Finite Group Databasebeta

← #86291

Subgroups — #86291

Order 384.

1032 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5733C212yesmaximal, complemented
3#5724C212yesmaximal, complemented
4#5700C212yesmaximal, complemented
5#5699C212yesmaximal, complemented
6#5644C212yesmaximal, complemented
7#5634C212yesmaximal, complemented
8#5632C212yesmaximal, complemented
9#5630C212yesmaximal, complemented
10#5582C214yesmaximal, complemented
11#5576C212yesmaximal, complemented
12#5568C212yesmaximal, complemented
13#5527C214yesmaximal, complemented
14#5479C211yesmaximal, characteristic, complemented
15C2×#875C211yesmaximal, characteristic, complemented
16#3561—31nomaximal, Sylow
17C2×#239C2212yes—
18C2×D3×Q8C2212yescomplemented
19C22×D12C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic, complemented
22C3×C4⋊Q8C2211yescharacteristic, complemented
23C3×#125C2212yescomplemented
24C3×C22⋊Q8C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C4×Q8C2212yescomplemented
27C3×C4×Q8C2212yescomplemented
28C3×#117C2212yes—
29C2×C3×C42C2211yescharacteristic
30#947C2212yes—
31#947C2214yescomplemented
32#946C2214yescomplemented
33#944C2212yescomplemented
34#930C2211yescharacteristic
35#929C2212yes—
36C4×#67C2212yes—
37C2×(C2×D3)⋊C4C2212yes—
38C2×(C2×D3)⋊C4C2212yes—
39#924C2211yescharacteristic
40#898C2214yescomplemented
41#898C2214yescomplemented
42#897C2212yescomplemented
43#897C2214yescomplemented
44#896C2214yescomplemented
45#896C2214yescomplemented
46#896C2214yescomplemented
47#896C2214yescomplemented
48#895C2214yes—
49#895C2212yes—
50#895C2214yescomplemented
51#894C2214yes—
52#894C2214yescomplemented
53#894C2214yescomplemented
54#894C2214yescomplemented
55#893C2212yes—
56#888C2214yescomplemented
57#887C2212yescomplemented
58#882C2214yes—
59#67⋊C4C2214yes—
60D3×C22⋊C4C2214yes—
61#878C2214yes—
62#876C2212yescomplemented
63#876C2214yescomplemented
64#875C2212yes—
65#875C2212yes—
66#875C2212yes—
67#875C2212yes—
68#875C2211yescharacteristic, complemented
69#875C2214yescomplemented
70C4×D12C2211yescharacteristic, complemented
71C4×D12C2214yescomplemented
72#872C2212yescomplemented
73#872C2214yescomplemented
74#870C2211yescharacteristic, complemented
75C4×Dic6C2211yescharacteristic, complemented
76#563—32no—
77#562—32no—
78#556—32no—
79#551—34no—
80D4×Q8—32no—
81#548—32no—
82#546—32no—
83#543—32no—
84#542—32no—
85#542—34no—
86#538—32no—
87#536—32no—
88#533D311yescharacteristic, complemented
89#529—31no—
90C2×#124—31no—
91C23×D3C2312yes—
92C2×C3×Q8C2312yes—
93C2×C3×Q8C2312yes—
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C2×#67C2312yes—
98C22×Dic3C2312yes—
99#239—44no—
100D3×Q8—44no—
101C2×D12—42no—
102C2×D12C2312yes—
103C2×D12C2311yescharacteristic
104C2×D12C2314yes—
105C2×D12C2312yes—
106C2×D12C2314yes—
107C2×C4×D3C2312yes—
108C2×C4×D3C2314yes—
109C2×C4×D3C2314yes—
110C2×C4×D3C2314yes—
111C2×Dic6—42no—
112C2×Dic6C2312yes—
113C2×Dic6C2311yescharacteristic
114C2×Dic6C2312yes—
115C2×Dic6C2314yes—
116C3×C4⋊C4C2312yes—
117C3×C4⋊C4C2314yes—
118C3×C4⋊C4C2312yes—
119C3×C4⋊C4C2312yes—
120C3×C4⋊C4C2312yes—
121C3×C4⋊C4C2314yes—
122C3×C22⋊C4C2314yes—
123C3×C42C2311yescharacteristic
124C3×C42C2311yescharacteristic
125C3×C42C2311yescharacteristic
126C3×C42C2314yes—
127C3×C42C2311yescharacteristic
128#217C2312yes—
129(C2×D3)⋊C4C2314yes—
