Finite Group Databasebeta

← #80954

Subgroups — #80954

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5733C211yesmaximal, characteristic, complemented
2#5562C211yesmaximal, characteristic, complemented
3C3×#474C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5016C211yesmaximal, characteristic, complemented
6#4891C211yesmaximal, characteristic, complemented
7#4886C211yesmaximal, characteristic, complemented
8#4883C211yesmaximal, characteristic, complemented
9#4877C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4782C211yesmaximal, characteristic, complemented
14#4778C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3257—31nomaximal, Sylow
17C3×C4⋊Q8C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic
19#947—21no—
20#946—21no—
21#940C2211yescharacteristic
22#938C2211yescharacteristic
23C2×#214C2211yescharacteristic
24C2×#213C2211yescharacteristic
25#930—21no—
26#929—21no—
27C2×D24C2211yescharacteristic
28C2×#204C2211yescharacteristic
29C2×ΓL(1,𝔽25)C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#898—21no—
32#897C2211yescharacteristic
33#895C2211yescharacteristic
34#895C2211yescharacteristic
35#894—21no—
36#893C2211yescharacteristic
37#890C2211yescharacteristic
38#887C2211yescharacteristic
39#884—21no—
40#883—21no—
41#875C2211yescharacteristic
42C4×D12C2211yescharacteristic
43#872C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#107C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×#102C2211yescharacteristic
49#821C2211yescharacteristic
50#816C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#542—31no—
61#540—31no—
62#479—31no—
63#478—31no—
64#476—31no—
65#474D311yescharacteristic, complemented
66#469—31no—
67#468—31no—
68#466—31no—
69#465—31no—
70#463—31no—
71#463—31no—
72#460—31no—
73#459—31no—
74#424—31no—
75C2×C3×Q8—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12C2311yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3C2311yescharacteristic
88C2×Dic6C2311yescharacteristic
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96C3×C42C2311yescharacteristic
97#217—21no—
98#214—41no—
99#213—41no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4D411yescharacteristic
108(C2×D3)⋊C4D411yescharacteristic
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4D411yescharacteristic
115Dic3⋊C4D411yescharacteristic
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120D24—41no—
121#204—41no—
122ΓL(1,𝔽25)—41no—
123D3×C8—41no—
124C2×SD16—31no—
125C2×SD16—31no—
126C2×D8—31no—
127C2×D8—31no—
128C2×M16—31no—
129C22×C8—31no—
130C4⋊Q8C2×D311yescharacteristic
131#127—31no—
132#126—61no—
133#125—31no—
134#124—31no—
135#124—61no—
136#123—61no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—31no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#107C2×D311yescharacteristic
152#107C2×D311yescharacteristic
153#106—31no—
154#106—31no—
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8C2×D311yescharacteristic
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C23×C3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3C2×D411yescharacteristic
171C3×Q8—81no—
172C3×D4—21no—
173C3×D4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—81no—
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181#67—81no—
182#67—81no—
183#67—81no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189D12—21no—
190D12—21no—
191D12—21no—
192D12—21no—
193D12—81no—
194D12—81no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—81no—
198C4×D3—81no—
199Dic6—21no—
200Dic6—21no—
201C3×C8—21no—
202C3×C8—21no—
203C3⋊C8—21no—
204C3⋊C8—21no—
205C2×Q8—21no—
206C2×Q8—31no—
207C2×D4—31no—
208C2×D4—31no—
209C2×D4—61no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4C22×D311yescharacteristic
215C22×C4—21no—
216C22×C4—31no—
217C22×C4—61no—
218C22×C4—61no—
219SD16—121no—
220SD16—121no—
221D8—121no—
222D8—121no—
223M16—121no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—121no—
227C2×C8C22×D311yescharacteristic
228C2×C8C22×D311yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—31no—
242C22⋊C4—31no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C42—31no—
252C42—61no—
253C42C22×D311yescharacteristic
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3C22×D411yescharacteristic
258C2×D3—21no—
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4H(5,𝔽2)11yescharacteristic
273C3×C42−1+411yescharacteristic
274Dic3—21no—
275Dic3—21no—
276Dic3—21no—
277Dic3—41no—
278Dic3—41no—
279C23—21no—
280C23—31no—
281C23—61no—
282C23—61no—
283Q8—61no—
284Q8—61no—
285Q8—81no—
286D4—21no—
287D4—21no—
288D4—61no—
289D4—61no—
290D4—61no—
291D4—61no—
292D4—241no—
293D4—241no—
294D4—241no—
295D4—241no—
296D4—241no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—81no—
308C2×C4—241no—
309C2×C4—241no—
310C2×C4D3×D411yescharacteristic
311C2×C4D3×D411yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—21no—
314C8—21no—
315C8—61no—
316C8—61no—
317C2×C3—81no—
318C2×C3#53711yescharacteristic
319C2×C3#57611yescharacteristic
320C2×C3#57711yescharacteristic
321D3—41no—
322D3—81no—
323D3—81no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22—121no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—61no—
343C4—61no—
344C4—61no—
345C4—121no—
346C4—121no—
347C4#100711yescharacteristic
348C4#100911yescharacteristic
349C3#325711yescharacteristic, complemented, Sylow
350C2—81no—
351C2—121no—
352C2—241no—
353C2—241no—
354C2#562911yescharacteristic
355C2#575711yescharacteristic
356C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
hypercenter#474D3
hypocenterC3#3257
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5733C2semidirect
2. #5562C2semidirect
3. C3×#474C2semidirect
4. #5084C2semidirect
5. #5016C2semidirect
6. #4891C2semidirect
7. #4886C2semidirect
8. #4883C2semidirect
9. #4877C2semidirect
10. #4844C2semidirect
11. #4839C2semidirect
12. #4786C2semidirect
13. #4782C2semidirect
14. #4778C2semidirect
15. #4762C2semidirect
65. #474D3semidirect
349. C3#3257semidirect

Maximal subgroups

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

subgroupindex
#57332
#55622
C3×#4742
#50842
#50162
#48912
#48862
#48832
#48772
#48442
#48392
#47862
#47822
#47782
#47622
#32573