Finite Group Databasebeta

← #85986

Subgroups — #85986

Order 384.

1100 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2Q8×#67C212yesmaximal, complemented
3C2×C4×#67C211yesmaximal, characteristic
4C2×#924C211yesmaximal, characteristic, complemented
5#5636C214yesmaximal, complemented
6#5635C212yesmaximal, complemented
7#5634C212yesmaximal, complemented
8#5632C212yesmaximal, complemented
9D3×C22⋊Q8C212yesmaximal, complemented
10#5623C212yesmaximal, complemented
11#5622C212yesmaximal, complemented
12#5511C212yesmaximal, complemented
13#5511C214yesmaximal, complemented
14#5486C212yesmaximal
15#5483C212yesmaximal, complemented
16#3501—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic, complemented
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C2×D3×Q8C2212yescomplemented
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic, complemented
23C3×C22⋊Q8C2212yes—
24C3×C22⋊Q8C2212yes—
25C3×C22⋊Q8C2212yescomplemented
26C3×C22⋊Q8C2212yescomplemented
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C4⋊C4C2212yescomplemented
29C2×C3×C22⋊C4C2212yes—
30C2×#217C2211yescharacteristic
31C2×#217C2212yes—
32#946C2214yescomplemented
33#946C2212yescomplemented
34Q8×Dic3C2212yescomplemented
35#944C2212yescomplemented
36#944C2214yescomplemented
37C4×#67C2212yes—
38C4×#67C2212yes—
39C4×#67C2214yes—
40C4×#67C2214yes—
41C4×#67C2212yes—
42C4×#67C2212yes—
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×C4⋊Dic3C2211yescharacteristic, complemented
45#924C2212yes—
46#924C2212yes—
47#924C2214yescomplemented
48#924C2214yescomplemented
49#924C2212yescomplemented
50#924C2212yescomplemented
51C2×Dic3⋊C4C2211yescharacteristic
52C2×Dic3⋊C4C2212yescomplemented
53C2×C4×Dic3C2211yescharacteristic
54#897C2212yescomplemented
55#897C2214yescomplemented
56#896C2214yescomplemented
57#896C2214yescomplemented
58#896C2214yescomplemented
59D3×C4⋊C4C2212yescomplemented
60D3×C4⋊C4C2214yescomplemented
61#889C2214yescomplemented
62#889C2214yescomplemented
63#889C2214yescomplemented
64#888C2214yescomplemented
65#888C2212yescomplemented
66#887C2212yescomplemented
67#887C2214yescomplemented
68#67⋊C4C2214yes—
69#67⋊C4C2214yes—
70D3×C22⋊C4C2214yes—
71#878C2214yes—
72#878C2214yes—
73#878C2214yescomplemented
74#878C2214yescomplemented
75#877C2214yes—
76#554—32no—
77#554—34no—
78#554—32no—
79#554—34no—
80D4×Q8—32no—
81D4×Q8—32no—
82#544—32no—
83#543—32no—
84#543—32no—
85#533—32no—
86C2×C22⋊Q8—31no—
87C2×C22⋊Q8—32no—
88C2×C22⋊Q8D311yescharacteristic, complemented
89#518—32no—
90C2×C4×D4—31no—
91C24×C3C2311yescharacteristic
92C23×D3C2311yescharacteristic
93C2×C3×Q8—42no—
94C2×C3×Q8C2312yes—
95C2×C3×Q8C2312yes—
96C22×C3×C4—42no—
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2312yes—
103C22×C3×C4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C2×#67Q818yescomplemented
106C2×#67C2312yes—
107C2×#67C2312yes—
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2312yes—
111C22×Dic3C2312yes—
112C22×Dic3C2311yescharacteristic
113D3×Q8—44no—
114C2×C4×D3—42no—
115C2×C4×D3C2312yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2312yes—
119C2×Dic6—42no—
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C2×Dic6C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2314yes—
127C3×C4⋊C4C2312yes—
128C3×C22⋊C4C2314yes—
129C3×C22⋊C4C2314yes—
130#217C2314yes—
131#217C2312yes—
132#217C2312yes—
133#217C2314yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2312yes—
138C4⋊Dic3C2314yes—
139C4⋊Dic3C2314yes—
140C4⋊Dic3C2312yes—
141C4⋊Dic3C2312yes—