130(C2×D3)⋊C4C2312yes—
131(C2×D3)⋊C4C2312yes—
132(C2×D3)⋊C4C2314yes—
133(C2×D3)⋊C4C2312yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2314yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142C4⋊Dic3C2311yescharacteristic
143C4⋊Dic3C2311yescharacteristic
144C4⋊Dic3C2314yes—
145Dic3⋊C4C2312yes—
146Dic3⋊C4C2312yes—
147Dic3⋊C4C2312yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152C4×Dic3C2312yes—
153C4×Dic3C2314yes—
154C2×#41—32no—
155C22×Q8—32no—
156C22×Q8—31no—
157C22×D4—31no—
158C4⋊Q8—32no—
159C4⋊Q8—34no—
160C4⋊Q8C2×D311yescharacteristic, complemented
161C4⋊Q8C2×D311yescharacteristic, complemented
162#126—34no—
163#126—34no—
164#126—32no—
165#126—34no—
166#125—31no—
167#125C2×D312yescomplemented
168#124—32no—
169#124—34no—
170#124—32no—
171#124—32no—
172#124—32no—
173#124—32no—
174#124—31no—
175#124—34no—
176#123—32no—
177#123—34no—
178#123—34no—
179#123—34no—
180#123—34no—
181C22⋊Q8—34no—
182C22⋊Q8—31no—
183C22⋊Q8—34no—
184C22⋊Q8—34no—
185C22⋊Q8—32no—
186C22⋊Q8—34no—
187C22⋊Q8—34no—
188C22⋊Q8—34no—
189C22⋊Q8—34no—
190C22⋊Q8C2×D312yes—
191C22⋊Q8C2×D312yes—
192#121—31no—
193#121—34no—
194#121—32no—
195#121—34no—
196#120—34no—
197C4×Q8—32no—
198C4×Q8—31no—
199C4×Q8C2×D312yescomplemented
200C4×Q8C2×D312yescomplemented
201C4×D4—32no—
202C4×D4—32no—
203C4×D4—34no—
204C4×D4—31no—
205C4×D4—34no—
206#117—32no—
207#117—34no—
208#117C2×D312yes—
209C2×C22⋊C4—32no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—34no—
212C2×C42C2×D311yescharacteristic
213C23×C3C2411yescharacteristic
214C22×D3—24no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3C2414yes—
218C22×D3C2412yes—
219C3×Q8—24no—
220C3×Q8—42no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4C2414yes—
225C2×C3×C4C2411yescharacteristic
226C2×C3×C4C2411yescharacteristic
227C2×C3×C4C2411yescharacteristic
228C2×C3×C4C2411yescharacteristic
229C2×C3×C4C2414yes—
230C2×C3×C4C2411yescharacteristic
231C2×C3×C4C2411yescharacteristic
232#67—24no—
233C2×Dic3—42no—
234C2×Dic3C2412yes—
235C2×Dic3C2412yes—
236C2×Dic3C2414yes—
237C2×Dic3C2412yes—
238D12—24no—
239D12—42no—
240D12—41no—
241D12—44no—
242D12—44no—
243C4×D3—24no—
244C4×D3—44no—
245C4×D3—44no—
246C4×D3—44no—
247Dic6—24no—
248Dic6—42no—
249Dic6—41no—
250Dic6—44no—
251C24—32no—
252#41—124no—
253C2×Q8—32no—
254C2×Q8—31no—
255C2×Q8—34no—
256C2×Q8—32no—
257C2×Q8—122no—
258C2×Q8—124no—
259C2×Q8C22×D312yes—
260C2×Q8C22×D312yes—
261C2×D4—32no—
262C2×D4—32no—
263C2×D4—34no—
264C2×D4—31no—
265C2×D4—34no—
266C2×D4—32no—
267C2×D4—122no—
268C22×C4—32no—
269C22×C4—32no—
270C22×C4—34no—
271C22×C4—34no—
272C22×C4—34no—
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C22×C4C22×D311yescharacteristic
276C4⋊C4—32no—
277C4⋊C4—32no—
278C4⋊C4—34no—
279C4⋊C4—32no—
280C4⋊C4—31no—
281C4⋊C4—34no—
282C4⋊C4—34no—
283C4⋊C4—32no—
284C4⋊C4—31no—
285C4⋊C4—34no—
286C4⋊C4C22×D312yes—
287C4⋊C4C22×D312yes—
288C4⋊C4C22×D312yes—
289C4⋊C4C22×D314yes—
290C4⋊C4C22×D314yes—
291C4⋊C4C22×D312yes—
292C22⋊C4—32no—
293C22⋊C4—34no—
294C22⋊C4—34no—
295C22⋊C4—34no—
296C22⋊C4—34no—
297C22⋊C4—34no—
298C22⋊C4—34no—
299C22⋊C4—34no—
300C22⋊C4—32no—
301C22⋊C4—32no—
302C22⋊C4—32no—
303C22⋊C4—34no—
304C22⋊C4—32no—
305C22⋊C4—34no—
306C22⋊C4C22×D314yes—
307C42—32no—
308C42—34no—
309C42C22×D311yescharacteristic
310C42C22×D311yescharacteristic
