Finite Group Databasebeta

← #81307

Subgroups — #81307

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5599C211yesmaximal, characteristic
2#5511C211yesmaximal, characteristic, complemented
3C3×#480C211yesmaximal, characteristic, complemented
4#5042C211yesmaximal, characteristic, complemented
5#4966C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4882C211yesmaximal, characteristic, complemented
8#4875C211yesmaximal, characteristic, complemented
9#4873C211yesmaximal, characteristic, complemented
10#4783C211yesmaximal, characteristic, complemented
11#4771C211yesmaximal, characteristic, complemented
12#4762C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3333—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#940—21no—
22#940C2211yescharacteristic
23#939—21no—
24C2×#214C2211yescharacteristic
25C4×#67—21no—
26C4×#67C2211yescharacteristic
27#924—21no—
28#924C2211yescharacteristic
29C2×#208C2211yescharacteristic
30C2×Dic12C2211yescharacteristic
31C2×#204C2211yescharacteristic
32C2×ΓL(1,𝔽25)C2211yescharacteristic
33#897C2211yescharacteristic
34#896—21no—
35#895C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic
37#889—21no—
38#888C2211yescharacteristic
39#887C2211yescharacteristic
40#887C2211yescharacteristic
41#884—21no—
42#67⋊C4—21no—
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47C3×C22⋊C8C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59#807C2211yescharacteristic
60#554—31no—
61#550—31no—
62#486—31no—
63#480—31no—
64#480D311yescharacteristic, complemented
65#477—31no—
66#468—31no—
67#468—31no—
68#460—31no—
69#452—31no—
70#452—31no—
71C4×Q16—31no—
72#436—31no—
73#417—31no—
74#417—31no—
75C2×C3×D4—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67D411yescharacteristic
84C22×Dic3—21no—
85#237—22no—
86#237—41no—
87#235—22no—
88#235—41no—
89C2×D12—21no—
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×Dic6—21no—
95C2×Dic6C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104#217—21no—
105#217D411yescharacteristic
106#214—41no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121#208—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124Dic12—22no—
125#204—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C2×#41—31no—
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C2×M16—31no—
134C4⋊Q8—31no—
135#127—31no—
136#127—61no—
137#125—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142#121—31no—
143#121—61no—
144#121—61no—
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4—61no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4C2×D311yescharacteristic
153#107—31no—
154#107C2×D311yescharacteristic
155#107C2×D311yescharacteristic
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—81no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—22no—
186#67—42no—
187#67—42no—
188#67—42no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—42no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199D12—81no—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—42no—
203C4×D3—81no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—81no—
208Dic6#4112yes—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—62no—
214#41—62no—
215#41—121no—
216#41—121no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×Q8—61no—
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235Q16—62no—
236SD16—121no—
237SD16—121no—
238M16—121no—
239M16—121no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4C22×D311yescharacteristic
254C4⋊C4C22×D311yescharacteristic
255C4⋊C4C22×D311yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—31no—
258C22⋊C4—31no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C42—31no—
263C42—61no—
264C42—61no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C22×D411yescharacteristic
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C3×C4—21no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—22no—
285Dic3—21no—
286Dic3—21no—
287Dic3—41no—
288Dic3—41no—
289Dic3#13712yes—
290C23—21no—
291C23—31no—
292C23—61no—
293C23D3×D411yescharacteristic
294Q8—32no—
295Q8—61no—
296Q8—61no—
297Q8—61no—
298Q8—241no—
299D4—21no—
300D4—21no—
301D4—62no—
302D4—61no—
303D4—61no—
304D4—81no—
305D4—122no—
306D4—122no—
307D4—122no—
308D4—241no—
309D4—241no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—81no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—241no—
327C2×C4D3×D411yescharacteristic
328C2×C4C23×D311yescharacteristic
329C8—21no—
330C8—21no—
331C8—61no—
332C8—61no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3#54611yescharacteristic
336C2×C3C2×#13711yescharacteristic
337C2×C3#57611yescharacteristic
338D3—41no—
339D3—81no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—61no—
347C22—61no—
348C22—61no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22C2×D3×D411yescharacteristic
353C4—21no—
354C4—32no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—61no—
359C4—62no—
360C4—61no—
361C4—121no—
362C4—121no—
363C4D3×#4111yescharacteristic
364C4#100911yescharacteristic
365C3#333311yescharacteristic, complemented, Sylow
366C2—41no—
367C2—81no—
368C2—121no—
369C2—241no—
370C2#565411yescharacteristic
371C2#575711yescharacteristic
372C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#480C2
socleC22×C3C22×D4
prime radical#5599C2
hypercenter#480D3
hypocenterC3#3333
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #5511C2semidirect
3. C3×#480C2semidirect
4. #5042C2semidirect
5. #4966C2semidirect
6. #4886C2semidirect
7. #4882C2semidirect
8. #4875C2semidirect
9. #4873C2semidirect
10. #4783C2semidirect
11. #4771C2semidirect
12. #4762C2semidirect
13. #4760C2semidirect
14. #4733C2semidirect
15. #4729C2semidirect
64. #480D3semidirect
365. C3#3333semidirect

Maximal subgroups

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

subgroupindex
#55992
#55112
C3×#4802
#50422
#49662
#48862
#48822
#48752
#48732
#47832
#47712
#47622
#47602
#47332
#47292
#33333