556102C23124yes—
84#14556101C23196yes—
85#14556099C23148yes—
86Q8×Dic71C23116yes—
87C4×Dic142C23124yes—
88C4×D284C23148yes—
89C4×D284C23124yes—
90C4×D284C23148yes—
91#14556090C23148yes—
92C4×C22⋊D71C23148yes—
93#14556085C23148yes—
94C2×C4⋊Dic71C2316yes—
95C2×C4⋊Dic71C2311yescharacteristic
96C2×C4⋊Dic71C2316yes—
97C2×C4⋊Dic71C2311yescharacteristic
98C2×Dic71⋊C4C2316yes—
99C2×Dic71⋊C4C2312yes—
100C2×Dic71⋊C4C2312yes—
101C2×Dic71⋊C4C2316yes—
102C4⋊C4×D71C23148yes—
103C2×C22⋊Dic71C2316yes—
104C2×(C2×D71)⋊C4C2316yes—
105C2×(C2×D71)⋊C4C2316yes—
106C2×(C2×D71)⋊C4C2316yes—
107C2×(C2×D71)⋊C4C2316yes—
108C2×C4×Dic71C2316yes—
109C2×#41×C71—28no—
110C2×#41×C71C2318yes—
111C22×Q8×C71C2311yescharacteristic
112C22×D4×C71C2313yes—
113C23×C4×C71C2313yes—
114C23×C4×C71C2313yes—
115C23×C4×C71C2313yes—
116C4×Q8×C71C2318yes—
117C4×Q8×C71C2318yes—
118C4×D4×C71C23124yes—
119C4×D4×C71C23124yes—
120#117×C71C23124yes—
121C2×C4⋊C4×C71C2313yes—
122C2×C4⋊C4×C71C2313yes—
123C2×C22⋊C4×C71C2316yes—
124C2×C42×C71C2311yescharacteristic
125C2×C42×C71C2313yes—
126C2×C42×C71C2313yes—
127C2×C42×C71C2313yes—
128C2×C42×C71C23124yes—
129C23×D71C2416yes—
130#11765866—264no—
131#11765865—496no—
132C2×Dic142—212no—
133C2×Dic142C24112yes—
134C2×D284—224no—
135C2×D284—212no—
136C2×D284—412no—
137C2×D284C24112yes—
138C2×D284C24124yes—
139C2×D284C24124yes—
140C2×C22⋊D71—412no—
141C2×C22⋊D71—424no—
142C2×C22⋊D71C24148yes—
143C2×C22⋊D71C24124yes—
144C22×Dic71C2412yes—
145C22×Dic71C2416yes—
146C22×Dic71C2412yes—
147C2×C4×D71—224no—
148C2×C4×D71—424no—
149C2×C4×D71C24124yes—
150C2×C4×D71C24148yes—
151C4⋊Dic71C24124yes—
152C4⋊Dic71C2414yes—
153C4⋊Dic71C24124yes—
154C4⋊Dic71C2414yes—
155Dic71⋊C4C24124yes—
156Dic71⋊C4C2418yes—
157Dic71⋊C4C2418yes—
158Dic71⋊C4C24124yes—
159C22⋊Dic71C24124yes—
160(C2×D71)⋊C4C24124yes—
161(C2×D71)⋊C4C24124yes—
162(C2×D71)⋊C4C24124yes—
163(C2×D71)⋊C4C24124yes—
164C4×Dic71C24124yes—
165C24×C71C2413yes—
166#41×C71—232no—
167C2×Q8×C71#4118yescomplemented
168C2×Q8×C71C2414yes—
169C2×D4×C71—212no—
170C2×D4×C71C24112yes—
171C22×C4×C71—212no—
172C22×C4×C71—212no—
173C22×C4×C71—46no—
174C22×C4×C71C2413yes—
175C22×C4×C71C2413yes—
176C22×C4×C71C24112yes—
177C22×C4×C71C2411yescharacteristic
178C22×C4×C71C2411yescharacteristic
179C22×C4×C71C24112yes—
180C22×C4×C71C24112yes—
181C22×C4×C71C2413yes—
182C22×C4×C71C2411yescharacteristic
183C4⋊C4×C71C24112yes—
184C4⋊C4×C71C24112yes—
185C22⋊C4×C71C24124yes—
186C42×C71C2414yes—
187C42×C71C24112yes—
188C42×C71C24112yes—
189C42×C71C24112yes—
190C23×C71—26no—
191C23×C71—26no—
192C23×C71—26no—
193C23×C71C2516yes—
194C23×C71C2511yescharacteristic
