Finite Group Databasebeta

← #85902

Subgroups — #85902

Order 384.

1124 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#520C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic, complemented
3#5828C211yesmaximal, characteristic, complemented
4#5581C212yesmaximal, complemented
5#5579C212yesmaximal, complemented
6#5564C214yesmaximal, complemented
7#5558C212yesmaximal, complemented
8#5551C212yesmaximal, complemented
9#5547C214yesmaximal, complemented
10#5530C212yesmaximal, complemented
11#5529C212yesmaximal, complemented
12#5524C211yesmaximal, characteristic, complemented
13#5519C211yesmaximal, characteristic, complemented
14#5511C212yesmaximal, complemented
15#5507C212yesmaximal, complemented
16#5504C211yesmaximal, characteristic, complemented
17C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
18#3492—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×#235C2214yes—
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2212yes—
24C3×C4×Q8C2212yescomplemented
25C3×C4×D4C2214yes—
26C3×C4×D4C2212yescomplemented
27C3×#117C2211yescharacteristic
28C3×#117C2212yescomplemented
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C4⋊C4C2212yescomplemented
31#946C2211yescharacteristic
32#946C2211yescharacteristic
33#946C2212yescomplemented
34#939C2214yes—
35#938C2211yescharacteristic
36#938C2211yescharacteristic
37#938C2212yescomplemented
38#937C2212yes—
39#935C2212yes—
40#930C2212yescomplemented
41#929C2214yescomplemented
42C4×#67C2214yes—
43C4×#67C2212yes—
44C4×#67C2212yes—
45C4×#67C2214yes—
46#926C2211yescharacteristic
47C2×C4⋊Dic3C2211yescharacteristic
48#924C2212yescomplemented
49#924C2214yescomplemented
50C2×Dic3⋊C4C2212yes—
51#897C2211yescharacteristic
52#897C2211yescharacteristic
53#896C2214yes—
54#896C2214yescomplemented
55#895C2211yescharacteristic
56#895C2211yescharacteristic
57#894C2214yes—
58#893C2212yescomplemented
59#892C2212yescomplemented
60D3×C4⋊C4C2212yes—
61D3×C4⋊C4C2212yes—
62D3×C4⋊C4C2212yes—
63D3×C4⋊C4C2212yes—
64D3×C4⋊C4C2212yescomplemented
65#890C2212yescomplemented
66#889C2214yescomplemented
67#888C2212yescomplemented
68#887C2212yescomplemented
69#886C2212yes—
70#884C2214yes—
71#883C2214yes—
72#883C2214yescomplemented
73#882C2212yes—
74#879C2214yescomplemented
75#878C2214yes—
76#876C2214yescomplemented
77C4×D12C2212yes—
78C4×D12C2212yes—
79C4×D12C2214yes—
80#872C2214yes—
81#870C2212yescomplemented
82#870C2214yescomplemented
83#869C2212yescomplemented
84#554—32no—
85#554—32no—
86#553—32no—
87#553—32no—
88#552—32no—
89#552—32no—
90#547—32no—
91#547—32no—
92#544—31no—
93#539—34no—
94#537—31no—
95#525—31no—
96#525—34no—
97#525—31no—
98#520—31no—
99#520D311yescharacteristic, complemented
100C22×C4⋊C4—31no—
101C23×D3C2311yescharacteristic
102C3×#41—24no—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2312yes—
105C2×C3×D4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C22×C3×C4C2312yes—
108C22×C3×C4C2314yes—
109C22×C3×C4C2311yescharacteristic
110C2×#67C2312yes—
111C2×#67C2312yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2312yes—
116#235—24no—
117#235—48no—
118C2×D12C2312yes—
119C2×D12C2312yes—
120C2×D12C2311yescharacteristic
121C2×C4×D3—24no—
122C2×C4×D3—44no—
123C2×C4×D3C2314yes—
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2312yes—
127C2×C4×D3C2311yescharacteristic
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2314yes—
130C2×C4×D3C2314yes—
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2314yes—
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2311yescharacteristic
135C3×C4⋊C4C2314yes—
