Finite Group Databasebeta

← #80680

Subgroups — #80680

Order 384.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5635C211yesmaximal, characteristic, complemented
2#5609C211yesmaximal, characteristic
3C3×#465C211yesmaximal, characteristic, complemented
4#5169C211yesmaximal, characteristic, complemented
5#5163C211yesmaximal, characteristic, complemented
6C2×#821C211yesmaximal, characteristic, complemented
7C8×#67C211yesmaximal, characteristic, complemented
8#5046C211yesmaximal, characteristic, complemented
9#5034C211yesmaximal, characteristic, complemented
10#4870C211yesmaximal, characteristic, complemented
11#4863C211yesmaximal, characteristic, complemented
12#4807C211yesmaximal, characteristic, complemented
13#4788C211yesmaximal, characteristic, complemented
14D3×#102C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3295—31nomaximal, Sylow
17C2×D3×D4—21no—
18C2×D3×D4C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic, complemented
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic, complemented
28#940C2211yescharacteristic
29#937—21no—
30C4×#67C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic, complemented
32C2×D3×C8C2211yescharacteristic, complemented
33#897—21no—
34#895C2211yescharacteristic
35#893C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic, complemented
37#889—21no—
38#888C2211yescharacteristic, complemented
39#884—21no—
40#882—21no—
41#67⋊C4—21no—
42#878—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#832C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#830C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#554—31no—
61D42—31no—
62#486—31no—
63#481—31no—
64#481—31no—
65#476—31no—
66#465—31no—
67#465D311yescharacteristic, complemented
68#460—31no—
69#450—31no—
70#450—31no—
71C4×SD16—31no—
72C8×D4—31no—
73C2×#102—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67D411yescharacteristic
85C22×Dic3—21no—
86D3×D4—22no—
87D3×D4—41no—
88D3×D4—44no—
89C2×D12—22no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—41no—
93C2×D12C2311yescharacteristic
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—21no—
99C3×SD16—41no—
100C2×C3×C8—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4—21no—
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108#217D411yescharacteristic
109#215—22no—
110#215—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16C2×D311yescharacteristic, complemented
134C22×C8—31no—
135C22×C8C2×D311yescharacteristic, complemented
136C4⋊Q8—31no—
137#127—31no—
138#125—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C22⋊Q8C2×D311yescharacteristic, complemented
143#121—31no—
144#121—61no—
145#121C2×D311yescharacteristic
146#120—61no—
147#120—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154#106—31no—
155#106—31no—
156#106C2×D311yescharacteristic, complemented
157C4⋊C8—31no—
158#103C2×D311yescharacteristic, complemented
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170C4×C8—31no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—81no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—42no—
196#67—81no—
197#67SD1614yescomplemented
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—81no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—42no—
208D12—42no—
209D12#4112yes—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—42no—
213C4×D3—81no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220C24—61no—
221C2×Q8—61no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—62no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—62no—
233C2×D4—61no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—124no—
237C2×D4C22×D311yescharacteristic
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4C22×D311yescharacteristic
243SD16—41no—
244SD16—62no—
245SD16—121no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—41no—
249C2×C8—121no—
250C2×C8C22×D311yescharacteristic
251C2×C8C22×D311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C4⋊C4C22×D311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—31no—
266C22⋊C4—31no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C42—31no—
271C42—61no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C2×SD1612yes—
278C22×C3C22×D411yescharacteristic
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—22no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—84no—
288C2×D3—84no—
289C2×D3C2×SD1612yes—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4C2×#4111yescharacteristic
295C3×C4H(5,𝔽2)11yescharacteristic
296Dic3—21no—
297Dic3—21no—
298Dic3—41no—
299Dic3—41no—
300Dic3C2×SD1612yes—
301C23—21no—
302C23—31no—
303C23—61no—
304C23—61no—
305C23—122no—
306C23—122no—
307C23—122no—
308C23—122no—
309C23—241no—
310C23—241no—
311C23D3×D411yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—34no—
318D4—32no—
319D4—42no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—122no—
324D4—122no—
325D4—122no—
326D4—122no—
327D4—241no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4D3×D411yescharacteristic
344C2×C4C23×D311yescharacteristic
345C8—21no—
346C8—21no—
347C8—61no—
348C8—61no—
349C2×C3—22no—
350C2×C3—81no—
351C2×C3#54611yescharacteristic
352C2×C3C22×SD1611yescharacteristic
353C2×C3#57611yescharacteristic
354D3—22no—
355D3—42no—
356D3—81no—
357C22—21no—
358C22—21no—
359C22—32no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—61no—
364C22—61no—
365C22—62no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—244no—
372C22—244no—
373C22D3×SD1612yes—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—32no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—61no—
381C4—61no—
382C4—121no—
383C4—121no—
384C4#100411yescharacteristic
385C4D3×#4111yescharacteristic
386C3#329511yescharacteristic, complemented, Sylow
387C2—22no—
388C2—62no—
389C2—81no—
390C2—122no—
391C2—241no—
392C2#561011yescharacteristic
393C2C2×D3×SD1611yescharacteristic
394C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
prime radical#5609C2
hypercenter#465D3
hypocenterC3#3295
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5635C2semidirect
3. C3×#465C2semidirect
4. #5169C2semidirect
5. #5163C2semidirect
6. C2×#821C2semidirect
7. C8×#67C2semidirect
8. #5046C2semidirect
9. #5034C2semidirect
10. #4870C2semidirect
11. #4863C2semidirect
12. #4807C2semidirect
13. #4788C2semidirect
14. D3×#102C2semidirect
15. #4760C2semidirect
20. C2×C3×SD16C22semidirect
21. C22×C3×C8C22semidirect
22. C3×C22⋊Q8C22semidirect
24. #946C22semidirect
25. Q8×Dic3C22semidirect
27. C2×#215C22semidirect
31. C2×C4⋊Dic3C22semidirect
32. C2×D3×C8C22semidirect
36. D3×C4⋊C4C22semidirect
38. #888C22semidirect
43. C3×#106C22semidirect
44. C3×#103C22semidirect
45. C3×#102C22semidirect
46. #832C22semidirect
47. #832C22semidirect
48. #830C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. Dic3⋊C8C22semidirect
55. C8×Dic3C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #808C22semidirect
67. #465D3semidirect
133. C2×SD16C2×D3semidirect
135. C22×C8C2×D3semidirect
142. C22⋊Q8C2×D3semidirect
156. #106C2×D3semidirect
158. #103C2×D3semidirect
167. #102C2×D3semidirect
197. #67SD16semidirect
386. C3#3295semidirect

Maximal subgroups

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

subgroupindex
#56352
#56092
C3×#4652
#51692
#51632
C2×#8212
C8×#672
#50462
#50342
#48702
#48632
#48072
#47882
D3×#1022
#47602
#32953