Finite Group Databasebeta

← #81521

Subgroups — #81521

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5629C211yesmaximal, characteristic
2#5523C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4#5043C211yesmaximal, characteristic, complemented
5#5003C211yesmaximal, characteristic, complemented
6#4885C211yesmaximal, characteristic, complemented
7#4881C211yesmaximal, characteristic, complemented
8#4863C211yesmaximal, characteristic, complemented
9D3×#106C211yesmaximal, characteristic, complemented
10#4815C211yesmaximal, characteristic, complemented
11#4805C211yesmaximal, characteristic, complemented
12#4804C211yesmaximal, characteristic, complemented
13#4803C211yesmaximal, characteristic, complemented
14#4735C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3099—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#946C2211yescharacteristic
22#930C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×ΓL(1,𝔽25)C2211yescharacteristic
25C2×D3×C8C2211yescharacteristic
26#896—21no—
27#895—21no—
28#895C2211yescharacteristic
29#892C2211yescharacteristic
30#892C2211yescharacteristic
31D3×C4⋊C4C2211yescharacteristic
32D3×C4⋊C4C2211yescharacteristic
33D3×C4⋊C4C2211yescharacteristic
34#884—21no—
35#884—21no—
36#67⋊C4—21no—
37#67⋊C4—21no—
38C4×D12—21no—
39C4×Dic6—21no—
40C3×#107C2211yescharacteristic
41C3×#106C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#830C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#537—31no—
61#520—31no—
62#485—31no—
63#485D311yescharacteristic, complemented
64#483—31no—
65#483—31no—
66#482—31no—
67#482—31no—
68#481—31no—
69#481—31no—
70#428—31no—
71C2×#106—31no—
72#417—31no—
73#416—31no—
74#408—31no—
75C2×C3×Q8C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C2×#67—21no—
78C2×#67D411yescharacteristic
79C2×#67D411yescharacteristic
80C22×Dic3—21no—
81#239—22no—
82#239—22no—
83#235—22no—
84#235—22no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12D411yescharacteristic
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×C4×D3C2311yescharacteristic
95C2×Dic6D411yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122ΓL(1,𝔽25)—22no—
123D3×C8—22no—
124C2×#41—31no—
125C2×#41—31no—
126C2×M16—31no—
127C22×C8—31no—
128C22⋊Q8—31no—
129C22⋊Q8—61no—
130C22⋊Q8C2×D311yescharacteristic
131#121—31no—
132#121—31no—
133#121—61no—
134#121—61no—
135#121—61no—
136C4×Q8—61no—
137C4×D4—61no—
138C4×D4—61no—
139C4×D4—61no—
140#117—31no—
141#117—31no—
142#117C2×D311yescharacteristic
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147#107—31no—
148#107C2×D311yescharacteristic
149#106—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106—31no—
154#106C2×D311yescharacteristic
155#103—31no—
156#103—31no—
157#103C2×D311yescharacteristic
158#103C2×D311yescharacteristic
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165C22⋊C8—31no—
166C22⋊C8C2×D311yescharacteristic
167C23×C3C2×D411yescharacteristic
168C22×D3—21no—
169C22×D3C2×D411yescharacteristic
170C22×D3C2×D411yescharacteristic
171C3×Q8—21no—
172C3×Q8—21no—
173C2×C3×C4—21no—
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
181#67—22no—
182#67—22no—
183#67—81no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—81no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190D12—22no—
191D12—21no—
192D12—21no—
193D12—42no—
194D12—81no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—42no—
198C4×D3—81no—
199C4×D3—81no—
200C4×D3—81no—
201C4×D3#4112yes—
202C4×D3#4112yes—
203Dic6—22no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—62no—
209#41—62no—
210#41—62no—
211#41—62no—
212C2×Q8—31no—
213C2×Q8C22×D311yescharacteristic
214C2×D4—31no—
215C2×D4—31no—
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4—61no—
220C2×D4—61no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229M16—62no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—62no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C42—21no—
255C42—61no—
256C42—61no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3C22×D411yescharacteristic
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3#12312yes—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4C2×#4111yescharacteristic
276C3×C4C2×#4111yescharacteristic
277Dic3—21no—
278Dic3—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3#12312yes—
282C23—31no—
283C23—31no—
284C23—61no—
285C23D3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—62no—
289D4—62no—
290D4—62no—
291D4—62no—
292D4—61no—
293D4—61no—
294D4—122no—
295D4—241no—
296D4—241no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—21no—
321C8—61no—
322C8—61no—
323C2×C3—41no—
324C2×C3C2×#12311yescharacteristic
325C2×C3#57711yescharacteristic
326C2×C3#57711yescharacteristic
327D3—22no—
328D3—41no—
329D3—81no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—32no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—61no—
338C22—61no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—32no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—61no—
350C4—121no—
351C4—121no—
352C4—121no—
353C4D3×#4111yescharacteristic
354C4D3×#4111yescharacteristic
355C3#309911yescharacteristic, complemented, Sylow
356C2—41no—
357C2—62no—
358C2—121no—
359C2—241no—
360C2D3×#12311yescharacteristic
361C2#577011yescharacteristic
362C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radical#5629C2
hypercenter#485D3
hypocenterC3#3099
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #5523C2semidirect
3. C3×#485C2semidirect
4. #5043C2semidirect
5. #5003C2semidirect
6. #4885C2semidirect
7. #4881C2semidirect
8. #4863C2semidirect
9. D3×#106C2semidirect
10. #4815C2semidirect
11. #4805C2semidirect
12. #4804C2semidirect
13. #4803C2semidirect
14. #4735C2semidirect
15. #4727C2semidirect
63. #485D3semidirect
355. C3#3099semidirect

Maximal subgroups

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

subgroupindex
#56292
#55232
C3×#4852
#50432
#50032
#48852
#48812
#48632
D3×#1062
#48152
#48052
#48042
#48032
#47352
#47272
#30993