Finite Group Databasebeta

← #78908

Subgroups — #78908

Order 384.

430 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5834C211yesmaximal, characteristic, complemented
2#5507C211yesmaximal, characteristic, complemented
3C3×#417C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5135C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#4968C211yesmaximal, characteristic, complemented
8#4810C211yesmaximal, characteristic, complemented
9#4803C211yesmaximal, characteristic, complemented
10#4791C211yesmaximal, characteristic, complemented
11#4788C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4782C211yesmaximal, characteristic, complemented
14#4777C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3332—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#235—21no—
20C2×C3×M16C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23C2×#215C2211yescharacteristic
24#940—21no—
25#940C2211yescharacteristic
26#938—21no—
27#938C2211yescharacteristic
28C2×#213C2211yescharacteristic
29#930—21no—
30#930C2211yescharacteristic
31#930C2211yescharacteristic
32#929—21no—
33C4×#67—21no—
34C4×#67C2211yescharacteristic
35C2×D24C2211yescharacteristic
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#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#817C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#550—31no—
61#547—31no—
62#486—31no—
63#481—31no—
64#481—31no—
65#475—31no—
66#469—31no—
67#469—31no—
68#459—31no—
69#449—31no—
70#449—31no—
71C4×SD16—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×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67D411yescharacteristic
89#239—22no—
90#239—41no—
91#235—44no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3C2311yescharacteristic
100C2×Dic6—21no—
101C3×M16—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4—21no—
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107#217—21no—
108#217D411yescharacteristic
109#215—22no—
110#213—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D24—41no—
127ΓL(1,𝔽25)—41no—
128C2×#41—31no—
129C2×#41—61no—
130C2×#41C2×D311yescharacteristic
131C2×SD16—31no—
132C2×D8—31no—
133C2×D8—31no—
134C2×M16—31no—
135C2×M16C2×D311yescharacteristic
136C4⋊Q8—31no—
137#127—31no—
138#127—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—61no—
142#121—31no—
143#121—31no—
144#121—31no—
145#121—31no—
146#121—61no—
147#121—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4C2×D311yescharacteristic
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#103C2×D311yescharacteristic
161#102—31no—
162#102—31no—
163#102—31no—
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×D3—21no—
175C22×D3C2×D411yescharacteristic
176C3×Q8—21no—
177C3×Q8#4112yescomplemented
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—42no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—22no—
191#67—42no—
192#67—42no—
193#67—81no—
194#67—81no—
195#67—81no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—42no—
206D12—42no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—42no—
210C4×D3—42no—
211C4×D3—42no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217#41—22no—
218#41—41no—
219#41—62no—
220#41—124no—
221#41—121no—
222C2×Q8—61no—
223C2×Q8C22×D311yescharacteristic
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4C22×D311yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4C22×D311yescharacteristic
243SD16—62no—
244D8—121no—
245D8—121no—
246M16—41no—
247M16—121no—
248C2×C8—31no—
249C2×C8—31no—
250C2×C8C22×D311yescharacteristic
251C2×C8C22×D311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C4⋊C4C22×D311yescharacteristic
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C42—31no—
269C42—61no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4C2×#4111yescharacteristic
291C3×C4H(5,𝔽2)11yescharacteristic
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296Dic3AGL(1,ℤ/8)12yes—
297C23—21no—
298C23—31no—
299C23—61no—
300C23—61no—
301C23D3×D411yescharacteristic
302Q8—21no—
303Q8—122no—
304Q8#23512yescomplemented
305D4—21no—
306D4—21no—
307D4—42no—
308D4—62no—
309D4—61no—
310D4—61no—
311D4—61no—
312D4—61no—
313D4—122no—
314D4—122no—
315D4—122no—
316D4—122no—
317D4—241no—
318D4—241no—
319D4—241no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4D3×D411yescharacteristic
338C2×C4C23×D311yescharacteristic
339C8—21no—
340C8—21no—
341C8—61no—
342C8—61no—
343C2×C3—41no—
344C2×C3—81no—
345C2×C3#54611yescharacteristic
346C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
347C2×C3#57711yescharacteristic
348D3—41no—
349D3—81no—
350D3—81no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22C2×D3×D411yescharacteristic
367C4—21no—
368C4—22no—
369C4—32no—
370C4—41no—
371C4—41no—
372C4—61no—
373C4—61no—
374C4—121no—
375C4—121no—
376C4C2×#23511yescharacteristic
377C4#100411yescharacteristic
378C3#333211yescharacteristic, complemented, Sylow
379C2—41no—
380C2—81no—
381C2—121no—
382C2—241no—
383C2—241no—
384C2#549011yescharacteristic
385C2D3×AGL(1,ℤ/8)11yescharacteristic
386C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
hypercenter#417D3
hypocenterC3#3332
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. #5834C2semidirect
2. #5507C2semidirect
3. C3×#417C2semidirect
4. #5238C2semidirect
5. #5135C2semidirect
6. #5128C2semidirect
7. #4968C2semidirect
8. #4810C2semidirect
9. #4803C2semidirect
10. #4791C2semidirect
11. #4788C2semidirect
12. #4786C2semidirect
13. #4782C2semidirect
14. #4777C2semidirect
15. #4760C2semidirect
74. #417D3semidirect
177. C3×Q8#41semidirect
304. Q8#235semidirect
378. C3#3332semidirect

Maximal subgroups

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

subgroupindex
#58342
#55072
C3×#4172
#52382
#51352
#51282
#49682
#48102
#48032
#47912
#47882
#47862
#47822
#47772
#47602
#33323