Finite Group Databasebeta

← #86948

Subgroups — #86948

Order 384.

1100 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#554C211yesmaximal, characteristic, complemented
2#5730C211yesmaximal, characteristic, complemented
3#5728C211yesmaximal, characteristic, complemented
4#5690C212yesmaximal, complemented
5#5689C212yesmaximal, complemented
6#5635C214yesmaximal, complemented
7#5634C214yesmaximal, complemented
8#5632C212yesmaximal, complemented
9D3×C22⋊Q8C212yesmaximal, complemented
10#5578C211yesmaximal, characteristic, complemented
11Q8×D12C211yesmaximal, characteristic, complemented
12#5545C212yesmaximal
13C4×D3×D4C211yesmaximal, characteristic
14#5543C212yesmaximal, complemented
15D4×Dic6C211yesmaximal, characteristic, complemented
16#5511C212yesmaximal, complemented
17C2×#897C212yesmaximal, complemented
18#3501—31nomaximal, Sylow
19C2×D3×Q8C2212yescomplemented
20C2×D3×D4C2211yescharacteristic
21C22×C4×D3C2212yes—
22C22×Dic6C2212yescomplemented
23C3×C4⋊Q8C2211yescharacteristic, complemented
24C3×#125C2212yescomplemented
25C3×C22⋊Q8C2212yescomplemented
26C3×C22⋊Q8C2214yescomplemented
27C3×C4×Q8C2211yescharacteristic, complemented
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2212yes—
30C2×C3×C4⋊C4C2212yescomplemented
31#946C2212yes—
32#946C2211yescharacteristic
33#946C2214yescomplemented
34Q8×Dic3C2212yescomplemented
35#944C2211yescharacteristic, complemented
36D4×Dic3C2212yes—
37D4×Dic3C2211yescharacteristic
38C4×#67C2212yes—
39C4×#67C2214yes—
40C2×(C2×D3)⋊C4C2214yes—
41C2×C4⋊Dic3C2214yescomplemented
42#924C2214yescomplemented
43#924C2212yescomplemented
44#897C2214yes—
45#897C2212yes—
46#897C2211yescharacteristic
47#897C2212yescomplemented
48#897C2214yescomplemented
49#897C2214yescomplemented
50#897C2212yescomplemented
51#896C2212yes—
52#896C2214yes—
53#896C2214yescomplemented
54#896C2214yescomplemented
55#893C2211yescharacteristic
56#893C2212yes—
57#892C2212yes—
58D3×C4⋊C4C2211yescharacteristic
59D3×C4⋊C4C2214yescomplemented
60D3×C4⋊C4C2212yescomplemented
61#890C2212yescomplemented
62#890C2212yescomplemented
63#889C2212yescomplemented
64#889C2214yescomplemented
65#888C2214yescomplemented
66#888C2212yescomplemented
67#888C2211yescharacteristic, complemented
68#887C2212yescomplemented
69#67⋊C4C2214yes—
70#67⋊C4C2212yes—
71D3×C22⋊C4C2212yes—
72D3×C22⋊C4C2214yes—
73#878C2214yescomplemented
74#878C2214yescomplemented
75#878C2214yescomplemented
76C4×D12C2212yes—
77C4×D12C2211yescharacteristic
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#869C2212yescomplemented
81#869C2211yescharacteristic, complemented
82C4×Dic6C2212yescomplemented
83C4×Dic6C2211yescharacteristic, complemented
84#554—31no—
85#554—32no—
86#554—34no—
87#554—32no—
88#554—32no—
89#554D311yescharacteristic, complemented
90D4×Q8—32no—
91D4×Q8—31no—
92D4×Q8—31no—
93#544—32no—
94#543—34no—
95#533—31no—
96#533—31no—
97C2×C22⋊Q8—32no—
98C2×C22⋊Q8—32no—
99#518—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C22×C3×C4C2314yes—
107C2×#67C2312yes—
108C22×Dic3C2312yes—
109C22×Dic3C2314yes—
110D3×Q8—44no—
111D3×D4Q818yescomplemented
112C2×D12C2311yescharacteristic
113C2×C4×D3—44no—
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2312yes—
116C2×C4×D3C2311yescharacteristic
117C2×C4×D3C2314yes—
118C2×C4×D3C2312yes—
119C2×C4×D3C2314yes—
120C2×C4×D3C2312yes—
121C2×C4×D3C2311yescharacteristic
122C2×Dic6—44no—
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2312yes—
125C2×Dic6C2312yes—
126C2×Dic6C2314yes—
127C2×Dic6C2314yes—
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2311yescharacteristic
130C3×C4⋊C4C2314yes—
131C3×C4⋊C4C2311yescharacteristic
132C3×C4⋊C4C2312yes—
133C3×C4⋊C4C2314yes—
134C3×C4⋊C4C2312yes—
