Finite Group Databasebeta

← #80814

Subgroups — #80814

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
1#5627C211yesmaximal, characteristic, complemented
2#5606C211yesmaximal, characteristic, complemented
3C3×#468C211yesmaximal, characteristic, complemented
4#5173C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic, complemented
7#5126C211yesmaximal, characteristic, complemented
8#5052C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#4888C211yesmaximal, characteristic, complemented
11#4885C211yesmaximal, characteristic, complemented
12#4814C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4784C211yesmaximal, characteristic, complemented
15#4781C211yesmaximal, characteristic, complemented
16#3197—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic, complemented
26#938C2211yescharacteristic
27#937—21no—
28D4×Dic3—21no—
29C2×#213C2211yescharacteristic, complemented
30#930C2211yescharacteristic
31#924C2211yescharacteristic
32C2×#208C2211yescharacteristic
33C2×#204C2211yescharacteristic, complemented
34#897C2211yescharacteristic
35#895C2211yescharacteristic
36#894—21no—
37#893—21no—
38#892C2211yescharacteristic
39#892C2211yescharacteristic
40#883—21no—
41#882—21no—
42#67⋊C4—21no—
43#67⋊C4—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#821C2211yescharacteristic, complemented
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic, complemented
55#816C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#525—31no—
61#525—31no—
62#485—31no—
63#485—31no—
64#482—31no—
65#482—31no—
66#470—31no—
67#469—31no—
68#468—31no—
69#468D311yescharacteristic, complemented
70#452—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#407—31no—
75C23×D3D411yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3D411yescharacteristic
85#239—41no—
86#239—41no—
87#237—41no—
88#237—41no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×C4×D3—41no—
94C2×C4×D3—41no—
95C2×C4×D3D411yescharacteristic
96C2×C4×D3D411yescharacteristic
97C2×C4×D3C2311yescharacteristic
98C2×C4×D3C2311yescharacteristic
99C2×Dic6C2311yescharacteristic
100C3×SD16—41no—
101C3×M16—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4—21no—
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109#217—21no—
110#217—21no—
111#216—41no—
112#213—41no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C4×Dic3—21no—
125C4×Dic3—21no—
126#208—41no—
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129#204—41no—
130C2×#41—31no—
131C2×#41—31no—
132C23×C4—31no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16C2×D311yescharacteristic, complemented
136C2×D8—31no—
137C2×M16—31no—
138C2×M16C2×D311yescharacteristic
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8C2×D311yescharacteristic
145#121—31no—
146#121—31no—
147#121—31no—
148#121C2×D311yescharacteristic
149#120—61no—
150#120—61no—
151C4×D4—61no—
152C4×D4—61no—
153C4×D4—61no—
154C4×D4—61no—
155#117—31no—
156#117—31no—
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic, complemented
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3C2×D411yescharacteristic
180C22×D3C2×D411yescharacteristic
181C3×Q8—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—42no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—42no—
195#67—42no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—42no—
206D12—42no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—41no—
212C4×D3—41no—
213C4×D3—42no—
214Dic6—21no—
215Dic6—21no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—31no—
221#41—121no—
222#41—121no—
223#41—121no—
224#41—121no—
225C2×Q8—31no—
226C2×Q8C22×D311yescharacteristic
227C2×D4—21no—
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4C22×D311yescharacteristic
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4C22×D311yescharacteristic
245Q16—121no—
246SD16—41no—
247SD16—121no—
248D8—121no—
249M16—41no—
250M16—121no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8C22×D311yescharacteristic
254C2×C8C22×D311yescharacteristic
255C4⋊C4—21no—
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C4⋊C4C22×D311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C42—61no—
274C42—61no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3C22×D411yescharacteristic
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—44no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4H(5,𝔽2)11yescharacteristic
297C3×C4H(5,𝔽2)11yescharacteristic
298Dic3—21no—
299Dic3—21no—
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303C23—21no—
304C23—31no—
305C23—31no—
306C23—61no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23D3×D411yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—61no—
319D4—61no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—122no—
324D4—122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4D3×D411yescharacteristic
345C2×C4C23×D311yescharacteristic
346C8—21no—
347C8—21no—
348C8—61no—
349C8—61no—
350C2×C3—41no—
351C2×C3—81no—
352C2×C3#53411yescharacteristic
353C2×C3#57511yescharacteristic
354C2×C3#57511yescharacteristic
355D3—41no—
356D3—41no—
357D3—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—124no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—61no—
380C4—61no—
381C4—121no—
382C4—121no—
383C4—121no—
384C4#100411yescharacteristic
385C4#100911yescharacteristic
386C3#319711yescharacteristic, complemented, Sylow
387C2—41no—
388C2—81no—
389C2—121no—
390C2—121no—
391C2—241no—
392C2#560511yescharacteristic
393C2#577311yescharacteristic
394C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
hypercenter#468D3
hypocenterC3#3197
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #5627C2semidirect
2. #5606C2semidirect
3. C3×#468C2semidirect
4. #5173C2semidirect
5. #5172C2semidirect
6. #5134C2semidirect
7. #5126C2semidirect
8. #5052C2semidirect
9. #5039C2semidirect
10. #4888C2semidirect
11. #4885C2semidirect
12. #4814C2semidirect
13. #4808C2semidirect
14. #4784C2semidirect
15. #4781C2semidirect
20. C2×C3×SD16C22semidirect
25. C2×#216C22semidirect
29. C2×#213C22semidirect
33. C2×#204C22semidirect
44. C3×#107C22semidirect
45. C3×#103C22semidirect
46. C3×#102C22semidirect
47. #835C22semidirect
48. #832C22semidirect
49. #821C22semidirect
54. #818C22semidirect
55. #816C22semidirect
56. #810C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #808C22semidirect
69. #468D3semidirect
135. C2×SD16C2×D3semidirect
158. #107C2×D3semidirect
164. #103C2×D3semidirect
168. #102C2×D3semidirect
386. C3#3197semidirect

Maximal subgroups

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

subgroupindex
#56272
#56062
C3×#4682
#51732
#51722
#51342
#51262
#50522
#50392
#48882
#48852
#48142
#48082
#47842
#47812
#31973