Finite Group Databasebeta

← #81192

Subgroups — #81192

Order 384.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5695C211yesmaximal, characteristic, complemented
2#5554C211yesmaximal, characteristic, complemented
3C3×#478C211yesmaximal, characteristic, complemented
4#5068C211yesmaximal, characteristic, complemented
5C4×#213C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4882C211yesmaximal, characteristic, complemented
8#4868C211yesmaximal, characteristic, complemented
9#4865C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4838C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4771C211yesmaximal, characteristic, complemented
15#4770C211yesmaximal, characteristic, complemented
16#3322—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C3×#125C2211yescharacteristic, complemented
20C3×C4×D4C2211yescharacteristic
21C2×#213C2211yescharacteristic
22#930—21no—
23C2×(C2×D3)⋊C4—21no—
24C2×D24C2211yescharacteristic
25C2×#204C2211yescharacteristic, complemented
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
28#898—21no—
29#897C2211yescharacteristic
30#895—21no—
31#895C2211yescharacteristic
32#895C2211yescharacteristic
33#895C2211yescharacteristic, complemented
34#894—21no—
35#893—21no—
36#892C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic, complemented
38#886—21no—
39#882—21no—
40#875C2211yescharacteristic
41C4×D12C2211yescharacteristic
42C4×D12C2211yescharacteristic, complemented
43C3×#107C2211yescharacteristic, complemented
44C3×#106C2211yescharacteristic, complemented
45C3×C4⋊C8C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53#810C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59C4×C3⋊C8C2211yescharacteristic, complemented
60#553—31no—
61#546—31no—
62#488—31no—
63#485—31no—
64#480—31no—
65#478D311yescharacteristic, complemented
66#469—31no—
67#468—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71C4×D8—31no—
72#436—31no—
73#418—31no—
74#417—31no—
75C23×D3—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81#237—22no—
82#237—41no—
83D3×D4—22no—
84D3×D4—41no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12D411yescharacteristic
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×C4×D3C2311yescharacteristic
95C2×Dic6C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104C3×C42C2311yescharacteristic
105#213—22no—
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)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3D411yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123D24—41no—
124#204—41no—
125ΓL(1,𝔽25)—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×SD16—31no—
130C2×D8—31no—
131C2×D8—31no—
132C2×M16—31no—
133C2×M16—31no—
134#126—61no—
135#125C2×D311yescharacteristic, complemented
136#124—31no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121—61no—
145#120—61no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—61no—
153#107—31no—
154#107C2×D311yescharacteristic, complemented
155#106C2×D311yescharacteristic, complemented
156C4⋊C8C2×D311yescharacteristic, complemented
157#103—31no—
158#103—31no—
159#103—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—42no—
174C22×D3—42no—
175C22×D3—81no—
176C22×D3C2×D411yescharacteristic
177C22×D3C2×D411yescharacteristic
178C3×D4—21no—
179C3×D4#4112yes—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—81no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—42no—
188#67—42no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—42no—
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194D12—21no—
195D12—21no—
196D12—21no—
197D12—21no—
198D12—41no—
199D12—42no—
200D12—81no—
201D12—81no—
202C4×D3—21no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—81no—
207C4×D3—81no—
208Dic6—21no—
209Dic6—21no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C24—61no—
215#41—62no—
216#41—121no—
217C2×Q8—31no—
218C2×D4—31no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—62no—
226C2×D4—61no—
227C2×D4—121no—
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235SD16—121no—
236D8—62no—
237D8—121no—
238M16—121no—
239M16—121no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4C22×D311yescharacteristic
251C4⋊C4C22×D311yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—31no—
255C22⋊C4—31no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—22no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—84no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4C2×#4111yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—21no—
291Dic3—21no—
292Dic3—41no—
293Dic3—41no—
294C23—21no—
295C23—31no—
296C23—31no—
297C23—61no—
298C23—61no—
299C23—122no—
300C23—122no—
301C23—241no—
302Q8—61no—
303Q8—61no—
304D4—21no—
305D4—61no—
306D4—61no—
307D4—61no—
308D4—61no—
309D4—121no—
310D4—122no—
311D4—122no—
312D4—122no—
313D4—241no—
314D4—241no—
315D4#23912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—81no—
328C2×C4—122no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4D3×D411yescharacteristic
332C2×C4D3×D411yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—21no—
335C8—21no—
336C8—61no—
337C8—61no—
338C2×C3—42no—
339C2×C3#54611yescharacteristic
340C2×C3#57511yescharacteristic
341C2×C3#57611yescharacteristic
342D3—41no—
343D3—41no—
344D3—81no—
345D3—81no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—61no—
350C22—61no—
351C22—61no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—244no—
362C22C2×D3×D411yescharacteristic
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—61no—
369C4—61no—
370C4—121no—
371C4—121no—
372C4C2×#23911yescharacteristic
373C4#100911yescharacteristic
374C3#332211yescharacteristic, complemented, Sylow
375C2—42no—
376C2—121no—
377C2—121no—
378C2—241no—
379C2—241no—
380C2#563011yescharacteristic
381C2#576811yescharacteristic
382C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#478C2
socleC22×C3C22×D4
hypercenter#478D3
hypocenterC3#3322
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #5695C2semidirect
2. #5554C2semidirect
3. C3×#478C2semidirect
4. #5068C2semidirect
5. C4×#213C2semidirect
6. #4886C2semidirect
7. #4882C2semidirect
8. #4868C2semidirect
9. #4865C2semidirect
10. #4844C2semidirect
11. #4838C2semidirect
12. #4781C2semidirect
13. #4773C2semidirect
14. #4771C2semidirect
15. #4770C2semidirect
19. C3×#125C22semidirect
25. C2×#204C22semidirect
27. C2×ΓL(1,𝔽25)C22semidirect
33. #895C22semidirect
37. D3×C4⋊C4C22semidirect
42. C4×D12C22semidirect
43. C3×#107C22semidirect
44. C3×#106C22semidirect
45. C3×C4⋊C8C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #816C22semidirect
53. #810C22semidirect
54. #810C22semidirect
57. #809C22semidirect
58. #807C22semidirect
59. C4×C3⋊C8C22semidirect
65. #478D3semidirect
135. #125C2×D3semidirect
154. #107C2×D3semidirect
155. #106C2×D3semidirect
156. C4⋊C8C2×D3semidirect
374. C3#3322semidirect

Maximal subgroups

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

subgroupindex
#56952
#55542
C3×#4782
#50682
C4×#2132
#48862
#48822
#48682
#48652
#48442
#48382
#47812
#47732
#47712
#47702
#33223