135C3×C22⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C42C2311yescharacteristic
138C3×C42C2312yes—
139#217C2312yes—
140#217C2314yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2314yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2311yescharacteristic
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2314yes—
154C4⋊Dic3C2314yes—
155C4⋊Dic3C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2312yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2312yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2314yes—
170C22×Q8—32no—
171C22×Q8—32no—
172C22×D4—31no—
173C23×C4—32no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8—34no—
177C4⋊Q8—32no—
178C4⋊Q8—31no—
179C4⋊Q8—32no—
180C4⋊Q8C2×D311yescharacteristic, complemented
181#125—32no—
182#125—32no—
183#125—34no—
184#125—32no—
185#125C2×D312yescomplemented
186C22⋊Q8—31no—
187C22⋊Q8—32no—
188C22⋊Q8—34no—
189C22⋊Q8—32no—
190C22⋊Q8—34no—
191C22⋊Q8—32no—
192C22⋊Q8—31no—
193C22⋊Q8—34no—
194C22⋊Q8—34no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8C2×D312yescomplemented
206C22⋊Q8C2×D314yescomplemented
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8C2×D311yescharacteristic, complemented
212C4×D4—31no—
213C4×D4—32no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—31no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—32no—
222C4×D4C2×D311yescharacteristic
223C4×D4C2×D312yes—
224#117—32no—
225#117—32no—
226C2×C4⋊C4—34no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4C2×D312yescomplemented
231C2×C22⋊C4—34no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—34no—
234C2×C42—31no—
235C23×C3C2412yes—
236C22×D3—42no—
237C22×D3—42no—
238C22×D3C2×Q818yescomplemented
239C22×D3C2411yescharacteristic
240C22×D3C2412yes—
241C3×Q8—22no—
242C3×Q8—42no—
243C3×D4C2×Q814yescomplemented
244C2×C3×C4—24no—
245C2×C3×C4—44no—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2414yes—
248C2×C3×C4C2412yes—
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253#67C2×Q818yescomplemented
254C2×Dic3—44no—
255C2×Dic3—42no—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260C2×Dic3C2414yes—
261C2×Dic3C2412yes—
262C2×Dic3C2411yescharacteristic
263D12C2×Q814yescomplemented
264C4×D3—22no—
265C4×D3—24no—
266C4×D3—42no—
267C4×D3—42no—
268C4×D3—41no—
269C4×D3—44no—
270C4×D3—44no—
271C4×D3C2×Q814yescomplemented
272Dic6—22no—
273Dic6—24no—
274Dic6—44no—
275Dic6—42no—
276Dic6—44no—
277C24—32no—
278C2×Q8—32no—
279C2×Q8—31no—
280C2×Q8—32no—
281C2×Q8—34no—
282C2×Q8—34no—
283C2×Q8—124no—
284C2×Q8—124no—
285C2×Q8C22×D311yescharacteristic
286C2×Q8C22×D312yes—
287C2×D4—32no—
288C2×D4—38no—
289C2×D4—31no—
290C2×D4C22×D311yescharacteristic
291C22×C4—31no—
292C22×C4—34no—
293C22×C4—32no—
294C22×C4—31no—
295C22×C4—32no—
296C22×C4—31no—
297C22×C4—34no—
298C22×C4—34no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—124no—
302C22×C4C22×D312yes—
303C22×C4C22×D314yes—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—32no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—31no—
310C4⋊C4—34no—
311C4⋊C4—34no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—32no—
317C4⋊C4—34no—
318C4⋊C4—32no—
319C4⋊C4—32no—
320C4⋊C4—32no—
321C4⋊C4C22×D312yes—
322C4⋊C4C22×D311yescharacteristic
323C4⋊C4C22×D314yes—
324C4⋊C4C22×D311yescharacteristic
325C4⋊C4C22×D314yes—
326C4⋊C4C22×D312yes—
327C4⋊C4C22×D312yes—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—34no—
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4—32no—
337C22⋊C4C22×D312yes—
338C22⋊C4C22×D314yes—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—34no—
343C42—32no—
344C42C22×D311yescharacteristic
345C42C22×D312yes—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3C22×Q814yes—
349C22×C3C2511yescharacteristic
350C2×D3—22no—
351C2×D3—21no—
