Finite Group Databasebeta

← #78906

Subgroups — #78906

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5824C211yesmaximal, characteristic, complemented
2#5507C211yesmaximal, characteristic, complemented
3C3×#417C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5134C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#4970C211yesmaximal, characteristic, complemented
8#4818C211yesmaximal, characteristic, complemented
9#4812C211yesmaximal, characteristic, complemented
10#4811C211yesmaximal, characteristic, complemented
11#4792C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4771C211yesmaximal, characteristic, complemented
14#4759C211yesmaximal, characteristic, complemented
15#4756C211yesmaximal, characteristic, complemented
16#3330—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#235—21no—
20C2×C3×M16C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22#946C2211yescharacteristic
23#944C2211yescharacteristic
24C2×#215C2211yescharacteristic
25#939—21no—
26#935—21no—
27D4×Dic3C2211yescharacteristic
28C2×#213C2211yescharacteristic
29#930C2211yescharacteristic
30#929—21no—
31#929—21no—
32C4×#67C2211yescharacteristic
33#924C2211yescharacteristic
34C2×Dic3⋊C4—21no—
35C2×#204C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#896—21no—
38#895C2211yescharacteristic
39#894—21no—
40#893C2211yescharacteristic
41D3×C4⋊C4C2211yescharacteristic
42#888C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#818C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#547—31no—
61#547—31no—
62#487—31no—
63#484—31no—
64#484—31no—
65#474—31no—
66#468—31no—
67#468—31no—
68#461—31no—
69#449—31no—
70#449—31no—
71C4×D8—31no—
72#436—31no—
73#417—31no—
74#417D311yescharacteristic, complemented
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67D411yescharacteristic
86C22×Dic3—21no—
87C22×Dic3—21no—
88#237—22no—
89#237—41no—
90#235—44no—
91C2×D12—21no—
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×Dic6—21no—
97C2×Dic6C2311yescharacteristic
98C3×M16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104#217—21no—
105#217—21no—
106#217D411yescharacteristic
107#215—41no—
108#213—22no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127ΓL(1,𝔽25)—41no—
128C2×#41—31no—
129C2×#41—61no—
130C2×#41C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16—31no—
133C2×D8—31no—
134C2×M16—31no—
135C2×M16C2×D311yescharacteristic
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138#123—61no—
139#123—61no—
140#123—61no—
141#123—61no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8—61no—
145#121—31no—
146#121—31no—
147#121—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4C2×D311yescharacteristic
154#107—31no—
155#107—31no—
156#107—31no—
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167#102C2×D311yescharacteristic
168C22⋊C8—31no—
169C22⋊C8—31no—
170C4×C8—31no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4—42no—
179C3×D4#4112yescomplemented
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×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—22no—
190#67—42no—
191#67—42no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202D12—42no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—42no—
206C4×D3—42no—
207Dic6—21no—
208Dic6—21no—
209Dic6—42no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214#41—22no—
215#41—41no—
216#41—62no—
217#41—124no—
218#41—121no—
219C2×Q8—31no—
220C2×Q8—61no—
221C2×Q8C22×D311yescharacteristic
222C2×D4—21no—
223C2×D4—31no—
224C2×D4—31no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4C22×D311yescharacteristic
237SD16—121no—
238SD16—121no—
239D8—62no—
240M16—41no—
241M16—121no—
242C2×C8—31no—
243C2×C8—31no—
244C2×C8C22×D311yescharacteristic
245C2×C8C22×D311yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4C22×D311yescharacteristic
257C4⋊C4C22×D311yescharacteristic
258C22⋊C4—31no—
259C22⋊C4—31no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C42—31no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3C22×D411yescharacteristic
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4C2×#4111yescharacteristic
285C3×C4H(5,𝔽2)11yescharacteristic
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291Dic3AGL(1,ℤ/8)12yes—
292C23—21no—
293C23—31no—
294C23—61no—
295C23D3×D411yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—61no—
299Q8—61no—
300Q8—122no—
301D4—21no—
302D4—42no—
303D4—62no—
304D4—61no—
305D4—61no—
306D4—122no—
307D4—122no—
308D4—122no—
309D4#23512yescomplemented
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—42no—
319C2×C4—42no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4D3×D411yescharacteristic
330C2×C4C23×D311yescharacteristic
331C8—21no—
332C8—21no—
333C8—61no—
334C8—61no—
335C2×C3—41no—
336C2×C3—42no—
337C2×C3#54611yescharacteristic
338C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
339C2×C3#57811yescharacteristic
340D3—41no—
341D3—81no—
342C22—21no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—61no—
349C22—61no—
350C22—61no—
351C22—121no—
352C22—121no—
353C22—121no—
354C22C2×D3×D411yescharacteristic
355C4—21no—
356C4—32no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—61no—
361C4—61no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4C2×#23511yescharacteristic
366C4#100411yescharacteristic
367C3#333011yescharacteristic, complemented, Sylow
368C2—41no—
369C2—42no—
370C2—121no—
371C2—241no—
372C2#549011yescharacteristic
373C2D3×AGL(1,ℤ/8)11yescharacteristic
374C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
hypercenter#417D3
hypocenterC3#3330
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #5824C2semidirect
2. #5507C2semidirect
3. C3×#417C2semidirect
4. #5238C2semidirect
5. #5134C2semidirect
6. #5128C2semidirect
7. #4970C2semidirect
8. #4818C2semidirect
9. #4812C2semidirect
10. #4811C2semidirect
11. #4792C2semidirect
12. #4781C2semidirect
13. #4771C2semidirect
14. #4759C2semidirect
15. #4756C2semidirect
74. #417D3semidirect
179. C3×D4#41semidirect
309. D4#235semidirect
367. C3#3330semidirect

Maximal subgroups

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

subgroupindex
#58242
#55072
C3×#4172
#52382
#51342
#51282
#49702
#48182
#48122
#48112
#47922
#47812
#47712
#47592
#47562
#33303