Finite Group Databasebeta

← #80063

Subgroups — #80063

Order 384.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2C22×#215C211yesmaximal, characteristic, complemented
3C2×D3×SD16C211yesmaximal, characteristic, complemented
4#5609C211yesmaximal, characteristic
5C3×#448C211yesmaximal, characteristic, complemented
6#5175C211yesmaximal, characteristic, complemented
7#5169C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4818C211yesmaximal, characteristic, complemented
11#4807C211yesmaximal, characteristic, complemented
12#4806C211yesmaximal, characteristic, complemented
13#4788C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16D4×SD16—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic, complemented
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×D3×D4—21no—
20C2×D3×D4C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic, complemented
23C2×C3×SD16C2211yescharacteristic, complemented
24C3×#121C2211yescharacteristic
25#946—21no—
26#946C2211yescharacteristic, complemented
27Q8×Dic3C2211yescharacteristic, complemented
28#944—21no—
29#944C2211yescharacteristic, complemented
30C2×#215—24no—
31C2×#215C2211yescharacteristic
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic, complemented
34C2×#215C2211yescharacteristic, complemented
35#940C2211yescharacteristic
36#937—21no—
37C2×#214C2211yescharacteristic, complemented
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40#924—21no—
41#924C2211yescharacteristic, complemented
42C22×C3⋊C8C2211yescharacteristic, complemented
43D3×SD16—24no—
44C2×#204C2211yescharacteristic
45C2×#204C2211yescharacteristic, complemented
46C2×D3×C8C2211yescharacteristic, complemented
47#895C2211yescharacteristic
48#893C2211yescharacteristic
49#884—21no—
50#882—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#810C2211yescharacteristic, complemented
61#808C2211yescharacteristic, complemented
62C22×SD16—31no—
63C22×SD16—31no—
64D4×Q8—31no—
65D42—31no—
66#492—31no—
67#465—31no—
68#465—31no—
69#461—31no—
70#460—31no—
71#450—31no—
72#450—31no—
73#448—31no—
74#448D311yescharacteristic, complemented
75C4×SD16—31no—
76C8×D4—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×Q8—21no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×D4—21no—
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4C2311yescharacteristic
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67D411yescharacteristic
92D3×Q8—22no—
93D3×Q8—41no—
94D3×D4—22no—
95D3×D4—41no—
96D3×D4—44no—
97C2×D12—22no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—41no—
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C2×Dic6—21no—
107C2×Dic6C2311yescharacteristic
108C3×SD16—22no—
109C3×SD16—41no—
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C22⋊C4—21no—
114#217—21no—
115#217D411yescharacteristic
116#215—22no—
117#215—22no—
118#215—22no—
119#215D414yescomplemented
120#214—22no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4D411yescharacteristic
125C4⋊Dic3—21no—
126C4⋊Dic3C2311yescharacteristic
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3—21no—
133C4×Dic3C2311yescharacteristic
134C2×C3⋊C8—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137#204—22no—
138#204—41no—
139D3×C8—22no—
140C22×Q8—31no—
141C22×Q8C2×D311yescharacteristic, complemented
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145C2×SD16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16—64no—
153C2×SD16—64no—
154C2×SD16C2×D311yescharacteristic, complemented
155C2×SD16C2×D311yescharacteristic, complemented
156C22×C8—31no—
157C22×C8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160#127—31no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165#121—31no—
166#121—61no—
167#121C2×D311yescharacteristic
168#120—61no—
169#120—61no—
170C4×Q8—31no—
171C4×D4—31no—
172C4×D4—31no—
173C4×D4—61no—
174#106—31no—
175C4⋊C8—31no—
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#103C2×D311yescharacteristic, complemented
179#102—31no—
180#102—31no—
181#102—31no—
182C22⋊C8—31no—
183C22⋊C8C2×D311yescharacteristic, complemented
184C4×C8—31no—
185C23×C3—21no—
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3C2×D411yescharacteristic
196C3×Q8—21no—
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—42no—
200C3×Q8C2×D412yescomplemented
201C3×D4—21no—
202C3×D4—21no—
203C3×D4—42no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2411yescharacteristic
212#67—42no—
213#67—42no—
214#67—81no—
215#67SD1614yescomplemented
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220C2×Dic3C2×D411yescharacteristic
221D12—21no—
222D12—21no—
223D12—21no—
224D12—42no—
225D12—42no—
226D12C2×D412yes—
227C4×D3—21no—
228C4×D3—21no—
229C4×D3—42no—
230C4×D3—42no—
231Dic6—21no—
232Dic6—21no—