142Dic3⋊C4C2314yes—
143Dic3⋊C4C2314yes—
144Dic3⋊C4C2314yes—
145Dic3⋊C4C2314yes—
146Dic3⋊C4C2314yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2314yes—
152C4×Dic3C2314yes—
153C4×Dic3C2314yes—
154C4×Dic3C2312yes—
155C4×Dic3C2312yes—
156C22×Q8—32no—
157C22×Q8—31no—
158C22×Q8C2×D311yescharacteristic, complemented
159C22×D4—31no—
160C23×C4—31no—
161C23×C4C2×D311yescharacteristic
162C4⋊Q8—32no—
163C4⋊Q8—34no—
164C4⋊Q8—32no—
165C4⋊Q8—34no—
166#125—34no—
167#125—34no—
168#125—34no—
169C22⋊Q8—32no—
170C22⋊Q8—34no—
171C22⋊Q8—32no—
172C22⋊Q8—32no—
173C22⋊Q8—32no—
174C22⋊Q8—32no—
175C22⋊Q8—34no—
176C22⋊Q8—34no—
177C22⋊Q8—32no—
178C22⋊Q8—34no—
179C22⋊Q8—34no—
180C22⋊Q8—34no—
181C22⋊Q8—34no—
182C22⋊Q8—34no—
183C22⋊Q8—34no—
184C22⋊Q8—34no—
185C22⋊Q8—34no—
186C22⋊Q8C2×D312yes—
187C22⋊Q8C2×D312yes—
188C22⋊Q8C2×D312yescomplemented
189C22⋊Q8C2×D312yescomplemented
190C4×Q8—32no—
191C4×Q8—32no—
192C4×Q8—34no—
193C4×D4—32no—
194C4×D4—32no—
195C4×D4—32no—
196C4×D4—32no—
197C4×D4—34no—
198C4×D4—34no—
199C4×D4—34no—
200C4×D4—34no—
201#117—34no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—34no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4C2×D312yescomplemented
208C2×C4⋊C4C2×D311yescharacteristic, complemented
209C2×C22⋊C4—31no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—31no—
212C2×C22⋊C4—34no—
213C2×C22⋊C4C2×D312yes—
214C2×C42—31no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C2×Q814yescomplemented
218C23×C3C2411yescharacteristic
219C23×C3C2411yescharacteristic
220C23×C3C2411yescharacteristic
221C22×D3—41no—
222C22×D3—41no—
223C22×D3C2×Q814yescomplemented
224C22×D3C2412yes—
225C3×Q8—24no—
226C3×Q8—42no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4C2411yescharacteristic
236C2×C3×C4C2414yes—
237C2×C3×C4C2414yes—
238C2×C3×C4C2411yescharacteristic
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2411yescharacteristic
241#67C2×Q818yescomplemented
242#67C2×Q818yescomplemented
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3C2×Q814yescomplemented
248C2×Dic3C2412yes—
249C2×Dic3C2414yes—
250C2×Dic3C2414yes—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C4×D3—24no—
254C4×D3—42no—
255C4×D3—44no—
256C4×D3—44no—
257Dic6—24no—
258Dic6—42no—
259Dic6—44no—
260Dic6—44no—
261C24—31no—
262C24C22×D311yescharacteristic
263C2×Q8—32no—
264C2×Q8—34no—
265C2×Q8—32no—
266C2×Q8—34no—
267C2×Q8—42no—
268C2×Q8—122no—
269C2×Q8—124no—
270C2×Q8C22×D312yes—
271C2×Q8C22×D312yes—
272C2×D4—38no—
273C2×D4—32no—
274C2×D4—32no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—34no—
282C22×C4—34no—
283C22×C4—32no—
284C22×C4—42no—
285C22×C4—122no—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C22×C4C22×D312yes—
291C22×C4C22×D312yes—
292C22×C4C22×D311yescharacteristic
293C22×C4C22×D311yescharacteristic
294C4⋊C4—34no—
295C4⋊C4—32no—
296C4⋊C4—32no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—32no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—34no—
304C4⋊C4—34no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4C22×D314yes—
309C4⋊C4C22×D312yes—
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D314yes—
312C22⋊C4—34no—
313C22⋊C4—32no—
314C22⋊C4—32no—
315C22⋊C4—34no—
316C22⋊C4—32no—
317C22⋊C4—32no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4C22×D314yes—
321C22⋊C4C22×D314yes—
322C42—32no—
323C42—32no—
324C42—34no—
325C42—34no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—44no—
333C22×C3C22×Q812yes—
334C22×C3C22×Q812yes—
335C22×C3C22×Q812yes—
336C22×C3C2511yescharacteristic