311C42C22×D311yescharacteristic
312C42C22×D311yescharacteristic
313C42C22×D314yes—
314C22×C3—21no—
315C22×C3—21no—
316C22×C32−1+412yes—
317C22×C3C2511yescharacteristic
318C2×D3—22no—
319C2×D3—24no—
320C2×D3—22no—
321C2×D3—24no—
322C2×D3—24no—
323C2×D3—22no—
324C2×D3—84no—
325C3×C4—21no—
326C3×C4—24no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—24no—
331C3×C4—21no—
332C3×C4H(5,𝔽2)12yes—
333Dic3—24no—
334Dic3—22no—
335Dic3—22no—
336Dic3—22no—
337C23—34no—
338C23—32no—
339C23—64no—
340C23—122no—
341C23—122no—
342C23C23×D311yescharacteristic
343Q8—24no—
344Q8—42no—
345Q8—64no—
346Q8—122no—
347Q8—124no—
348Q8—121no—
349D4—64no—
350D4—64no—
351D4—122no—
352D4—124no—
353D4—121no—
354D4—124no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—34no—
360C2×C4—32no—
361C2×C4—41no—
362C2×C4—64no—
363C2×C4—124no—
364C2×C4—124no—
365C2×C4—124no—
366C2×C4—122no—
367C2×C4C23×D314yes—
368C2×C4C23×D311yescharacteristic
369C2×C4C23×D311yescharacteristic
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D311yescharacteristic
372C2×C4C23×D311yescharacteristic
373C2×C4C23×D314yes—
374C2×C4C23×D311yescharacteristic
375C2×C3—22no—
376C2×C3C2×H(5,𝔽2)11yescharacteristic
377C2×C3C2×2−1+411yescharacteristic
378C2×C3#58511yescharacteristic
379D3—42no—
380D3—44no—
381C22—21no—
382C22—21no—
383C22—62no—
384C22—64no—
385C22—62no—
386C22—64no—
387C22—64no—
388C22—62no—
389C22—244no—
390C22#100712yes—
391C22C24×D311yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—24no—
396C4—21no—
397C4—21no—
398C4—24no—
399C4—62no—
400C4—62no—
401C4—64no—
402C4—62no—
403C4#100912yes—
404C3#356111yescharacteristic, complemented, Sylow
405C2—22no—
406C2—124no—
407C2—122no—
408C2C2×#100711yescharacteristic
409C2C2×#100911yescharacteristic
410C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#533C2
hypercenter#533D3
hypocenterC3#3561
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5733C2semidirect
3. #5724C2semidirect
4. #5700C2semidirect
5. #5699C2semidirect
6. #5644C2semidirect
7. #5634C2semidirect
8. #5632C2semidirect
9. #5630C2semidirect
10. #5582C2semidirect
11. #5576C2semidirect
12. #5568C2semidirect
13. #5527C2semidirect
14. #5479C2semidirect
15. C2×#875C2semidirect
18. C2×D3×Q8C22semidirect
21. C3×C4⋊Q8C22semidirect
22. C3×C4⋊Q8C22semidirect
23. C3×#125C22semidirect
26. C3×C4×Q8C22semidirect
27. C3×C4×Q8C22semidirect
31. #947C22semidirect
32. #946C22semidirect
33. #944C22semidirect
40. #898C22semidirect
41. #898C22semidirect
42. #897C22semidirect
43. #897C22semidirect
44. #896C22semidirect
45. #896C22semidirect
46. #896C22semidirect
47. #896C22semidirect
50. #895C22semidirect
52. #894C22semidirect
53. #894C22semidirect
54. #894C22semidirect
56. #888C22semidirect
57. #887C22semidirect
62. #876C22semidirect
63. #876C22semidirect
68. #875C22semidirect
69. #875C22semidirect
70. C4×D12C22semidirect
71. C4×D12C22semidirect
72. #872C22semidirect
73. #872C22semidirect
74. #870C22semidirect
75. C4×Dic6C22semidirect
88. #533D3semidirect
160. C4⋊Q8C2×D3semidirect
161. C4⋊Q8C2×D3semidirect
167. #125C2×D3semidirect
199. C4×Q8C2×D3semidirect
200. C4×Q8C2×D3semidirect
404. C3#3561semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57332
#57242
#57002
#56992
#56442
#56342
#56322
#56302
#55822
#55762
#55682
#55272
#54792
C2×#8752
#35613