195C22×D71—212no—
196C22×D71—212no—
197C22×D71—212no—
198C22×D71C25112yes—
199Q8×C71C2×#41116yescomplemented
200D4×C71—224no—
201C2×C4×C71—22no—
202C2×C4×C71—26no—
203C2×C4×C71—22no—
204C2×C4×C71—26no—
205C2×C4×C71—224no—
206C2×C4×C71—224no—
207C2×C4×C71—412no—
208C2×C4×C71C2×#41112yes—
209C2×C4×C71H(5,𝔽2)14yes—
210C2×C4×C71C2512yes—
211C2×C4×C71C2512yes—
212C2×C4×C71C2516yes—
213C2×C4×C71C2516yes—
214C2×C4×C71C2516yes—
215C2×C4×C71C2512yes—
216C22⋊D71—424no—
217C22⋊D71—448no—
218C2×Dic71—24no—
219C2×Dic71—212no—
220C2×Dic71—24no—
221C2×Dic71C2514yes—
222C2×Dic71C25112yes—
223C2×Dic71C2514yes—
224D284—248no—
225D284—224no—
226D284—424no—
227C4×D71—248no—
228C4×D71—448no—
229Dic142—224no—
230C22×C71—26no—
231C22×C71—26no—
232C22×C71—26no—
233C22×C71—412no—
234C22×C71C22×#4112yes—
235C22×C71C2×H(5,𝔽2)12yes—
236C22×C71#58512yes—
237C22×C71C2611yescharacteristic
238C2×D71—212no—
239C2×D71—212no—
240C2×D71—212no—
241C2×D71—424no—
242C4×C71—22no—
243C4×C71—26no—
244C4×C71—26no—
245C4×C71—22no—
246C4×C71C22×#41112yes—
247C4×C71C2×H(5,𝔽2)14yes—
248Dic71—24no—
249Dic71—212no—
250Dic71—24no—
251C2×#3517—711nomaximal, Sylow
252C2×C71—46no—
253C2×C71#351714yescomplemented
254C2×C71C23×#4111yescharacteristic
255C2×C71C22×H(5,𝔽2)11yescharacteristic
256C2×C71C2×#58511yescharacteristic
257D71—412no—
258#3517—7132no—
259C2×#557—711no—
260C2×#553—716no—
261C2×#550—712no—
262C2×#550—711no—
263C2×#550—713no—
264C2×#547—713no—
265C2×#540—716no—
266C2×#532—713no—
267C2×#529—713no—
268C2×#520—712no—
269C2×C4×#41D7111yescharacteristic, complemented
270C71C2×#351711yescharacteristic, complemented, Sylow
271C22×#41—712no—
272C22×#41—716no—
273C22×#41C2×D7111yescharacteristic
274#557—7116no—
275#553—7196no—
276#550—7132no—
277#550—7116no—
278#550—7148no—
279#547—7148no—
280#540—7196no—
281#532—7148no—
282C2×C4⋊Q8—713no—
283C2×#127—716no—
284C2×#127—713no—
285#529—7148no—
286C2×#126—716no—
287C2×#125—716no—
288C2×#125—711no—
289C2×#124—713no—
290C2×#123—716no—
291C2×#123—7112no—
292C2×C22⋊Q8—713no—
293C2×#121—716no—
294C2×#121—713no—
295C2×#121—716no—
296C2×#121—716no—
297C2×#121—716no—
298C2×#121—7112no—
299#520—7132no—
300C4×#41C2×D71116yescomplemented
301C2×C4×Q8—712no—
302C2×C4×Q8—713no—
303C2×C4×Q8C2×D7111yescharacteristic
304C2×C4×Q8C2×D7111yescharacteristic
305C2×C4×D4—716no—
306C2×C4×D4—716no—
307C2×C4×D4—716no—
308C2×C4×D4—713no—
309C2×C4×D4—716no—
310C2×C4×D4C2×D7113yes—
311C2×C4×D4C2×D7113yes—
312C2×#117—716no—
313C2×#117C2×D7113yes—
314C22×C4⋊C4—716no—
315C22×C42C2×D7113yes—
316C2×#41—28no—
317C2×#41—28424no—
318C2×#41—7116no—
319C2×#41—7148no—
