Finite Group Databasebeta

← #80816

Subgroups — #80816

Order 384.

410 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#5628C211yesmaximal, characteristic, complemented
2#5607C211yesmaximal, characteristic
3C3×#468C211yesmaximal, characteristic, complemented
4#5170C211yesmaximal, characteristic, complemented
5#5169C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic, complemented
7#5129C211yesmaximal, characteristic, complemented
8#5048C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4888C211yesmaximal, characteristic, complemented
11#4885C211yesmaximal, characteristic, complemented
12#4814C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4784C211yesmaximal, characteristic, complemented
15#4776C211yesmaximal, characteristic, complemented
16#3200—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic, complemented
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#215C2211yescharacteristic
26#940—21no—
27#938C2211yescharacteristic
28#937—21no—
29C2×#214C2211yescharacteristic
30#930C2211yescharacteristic
31#924C2211yescharacteristic, complemented
32C22×C3⋊C8C2211yescharacteristic, complemented
33C2×#204C2211yescharacteristic
34#897C2211yescharacteristic
35#895C2211yescharacteristic
36#894—21no—
37#892C2211yescharacteristic
38#892C2211yescharacteristic, complemented
39#888—21no—
40#885—21no—
41#885—21no—
42#883—21no—
43#882—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#818C2211yescharacteristic, complemented
55#816C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#536—31no—
61#535—31no—
62#485—31no—
63#485—31no—
64#482—31no—
65#482—31no—
66#468—31no—
67#468D311yescharacteristic, complemented
68#465—31no—
69#465—31no—
70#450—31no—
71#450—31no—
72#448—31no—
73#448—31no—
74#408—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67D411yescharacteristic
84C2×#67D411yescharacteristic
85D3×Q8—41no—
86D3×Q8—41no—
87D3×D4—41no—
88D3×D4—41no—
89#235—41no—
90#235—41no—
91C2×D12—21no—
92C2×D12D411yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6D411yescharacteristic
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#215—41no—
112#214—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—
126C2×C3⋊C8—41no—
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129#204—41no—
130C2×#41—31no—
131C22×Q8—31no—
132C22×D4—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic
137C2×M16C2×D311yescharacteristic, complemented
138C22×C8—31no—
139C4⋊Q8—61no—
140#127—61no—
141#124—61no—
142#124—61no—
143#123—61no—
144#123—61no—
145C22⋊Q8—31no—
146C22⋊Q8—31no—
147C22⋊Q8—31no—
148C22⋊Q8C2×D311yescharacteristic, complemented
149#121—31no—
150#121—31no—
151#121—31no—
152#121C2×D311yescharacteristic
153#120—61no—
154#120—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
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3—81no—
179C22×D3C2×D411yescharacteristic
180C22×D3C2×D411yescharacteristic
181C3×Q8—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—81no—
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—41no—
194#67—41no—
195#67—42no—
196#67—81no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—41no—
206D12—81no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212Dic6—21no—
213Dic6—21no—
214Dic6—41no—
215Dic6—42no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—61no—
221#41—121no—
222#41—121no—
223C2×Q8—31no—
224C2×Q8—31no—
225C2×Q8—61no—
226C2×Q8—121no—
227C2×Q8—121no—
228C2×Q8C22×D311yescharacteristic
229C2×D4—21no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4C22×D311yescharacteristic
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—61no—
244C22×C4C22×D311yescharacteristic
245SD16—41no—
246SD16—121no—
247SD16—121no—
248SD16—121no—
249M16—41no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—121no—
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—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—84no—
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—122no—
308C23—241no—
309C23—241no—
310C23D3×D411yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315Q8—121no—
316Q8—122no—
317D4—21no—
318D4—21no—
319D4—61no—
320D4—61no—
321D4—81no—
322D4—121no—
323D4—121no—
324D4—121no—
325D4—122no—
326D4—241no—
327D4—241no—
328D4—241no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
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#57711yescharacteristic
354C2×C3#57711yescharacteristic
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—121no—
371C22—121no—
372C22—121no—
373C22—244no—
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#320011yescharacteristic, complemented, Sylow
387C2—41no—
388C2—81no—
389C2—121no—
390C2—121no—
391C2—241no—
392C2#560511yescharacteristic
393C2#577511yescharacteristic
394C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radical#5607C2
hypercenter#468D3
hypocenterC3#3200
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. #5628C2semidirect
3. C3×#468C2semidirect
4. #5170C2semidirect
5. #5169C2semidirect
6. #5134C2semidirect
7. #5129C2semidirect
8. #5048C2semidirect
9. #5041C2semidirect
10. #4888C2semidirect
11. #4885C2semidirect
12. #4814C2semidirect
13. #4806C2semidirect
14. #4784C2semidirect
15. #4776C2semidirect
17. C2×D3×Q8C22semidirect
21. C2×C3×M16C22semidirect
22. C3×C22⋊Q8C22semidirect
31. #924C22semidirect
32. C22×C3⋊C8C22semidirect
38. #892C22semidirect
44. C3×#107C22semidirect
45. C3×#103C22semidirect
47. #835C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #818C22semidirect
55. #816C22semidirect
56. #810C22semidirect
58. #808C22semidirect
59. #808C22semidirect
67. #468D3semidirect
137. C2×M16C2×D3semidirect
148. C22⋊Q8C2×D3semidirect
158. #107C2×D3semidirect
164. #103C2×D3semidirect
386. C3#3200semidirect

Maximal subgroups

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

subgroupindex
#56282
#56072
C3×#4682
#51702
#51692
#51342
#51292
#50482
#50412
#48882
#48852
#48142
#48062
#47842
#47762
#32003