Finite Group Databasebeta

← #78970

Subgroups — #78970

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5817C211yesmaximal, characteristic, complemented
2#5530C211yesmaximal, characteristic, complemented
3C3×#419C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5135C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#5003C211yesmaximal, characteristic, complemented
8#4819C211yesmaximal, characteristic, complemented
9#4812C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4805C211yesmaximal, characteristic, complemented
12#4803C211yesmaximal, characteristic, complemented
13#4800C211yesmaximal, characteristic, complemented
14#4799C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3326—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#947—21no—
22#947C2211yescharacteristic
23#946—21no—
24#946C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic
28#930—21no—
29#930C2211yescharacteristic
30#930C2211yescharacteristic
31C4×#67—21no—
32C4×#67C2211yescharacteristic
33C2×D24C2211yescharacteristic
34C2×ΓL(1,𝔽25)C2211yescharacteristic
35#895C2211yescharacteristic
36#893C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic
38#890C2211yescharacteristic
39#884—21no—
40#883—21no—
41#876—21no—
42C4×D12—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#830C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#817C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#553—31no—
61#548—31no—
62#488—31no—
63#483—31no—
64#482—31no—
65#479—31no—
66#469—31no—
67#468—31no—
68#463—31no—
69#449—31no—
70#448—31no—
71C4×SD16—31no—
72#436—31no—
73#419D311yescharacteristic, complemented
74#417—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C2×#67D411yescharacteristic
85#239—22no—
86#239—41no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C3×M16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C42—21no—
102#217D411yescharacteristic
103#215—22no—
104#215—41no—
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—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125D24—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C22×Q8C2×D311yescharacteristic
129C2×SD16—31no—
130C2×SD16—31no—
131C2×D8—31no—
132C2×M16—31no—
133C2×M16C2×D311yescharacteristic
134#126—61no—
135#125—31no—
136#124—31no—
137#124—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×D4—61no—
150C4×D4—61no—
151#117C2×D311yescharacteristic
152C2×C4⋊C4—31no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#103C2×D311yescharacteristic
161#103C2×D311yescharacteristic
162#103C2×D311yescharacteristic
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—42no—
178C3×Q8#4112yes—
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×C4—42no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—41no—
189#67—42no—
190#67—81no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196D12—21no—
197D12—21no—
198D12—21no—
199D12—21no—
200D12—42no—
201D12—42no—
202C4×D3—21no—
203C4×D3—21no—
204C4×D3—42no—
205C4×D3—81no—
206C4×D3—81no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—62no—
212#41—121no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8C22×D311yescharacteristic
217C2×Q8C22×D311yescharacteristic
218C2×D4—31no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C22×C4—21no—
226C22×C4—31no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231SD16—62no—
232SD16—121no—
233D8—121no—
234M16—41no—
235M16—121no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8C22×D311yescharacteristic
239C2×C8C22×D311yescharacteristic
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4C22×D311yescharacteristic
249C4⋊C4C22×D311yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—31no—
252C22⋊C4—31no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C42—21no—
261C42—31no—
262C42—61no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4C2×#4111yescharacteristic
282C3×C4H(5,𝔽2)11yescharacteristic
283Dic3—21no—
284Dic3—21no—
285Dic3—21no—
286Dic3—41no—
287Dic3—41no—
288C23—31no—
289C23—61no—
290C23—61no—
291C23D3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8#23512yes—
297D4—61no—
298D4—61no—
299D4—61no—
300D4—61no—
301D4—122no—
302D4—122no—
303D4—121no—
304D4—122no—
305D4—241no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—42no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—122no—
321C2×C4—241no—
322C2×C4—241no—
323C2×C4D3×D411yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—21no—
326C8—21no—
327C8—61no—
328C8—61no—
329C2×C3—41no—
330C2×C3#54611yescharacteristic
331C2×C3#57511yescharacteristic
332C2×C3#57711yescharacteristic
333D3—41no—
334D3—81no—
335D3—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—61no—
340C22—61no—
341C22—61no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22C2×D3×D411yescharacteristic
349C4—21no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—61no—
355C4—61no—
356C4—61no—
357C4—121no—
358C4—121no—
359C4C2×#23511yescharacteristic
360C4#100411yescharacteristic
361C3#332611yescharacteristic, complemented, Sylow
362C2—41no—
363C2—121no—
364C2—241no—
365C2—241no—
366C2#549011yescharacteristic
367C2#577711yescharacteristic
368C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#419C2
socleC22×C3C22×D4
hypercenter#419D3
hypocenterC3#3326
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #5817C2semidirect
2. #5530C2semidirect
3. C3×#419C2semidirect
4. #5227C2semidirect
5. #5135C2semidirect
6. #5128C2semidirect
7. #5003C2semidirect
8. #4819C2semidirect
9. #4812C2semidirect
10. #4810C2semidirect
11. #4805C2semidirect
12. #4803C2semidirect
13. #4800C2semidirect
14. #4799C2semidirect
15. #4788C2semidirect
73. #419D3semidirect
361. C3#3326semidirect

Maximal subgroups

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

subgroupindex
#58172
#55302
C3×#4192
#52272
#51352
#51282
#50032
#48192
#48122
#48102
#48052
#48032
#48002
#47992
#47882
#33263