Finite Group Databasebeta

← #82932

Subgroups — #82932

Order 384.

418 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
1C3×#525C211yesmaximal, characteristic, complemented
2#5791C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5207C211yesmaximal, characteristic, complemented
6#5052C211yesmaximal, characteristic, complemented
7#5049C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5045C211yesmaximal, characteristic, complemented
10#5042C211yesmaximal, characteristic, complemented
11#5039C211yesmaximal, characteristic, complemented
12#5037C211yesmaximal, characteristic, complemented
13#5035C211yesmaximal, characteristic, complemented
14#5003C211yesmaximal, characteristic, complemented
15#4996C211yesmaximal, characteristic, complemented
16#3197—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C3×#123—21no—
21C3×C22⋊Q8C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×#120—21no—
26C3×C4×D4—21no—
27C3×C4×D4—21no—
28C3×#117C2211yescharacteristic
29#953—21no—
30C2×#216C2211yescharacteristic, complemented
31C2×#215C2211yescharacteristic, complemented
32C2×#214C2211yescharacteristic, complemented
33C2×#213C2211yescharacteristic, complemented
34#930C2211yescharacteristic
35#930C2211yescharacteristic
36#929—21no—
37C4×#67—21no—
38C4×#67—21no—
39#926C2211yescharacteristic
40#924C2211yescharacteristic
41#924C2211yescharacteristic
42C2×#208C2211yescharacteristic
43C2×#208C2211yescharacteristic
44#835C2211yescharacteristic, complemented
45#835C2211yescharacteristic, complemented
46#832C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#830C2211yescharacteristic
49#830C2211yescharacteristic
50#830C2211yescharacteristic
51#830C2211yescharacteristic
52#810C2211yescharacteristic, complemented
53#810C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#808C2211yescharacteristic, complemented
57#808C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#525—31no—
61#525D311yescharacteristic, complemented
62#485—31no—
63#485—31no—
64#482—31no—
65#482—31no—
66#470—31no—
67#469—31no—
68#468—31no—
69#468—31no—
70#452—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#407—31no—
75C24×C3D411yescharacteristic
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4D411yescharacteristic
87C22×C3×C4D411yescharacteristic
88C22×C3×C4D411yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95#235—41no—
96#235—41no—
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×Dic6C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109C3×C42—21no—
110#217—21no—
111#217—21no—
112#217—21no—
113#216—41no—
114#215—41no—
115#214—41no—
116#213—41no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124#208—41no—
125#208—41no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129C2×C3⋊C8C2311yescharacteristic
130C2×#41—31no—
131C2×#41C2×D311yescharacteristic
132C23×C4C2×D311yescharacteristic
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×D8—31no—
137C2×M16—31no—
138C2×M16—31no—
139#123—21no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8C2×D311yescharacteristic
144C22⋊Q8C2×D311yescharacteristic
145#121—31no—
146#121—31no—
147#121C2×D311yescharacteristic
148#121C2×D311yescharacteristic
149#120—21no—
150#120—61no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—61no—
154C4×D4—61no—
155#117—31no—
156#117C2×D311yescharacteristic
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3C2×D411yescharacteristic
178C23×C3C2×D411yescharacteristic
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—42no—
186C3×D4—42no—
187C3×D4—42no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2411yescharacteristic
205#67—42no—
206#67—42no—
207#67—42no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211D12—21no—
212D12—21no—
213C4×D3—42no—
214Dic6—21no—
215Dic6—21no—
216C3⋊C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24#6711yescharacteristic
221#41—41no—
222#41—41no—
223#41—121no—
224#41—121no—
225C2×Q8—31no—
226C2×Q8C22×D311yescharacteristic
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—31no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4C22×D311yescharacteristic
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—61no—
239C22×C4#6711yescharacteristic
240C22×C4#6711yescharacteristic
241C22×C4#6711yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245Q16—121no—
246SD16—121no—
247SD16—121no—
248D8—121no—
249M16—121no—
250M16—121no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—31no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—31no—
258C4⋊C4—31no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C4⋊C4C22×D311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C42—21no—
274C42—61no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—44no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3C22×D411yescharacteristic
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4H(5,𝔽2)11yescharacteristic
299C3×C4H(5,𝔽2)11yescharacteristic
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—61no—
308C23C2×#6711yescharacteristic
309C23C2×#6711yescharacteristic
310C23C2×#6711yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—61no—
321D4—61no—
322D4—122no—
323D4—122no—
324D4—122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—122no—
342C2×C4C2×#6711yescharacteristic
343C2×C4C2×#6711yescharacteristic
344C2×C4C2×#6711yescharacteristic
345C2×C4C23×D311yescharacteristic
346C8—61no—
347C8—61no—
348C8—61no—
349C8—61no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—81no—
354C2×C3#53411yescharacteristic
355C2×C3#57511yescharacteristic
356C2×C3#57511yescharacteristic
357D3—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22C22×#6711yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—121no—
382C4—121no—
383C4—121no—
384C4#100411yescharacteristic
385C4#100411yescharacteristic
386C3#319711yescharacteristic, complemented, Sylow
387C2—41no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—241no—
392C2#580411yescharacteristic
393C2#582211yescharacteristic
394C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#525C2
socleC22×C3C22×D4
hypercenter#525D3
hypocenterC3#3197
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C3×#525C2semidirect
2. #5791C2semidirect
3. #5239C2semidirect
4. #5238C2semidirect
5. #5207C2semidirect
6. #5052C2semidirect
7. #5049C2semidirect
8. #5047C2semidirect
9. #5045C2semidirect
10. #5042C2semidirect
11. #5039C2semidirect
12. #5037C2semidirect
13. #5035C2semidirect
14. #5003C2semidirect
15. #4996C2semidirect
30. C2×#216C22semidirect
31. C2×#215C22semidirect
32. C2×#214C22semidirect
33. C2×#213C22semidirect
44. #835C22semidirect
45. #835C22semidirect
46. #832C22semidirect
47. #832C22semidirect
52. #810C22semidirect
53. #810C22semidirect
54. #809C22semidirect
55. #809C22semidirect
56. #808C22semidirect
57. #808C22semidirect
58. #807C22semidirect
59. #807C22semidirect
61. #525D3semidirect
386. C3#3197semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
#57912
#52392
#52382
#52072
#50522
#50492
#50472
#50452
#50422
#50392
#50372
#50352
#50032
#49962
#31973