Finite Group Databasebeta

← #80683

Subgroups — #80683

Order 384.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5630C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic, complemented
3C3×#465C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5165C211yesmaximal, characteristic, complemented
6C2×#821C211yesmaximal, characteristic, complemented
7#5111C211yesmaximal, characteristic, complemented
8#5043C211yesmaximal, characteristic, complemented
9#5036C211yesmaximal, characteristic, complemented
10#4870C211yesmaximal, characteristic, complemented
11#4863C211yesmaximal, characteristic, complemented
12#4803C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4785C211yesmaximal, characteristic, complemented
15#4782C211yesmaximal, characteristic, complemented
16#3306—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237—21no—
19C22×D12C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#947C2211yescharacteristic
25#938C2211yescharacteristic, complemented
26#935—21no—
27D4×Dic3C2211yescharacteristic, complemented
28C2×#213C2211yescharacteristic
29C2×#213C2211yescharacteristic
30C4×#67C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
33#895—21no—
34#895C2211yescharacteristic
35#894—21no—
36#893C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic, complemented
38#888C2211yescharacteristic, complemented
39#883—21no—
40#882—21no—
41D3×C22⋊C4—21no—
42#878—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#830C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#547—31no—
61#546—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#476—31no—
66#469—31no—
67#465D311yescharacteristic, complemented
68#463—31no—
69#449—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#417—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67D411yescharacteristic
82C22×Dic3—21no—
83C22×Dic3—21no—
84#239—22no—
85#239—41no—
86#237—44no—
87C2×D12—22no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—41no—
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—21no—
99C3×SD16—41no—
100C2×C3×C8—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4—21no—
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108#217—21no—
109#217—21no—
110#217D411yescharacteristic
111#213—22no—
112#213—41no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4D411yescharacteristic
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127ΓL(1,𝔽25)—41no—
128C2×#41—31no—
129C2×#41—61no—
130C22×D4—31no—
131C2×SD16C2×D311yescharacteristic, complemented
132C2×D8—31no—
133C2×D8—31no—
134C2×M16—31no—
135C22×C8C2×D311yescharacteristic
136C4⋊Q8—31no—
137#124—31no—
138#123—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—61no—
142C22⋊Q8C2×D311yescharacteristic
143#121—31no—
144#121—31no—
145#121—61no—
146#121C2×D311yescharacteristic
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C22⋊C4—61no—
154#106—31no—
155#106—31no—
156#106C2×D311yescharacteristic, complemented
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103C2×D311yescharacteristic, complemented
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3C2×D411yescharacteristic
179C3×Q8—21no—
180C3×Q8—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—42no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—22no—
192#67—42no—
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—21no—
203D12—42no—
204D12—42no—
205D12—81no—
206D12#4112yescomplemented
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—42no—
210C4×D3—42no—
211C4×D3—81no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—61no—
218#41—62no—
219#41—121no—
220#41—124no—
221C2×Q8—61no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—62no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—121no—
233C2×D4C22×D311yescharacteristic
234C22×C4—31no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241SD16—41no—
242D8—62no—
243D8—121no—
244M16—121no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—41no—
248C2×C8C22×D311yescharacteristic
249C2×C8C22×D311yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4C22×D311yescharacteristic
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C42—31no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3AGL(1,ℤ/8)12yes—
276C22×C3C22×D411yescharacteristic
277C2×D3—21no—
278C2×D3—22no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—84no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4C2×#4111yescharacteristic
292C3×C4H(5,𝔽2)11yescharacteristic
293Dic3—21no—
294Dic3—21no—
295Dic3—21no—
296Dic3—41no—
297Dic3—41no—
298C23—21no—
299C23—31no—
300C23—61no—
301C23—61no—
302C23—122no—
303C23—122no—
304C23—241no—
305C23D3×D411yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—122no—
309D4—21no—
310D4—21no—
311D4—32no—
312D4—42no—
313D4—62no—
314D4—61no—
315D4—61no—
316D4—61no—
317D4—122no—
318D4—122no—
319D4—122no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—241no—
338C2×C4D3×D411yescharacteristic
339C2×C4C23×D311yescharacteristic
340C8—21no—
341C8—21no—
342C8—61no—
343C8—61no—
344C2×C3—22no—
345C2×C3—81no—
346C2×C3#54611yescharacteristic
347C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C3#57711yescharacteristic
349D3—41no—
350D3—42no—
351D3—81no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22—62no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—244no—
367C22#91412yes—
368C22C2×D3×D411yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—61no—
374C4—61no—
375C4—61no—
376C4—121no—
377C4—121no—
378C4#100411yescharacteristic
379C4D3×#4111yescharacteristic
380C3#330611yescharacteristic, complemented, Sylow
381C2—22no—
382C2—81no—
383C2—121no—
384C2—122no—
385C2—241no—
386C2#561011yescharacteristic
387C2C2×#91411yescharacteristic
388C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3306
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5630C2semidirect
2. #5611C2semidirect
3. C3×#465C2semidirect
4. #5172C2semidirect
5. #5165C2semidirect
6. C2×#821C2semidirect
7. #5111C2semidirect
8. #5043C2semidirect
9. #5036C2semidirect
10. #4870C2semidirect
11. #4863C2semidirect
12. #4803C2semidirect
13. #4794C2semidirect
14. #4785C2semidirect
15. #4782C2semidirect
20. C2×C3×SD16C22semidirect
25. #938C22semidirect
27. D4×Dic3C22semidirect
32. C2×ΓL(1,𝔽25)C22semidirect
37. D3×C4⋊C4C22semidirect
38. #888C22semidirect
43. C3×#106C22semidirect
44. C3×#103C22semidirect
45. C3×#102C22semidirect
46. #835C22semidirect
47. #835C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
58. #808C22semidirect
59. #808C22semidirect
67. #465D3semidirect
131. C2×SD16C2×D3semidirect
156. #106C2×D3semidirect
160. #103C2×D3semidirect
167. #102C2×D3semidirect
206. D12#41semidirect
380. C3#3306semidirect

Maximal subgroups

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

subgroupindex
#56302
#56112
C3×#4652
#51722
#51652
C2×#8212
#51112
#50432
#50362
#48702
#48632
#48032
#47942
#47852
#47822
#33063