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—44no—
340C2×D3C22×Q814yes—
341C3×C4—21no—
342C3×C4—24no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—24no—
346C3×C4H(5,𝔽2)12yes—
347Dic3—24no—
348Dic3—22no—
349Dic3—22no—
350Dic3—24no—
351Dic3C22×Q814yes—
352C23—32no—
353C23—34no—
354C23—41no—
355C23—41no—
356C23—121no—
357C23—121no—
358C23D3×Q814yescomplemented
359C23C23×D311yescharacteristic
360C23C23×D311yescharacteristic
361C23C23×D311yescharacteristic
362Q8—24no—
363Q8—42no—
364Q8—64no—
365Q8—122no—
366Q8—124no—
367Q8—124no—
368D4—38no—
369D4—38no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—34no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—34no—
377C2×C4—34no—
378C2×C4—32no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—64no—
385C2×C4—122no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4—121no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D314yes—
394C2×C4C23×D314yes—
395C2×C4C23×D311yescharacteristic
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—22no—
399C2×C3—22no—
400C2×C3—22no—
401C2×C3C23×Q811yescharacteristic
402C2×C3C2×H(5,𝔽2)11yescharacteristic
403C2×C3#58511yescharacteristic
404D3—24no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—34no—
412C22—44no—
413C22—62no—
414C22—62no—
415C22—124no—
416C22C2×D3×Q812yes—
417C22C2×D3×Q812yes—
418C22C2×D3×Q812yes—
419C22C24×D311yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—24no—
424C4—24no—
425C4—34no—
426C4—62no—
427C4—64no—
428C4—64no—
429C4—62no—
430C4#100412yes—
431C3#350111yescharacteristic, complemented, Sylow
432C2—22no—
433C2—22no—
434C2—22no—
435C2—64no—
436C2C2×#100411yescharacteristic
437C2C22×D3×Q811yescharacteristic
438C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×C22⋊Q8C2
prime radicalC22×#67C22
hypercenterC2×C22⋊Q8D3
hypocenterC3#3501
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
2. Q8×#67C2semidirect
4. C2×#924C2semidirect
5. #5636C2semidirect
6. #5635C2semidirect
7. #5634C2semidirect
8. #5632C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5623C2semidirect
11. #5622C2semidirect
12. #5511C2semidirect
13. #5511C2semidirect
15. #5483C2semidirect
17. C22×C3×Q8C22semidirect
20. C2×D3×Q8C22semidirect
22. C22×Dic6C22semidirect
25. C3×C22⋊Q8C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C2×C3×C4⋊C4C22semidirect
28. C2×C3×C4⋊C4C22semidirect
32. #946C22semidirect
33. #946C22semidirect
34. Q8×Dic3C22semidirect
35. #944C22semidirect
36. #944C22semidirect
44. C2×C4⋊Dic3C22semidirect
47. #924C22semidirect
48. #924C22semidirect
49. #924C22semidirect
50. #924C22semidirect
52. C2×Dic3⋊C4C22semidirect
54. #897C22semidirect
55. #897C22semidirect
56. #896C22semidirect
57. #896C22semidirect
58. #896C22semidirect
59. D3×C4⋊C4C22semidirect
60. D3×C4⋊C4C22semidirect
61. #889C22semidirect
62. #889C22semidirect
63. #889C22semidirect
64. #888C22semidirect
65. #888C22semidirect
66. #887C22semidirect
67. #887C22semidirect
73. #878C22semidirect
74. #878C22semidirect
88. C2×C22⋊Q8D3semidirect
105. C2×#67Q8semidirect
158. C22×Q8C2×D3semidirect
188. C22⋊Q8C2×D3semidirect
189. C22⋊Q8C2×D3semidirect
207. C2×C4⋊C4C2×D3semidirect
208. C2×C4⋊C4C2×D3semidirect
217. C23×C3C2×Q8semidirect
223. C22×D3C2×Q8semidirect
241. #67C2×Q8semidirect
242. #67C2×Q8semidirect
247. C2×Dic3C2×Q8semidirect
358. C23D3×Q8semidirect
431. C3#3501semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
Q8×#672
C2×C4×#672
C2×#9242
#56362
#56352
#56342
#56322
D3×C22⋊Q82
#56232
#56222
#55112
#54862
#54832
#35013