Finite Group Databasebeta

← C2×#87291 #11703360

Subgroups — C2×#87291 #11703360

Order 768.

11210 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic, complemented
2C22×#1007C211yesmaximal, characteristic, complemented
3#87291C2132yesmaximal, direct product factor
4C2×#5834C214yesmaximal, complemented
5C2×#5831C214yesmaximal, complemented
6C2×#5833C216yesmaximal, complemented
7C2×#5832C216yesmaximal, complemented
8C2×#5817C214yesmaximal, complemented
9C2×Q8×#67C214yesmaximal, complemented
10C22×Q8×Dic3C211yesmaximal, characteristic, complemented
11C2×#3497—31nomaximal, Sylow
12C2×C3×2−1+4—216no—
13C2×C3×2−1+4C22116yescomplemented
14C22×C3×#41C2216yes—
15C22×C3×#41C2214yes—
16C23×C3×Q8C2214yes—
17C23×C3×Q8C2211yescharacteristic
18C2×#1007—216no—
19C2×#1007C22116yescomplemented
20C22×#239C2214yes—
21C22×D3×Q8C2214yes—
22C22×#235C2216yes—
23#5834C22164yescomplemented
24#5833C22196yescomplemented
25#5832C22196yescomplemented
26#5831C22164yescomplemented
27#5817C22164yescomplemented
28Q8×#67C22164yescomplemented
29C2×#947C2216yes—
30C2×#946C22112yes—
31C2×Q8×Dic3C2214yes—
32C2×Q8×Dic3C2214yes—
33C2×Q8×Dic3C22116yescomplemented
34C2×#944C2216yes—
35C2×#940C2216yes—
36C2×#939C2214yes—
37C2×#939C22112yes—
38C2×#938C22112yes—
39C2×#936C2216yes—
40C2×#930C22112yes—
41C2×C4×#67C22112yes—
42C2×C4×#67C22112yes—
43C22×C4⋊Dic3C2213yes—
44C2×#924C22112yes—
45C22×C4×Dic3C2213yes—
46C22×2−1+4—31no—
47C22×2−1+4D311yescharacteristic, complemented
48#3497—332no—
49C2×#550—34no—
50C2×D4×Q8—34no—
51C2×#548—34no—
52C2×#548—34no—
53C2×#537—36no—
54C2×#532—36no—
55C22×C4×Q8—31no—
56C3×2−1+4—2128no—
57C2×C3×#41—232no—
58C2×C3×#41—248no—
59C2×C3×#41C23132yes—
60C2×C3×#41C23148yes—
61C22×C3×Q8—232no—
62C22×C3×Q8D4116yescomplemented
63C22×C3×Q8C2316yes—
64C22×C3×Q8C23132yes—
65C22×C3×Q8C2314yes—
66C22×C3×Q8C2318yes—
67C22×C3×D4C2316yes—
68C22×C3×D4C2314yes—
69C23×C3×C4C23112yes—
70C23×C3×C4C2313yes—
71C22×#67C2314yes—
72C22×#67C2314yes—
73C22×#67C23112yes—
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2313yes—
76#1007—2128no—
77C2×#239—232no—
78C2×#239C23132yes—
79C2×D3×Q8—232no—
80C2×D3×Q8C23132yes—
81C2×#235—248no—
82C2×#235C23148yes—
83C22×D12C2316yes—
84C22×C4×D3C23112yes—
85C22×Dic6C2316yes—
86C2×#217C23112yes—
87C2×#217C2314yes—
88#947C23148yes—
89#946C23196yes—
90Q8×Dic3C23132yes—
91Q8×Dic3C23132yes—
92#944C23148yes—
93#940C23148yes—
94#939C23132yes—
95#939C23196yes—
96#938C23196yes—
97#936C23148yes—
98#930C23196yes—
99C4×#67C23196yes—
100C4×#67C23196yes—
101C2×(C2×D3)⋊C4C2314yes—
102C2×(C2×D3)⋊C4C23112yes—
103C2×C4⋊Dic3C23112yes—
104C2×C4⋊Dic3C23124yes—
105#924C23196yes—
106C2×Dic3⋊C4C23112yes—
107C2×Dic3⋊C4C2314yes—
108C2×C4×Dic3C2316yes—
109C2×C4×Dic3C2316yes—
110C2×C4×Dic3C23124yes—
111C2×2−1+4—216no—
112C2×2−1+4—316no—
113C2×2−1+4—616no—
114C2×2−1+4C2×D3116yescomplemented
115C22×#41—34no—
116C22×#41—36no—
117C22×#41C2×D316yes—
118C22×#41C2×D314yes—
119C23×Q8—34no—
120C23×Q8C2×D314yes—
121C23×Q8C2×D311yescharacteristic
122#550—364no—
123D4×Q8—364no—
124#548—364no—
125#548—364no—
126#537—396no—
