Finite Group Databasebeta

← C2×#85804 #11703410

Subgroups — C2×#85804 #11703410

Order 768.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×Q8×#67C212yesmaximal, complemented
3C2×Q8×D12C211yesmaximal, characteristic, complemented
4C2×#5583C216yesmaximal, complemented
5C2×#5572C212yesmaximal, complemented
6#85804C2132yesmaximal, direct product factor
7C2×#5567C213yesmaximal, complemented
8C2×#5570C211yesmaximal, characteristic, complemented
9C2×#5584C213yesmaximal, complemented
10C2×#5513C216yesmaximal, complemented
11C2×#5479C213yesmaximal
12C2×#5470C213yesmaximal, complemented
13C2×#3523—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C22×D3×Q8C2212yes—
16C2×C3×C4×Q8C2213yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2213yes—
20C2×C3×C4×Q8C22116yescomplemented
21C22×C3×C4⋊C4C2213yes—
22C22×C3×C42C2213yes—
23Q8×#67C22132yescomplemented
24C2×#946C2216yes—
25C2×Q8×Dic3C2212yes—
26C2×#944C2216yes—
27C2×#930C2211yescharacteristic
28C2×#929C2216yes—
29C2×C4×#67C2216yes—
30C2×#926C2216yes—
31C2×#924C2213yes—
32C2×#924C2216yes—
33#5584C22148yescomplemented
34#5583C22196yescomplemented
35Q8×D12C22116yescomplemented
36#5572C22132yescomplemented
37#5570C22116yescomplemented
38#5567C22148yescomplemented
39#5513C22196yescomplemented
40C2×#898C22112yes—
41C2×#897C2216yes—
42C2×#892C2216yes—
43C2×#890C2216yes—
44C2×#889C22112yes—
45C2×#887C2216yes—
46#5479C22148yes—
47C2×#876C2216yes—
48C2×#875C2213yes—
49C2×C4×D12C2213yes—
50C2×#872C2216yes—
51#5470C22148yescomplemented
52C2×#870C2213yes—
53C2×#870C2216yes—
54C2×#869C2213yes—
55C2×#869C2213yes—
56C2×C4×Dic6C2216yes—
57C2×C4×Dic6C2213yes—
58C2×C4×Dic6C2216yes—
59#3523—332no—
60C2×#557—31no—
61C2×#557—33no—
62C2×#556—33no—
63C2×#552—36no—
64C2×D4×Q8—32no—
65C2×D4×Q8—31no—
66C2×#541—36no—
67C2×#533—33no—
68C2×#529—33no—
69C2×#519—32no—
70C22×C4×Q8D311yescharacteristic, complemented
71C22×C3×Q8—28no—
72C22×C3×Q8C2313yes—
73C22×C3×Q8C2318yes—
74C22×C3×Q8C2311yescharacteristic
75C23×C3×C4C2313yes—
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2313yes—
78C22×#67C2312yes—
79C2×D3×Q8—216no—
80C2×D3×Q8C23116yes—
81C22×D12C2311yescharacteristic
82C22×C4×D3C2316yes—
83C22×Dic6C2316yes—
84C22×Dic6C2313yes—
85C3×C4×Q8C23124yes—
86C3×C4×Q8C2318yes—
87C3×C4×Q8C2318yes—
88C3×C4×Q8C23124yes—
89C2×C3×C4⋊C4C2313yes—
90C2×C3×C4⋊C4C2316yes—
91C2×C3×C4⋊C4C2313yes—
92C2×C3×C4⋊C4C23124yes—
93C2×C3×C42C2313yes—
94C2×C3×C42C2313yes—
95C2×C3×C42C2313yes—
96C2×C3×C42C2313yes—
97C2×C3×C42C23124yes—
98C2×#217C2316yes—
99#946C23148yes—
100Q8×Dic3C23116yes—
101#944C23148yes—
102#930C2318yes—
103#929C23148yes—
104C4×#67C23148yes—
105C2×(C2×D3)⋊C4C2312yes—
106C2×(C2×D3)⋊C4C2316yes—
107C2×(C2×D3)⋊C4C2316yes—
108C2×(C2×D3)⋊C4C2316yes—
109#926C23148yes—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2313yes—
112C2×C4⋊Dic3C2316yes—