136C3×C4⋊C4C2311yescharacteristic
137C3×C4⋊C4C2312yes—
138C3×C22⋊C4C2312yes—
139C3×C22⋊C4C2314yes—
140C3×C42C2312yes—
141C3×C42C2314yes—
142#217C2312yes—
143#217C2312yes—
144#217C2314yes—
145#217C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2312yes—
153(C2×D3)⋊C4C2314yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2314yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2312yes—
160C4⋊Dic3C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166Dic3⋊C4C2314yes—
167Dic3⋊C4C2312yes—
168Dic3⋊C4C2314yes—
169C4×Dic3C2314yes—
170C4×Dic3C2312yes—
171C2×#41—31no—
172C2×#41—34no—
173C2×#41C2×D311yescharacteristic
174C23×C4—31no—
175C23×C4—32no—
176C4⋊Q8—32no—
177C4⋊Q8—32no—
178#126—34no—
179#126—34no—
180#125—34no—
181#125—32no—
182#125—34no—
183#125—32no—
184#123—32no—
185#123—34no—
186#123—34no—
187#123—32no—
188#123—34no—
189#123—34no—
190C22⋊Q8—31no—
191C22⋊Q8—31no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—34no—
195C22⋊Q8—31no—
196C22⋊Q8—34no—
197C22⋊Q8—31no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200#121—34no—
201#121—31no—
202#121—31no—
203#121—32no—
204#121—32no—
205#121—31no—
206#121—31no—
207#121—34no—
208#120—32no—
209#120—32no—
210C4×Q8—32no—
211C4×Q8C2×D312yescomplemented
212C4×D4—34no—
213C4×D4—32no—
214C4×D4—34no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—32no—
220C4×D4C2×D314yes—
221C4×D4C2×D312yescomplemented
222#117—31no—
223#117—32no—
224#117—34no—
225#117C2×D311yescharacteristic
226#117C2×D312yescomplemented
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4C2×D311yescharacteristic
235C2×C4⋊C4C2×D312yescomplemented
236C23×C3C2412yes—
237C23×C3C2411yescharacteristic
238C22×D3—22no—
239C22×D3—22no—
240C22×D3—22no—
241C22×D3C2412yes—
242C22×D3C2411yescharacteristic
243C22×D3C2411yescharacteristic
244C3×Q8—22no—
245C3×D4—24no—
246C3×D4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4#4114yescomplemented
251C2×C3×C4C2414yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2412yes—
255C2×C3×C4C2411yescharacteristic
256C2×C3×C4C2411yescharacteristic
257#67—24no—
258#67—24no—
259#67—24no—
260#67—44no—
261C2×Dic3—22no—
262C2×Dic3—42no—
263C2×Dic3C2412yes—
264C2×Dic3C2412yes—
265C2×Dic3C2412yes—
266C2×Dic3C2411yescharacteristic
267C2×Dic3C2411yescharacteristic
268C2×Dic3C2414yes—
269D12—24no—
270D12—24no—
271D12—22no—
272C4×D3—28no—
273C4×D3—22no—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—24no—
278C4×D3—44no—
279C4×D3—44no—
280Dic6—22no—
281Dic6—44no—
282C24—31no—
283#41—24no—
284#41—64no—
285#41—128no—
286C2×Q8—31no—
287C2×Q8—34no—
288C2×Q8C22×D311yescharacteristic
289C2×D4—32no—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—32no—
294C2×D4—31no—
295C2×D4—32no—
296C2×D4C22×D312yes—
297C2×D4C22×D311yescharacteristic
298C22×C4—31no—
299C22×C4—32no—
300C22×C4—34no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—32no—
306C22×C4—34no—
307C22×C4—34no—
308C22×C4—64no—
309C22×C4—124no—
310C22×C4C22×D312yes—
311C22×C4C22×D312yes—
312C22×C4C22×D314yes—
313C22×C4C22×D311yescharacteristic
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—34no—
317C4⋊C4—34no—
318C4⋊C4—31no—
319C4⋊C4—34no—
320C4⋊C4—31no—
321C4⋊C4—34no—
322C4⋊C4—34no—
323C4⋊C4—32no—
324C4⋊C4—32no—
325C4⋊C4—34no—
326C4⋊C4—34no—
327C4⋊C4—32no—
328C4⋊C4—32no—
329C4⋊C4C22×D312yes—
330C4⋊C4C22×D312yes—
331C4⋊C4C22×D311yescharacteristic
332C4⋊C4C22×D314yes—
333C4⋊C4C22×D311yescharacteristic
334C22⋊C4—32no—
335C22⋊C4—34no—
336C22⋊C4—32no—
337C22⋊C4—32no—
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—34no—
341C22⋊C4—32no—
