Finite Group Databasebeta

← #10830436

Subgroups — #10830436

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85786C211yesmaximal, characteristic
2C3×#2326C211yesmaximal, characteristic, complemented
3C2×#5061C212yesmaximal, complemented
4#82441C212yesmaximal, complemented
5#71617C211yesmaximal, characteristic, complemented
6#69694C212yesmaximal, complemented
7#69662C212yesmaximal, complemented
8#69593C212yesmaximal, complemented
9#69440C212yesmaximal, complemented
10#24228—31nomaximal, Sylow
11C2×#1004C2212yes—
12C2×C3×#124C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic, complemented
14C2×C3×C4×D4C2211yescharacteristic
15C2×#942C2214yescomplemented
16C2×#930—22no—
17#5555—22no—
18#5554—24no—
19D4×D12—22no—
20#5510C2212yes—
21#5493—22no—
22#5477C2212yes—
23C3×#391C2211yescharacteristic, complemented
24C3×#390C2211yescharacteristic
25C3×#388—22no—
26C3×#385—22no—
27C3×#384C2211yescharacteristic, complemented
28C3×#376C2211yescharacteristic, complemented
29C2×#833C2212yes—
30#5061—28no—
31#5031—22no—
32#5028—24no—
33#5027—22no—
34#4966C2214yescomplemented
35C2×#805C2214yescomplemented
36C2×#805C2214yescomplemented
37#4924C2212yescomplemented
38#4530C2212yescomplemented
39#3496—31no—
40#3037—32no—
41C2×#455—32no—
42#2326D311yescharacteristic, complemented
43#2039—32no—
44#2007—32no—
45#1963—32no—
46#1800—31no—
47#1792—32no—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4D412yes—
51C22×#67—24no—
52#1004—24no—
53#1004—24no—
54#1004—44no—
55C2×#237—24no—
56C2×D3×D4—24no—
57C2×#235C2314yes—
58C22×D12—22no—
59C3×#124—22no—
60C3×#124D414yescomplemented
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#216—24no—
77#942—28no—
78#942—44no—
79C2×#215—24no—
80#930—24no—
81#930—24no—
82#930—22no—
83#930—22no—
84#930—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×C4⋊Dic3—22no—
87C2×#208C2314yes—
88#897—24no—
89#895—24no—
90#886—28no—
91#882—28no—
92#875D414yes—
93#874D412yes—
94C4×D12—22no—
95C4×D12—22no—
96#869D412yes—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—22no—
100#833—24no—
101#810—24no—
102#809—24no—
103#805—28no—
104#805—44no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×H(5,𝔽2)—32no—
108C2×#137—34no—
109#547—62no—
110#546—64no—
111D42—62no—
112#537—32no—
113#534—62no—
114#532—32no—
115C2×#124C2×D311yescharacteristic
116C2×#121—62no—
117C2×C4×Q8C2×D311yescharacteristic, complemented
118C2×C4×D4C2×D311yescharacteristic
119#464—64no—
120#462—62no—
121#460—62no—
122#455—68no—
123#423—32no—
124C2×#104—34no—
125C2×#104—34no—
126#417—34no—
127C2×#100—32no—
128#391C2×D311yescharacteristic, complemented
129#390C2×D311yescharacteristic
130#388—22no—
131#385—22no—
132#384C2×D311yescharacteristic, complemented
133#376C2×D311yescharacteristic, complemented
134#343—32no—
135C24×C3C2×D412yes—
136C23×D3—24no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×D412yescomplemented
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4C2×D412yes—
144C2×C3×D4#4112yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—82no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C2×#67—28no—
157C2×#67—24no—
158C2×#67—48no—
159C2×#67—48no—
160C22×Dic3—24no—
161#237—48no—
162#237—48no—
163D3×D4—48no—
164D3×D4—48no—
165#235—24no—
166#235—24no—
167#235—24no—
168#235—24no—
169C2×D12—22no—
170C2×D12—24no—
171C2×D12—22no—
172C2×D12—84no—
173C2×D12C2×D414yes—
174C2×C4×D3—24no—
175C2×Dic6C2×D414yes—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—41no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4—82no—
181C3×C22⋊C4—28no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—82no—
184C3×C42—21no—
185C3×C42—22no—
186C3×C42—42no—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189C3×C42C2×D412yescomplemented
190#216—48no—
191#215—48no—
192(C2×D3)⋊C4—28no—
193(C2×D3)⋊C4—24no—
194(C2×D3)⋊C4—24no—
195C4⋊Dic3—24no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198#208—24no—
199#208—24no—
200C2×C3⋊C8—24no—
201H(5,𝔽2)—64no—
202H(5,𝔽2)—64no—
203H(5,𝔽2)—124no—
204C2×#41—34no—
205C2×#41—64no—
206C22×Q8C22×D311yescharacteristic
207C22×D4—62no—
208C22×D4—64no—
209C22×D4—64no—
210C22×D4C22×D311yescharacteristic
211C23×C4#6712yes—
212#137—68no—
213#137—124no—
214C2×Q16—64no—
215C2×SD16—64no—
216C2×M16—34no—
217C4⋊Q8—32no—
218#127—32no—
219#124—22no—
220#124—34no—
221#124D1214yescomplemented
222#123—68no—
223C22⋊Q8—64no—
224#121—64no—
225#121—64no—
226#121—64no—
227#121—62no—
228#121—64no—
229#121—62no—
230#120—68no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—62no—
236C4×D4—62no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4#6712yes—
246C2×C42—21no—
247C2×C42C22×D311yescharacteristic
248C2×C42C22×D311yescharacteristic
249C2×C42C22×D311yescharacteristic
250C4⋊C8—62no—
251C4⋊C8—62no—
252#104—68no—
253#104—124no—
254#103—64no—
255#102—64no—