113C2×C4⋊Dic3C2316yes—
114C2×C4⋊Dic3C2316yes—
115#924C23124yes—
116#924C23148yes—
117C2×Dic3⋊C4C2316yes—
118C2×Dic3⋊C4C23112yes—
119C2×Dic3⋊C4C2316yes—
120C2×Dic3⋊C4C2316yes—
121C2×Dic3⋊C4C2316yes—
122C2×C4×Dic3C23112yes—
123C2×C4×Dic3C2316yes—
124#898C23196yes—
125#897C23148yes—
126#892C23148yes—
127#890C23148yes—
128#889C23196yes—
129#887C23148yes—
130#876C23148yes—
131#875C23124yes—
132C4×D12C23124yes—
133#872C23148yes—
134#870C23148yes—
135#870C23124yes—
136#869C23124yes—
137#869C23124yes—
138C4×Dic6C23148yes—
139C4×Dic6C23148yes—
140C4×Dic6C23124yes—
141C23×Q8—32no—
142C23×Q8C2×D311yescharacteristic
143#557—348no—
144#557—316no—
145#556—348no—
146#552—396no—
147D4×Q8—332no—
148D4×Q8—316no—
149#541—396no—
150#533—348no—
151C2×C4⋊Q8—36no—
152C2×C4⋊Q8—33no—
153C2×C4⋊Q8—33no—
154#529—348no—
155C2×#126—312no—
156C2×#126—36no—
157C2×#125—312no—
158C2×#125—36no—
159C2×#125—33no—
160C2×#125—36no—
161C2×#124—33no—
162C2×#123—36no—
163C2×C22⋊Q8—36no—
164C2×C22⋊Q8—33no—
165C2×C22⋊Q8—36no—
166C2×C22⋊Q8—36no—
167C2×#121—31no—
168#519—332no—
169C2×C4×Q8—32no—
170C2×C4×Q8—36no—
171C2×C4×Q8—36no—
172C2×C4×Q8—33no—
173C2×C4×Q8—36no—
174C2×C4×Q8C2×D311yescharacteristic
175C2×C4×Q8C2×D313yes—
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×Q8C2×D313yes—
178C2×C4×Q8C2×D3116yescomplemented
179C2×C4×D4—36no—
180C2×C4×D4—33no—
181C2×#117—36no—
182C2×#117—36no—
183C2×#117—36no—
184C22×C4⋊C4C2×D313yes—
185C22×C42C2×D313yes—
186C24×C3C2411yescharacteristic
187C23×D3C2412yes—
188C2×C3×Q8—232no—
189C2×C3×Q8—212no—
190C2×C3×Q8#4118yes—
191C2×C3×Q8C24112yes—
192C2×C3×Q8C2414yes—
193C22×C3×C4—212no—
194C22×C3×C4—212no—
195C22×C3×C4—24no—
196C22×C3×C4C2413yes—
197C22×C3×C4C2411yescharacteristic
198C22×C3×C4C2413yes—
199C22×C3×C4C2413yes—
200C22×C3×C4C2414yes—
201C22×C3×C4C24112yes—
202C22×C3×C4C24112yes—
203C22×C3×C4C2413yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—28no—
206C2×#67C2418yes—
207C22×Dic3C2412yes—
208C22×Dic3C2416yes—
209C22×Dic3C2416yes—
210D3×Q8—264no—
211C2×D12—24no—
212C2×D12C2414yes—
213C2×C4×D3—224no—
214C2×C4×D3C24124yes—
215C2×Dic6—224no—
216C2×Dic6—212no—
217C2×Dic6C24112yes—
218C2×Dic6C24124yes—
219C3×C4⋊C4C24112yes—
220C3×C4⋊C4C24124yes—
221C3×C4⋊C4C24112yes—
222C3×C42C24112yes—
223C3×C42C24112yes—
224C3×C42C24112yes—
225C3×C42C24112yes—
226#217C24124yes—
227(C2×D3)⋊C4C2418yes—
228(C2×D3)⋊C4C24124yes—
229(C2×D3)⋊C4C24124yes—
230(C2×D3)⋊C4C24124yes—
231C4⋊Dic3C2414yes—
232C4⋊Dic3C24112yes—
233C4⋊Dic3C24124yes—
234C4⋊Dic3C24124yes—
235C4⋊Dic3C24124yes—
236Dic3⋊C4C24124yes—
237Dic3⋊C4C24148yes—
238Dic3⋊C4C24124yes—
239Dic3⋊C4C24124yes—
240Dic3⋊C4C24124yes—
241C4×Dic3C24148yes—
242C4×Dic3C24124yes—
243C22×Q8—28no—
244C22×Q8—316no—
245C22×Q8—33no—
246C22×Q8—36no—
247C22×Q8—616no—
248C22×Q8C22×D313yes—
249C22×Q8C22×D318yes—
250C22×Q8C22×D311yescharacteristic