342C22⋊C4—32no—
343C22⋊C4—34no—
344C22⋊C4—32no—
345C22⋊C4—32no—
346C22⋊C4C22×D312yes—
347C22⋊C4C22×D314yes—
348C42—34no—
349C42—32no—
350C42C22×D314yes—
351C42C22×D312yes—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3C2×#4112yes—
358C22×C3C2511yescharacteristic
359C2×D3—28no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—22no—
367C2×D3H(5,𝔽2)12yes—
368C2×D32−1+412yes—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—24no—
372C3×C4—22no—
373C3×C4C2×#4112yes—
374C3×C4C2×#4112yes—
375Dic3—22no—
376Dic3—21no—
377Dic3—21no—
378Dic3—22no—
379Dic3—24no—
380Dic3—22no—
381C23—31no—
382C23—31no—
383C23—32no—
384C23—62no—
385C23—62no—
386C23—62no—
387C23C23×D312yes—
388C23C23×D311yescharacteristic
389Q8—22no—
390Q8—62no—
391Q8—124no—
392D4—24no—
393D4—22no—
394D4—64no—
395D4—64no—
396D4—64no—
397D4—64no—
398D4—64no—
399D4—62no—
400D4—124no—
401C2×C4—24no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—34no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—68no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—64no—
417C2×C4—124no—
418C2×C4—124no—
419C2×C4—122no—
420C2×C4#23514yescomplemented
421C2×C4C23×D314yes—
422C2×C4C23×D312yes—
423C2×C4C23×D312yes—
424C2×C4C23×D312yes—
425C2×C4C23×D311yescharacteristic
426C2×C4C23×D311yescharacteristic
427C2×C3—22no—
428C2×C3—42no—
429C2×C3C22×#4111yescharacteristic
430C2×C3C2×H(5,𝔽2)11yescharacteristic
431C2×C3C2×2−1+411yescharacteristic
432D3—22no—
433D3—22no—
434D3—42no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—32no—
441C22—32no—
442C22—68no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—62no—
448C22—62no—
449C22—62no—
450C22C2×#23512yes—
451C22C24×D311yescharacteristic
452C4—22no—
453C4—24no—
454C4—22no—
455C4—22no—
456C4—61no—
457C4—61no—
458C4—62no—
459C4—64no—
460C4—62no—
461C4—62no—
462C4C2×#23512yes—
463C4C2×#23512yes—
464C3#349211yescharacteristic, complemented, Sylow
465C2—22no—
466C2—42no—
467C2—62no—
468C2—62no—
469C2—122no—
470C2C22×#23511yescharacteristic
471C2D3×H(5,𝔽2)11yescharacteristic
472C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#520C2
hypercenter#520D3
hypocenterC3#3492
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C3×#520C2semidirect
2. #5833C2semidirect
3. #5828C2semidirect
4. #5581C2semidirect
5. #5579C2semidirect
6. #5564C2semidirect
7. #5558C2semidirect
8. #5551C2semidirect
9. #5547C2semidirect
10. #5530C2semidirect
11. #5529C2semidirect
12. #5524C2semidirect
13. #5519C2semidirect
14. #5511C2semidirect
15. #5507C2semidirect
16. #5504C2semidirect
17. C2×D3×C4⋊C4C2semidirect
24. C3×C4×Q8C22semidirect
26. C3×C4×D4C22semidirect
28. C3×#117C22semidirect
30. C2×C3×C4⋊C4C22semidirect
33. #946C22semidirect
37. #938C22semidirect
40. #930C22semidirect
41. #929C22semidirect
48. #924C22semidirect
49. #924C22semidirect
54. #896C22semidirect
58. #893C22semidirect
59. #892C22semidirect
64. D3×C4⋊C4C22semidirect
65. #890C22semidirect
66. #889C22semidirect
67. #888C22semidirect
68. #887C22semidirect
72. #883C22semidirect
74. #879C22semidirect
76. #876C22semidirect
81. #870C22semidirect
82. #870C22semidirect
83. #869C22semidirect
99. #520D3semidirect
211. C4×Q8C2×D3semidirect
221. C4×D4C2×D3semidirect
226. #117C2×D3semidirect
235. C2×C4⋊C4C2×D3semidirect
250. C2×C3×C4#41semidirect
420. C2×C4#235semidirect
464. C3#3492semidirect

Maximal subgroups

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

subgroupindex
C3×#5202
#58332
#58282
#55812
#55792
#55642
#55582
#55512
#55472
#55302
#55292
#55242
#55192
#55112
#55072
#55042
C2×D3×C4⋊C42
#34923