Finite Group Databasebeta

← #81051

Subgroups — #81051

Order 384.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5721C211yesmaximal, characteristic, complemented
2#5559C211yesmaximal, characteristic
3C3×#476C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5018C211yesmaximal, characteristic, complemented
6#4871C211yesmaximal, characteristic, complemented
7#4857C211yesmaximal, characteristic, complemented
8#4853C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4768C211yesmaximal, characteristic, complemented
13#4767C211yesmaximal, characteristic, complemented
14#4764C211yesmaximal, characteristic
15#4762C211yesmaximal, characteristic
16#3371—31nomaximal, Sylow
17C3×C4⋊Q8C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic
19Q8×Dic3—21no—
20#944—21no—
21#940C2211yescharacteristic
22#936C2211yescharacteristic
23C2×#214C2211yescharacteristic
24#929—21no—
25C4×#67—21no—
26C2×Dic12C2211yescharacteristic
27C2×#204C2211yescharacteristic
28#893C2211yescharacteristic
29#890C2211yescharacteristic
30#890C2211yescharacteristic
31#889—21no—
32#889—21no—
33#887C2211yescharacteristic
34#887C2211yescharacteristic
35#887C2211yescharacteristic
36#885—21no—
37#883—21no—
38#875C2211yescharacteristic
39C4×Dic6C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×#106C2211yescharacteristic
42C3×#106C2211yescharacteristic
43C3×C4⋊C8C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic
46#832C2211yescharacteristic
47#821C2211yescharacteristic
48#816C2211yescharacteristic
49#815C2211yescharacteristic
50Dic3⋊C8C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52C8×Dic3C2211yescharacteristic
53#810C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#559—31no—
61#557—31no—
62#490—31no—
63#489—31no—
64#488—31no—
65#479—31no—
66#478—31no—
67#476D311yescharacteristic, complemented
68#464—31no—
69#464—31no—
70#460—31no—
71#446—31no—
72#441—31no—
73#440—31no—
74C4×SD16—31no—
75C2×C3×Q8—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C2×D12C2311yescharacteristic
81C2×C4×D3—21no—
82C2×Dic6D411yescharacteristic
83C2×Dic6C2311yescharacteristic
84C2×Dic6C2311yescharacteristic
85C2×C3×C8C2311yescharacteristic
86C2×C3×C8C2311yescharacteristic
87C3×C4⋊C4—21no—
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C22⋊C4—21no—
92C3×C42C2311yescharacteristic
93#217—21no—
94#217—21no—
95#214—22no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99C4⋊Dic3—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3D411yescharacteristic
102C4⋊Dic3C2311yescharacteristic
103Dic3⋊C4—21no—
104Dic3⋊C4—21no—
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4D411yescharacteristic
110Dic3⋊C4D411yescharacteristic
111C4×Dic3—21no—
112C4×Dic3—21no—
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3C2311yescharacteristic
116C4×Dic3C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119Dic12—22no—
120#204—22no—
121C2×Q16—31no—
122C2×SD16—31no—
123C2×SD16—31no—
124C4⋊Q8—61no—
125C4⋊Q8C2×D311yescharacteristic
126#127—31no—
127#125—31no—
128#125—31no—
129#125—61no—
130#125—61no—
131#124—31no—
132#124—31no—
133#124—61no—
134#123—61no—
135#123—61no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×Q8—61no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4C2×D311yescharacteristic
145#107—31no—
146#106C2×D311yescharacteristic
147#106C2×D311yescharacteristic
148C4⋊C8—31no—
149C4⋊C8—31no—
150C4⋊C8C2×D311yescharacteristic
151#103—31no—
152#103—31no—
153#103—31no—
154#103—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102C2×D311yescharacteristic
160#102C2×D311yescharacteristic
161#97—31no—
162#97—31no—
163C4×C8—31no—
164C23×C3—21no—
165C22×D3—21no—
166C3×Q8—42no—
167C3×D4—21no—
168C3×D4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—81no—
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2411yescharacteristic
176#67—42no—
177#67—81no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3C2×D411yescharacteristic
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185D12—21no—
186D12—21no—
187C4×D3—81no—
188Dic6—21no—
189Dic6—21no—
190Dic6—21no—
191Dic6—41no—
192Dic6#4112yes—
193C3×C8—21no—
194C3×C8—21no—
195C3⋊C8—21no—
196C3⋊C8—21no—
197C2×Q8—21no—
198C2×Q8—31no—
199C2×Q8—31no—
200C2×Q8—31no—
201C2×D4—31no—
202C2×D4—61no—
203C2×D4—61no—
204C2×D4C22×D311yescharacteristic
205C22×C4—21no—
206C22×C4—61no—
207Q16—62no—
208SD16—62no—
209SD16—62no—
210C2×C8—31no—
211C2×C8—31no—
212C2×C8C22×D311yescharacteristic
213C2×C8C22×D311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—31no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4C22×D311yescharacteristic
228C4⋊C4C22×D311yescharacteristic
229C4⋊C4C22×D311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C42—31no—
237C42—31no—
238C42—61no—
239C42—61no—
240C42—61no—
241C42—61no—
242C42C22×D311yescharacteristic
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3C22×D411yescharacteristic
247C2×D3—41no—
248C2×D3—41no—
249C2×D3—41no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4C2×#4111yescharacteristic
256C3×C42−1+411yescharacteristic
257Dic3—22no—
258Dic3—21no—
259Dic3—21no—
260Dic3—21no—
261Dic3—21no—
262Dic3—41no—
263Dic3—41no—
264C23—21no—
265C23—61no—
266Q8—32no—
267Q8—42no—
268Q8—61no—
269Q8—61no—
270Q8—61no—
271Q8—121no—
272D4—21no—
273D4—21no—
274D4—61no—
275D4—61no—
276D4—122no—
277D4—241no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—81no—
289C2×C4—241no—
290C2×C4D3×D411yescharacteristic
291C2×C4D3×D411yescharacteristic
292C2×C4C23×D311yescharacteristic
293C8—21no—
294C8—21no—
295C8—61no—
296C8—61no—
297C2×C3—81no—
298C2×C3#54811yescharacteristic
299C2×C3#57511yescharacteristic
300C2×C3#57711yescharacteristic
301D3—81no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—121no—
306C22—121no—
307C22—121no—
308C22C2×D3×D411yescharacteristic
309C4—21no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—62no—
315C4—61no—
316C4—61no—
317C4—61no—
318C4—61no—
319C4—121no—
320C4—121no—
321C4#100711yescharacteristic
322C4D3×#4111yescharacteristic
323C3#337111yescharacteristic, complemented, Sylow
324C2—81no—
325C2—241no—
326C2#563311yescharacteristic
327C2#576211yescharacteristic
328C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#476C2
socleC22×C3C22×D4
prime radical#940C22
hypercenter#476D3
hypocenterC3#3371
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #5721C2semidirect
3. C3×#476C2semidirect
4. #5089C2semidirect
5. #5018C2semidirect
6. #4871C2semidirect
7. #4857C2semidirect
8. #4853C2semidirect
9. #4852C2semidirect
10. #4845C2semidirect
11. #4828C2semidirect
12. #4768C2semidirect
13. #4767C2semidirect
67. #476D3semidirect
323. C3#3371semidirect

Maximal subgroups

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

subgroupindex
#57212
#55592
C3×#4762
#50892
#50182
#48712
#48572
#48532
#48522
#48452
#48282
#47682
#47672
#47642
#47622
#33713