127#532—396no—
128C2×C4⋊Q8—36no—
129C2×#127—36no—
130C2×#124—36no—
131C2×#124—36no—
132C2×C22⋊Q8—312no—
133C2×C22⋊Q8—312no—
134C2×#121—34no—
135C2×#121—312no—
136C2×#121—312no—
137C2×#121—312no—
138C2×C4×Q8—34no—
139C2×C4×Q8—34no—
140C2×C4×Q8—316no—
141C2×C4×D4—312no—
142C2×C4×D4—312no—
143C22×C4⋊C4—33no—
144C22×C42—33no—
145C24×C3C2414yes—
146C24×C3C2411yescharacteristic
147C23×D3C2414yes—
148C3×#41—2128no—
149C3×#41—2192no—
150C2×C3×Q8—2128no—
151C2×C3×Q8—224no—
152C2×C3×Q8C2×D4132yes—
153C2×C3×Q8C2×D4164yescomplemented
154C2×C3×Q8C24124yes—
155C2×C3×Q8C24116yes—
156C2×C3×D4—216no—
157C2×C3×D4—224no—
158C2×C3×D4C24124yes—
159C2×C3×D4C24116yes—
160C22×C3×C4—248no—
161C22×C3×C4C2×D4124yes—
162C22×C3×C4C2414yes—
163C22×C3×C4C24148yes—
164C22×C3×C4C2416yes—
165C22×C3×C4C24112yes—
166C2×#67—216no—
167C2×#67—216no—
168C2×#67—424no—
169C2×#67C24116yes—
170C2×#67C24116yes—
171C2×#67C24148yes—
172C22×Dic3—24no—
173C22×Dic3—46no—
174C22×Dic3C2411yescharacteristic
175C22×Dic3C2414yes—
176C22×Dic3C2416yes—
177C22×Dic3C2414yes—
178C22×Dic3C24112yes—
179C22×Dic3C2411yescharacteristic
180#239—2128no—
181D3×Q8—2128no—
182#235—2192no—
183C2×D12—224no—
184C2×D12C24124yes—
185C2×C4×D3—248no—
186C2×C4×D3C24148yes—
187C2×Dic6—224no—
188C2×Dic6C24124yes—
189#217C24148yes—
190#217C24116yes—
191(C2×D3)⋊C4C24116yes—
192(C2×D3)⋊C4C24148yes—
193C4⋊Dic3C24148yes—
194Dic3⋊C4C24148yes—
195Dic3⋊C4C24116yes—
196C4×Dic3C24124yes—
197C4×Dic3C24124yes—
1982−1+4—2128no—
1992−1+4—6128no—
200C2×#41—232no—
201C2×#41—248no—
202C2×#41—332no—
203C2×#41—348no—
204C2×#41—632no—
205C2×#41—648no—
206C2×#41C22×D3148yes—
207C2×#41C22×D3132yes—
208C22×Q8—232no—
209C22×Q8—332no—
210C22×Q8—36no—
211C22×Q8—632no—
212C22×Q8#67116yescomplemented
213C22×Q8C22×D316yes—
214C22×Q8C22×D3132yes—
215C22×Q8C22×D314yes—
216C22×Q8C22×D318yes—
217C22×D4—34no—
218C22×D4—34no—
219C22×D4—312no—
220C22×D4—36no—
221C22×D4C22×D314yes—
222C22×D4C22×D316yes—
223C23×C4—31no—
224C23×C4—33no—
225C23×C4—312no—
226C23×C4C22×D3112yes—
227C23×C4C22×D313yes—
228C4⋊Q8—348no—
229#127—348no—
230#124—348no—
231#124—348no—
232C22⋊Q8—396no—
233C22⋊Q8—396no—
234#121—332no—
235#121—396no—
236#121—396no—
237#121—396no—
238C4×Q8—332no—
239C4×Q8—332no—
240C4×D4—396no—
241C4×D4—396no—
242C2×C4⋊C4—312no—
243C2×C4⋊C4—324no—
244C2×C4⋊C4—34no—
245C2×C4⋊C4—312no—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—312no—
248C2×C22⋊C4—34no—
249C2×C22⋊C4—312no—
250C2×C42—36no—
251C2×C42—36no—
252C2×C42—324no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—28no—
256C23×C3—28no—
257C23×C3—28no—
258C23×C3C22×D414yes—
259C23×C3C2518yes—
260C23×C3C2512yes—
261C23×C3C2511yescharacteristic
262C22×D3—28no—
263C22×D3—28no—
264C22×D3—28no—
265C22×D3C2518yes—
266C3×Q8—248no—
267C3×Q8C22×D4164yes—
268C3×D4—232no—
269C3×D4—248no—
270C2×C3×C4—28no—
271C2×C3×C4—296no—
272C2×C3×C4C22×D4124yes—
273C2×C3×C4C22×D4148yes—
274C2×C3×C4C2518yes—
275C2×C3×C4C25112yes—
276#67—232no—
277#67—232no—
278#67—448no—
279C2×Dic3—28no—
280C2×Dic3—22no—
281C2×Dic3—212no—
282C2×Dic3—28no—
283C2×Dic3—412no—
284C2×Dic32−1+414yescomplemented