251C22×D4—32no—
252C22×D4—31no—
253C23×C4—36no—
254C23×C4C22×D313yes—
255C23×C4C22×D313yes—
256C23×C4C22×D311yescharacteristic
257C4⋊Q8—348no—
258C4⋊Q8—324no—
259C4⋊Q8—324no—
260#126—396no—
261#126—348no—
262#125—396no—
263#125—348no—
264#125—324no—
265#125—348no—
266#124—324no—
267#123—348no—
268C22⋊Q8—348no—
269C22⋊Q8—324no—
270C22⋊Q8—348no—
271C22⋊Q8—348no—
272#121—38no—
273C4×Q8—316no—
274C4×Q8—348no—
275C4×Q8—348no—
276C4×Q8—324no—
277C4×Q8—348no—
278C4×Q8C22×D318yes—
279C4×Q8C22×D3124yes—
280C4×Q8C22×D318yes—
281C4×Q8C22×D3124yes—
282C4×D4—348no—
283C4×D4—324no—
284#117—348no—
285#117—348no—
286#117—348no—
287C2×C4⋊C4—36no—
288C2×C4⋊C4—36no—
289C2×C4⋊C4—36no—
290C2×C4⋊C4—36no—
291C2×C4⋊C4—36no—
292C2×C4⋊C4—33no—
293C2×C4⋊C4—36no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—312no—
296C2×C4⋊C4—36no—
297C2×C4⋊C4C22×D313yes—
298C2×C4⋊C4C22×D313yes—
299C2×C4⋊C4C22×D316yes—
300C2×C4⋊C4C22×D3124yes—
301C2×C22⋊C4—36no—
302C2×C22⋊C4—32no—
303C2×C22⋊C4—36no—
304C2×C22⋊C4—36no—
305C2×C22⋊C4—36no—
306C2×C42—312no—
307C2×C42—36no—
308C2×C42C22×D313yes—
309C2×C42C22×D313yes—
310C2×C42C22×D313yes—
311C2×C42C22×D313yes—
312C2×C42C22×D3124yes—
313C23×C3—22no—
314C23×C3—22no—
315C23×C32−1+414yes—
316C23×C3C2512yes—
317C23×C3C2511yescharacteristic
318C22×D3—24no—
319C22×D3—24no—
320C22×D3—24no—
321C22×D3C2514yes—
322C3×Q8—224no—
323C3×Q8C2×#41116yes—
324C2×C3×C4—26no—
325C2×C3×C4—22no—
326C2×C3×C4—224no—
327C2×C3×C4—26no—
328C2×C3×C4—26no—
329C2×C3×C4—224no—
330C2×C3×C4—28no—
331C2×C3×C4C2×#41112yes—
332C2×C3×C42−1+414yes—
333C2×C3×C4C2516yes—
334C2×C3×C4C2512yes—
335C2×C3×C4C2516yes—
336C2×C3×C4C2516yes—
337C2×C3×C4C2516yes—
338C2×C3×C4C2512yes—
339#67—216no—
340C2×Dic3—212no—
341C2×Dic3—212no—
342C2×Dic3—24no—
343C2×Dic3C2514yes—
344C2×Dic3C25112yes—
345C2×Dic3C25112yes—
346D12—28no—
347C4×D3—248no—
348Dic6—248no—
349Dic6—224no—
350C24—32no—
351C24C23×D311yescharacteristic
352C2×Q8—232no—
353C2×Q8—212no—
354C2×Q8—312no—
355C2×Q8—324no—
356C2×Q8—664no—
357C2×Q8—624no—
358C2×Q8—612no—
359C2×Q8#23518yes—
360C2×Q8C23×D3112yes—
361C2×Q8C23×D314yes—
362C2×D4—38no—
363C2×D4—34no—
364C2×D4—68no—
365C2×D4—64no—
366C22×C4—212no—
367C22×C4—212no—
368C22×C4—24no—
369C22×C4—36no—
370C22×C4—32no—
371C22×C4—36no—
372C22×C4—324no—
373C22×C4—624no—
374C22×C4C23×D314yes—
375C22×C4C23×D3112yes—
376C22×C4C23×D3112yes—
377C22×C4C23×D313yes—
378C22×C4C23×D311yescharacteristic
379C22×C4C23×D313yes—
380C22×C4C23×D313yes—
381C22×C4C23×D313yes—
382C22×C4C23×D311yescharacteristic
383C4⋊C4—324no—
384C4⋊C4—324no—
385C4⋊C4—324no—
386C4⋊C4—324no—
387C4⋊C4—324no—
388C4⋊C4—312no—
389C4⋊C4—324no—
390C4⋊C4—34no—
391C4⋊C4—348no—
392C4⋊C4—324no—
393C4⋊C4C23×D3112yes—
394C4⋊C4C23×D3112yes—
395C4⋊C4C23×D3124yes—
396C22⋊C4—324no—
397C22⋊C4—38no—
398C22⋊C4—324no—
399C22⋊C4—324no—
400C22⋊C4—324no—
