Finite Group Databasebeta

← #79509

Subgroups — #79509

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#5555C211yesmaximal, characteristic, complemented
2#5552C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5134C211yesmaximal, characteristic, complemented
5#5133C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7C2×#816C211yesmaximal, characteristic, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4844C211yesmaximal, characteristic, complemented
10#4738C211yesmaximal, characteristic, complemented
11#4736C211yesmaximal, characteristic, complemented
12#4734C211yesmaximal, characteristic, complemented
13#4722C211yesmaximal, characteristic, complemented
14#4714C211yesmaximal, characteristic, complemented
15#4710C211yesmaximal, characteristic, complemented
16#3308—31nomaximal, Sylow
17C2×#237—21no—
18C2×#235C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic
21C22×C3×C8C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic
23#939—21no—
24#930C2211yescharacteristic
25#930C2211yescharacteristic
26#930C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29#924C2211yescharacteristic
30C2×D24C2211yescharacteristic
31C2×#204C2211yescharacteristic
32C2×#204C2211yescharacteristic
33#897—21no—
34#896—21no—
35#886—21no—
36#886—21no—
37#885—21no—
38#67⋊C4—21no—
39#875C2211yescharacteristic
40C4×D12C2211yescharacteristic
41#869C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43C3×C4⋊C8C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46C3×#97C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52#818C2211yescharacteristic
53#818C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#816C2211yescharacteristic
57#816C2211yescharacteristic
58#816C2211yescharacteristic
59#816C2211yescharacteristic
60#548—31no—
61#547—31no—
62#489—31no—
63#484—31no—
64#480—31no—
65#477—31no—
66#468—31no—
67#466—31no—
68#463—31no—
69#449—31no—
70#448—31no—
71#440—31no—
72#435D311yescharacteristic, complemented
73#417—31no—
74C2×#103—31no—
75C2×C3×D4D411yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85#237—44no—
86#235—22no—
87#235—41no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×Dic6—22no—
93C2×Dic6—21no—
94C2×Dic6—21no—
95C2×Dic6—41no—
96C2×Dic6C2311yescharacteristic
97C2×Dic6C2311yescharacteristic
98C3×M16—41no—
99C2×C3×C8—41no—
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4D411yescharacteristic
105C3×C22⋊C4D411yescharacteristic
106C3×C22⋊C4D411yescharacteristic
107C3×C42C2311yescharacteristic
108#217—21no—
109(C2×D3)⋊C4—21no—
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—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C4×Dic3—21no—
125D24—41no—
126#204—22no—
127#204—41no—
128C2×#41—31no—
129C2×#41—61no—
130C22×Q8—31no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×D8—31no—
134C2×M16C2×D311yescharacteristic
135C22×C8C2×D311yescharacteristic
136C4⋊Q8—31no—
137#124—31no—
138#124—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144#121—31no—
145#121—31no—
146#121—31no—
147#121—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—61no—
151C4×D4C2×D311yescharacteristic
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154#107—31no—
155#107—31no—
156#107—31no—
157C4⋊C8C2×D311yescharacteristic
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168C22⋊C8C2×D311yescharacteristic
169C22⋊C8C2×D311yescharacteristic
170#97C2×D311yescharacteristic
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C3×D4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—42no—
186#67—42no—
187#67—42no—
188#67—42no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—21no—
202C4×D3—42no—
203C4×D3—42no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—42no—
208Dic6—42no—
209Dic6#4112yescomplemented
210C3×C8—21no—
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—21no—
214#41—62no—
215#41—121no—
216#41—124no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×Q8—62no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—121no—
223C2×D4—31no—
224C2×D4—31no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4D1211yescharacteristic
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237SD16—62no—
238SD16—121no—
239D8—121no—
240M16—41no—
241C2×C8—41no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C2×C8C22×D311yescharacteristic
245C2×C8C22×D311yescharacteristic
246C4⋊C4—31no—
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⋊C4D1211yescharacteristic
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4D1211yescharacteristic
264C22⋊C4D1211yescharacteristic
265C42—61no—
266C42C22×D311yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3AGL(1,ℤ/8)12yes—
273C22×C3C22×D411yescharacteristic
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4C2×#4111yescharacteristic
285C3×C4H(5,𝔽2)11yescharacteristic
286Dic3—22no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291Dic3—41no—
292C23—61no—
293C23—61no—
294C23C2×D1211yescharacteristic
295C23C2×D1211yescharacteristic
296Q8—32no—
297Q8—61no—
298Q8—61no—
299Q8—61no—
300Q8—122no—
301Q8—122no—
302D4—22no—
303D4—61no—
304D4—61no—
305D4—61no—
306D4—61no—
307D4—122no—
308D4—122no—
309D4—122no—
310D4—122no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4C2×D1211yescharacteristic
327C2×C4C2×D1211yescharacteristic
328C2×C4C2×D1211yescharacteristic
329C2×C4C2×D1211yescharacteristic
330C2×C4C23×D311yescharacteristic
331C8—21no—
332C8—21no—
333C8—21no—
334C8—21no—
335C2×C3—22no—
336C2×C3—41no—
337C2×C3#54611yescharacteristic
338C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
339C2×C3#57811yescharacteristic
340D3—81no—
341D3—81no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22—121no—
353C22#90812yes—
354C22C22×D1211yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—62no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4#100411yescharacteristic
366C4D3×#4111yescharacteristic
367C3#330811yescharacteristic, complemented, Sylow
368C2—22no—
369C2—41no—
370C2—241no—
371C2—241no—
372C2#555411yescharacteristic
373C2C2×#90811yescharacteristic
374C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3308
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #5555C2semidirect
2. #5552C2semidirect
3. C3×#435C2semidirect
4. #5134C2semidirect
5. #5133C2semidirect
6. #5115C2semidirect
7. C2×#816C2semidirect
8. #4850C2semidirect
9. #4844C2semidirect
10. #4738C2semidirect
11. #4736C2semidirect
12. #4734C2semidirect
13. #4722C2semidirect
14. #4714C2semidirect
15. #4710C2semidirect
72. #435D3semidirect
209. Dic6#41semidirect
367. C3#3308semidirect

Maximal subgroups

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

subgroupindex
#55552
#55522
C3×#4352
#51342
#51332
#51152
C2×#8162
#48502
#48442
#47382
#47362
#47342
#47222
#47142
#47102
#33083