320C2×#41—14216no—
321C2×#41C22×D7118yes—
322C22×Q8—713no—
323C22×Q8C22×D7111yescharacteristic
324C22×D4—716no—
325C22×D4—7112no—
326C22×D4—716no—
327C22×D4—716no—
328C22×D4—713no—
329C22×D4C22×D7113yes—
330C23×C4—7112no—
331C23×C4—716no—
332C23×C4C22×D7113yes—
333C23×C4C22×D7113yes—
334C23×C4C22×D7113yes—
335C4⋊Q8—7124no—
336#127—7148no—
337#127—7124no—
338#126—7148no—
339#125—7148no—
340#125—718no—
341#124—7124no—
342#123—7148no—
343#123—7196no—
344C22⋊Q8—7124no—
345#121—7148no—
346#121—7124no—
347#121—7148no—
348#121—7148no—
349#121—7148no—
350#121—7196no—
351C4×Q8—7116no—
352C4×Q8—7124no—
353C4×Q8C22×D7118yes—
354C4×Q8C22×D7118yes—
355C4×D4—7148no—
356C4×D4—7148no—
357C4×D4—7148no—
358C4×D4—7124no—
359C4×D4—7148no—
360C4×D4C22×D71124yes—
361C4×D4C22×D71124yes—
362#117—7148no—
363#117C22×D71124yes—
364C2×C4⋊C4—716no—
365C2×C4⋊C4—716no—
366C2×C4⋊C4—712no—
367C2×C4⋊C4—711no—
368C2×C4⋊C4—712no—
369C2×C4⋊C4—716no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—716no—
372C2×C4⋊C4—7148no—
373C2×C4⋊C4C22×D7113yes—
374C2×C4⋊C4C22×D7113yes—
375C2×C22⋊C4—716no—
376C2×C22⋊C4—716no—
377C2×C22⋊C4—716no—
378C2×C22⋊C4—716no—
379C2×C22⋊C4—716no—
380C2×C22⋊C4C22×D7116yes—
381C2×C42—716no—
382C2×C42C22×D7113yes—
383C2×C42C22×D7113yes—
384C2×C42C22×D7113yes—
385C2×C42C22×D71124yes—
386C2×C42C22×D7111yescharacteristic
387C24—716no—
388C24C23×D7113yes—
389#41—232no—
390#41—28496no—
391#41—14264no—
392C2×Q8—7112no—
393C2×Q8—14212no—
394C2×Q8#1176586518yescomplemented
395C2×Q8C23×D7114yes—
396C2×D4—212no—
397C2×D4—28412no—
398C2×D4—28412no—
399C2×D4—28424no—
400C2×D4—7124no—
401C2×D4—7148no—
402C2×D4—7124no—
403C2×D4—7124no—
404C2×D4—7112no—
405C2×D4—14224no—
406C2×D4—14212no—
407C2×D4C23×D71112yes—
408C22×C4—212no—
409C22×C4—212no—
410C22×C4—46no—
411C22×C4—28424no—
412C22×C4—7148no—
413C22×C4—712no—
414C22×C4—716no—
415C22×C4—712no—
416C22×C4—7124no—
417C22×C4—14224no—
418C22×C4C23×D71112yes—
419C22×C4C23×D71112yes—
420C22×C4C23×D7111yescharacteristic
421C22×C4C23×D7111yescharacteristic
422C22×C4C23×D7113yes—
423C22×C4C23×D7113yes—
424C22×C4C23×D71112yes—
425C22×C4C23×D7113yes—
426C22×C4C23×D7111yescharacteristic
427C4⋊C4—7124no—
428C4⋊C4—718no—
429C4⋊C4—718no—
430C4⋊C4—7124no—
431C4⋊C4—7124no—
432C4⋊C4—714no—
433C4⋊C4—7124no—
434C4⋊C4—714no—
435C4⋊C4C23×D71112yes—
436C4⋊C4C23×D71112yes—
437C22⋊C4—7124no—
438C22⋊C4—7124no—
439C22⋊C4—7124no—
440C22⋊C4—7124no—
441C22⋊C4—7124no—
442C22⋊C4C23×D71124yes—
443C42—7124no—
444C42C23×D7114yes—
445C42C23×D71112yes—
446C42C23×D71112yes—
447C42C23×D71112yes—
448C23—26no—
449C23—26no—
450C23—26no—
451C23—7112no—