285C2×Dic3C2512yes—
286C2×Dic3C25112yes—
287C2×Dic3C2518yes—
288C2×Dic3C2512yes—
289D12—248no—
290C4×D3—296no—
291Dic6—248no—
292C24—34no—
293C24C23×D314yes—
294C24C23×D311yescharacteristic
295#41—2128no—
296#41—2192no—
297#41—6128no—
298#41—6192no—
299C2×Q8—2128no—
300C2×Q8—224no—
301C2×Q8—324no—
302C2×Q8—6128no—
303C2×Q8—624no—
304C2×Q8C2×#67132yes—
305C2×Q8C2×#67164yescomplemented
306C2×Q8C23×D3124yes—
307C2×Q8C23×D3116yes—
308C2×D4—216no—
309C2×D4—224no—
310C2×D4—316no—
311C2×D4—316no—
312C2×D4—348no—
313C2×D4—324no—
314C2×D4—616no—
315C2×D4—616no—
316C2×D4—624no—
317C2×D4—1224no—
318C2×D4C23×D3116yes—
319C2×D4C23×D3124yes—
320C22×C4—248no—
321C22×C4—31no—
322C22×C4—34no—
323C22×C4—31no—
324C22×C4—36no—
325C22×C4—34no—
326C22×C4—312no—
327C22×C4—348no—
328C22×C4—64no—
329C22×C4—648no—
330C22×C4—126no—
331C22×C4C2×#67124yes—
332C22×C4C23×D3148yes—
333C22×C4C23×D314yes—
334C22×C4C23×D3112yes—
335C22×C4C23×D316yes—
336C4⋊C4—348no—
337C4⋊C4—316no—
338C4⋊C4—348no—
339C22⋊C4—316no—
340C22⋊C4—348no—
341C22⋊C4—316no—
342C22⋊C4—348no—
343C42—324no—
344C42—324no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—28no—
348C22×C3—28no—
349C22×C3—28no—
350C22×C3—28no—
351C22×C3—416no—
352C22×C3C23×D414yes—
353C22×C3C23×D412yes—
354C22×C3C2×2−1+412yes—
355C22×C3#58512yes—
356C22×C3C2611yescharacteristic
357C2×D3—28no—
358C2×D3—28no—
359C2×D3—28no—
360C2×D3—416no—
361C3×C4—28no—
362C3×C4C23×D4124yes—
363Dic3—28no—
364Dic3—22no—
365Dic3—212no—
366Dic3C2×2−1+414yescomplemented
367C23—22no—
368C23—22no—
369C23—28no—
370C23—28no—
371C23—28no—
372C23—38no—
373C23—68no—
374C23—68no—
375C23—68no—
376C23C22×#6714yes—
377C23C24×D318yes—
378C23C24×D312yes—
379C23C24×D311yescharacteristic
380Q8—248no—
381Q8—648no—
382Q8C22×#67164yes—
383D4—232no—
384D4—248no—
385D4—632no—
386D4—632no—
387D4—648no—
388D4—1248no—
389C2×C4—28no—
390C2×C4—296no—
391C2×C4—34no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—312no—
395C2×C4—38no—
396C2×C4—68no—
397C2×C4—62no—
398C2×C4—612no—
399C2×C4—68no—
400C2×C4—696no—
401C2×C4—1212no—
402C2×C4C22×#67124yes—
403C2×C4C22×#67148yes—
404C2×C4C24×D318yes—
405C2×C4C24×D3112yes—
406C2×C3—24no—
407C2×C3—48no—
408C2×C3#349714yescomplemented
409C2×C3C24×D411yescharacteristic
410C2×C3C22×2−1+411yescharacteristic
411C2×C3C2×#58511yescharacteristic
412D3—48no—
413C22—22no—
414C22—22no—
415C22—28no—
416C22—28no—
417C22—28no—
418C22—28no—
419C22—416no—
420C22—68no—
421C22—68no—
422C22—68no—
423C22—1216no—
424C22D3×2−1+412yes—
425C22#596812yes—
426C22C23×#6714yes—
427C22C23×#6712yes—
428C22C25×D311yescharacteristic
429C4—28no—
430C4—34no—
431C4—612no—
432C4—62no—
433C4—68no—
434C4C23×#67124yes—
435C3C2×#349711yescharacteristic, complemented, Sylow
436C2—24no—
437C2—48no—
438C2—128no—
439C2#8729114yesdirect product factor
440C2C2×#596811yescharacteristic
441C2C2×D3×2−1+411yescharacteristic
442C2C24×#6711yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×2−1+4C2
socleC23×C3C25
hypercenterC22×2−1+4D3
hypocenterC3C2×#3497
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×2−1+4