352C2×D3—22no—
353C2×D3—21no—
354C2×D3—48no—
355C2×D3C22×Q812yes—
356C2×D3C22×Q814yes—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—24no—
361C3×C4—21no—
362C3×C4C22×Q812yes—
363C3×C4H(5,𝔽2)12yes—
364Dic3—24no—
365Dic3—22no—
366Dic3—22no—
367Dic3—21no—
368Dic3—21no—
369Dic3—22no—
370Dic3C22×Q812yes—
371C23—31no—
372C23—32no—
373C23—38no—
374C23—122no—
375C23—122no—
376C23C23×D312yes—
377Q8—22no—
378Q8—42no—
379Q8—62no—
380Q8—64no—
381Q8—124no—
382Q8—124no—
383Q8—122no—
384D4—38no—
385D4—34no—
386D4D3×Q814yescomplemented
387C2×C4—24no—
388C2×C4—31no—
389C2×C4—34no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—32no—
393C2×C4—34no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—44no—
397C2×C4—62no—
398C2×C4—64no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—124no—
402C2×C4—124no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—124no—
406C2×C4C23×D311yescharacteristic
407C2×C4C23×D314yes—
408C2×C4C23×D312yes—
409C2×C4C23×D312yes—
410C2×C4C23×D312yes—
411C2×C4C23×D311yescharacteristic
412C2×C4C23×D311yescharacteristic
413C2×C3—24no—
414C2×C3C23×Q811yescharacteristic
415C2×C3C2×H(5,𝔽2)11yescharacteristic
416C2×C3#58511yescharacteristic
417D3—22no—
418D3—24no—
419C22—22no—
420C22—22no—
421C22—32no—
422C22—34no—
423C22—62no—
424C22—61no—
425C22—62no—
426C22—61no—
427C22—128no—
428C22C2×D3×Q814yes—
429C22C24×D311yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—24no—
434C4—21no—
435C4—32no—
436C4—61no—
437C4—62no—
438C4—64no—
439C4—62no—
440C4—62no—
441C4—61no—
442C4C2×D3×Q812yes—
443C4#100912yes—
444C3#350111yescharacteristic, complemented, Sylow
445C2—24no—
446C2—62no—
447C2—64no—
448C2C22×D3×Q811yescharacteristic
449C2C2×#100911yescharacteristic
450C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#554C2
prime radicalC2×D3×D4C22
hypercenter#554D3
hypocenterC3#3501
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C3×#554C2semidirect
2. #5730C2semidirect
3. #5728C2semidirect
4. #5690C2semidirect
5. #5689C2semidirect
6. #5635C2semidirect
7. #5634C2semidirect
8. #5632C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5578C2semidirect
11. Q8×D12C2semidirect
14. #5543C2semidirect
15. D4×Dic6C2semidirect
16. #5511C2semidirect
17. C2×#897C2semidirect
19. C2×D3×Q8C22semidirect
22. C22×Dic6C22semidirect
23. C3×C4⋊Q8C22semidirect
24. C3×#125C22semidirect
25. C3×C22⋊Q8C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C3×C4×Q8C22semidirect
30. C2×C3×C4⋊C4C22semidirect
33. #946C22semidirect
34. Q8×Dic3C22semidirect
35. #944C22semidirect
41. C2×C4⋊Dic3C22semidirect
42. #924C22semidirect
43. #924C22semidirect
47. #897C22semidirect
48. #897C22semidirect
49. #897C22semidirect
50. #897C22semidirect
53. #896C22semidirect
54. #896C22semidirect
59. D3×C4⋊C4C22semidirect
60. D3×C4⋊C4C22semidirect
61. #890C22semidirect
62. #890C22semidirect
63. #889C22semidirect
64. #889C22semidirect
65. #888C22semidirect
66. #888C22semidirect
67. #888C22semidirect
68. #887C22semidirect
73. #878C22semidirect
74. #878C22semidirect
75. #878C22semidirect
80. #869C22semidirect
81. #869C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
89. #554D3semidirect
111. D3×D4Q8semidirect
180. C4⋊Q8C2×D3semidirect
185. #125C2×D3semidirect
205. C22⋊Q8C2×D3semidirect
206. C22⋊Q8C2×D3semidirect
211. C4×Q8C2×D3semidirect
230. C2×C4⋊C4C2×D3semidirect
238. C22×D3C2×Q8semidirect
243. C3×D4C2×Q8semidirect
253. #67C2×Q8semidirect
263. D12C2×Q8semidirect
271. C4×D3C2×Q8semidirect
386. D4D3×Q8semidirect
444. C3#3501semidirect

Maximal subgroups

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

subgroupindex
C3×#5542
#57302
#57282
#56902
#56892
#56352
#56342
#56322
D3×C22⋊Q82
#55782
Q8×D122
#55452
C4×D3×D42
#55432
D4×Dic62
#55112
C2×#8972
#35013