452C23—14212no—
453C23—14212no—
454C23—14212no—
455C23C24×D7116yes—
456C23C24×D7111yescharacteristic
457Q8—14224no—
458Q8C2×#11765865116yescomplemented
459D4—224no—
460D4—28424no—
461D4—28448no—
462D4—28424no—
463D4—14248no—
464D4—14224no—
465C2×C4—22no—
466C2×C4—26no—
467C2×C4—22no—
468C2×C4—26no—
469C2×C4—224no—
470C2×C4—224no—
471C2×C4—412no—
472C2×C4—28448no—
473C2×C4—714no—
474C2×C4—7112no—
475C2×C4—714no—
476C2×C4—1424no—
477C2×C4—14212no—
478C2×C4—1424no—
479C2×C4—14248no—
480C2×C4C2×#11765865112yes—
481C2×C4#1455617414yes—
482C2×C4C24×D7112yes—
483C2×C4C24×D7112yes—
484C2×C4C24×D7116yes—
485C2×C4C24×D7116yes—
486C2×C4C24×D7116yes—
487C2×C4C24×D7112yes—
488C22—26no—
489C22—26no—
490C22—26no—
491C22—412no—
492C22—28424no—
493C22—14212no—
494C22—14212no—
495C22—14212no—
496C22C22×#1176586512yes—
497C22C2×#1455617412yes—
498C22#1798683512yes—
499C22C25×D7111yescharacteristic
500C4—22no—
501C4—26no—
502C4—22no—
503C4—26no—
504C4—1424no—
505C4—14212no—
506C4—1424no—
507C4C22×#11765865112yes—
508C4C2×#1455617414yes—
509C2—46no—
510C2—28412no—
511C2#2260234614yesdirect product factor
512C2C23×#1176586511yescharacteristic
513C2C22×#1455617411yescharacteristic
514C2C2×#1798683511yescharacteristic
subgroupG/H
centerC23C24×D71
commutator, cosocleC22×C71C26
FrattiniC22C25×D71
FittingC2×C4×#41×C71C2
socleC23×C71C25
hypercenterC2×C4×#41D71
hypocenterC71C2×#3517
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #22602346C2direct (N × H)
2. C2×#17986728C2semidirect
3. C2×#17986693C2semidirect
4. C2×#17986671C2semidirect
5. C2×#17986670C2semidirect
6. C2×#17986668C2semidirect
7. C2×#17986654C2semidirect
8. C2×#17986600C2semidirect
9. C2×#17986549C2semidirect
10. C2×#17986533C2semidirect
11. C2×#17986481C2semidirect
12. C2×C4×#41×C71C2semidirect
15. #17986728C22semidirect
16. #17986693C22semidirect
17. #17986671C22semidirect
18. #17986670C22semidirect
19. #17986668C22semidirect
20. #17986654C22semidirect
21. #17986600C22semidirect
22. #17986549C22semidirect
26. #17986533C22semidirect
40. #17986481C22semidirect
51. C4×#41×C71C22semidirect
167. C2×Q8×C71#41semidirect
199. Q8×C71C2×#41semidirect
253. C2×C71#3517semidirect
269. C2×C4×#41D71semidirect
270. C71C2×#3517semidirect
300. C4×#41C2×D71semidirect
394. C2×Q8#11765865semidirect
458. Q8C2×#11765865semidirect
511. C2#22602346direct (N × H)

Maximal subgroups

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

subgroupindex
#226023462
C2×#179867282
C2×#179866932
C2×#179866712
C2×#179866702
C2×#179866682
C2×#179866542
C2×#179866002
C2×#179865492
C2×#179865332
C2×#179864812
C2×C4×#41×C712
C2×#351771