256#100—64no—
257#95—21no—
258#95—21no—
259#95—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3#12114yes—
265C23×C3C22×D411yescharacteristic
266C22×D3—24no—
267C22×D3—48no—
268C22×D3—84no—
269C22×D3—84no—
270C3×Q8—24no—
271C3×Q8—42no—
272C3×D4—24no—
273C3×D4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—44no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—82no—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12012yes—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D412yes—
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)12yes—
298#67—416no—
299#67—44no—
300#67—44no—
301C2×Dic3—24no—
302C2×Dic3—48no—
303D12—24no—
304D12—24no—
305D12—44no—
306D12—82no—
307D12—84no—
308C4×D3—44no—
309C4×D3—44no—
310Dic6—24no—
311Dic6—24no—
312C3⋊C8—44no—
313C24—64no—
314C24C2×#6712yes—
315#41—64no—
316#41—64no—
317#41—64no—
318#41—64no—
319#41—128no—
320#41—128no—
321C2×Q8—21no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—34no—
325C2×Q8C2×D1212yescomplemented
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—34no—
329C2×D4—64no—
330C2×D4—68no—
331C2×D4—64no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—128no—
335C2×D4—128no—
336C2×D4—128no—
337C2×D4—128no—
338C2×D4—244no—
339C2×D4C2×D1212yes—
340C2×D4#23512yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—64no—
348C22×C4—64no—
349C22×C4—82no—
350C22×C4C2×#6712yes—
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C22×C4C23×D311yescharacteristic
354Q16—128no—
355SD16—128no—
356M16—64no—
357M16—64no—
358C2×C8—64no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—64no—
364C4⋊C4—62no—
365C4⋊C4—62no—
366C4⋊C4—82no—
367C22⋊C4—28no—
368C22⋊C4—24no—
369C22⋊C4—68no—
370C22⋊C4—64no—
371C22⋊C4—64no—
372C22⋊C4—82no—
373C42—21no—
374C42—22no—
375C42—42no—
376C42C2×D1212yescomplemented
377C42D3×D412yes—
378C42D3×D412yes—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—24no—
382C22×C3—44no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—82no—
386C22×C3C2×#12011yescharacteristic
387C22×C3C2×#12111yescharacteristic
388C22×C3#53511yescharacteristic
389C2×D3—44no—
390C2×D3—44no—
391C2×D3—44no—
392C2×D3—168no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—22no—
396C3×C4—42no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4D4212yes—
402C3×C4#54612yes—
403Dic3—44no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—64no—
409C23—128no—
410C23—244no—
411C23—244no—
412C23#93014yes—
413C23C22×#6711yescharacteristic
414Q8—24no—
415Q8—42no—
416Q8—64no—
417Q8—64no—
418D4—24no—
419D4—24no—
420D4—64no—
421D4—64no—
422D4—124no—
423D4—1216no—
424D4—124no—
425D4—124no—
426D4—242no—
427D4—244no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—64no—
445C2×C4—82no—
446C2×C4—128no—
447C2×C4—124no—
448C2×C4—124no—
449C2×C4#93712yes—
450C2×C4#93711yescharacteristic
451C2×C4#93711yescharacteristic
452C2×C4C22×D1211yescharacteristic
453C2×C4C2×#23511yescharacteristic
454C2×C4C2×D3×D412yes—
455C2×C4#100912yes—
456C8—124no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—44no—
460C2×C3#240811yescharacteristic
461C2×C3#301911yescharacteristic
462C2×C3#302211yescharacteristic
463D3—84no—
464C22—21no—
465C22—21no—
466C22—24no—
467C22—44no—
468C22—42no—
469C22—42no—
470C22—82no—
471C22—124no—
472C22—124no—
473C22—124no—
474C22—488no—
475C22C2×#93011yescharacteristic
476C22C2×#93711yescharacteristic
477C22#583011yescharacteristic
478C4—21no—
479C4—21no—
480C4—22no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—124no—
487C4D4×D1212yes—
488C4#555012yes—
489C3#2422811yescharacteristic, complemented, Sylow
490C2—21no—
491C2—21no—
492C2—44no—
493C2—244no—
494C2#8240311yescharacteristic
495C2#8339411yescharacteristic
496C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2326C2
socleC22×C3C2×#120
prime radical#85786C2
hypercenter#2326D3
hypocenterC3#24228
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C3×#2326C2semidirect
3. C2×#5061C2semidirect
4. #82441C2semidirect
5. #71617C2semidirect
6. #69694C2semidirect
7. #69662C2semidirect
8. #69593C2semidirect
9. #69440C2semidirect
13. C2×C3×C4×Q8C22semidirect
15. C2×#942C22semidirect
23. C3×#391C22semidirect
27. C3×#384C22semidirect
28. C3×#376C22semidirect
34. #4966C22semidirect
35. C2×#805C22semidirect
36. C2×#805C22semidirect
37. #4924C22semidirect
38. #4530C22semidirect
42. #2326D3semidirect
60. C3×#124D4semidirect
117. C2×C4×Q8C2×D3semidirect
128. #391C2×D3semidirect
132. #384C2×D3semidirect
133. #376C2×D3semidirect
140. C2×C3×Q8C2×D4semidirect
189. C3×C42C2×D4semidirect
221. #124D12semidirect
325. C2×Q8C2×D12semidirect
376. C42C2×D12semidirect
489. C3#24228semidirect

Maximal subgroups

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

subgroupindex
#857862
C3×#23262
C2×#50612
#824412
#716172
#696942
#696622
#695932
#694402
#242283