Finite Group Databasebeta

← #81550

Subgroups — #81550

Order 384.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5629C211yesmaximal, characteristic
2#5526C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4#5050C211yesmaximal, characteristic, complemented
5#5003C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4884C211yesmaximal, characteristic, complemented
8#4864C211yesmaximal, characteristic, complemented
9#4862C211yesmaximal, characteristic, complemented
10#4816C211yesmaximal, characteristic, complemented
11#4813C211yesmaximal, characteristic, complemented
12#4808C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4738C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3201—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21C3×#117C2211yescharacteristic
22#946C2211yescharacteristic
23C2×#216C2211yescharacteristic
24C2×#216C2211yescharacteristic
25#930C2211yescharacteristic
26C2×#204C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29C2×D3×C8C2211yescharacteristic
30#897C2211yescharacteristic
31#897C2211yescharacteristic
32#897C2211yescharacteristic
33#897C2211yescharacteristic
34#896—21no—
35#895—21no—
36#895C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic
38#886—21no—
39#886—21no—
40#884—21no—
41#884—21no—
42#875—21no—
43#869—21no—
44C3×#107C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#537—31no—
61#536—31no—
62#485D311yescharacteristic, complemented
63#484—31no—
64#483—31no—
65#482—31no—
66#470—31no—
67#468—31no—
68#467—31no—
69#465—31no—
70#452—31no—
71#452—31no—
72#451—31no—
73#448—31no—
74#408—31no—
75C2×C3×Q8C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C2×#67—21no—
78C2×#67D411yescharacteristic
79C2×#67D411yescharacteristic
80C22×Dic3—21no—
81#239—41no—
82#239—41no—
83#235—41no—
84#235—41no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12D411yescharacteristic
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C2×C4×D3C2311yescharacteristic
93C2×Dic6—21no—
94C2×Dic6—41no—
95C2×Dic6—41no—
96C2×Dic6D411yescharacteristic
97C2×Dic6C2311yescharacteristic
98C2×Dic6C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C42—21no—
108#216—41no—
109#216—41no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127#204—41no—
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C2×#41—31no—
132C22×Q8—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×M16—31no—
138C22×C8—31no—
139C4⋊Q8—61no—
140#124—61no—
141#123—61no—
142#123—61no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145C22⋊Q8—31no—
146C22⋊Q8—31no—
147C22⋊Q8—31no—
148C22⋊Q8—61no—
149C22⋊Q8C2×D311yescharacteristic
150#121—31no—
151#121—31no—
152#121—61no—
153#121—61no—
154#121—61no—
155#117C2×D311yescharacteristic
156C2×C4⋊C4—31no—
157#107—31no—
158#107C2×D311yescharacteristic
159#106—31no—
160#106C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C22×D3C2×D411yescharacteristic
176C22×D3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—41no—
188#67—41no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—42no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196D12—21no—
197D12—21no—
198D12—41no—
199D12—42no—
200D12—81no—
201C4×D3—21no—
202C4×D3—21no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—42no—
206C4×D3—81no—
207Dic6—21no—
208Dic6—21no—
209Dic6—21no—
210Dic6—21no—
211Dic6—41no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217#41—121no—
218#41—121no—
219#41—121no—
220#41—121no—
221C2×Q8—31no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—61no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×Q8C22×D311yescharacteristic
228C2×D4—31no—
229C2×D4—31no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241Q16—121no—
242Q16—121no—
243SD16—121no—
244SD16—121no—
245M16—121no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—121no—
249C2×C8C22×D311yescharacteristic
250C2×C8C22×D311yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—31no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C4⋊C4C22×D311yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3C22×D411yescharacteristic
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4H(5,𝔽2)11yescharacteristic
291C3×C4H(5,𝔽2)11yescharacteristic
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—31no—
298C23—31no—
299C23—61no—
300C23D3×D411yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—61no—
304Q8—61no—
305Q8—61no—
306Q8—61no—
307Q8—121no—
308Q8—122no—
309D4—61no—
310D4—61no—
311D4—121no—
312D4—122no—
313D4—121no—
314D4—121no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—241no—
335C2×C4D3×D411yescharacteristic
336C2×C4C23×D311yescharacteristic
337C8—21no—
338C8—21no—
339C8—61no—
340C8—61no—
341C2×C3—41no—
342C2×C3#53411yescharacteristic
343C2×C3#57711yescharacteristic
344C2×C3#57811yescharacteristic
345D3—41no—
346D3—41no—
347D3—81no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—61no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—61no—
356C22—61no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—61no—
367C4—61no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100911yescharacteristic
372C4#100911yescharacteristic
373C3#320111yescharacteristic, complemented, Sylow
374C2—41no—
375C2—121no—
376C2—121no—
377C2—241no—
378C2#565311yescharacteristic
379C2#577011yescharacteristic
380C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radical#5629C2
hypercenter#485D3
hypocenterC3#3201
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. #5526C2semidirect
3. C3×#485C2semidirect
4. #5050C2semidirect
5. #5003C2semidirect
6. #4886C2semidirect
7. #4884C2semidirect
8. #4864C2semidirect
9. #4862C2semidirect
10. #4816C2semidirect
11. #4813C2semidirect
12. #4808C2semidirect
13. #4808C2semidirect
14. #4738C2semidirect
15. #4727C2semidirect
62. #485D3semidirect
373. C3#3201semidirect

Maximal subgroups

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

subgroupindex
#56292
#55262
C3×#4852
#50502
#50032
#48862
#48842
#48642
#48622
#48162
#48132
#48082
#47382
#47272
#32013