401C42—348no—
402C42—324no—
403C42C23×D3112yes—
404C42C23×D3112yes—
405C42C23×D3112yes—
406C42C23×D3112yes—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—28no—
410C22×C3C22×#4112yes—
411C22×C3C2×2−1+412yes—
412C22×C3C2×2−1+414yes—
413C22×C3C2×2−1+412yes—
414C22×C3C2611yescharacteristic
415C2×D3—24no—
416C2×D3—24no—
417C2×D3—24no—
418C2×D3—48no—
419C3×C4—26no—
420C3×C4—22no—
421C3×C4—26no—
422C3×C4—26no—
423C3×C4C22×#41112yes—
424C3×C4C2×2−1+414yes—
425Dic3—212no—
426Dic3—212no—
427Dic3—24no—
428C23—22no—
429C23—22no—
430C23—34no—
431C23—64no—
432C23—64no—
433C23—64no—
434C23#101014yes—
435C23C24×D312yes—
436C23C24×D311yescharacteristic
437Q8—224no—
438Q8—648no—
439Q8—624no—
440Q8C2×#235116yes—
441D4—616no—
442D4—68no—
443C2×C4—26no—
444C2×C4—22no—
445C2×C4—224no—
446C2×C4—26no—
447C2×C4—26no—
448C2×C4—224no—
449C2×C4—28no—
450C2×C4—312no—
451C2×C4—34no—
452C2×C4—312no—
453C2×C4—612no—
454C2×C4—612no—
455C2×C4—64no—
456C2×C4—648no—
457C2×C4C2×#235112yes—
458C2×C4#100714yes—
459C2×C4C24×D316yes—
460C2×C4C24×D312yes—
461C2×C4C24×D316yes—
462C2×C4C24×D316yes—
463C2×C4C24×D316yes—
464C2×C4C24×D312yes—
465C2×C3—24no—
466C2×C3#352314yescomplemented
467C2×C3C23×#4111yescharacteristic
468C2×C3C22×2−1+411yescharacteristic
469C2×C3C22×2−1+411yescharacteristic
470D3—44no—
471C22—22no—
472C22—22no—
473C22—28no—
474C22—64no—
475C22—64no—
476C22—64no—
477C22—128no—
478C22C22×#23512yes—
479C22C2×#100712yes—
480C22C2×#101014yes—
481C22C2×#101012yes—
482C22C25×D311yescharacteristic
483C4—26no—
484C4—22no—
485C4—26no—
486C4—26no—
487C4—612no—
488C4—612no—
489C4—64no—
490C4C22×#235112yes—
491C4C2×#100714yes—
492C3C2×#352311yescharacteristic, complemented, Sylow
493C2—24no—
494C2—124no—
495C2#8580414yesdirect product factor
496C2C23×#23511yescharacteristic
497C2C22×#100711yescharacteristic
498C2C22×#101011yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×C4×Q8C2
socleC23×C3C25
prime radicalC2×#930C22
hypercenterC22×C4×Q8D3
hypocenterC3C2×#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Q8C2semidirect
2. C2×Q8×#67C2semidirect
3. C2×Q8×D12C2semidirect
4. C2×#5583C2semidirect
5. C2×#5572C2semidirect
6. #85804C2direct (N × H)
7. C2×#5567C2semidirect
8. C2×#5570C2semidirect
9. C2×#5584C2semidirect
10. C2×#5513C2semidirect
12. C2×#5470C2semidirect
20. C2×C3×C4×Q8C22semidirect
23. Q8×#67C22semidirect
33. #5584C22semidirect
34. #5583C22semidirect
35. Q8×D12C22semidirect
36. #5572C22semidirect
37. #5570C22semidirect
38. #5567C22semidirect
39. #5513C22semidirect
51. #5470C22semidirect
70. C22×C4×Q8D3semidirect
178. C2×C4×Q8C2×D3semidirect
466. C2×C3#3523semidirect
492. C3C2×#3523semidirect
495. C2#85804direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×Q8×#672
C2×Q8×D122
C2×#55832
C2×#55722
#858042
C2×#55672
C2×#55702
C2×#55842
C2×#55132
C2×#54792
C2×#54702
C2×#35233