Finite Group Databasebeta

← #80137

Subgroups — #80137

Order 384.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5798C211yesmaximal, characteristic
2#5625C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic
4#5218C211yesmaximal, characteristic
5#5164C211yesmaximal, characteristic, complemented
6#5162C211yesmaximal, characteristic, complemented
7#5162C211yesmaximal, characteristic, complemented
8Dic3×SD16C211yesmaximal, characteristic, complemented
9#5045C211yesmaximal, characteristic, complemented
10#4766C211yesmaximal, characteristic, complemented
11#4763C211yesmaximal, characteristic, complemented
12#4758C211yesmaximal, characteristic, complemented
13#4757C211yesmaximal, characteristic, complemented
14#4723C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3123—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21C2×#217—21no—
22C2×#217—21no—
23Q8×Dic3C2211yescharacteristic, complemented
24#944C2211yescharacteristic, complemented
25D4×Dic3—21no—
26D4×Dic3—21no—
27D4×Dic3C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic
33#926C2211yescharacteristic
34#926C2211yescharacteristic
35#924C2211yescharacteristic
36#890—21no—
37#890C2211yescharacteristic, complemented
38#889—21no—
39#887C2211yescharacteristic, complemented
40#879—21no—
41#879—21no—
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45#835C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50#818C2211yescharacteristic, complemented
51#817C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#541—31no—
61#518—31no—
62#485—31no—
63#485—31no—
64#478—31no—
65#478—31no—
66#461—31no—
67#461—31no—
68#450—31no—
69#450D311yescharacteristic
70#440—31no—
71#440—31no—
72C4×SD16—31no—
73C4×SD16—31no—
74#433—31no—
75C24×C3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×Dic3—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86C22×Dic3—21no—
87C2×Dic6C2311yescharacteristic
88C3×SD16—22no—
89C3×SD16—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95#217—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217D411yescharacteristic
102#217D411yescharacteristic
103#214—22no—
104#214—22no—
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3D411yescharacteristic
108C4⋊Dic3D411yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C22×D4C2×D311yescharacteristic
125C2×SD16—31no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×SD16C2×D311yescharacteristic
129C4⋊Q8—31no—
130#126—61no—
131#126—61no—
132#125—31no—
133#125—61no—
134#125—61no—
135C22⋊Q8—31no—
136C22⋊Q8C2×D311yescharacteristic
137C4×Q8—31no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145#117—31no—
146#117—31no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic
162#102C2×D311yescharacteristic
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3—24no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×D4—24no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4#4112yes—
179C3×D4#4112yes—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—81no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198Dic6—21no—
199Dic6—21no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204C24—21no—
205C2×Q8—31no—
206C2×Q8C22×D311yescharacteristic
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—21no—
210C2×D4C22×D311yescharacteristic
211C2×D4C22×D311yescharacteristic
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4C22×D311yescharacteristic
217SD16—22no—
218SD16—22no—
219SD16—62no—
220SD16—62no—
221C2×C8—31no—
222C2×C8—31no—
223C2×C8C22×D311yescharacteristic
224C2×C8C22×D311yescharacteristic
225C4⋊C4—21no—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—31no—
241C22⋊C4—31no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—31no—
249C42—31no—
250C42—61no—
251C42—61no—
252C42—61no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—84no—
263C22×C3C22×D411yescharacteristic
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4C2×#4111yescharacteristic
268C3×C4C2×#4111yescharacteristic
269Dic3—21no—
270Dic3—21no—
271Dic3—21no—
272Dic3—21no—
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276Dic3—41no—
277C23—24no—
278C23—21no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23D3×D411yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—61no—
286Q8—61no—
287D4—24no—
288D4—21no—
289D4—21no—
290D4#23712yes—
291D4#23712yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—241no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4—241no—
308C2×C4D3×D411yescharacteristic
309C2×C4C23×D311yescharacteristic
310C8—21no—
311C8—21no—
312C8—61no—
313C8—61no—
314C2×C3—41no—
315C2×C3—42no—
316C2×C3—42no—
317C2×C3#52511yescharacteristic
318C2×C3#57711yescharacteristic
319C2×C3#57711yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—84no—
330C22C2×D3×D411yescharacteristic
331C4—21no—
332C4—41no—
333C4—41no—
334C4—61no—
335C4—61no—
336C4—61no—
337C4—61no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4—121no—
342C4C2×#23711yescharacteristic
343C4C2×#23711yescharacteristic
344C3#312311yescharacteristic, complemented, Sylow
345C2—41no—
346C2—42no—
347C2—42no—
348C2#558611yescharacteristic
349C2#576211yescharacteristic
350C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalC22×C3×D4C22
hypercenter#450D3
hypocenterC3#3123
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #5625C2semidirect
5. #5164C2semidirect
6. #5162C2semidirect
7. #5162C2semidirect
8. Dic3×SD16C2semidirect
9. #5045C2semidirect
10. #4766C2semidirect
11. #4763C2semidirect
12. #4758C2semidirect
13. #4757C2semidirect
14. #4723C2semidirect
15. #4720C2semidirect
23. Q8×Dic3C22semidirect
24. #944C22semidirect
37. #890C22semidirect
39. #887C22semidirect
45. #835C22semidirect
46. #835C22semidirect
50. #818C22semidirect
51. #817C22semidirect
52. #816C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #808C22semidirect
59. #807C22semidirect
344. C3#3123semidirect

Maximal subgroups

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

subgroupindex
#57982
#56252
C3×#4502
#52182
#51642
#51622
Dic3×SD162
#50452
#47662
#47632
#47582
#47572
#47232
#47202
#31233