Finite Group Databasebeta

← #80788

Subgroups — #80788

Order 384.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5625C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic
3C3×#468C211yesmaximal, characteristic
4#5167C211yesmaximal, characteristic, complemented
5Dic3×SD16C211yesmaximal, characteristic, complemented
6#5137C211yesmaximal, characteristic, complemented
7#5122C211yesmaximal, characteristic, complemented
8#5051C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic
10#4876C211yesmaximal, characteristic, complemented
11#4873C211yesmaximal, characteristic, complemented
12#4792C211yesmaximal, characteristic, complemented
13#4789C211yesmaximal, characteristic, complemented
14#4768C211yesmaximal, characteristic, complemented
15#4757C211yesmaximal, characteristic, complemented
16#3178—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#216C2211yescharacteristic
24#936—21no—
25#936C2211yescharacteristic
26#935—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29#926C2211yescharacteristic
30#924C2211yescharacteristic
31#924C2211yescharacteristic
32C22×C3⋊C8C2211yescharacteristic
33C2×Dic12C2211yescharacteristic
34#890—21no—
35#890C2211yescharacteristic
36#889—21no—
37#888C2211yescharacteristic
38#887C2211yescharacteristic
39#887C2211yescharacteristic
40#879—21no—
41#879—21no—
42#879—21no—
43#878—21no—
44C3×#107C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#817C2211yescharacteristic
50#816C2211yescharacteristic
51#816C2211yescharacteristic
52#815C2211yescharacteristic
53#815C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#542—31no—
61#541—31no—
62#501—31no—
63#497—31no—
64#488—31no—
65#487—31no—
66#470—31no—
67#468D311yescharacteristic
68#467—31no—
69#465—31no—
70#441—31no—
71#440—31no—
72C4×Q16—31no—
73C4×SD16—31no—
74#405—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×Dic3—21no—
80C2×Dic6—21no—
81C2×Dic6C2311yescharacteristic
82C2×Dic6C2311yescharacteristic
83C3×SD16—22no—
84C3×M16—22no—
85C2×C3×C8C2311yescharacteristic
86C2×C3×C8C2311yescharacteristic
87C3×C4⋊C4—21no—
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C22⋊C4—21no—
91C3×C22⋊C4—21no—
92#217—21no—
93#217—21no—
94#217—21no—
95#217—21no—
96#217—21no—
97#217D411yescharacteristic
98#217D411yescharacteristic
99#216—22no—
100#214—22no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3D411yescharacteristic
104C4⋊Dic3D411yescharacteristic
105C4⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8—22no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123Dic12—22no—
124C2×Q16—31no—
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×M16C2×D311yescharacteristic
129C22×C8—31no—
130C4⋊Q8—31no—
131C4⋊Q8—31no—
132#126—61no—
133#126—61no—
134#126—61no—
135#125—31no—
136#125—61no—
137#125—61no—
138#124—31no—
139#124—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144C22⋊Q8C2×D311yescharacteristic
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—31no—
149C4×D4—31no—
150#117—31no—
151#107—31no—
152#107C2×D311yescharacteristic
153#106—31no—
154#106—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103C2×D311yescharacteristic
161#102—31no—
162#102C2×D311yescharacteristic
163#97—31no—
164#97—31no—
165C4×C8—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C3×Q8—21no—
170C3×Q8—21no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—81no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—81no—
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190Dic6—21no—
191Dic6—21no—
192Dic6—21no—
193Dic6—21no—
194Dic6—81no—
195C3×C8—21no—
196C3×C8#4112yes—
197C3⋊C8—21no—
198C3⋊C8#4112yes—
199C2×Q8—31no—
200C2×Q8—31no—
201C2×Q8—61no—
202C2×Q8C22×D311yescharacteristic
203C2×D4—21no—
204C2×D4C22×D311yescharacteristic
205C22×C4—61no—
206C22×C4C22×D311yescharacteristic
207Q16—62no—
208Q16—62no—
209SD16—22no—
210SD16—62no—
211M16—22no—
212C2×C8—31no—
213C2×C8—31no—
214C2×C8—62no—
215C2×C8C22×D311yescharacteristic
216C2×C8C22×D311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4C22×D311yescharacteristic
233C4⋊C4C22×D311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—31no—
237C22⋊C4—31no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C42—31no—
244C42—31no—
245C42—61no—
246C42—61no—
247C42—61no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3C22×D411yescharacteristic
255C3×C4—21no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4C2×#4111yescharacteristic
260C3×C4C2×#4111yescharacteristic
261Dic3—21no—
262Dic3—21no—
263Dic3—21no—
264Dic3—21no—
265Dic3—41no—
266Dic3—41no—
267Dic3—41no—
268Dic3—41no—
269C23—21no—
270C23D3×D411yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—61no—
274Q8—61no—
275Q8—61no—
276Q8—61no—
277Q8—241no—
278D4—21no—
279D4—21no—
280D4—81no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—241no—
295C2×C4D3×D411yescharacteristic
296C2×C4C23×D311yescharacteristic
297C8—21no—
298C8—32no—
299C8—61no—
300C8#23712yes—
301C2×C3—41no—
302C2×C3—81no—
303C2×C3#53211yescharacteristic
304C2×C3#57711yescharacteristic
305C2×C3#57811yescharacteristic
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22C2×D3×D411yescharacteristic
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—61no—
318C4—61no—
319C4—61no—
320C4—61no—
321C4—121no—
322C4—121no—
323C4—121no—
324C4—121no—
325C4C2×#23711yescharacteristic
326C4D3×#4111yescharacteristic
327C3#317811yescharacteristic, complemented, Sylow
328C2—41no—
329C2—81no—
330C2#559611yescharacteristic
331C2#577511yescharacteristic
332C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radicalC3×#121C22
hypercenter#468D3
hypocenterC3#3178
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. #5625C2semidirect
4. #5167C2semidirect
5. Dic3×SD16C2semidirect
6. #5137C2semidirect
7. #5122C2semidirect
8. #5051C2semidirect
10. #4876C2semidirect
11. #4873C2semidirect
12. #4792C2semidirect
13. #4789C2semidirect
14. #4768C2semidirect
15. #4757C2semidirect
327. C3#3178semidirect

Maximal subgroups

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

subgroupindex
#56252
#56032
C3×#4682
#51672
Dic3×SD162
#51372
#51222
#50512
#50412
#48762
#48732
#47922
#47892
#47682
#47572
#31783