233Dic6—42no—
234C3×C8—21no—
235C3×C8—21no—
236C3⋊C8—21no—
237C3⋊C8C2×D412yescomplemented
238C24—61no—
239C24—61no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—31no—
243C2×Q8—41no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—121no—
247C2×Q8C22×D311yescharacteristic
248C2×Q8C22×D311yescharacteristic
249C2×D4—21no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—62no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—62no—
259C2×D4—61no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—124no—
263C2×D4C22×D311yescharacteristic
264C22×C4—21no—
265C22×C4—31no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4C22×D311yescharacteristic
269SD16—22no—
270SD16—34no—
271SD16—41no—
272SD16—62no—
273SD16—62no—
274SD16—62no—
275SD16—62no—
276SD16—62no—
277SD16—121no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—62no—
281C2×C8—62no—
282C2×C8C22×D311yescharacteristic
283C2×C8C22×D311yescharacteristic
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4C22×D311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—31no—
294C22⋊C4—31no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C42—31no—
300C42—61no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3C2×SD1612yescomplemented
307C22×C3C22×D411yescharacteristic
308C2×D3—21no—
309C2×D3—21no—
310C2×D3—22no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—84no—
317C2×D3—84no—
318C2×D3C2×SD1612yescomplemented
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4C22×D411yescharacteristic
324C3×C4H(5,𝔽2)11yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—41no—
328Dic3—41no—
329Dic3C2×SD1612yescomplemented
330C23—21no—
331C23—31no—
332C23—61no—
333C23—61no—
334C23—122no—
335C23—122no—
336C23—122no—
337C23—122no—
338C23—241no—
339C23—241no—
340C23D3×D411yescharacteristic
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—42no—
345Q8—61no—
346Q8—61no—
347Q8—122no—
348Q8D3×D412yescomplemented
349D4—21no—
350D4—21no—
351D4—34no—
352D4—32no—
353D4—42no—
354D4—61no—
355D4—61no—
356D4—61no—
357D4—122no—
358D4—122no—
359D4—122no—
360D4—122no—
361D4—241no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—42no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4D3×D411yescharacteristic
378C2×C4C23×D311yescharacteristic
379C8—21no—
380C8—21no—
381C8—32no—
382C8—61no—
383C2×C3—22no—
384C2×C3—81no—
385C2×C3D4211yescharacteristic
386C2×C3C22×SD1611yescharacteristic
387C2×C3#57711yescharacteristic
388D3—22no—
389D3—42no—
390D3—81no—
391C22—21no—
392C22—21no—
393C22—32no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—61no—
398C22—61no—
399C22—62no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—244no—
406C22—244no—
407C22D3×SD1612yescomplemented
408C22C2×D3×D411yescharacteristic
409C4—21no—
410C4—22no—
411C4—32no—
412C4—41no—
413C4—41no—
414C4—61no—
415C4—61no—
416C4—121no—
417C4—121no—
418C4C2×D3×D411yescharacteristic
419C4#100411yescharacteristic
420C3D4×SD1611yescharacteristic, complemented, Sylow
421C2—22no—
422C2—62no—
423C2—81no—
424C2—122no—
425C2—241no—
426C2#559011yescharacteristic
427C2C2×D3×SD1611yescharacteristic
428C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5609C2
hypercenter#448D3
hypocenterC3D4×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
2. C22×#215C2semidirect
3. C2×D3×SD16C2semidirect
5. C3×#448C2semidirect
6. #5175C2semidirect
7. #5169C2semidirect
8. #5163C2semidirect
9. #5041C2semidirect
10. #4818C2semidirect
11. #4807C2semidirect
12. #4806C2semidirect
13. #4788C2semidirect
14. #4731C2semidirect
15. #67⋊C8C2semidirect
17. C22×C3×Q8C22semidirect
18. C2×D3×Q8C22semidirect
22. C2×C3×SD16C22semidirect
23. C2×C3×SD16C22semidirect
26. #946C22semidirect
27. Q8×Dic3C22semidirect
29. #944C22semidirect
33. C2×#215C22semidirect
34. C2×#215C22semidirect
37. C2×#214C22semidirect
41. #924C22semidirect
42. C22×C3⋊C8C22semidirect
45. C2×#204C22semidirect
46. C2×D3×C8C22semidirect
51. C3×#103C22semidirect
52. C3×#103C22semidirect
53. C3×C22⋊C8C22semidirect
54. #832C22semidirect
56. #821C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #810C22semidirect
61. #808C22semidirect
74. #448D3semidirect
119. #215D4semidirect
141. C22×Q8C2×D3semidirect
154. C2×SD16C2×D3semidirect
155. C2×SD16C2×D3semidirect
177. #103C2×D3semidirect
178. #103C2×D3semidirect
183. C22⋊C8C2×D3semidirect
200. C3×Q8C2×D4semidirect
215. #67SD16semidirect
237. C3⋊C8C2×D4semidirect
306. C22×C3C2×SD16semidirect
318. C2×D3C2×SD16semidirect
329. Dic3C2×SD16semidirect
348. Q8D3×D4semidirect
407. C22D3×SD16semidirect
420. C3D4×SD16semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
C22×#2152
C2×D3×SD162
#56092
C3×#4482
#51752
#51692
#51632
#50412
#48182
#48072
#48062
#47882
#47312
#67⋊C82
D4×SD163