Complements

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

NH ≅ G/Ntype
1. C22×C3×2−1+4C2semidirect
2. C22×#1007C2semidirect
3. #87291C2direct (N × H)
4. C2×#5834C2semidirect
5. C2×#5831C2semidirect
6. C2×#5833C2semidirect
7. C2×#5832C2semidirect
8. C2×#5817C2semidirect
9. C2×Q8×#67C2semidirect
10. C22×Q8×Dic3C2semidirect
13. C2×C3×2−1+4C22semidirect
19. C2×#1007C22semidirect
23. #5834C22semidirect
24. #5833C22semidirect
25. #5832C22semidirect
26. #5831C22semidirect
27. #5817C22semidirect
28. Q8×#67C22semidirect
33. C2×Q8×Dic3C22semidirect
47. C22×2−1+4D3semidirect
62. C22×C3×Q8D4semidirect
114. C2×2−1+4C2×D3semidirect
153. C2×C3×Q8C2×D4semidirect
212. C22×Q8#67semidirect
284. C2×Dic32−1+4semidirect
305. C2×Q8C2×#67semidirect
366. Dic3C2×2−1+4semidirect
408. C2×C3#3497semidirect
435. C3C2×#3497semidirect
439. C2#87291direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C22×#10072
#872912
C2×#58342
C2×#58312
C2×#58332
C2×#58322
C2×#58172
C2×Q8×#672
C22×Q8×Dic32
C2×#34973