Finite Group Databasebeta

← #81528

Subgroups — #81528

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#5630C211yesmaximal, characteristic, complemented
2#5532C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4#5049C211yesmaximal, characteristic, complemented
5#5001C211yesmaximal, characteristic, complemented
6#4890C211yesmaximal, characteristic, complemented
7#4883C211yesmaximal, characteristic, complemented
8#4869C211yesmaximal, characteristic, complemented
9#4866C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4810C211yesmaximal, characteristic, complemented
12#4804C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3319—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#947C2211yescharacteristic
22C2×#215C2211yescharacteristic, complemented
23#930C2211yescharacteristic
24C4×#67C2211yescharacteristic
25C2×#208C2211yescharacteristic
26C2×D24C2211yescharacteristic
27C2×D24C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic, complemented
29#895—21no—
30#895C2211yescharacteristic
31#895C2211yescharacteristic, complemented
32#894—21no—
33#893C2211yescharacteristic
34#893C2211yescharacteristic, complemented
35#892C2211yescharacteristic, complemented
36#888C2211yescharacteristic, complemented
37#885—21no—
38#884—21no—
39#882—21no—
40D3×C22⋊C4—21no—
41#875—21no—
42#872—21no—
43C3×#107C2211yescharacteristic, complemented
44C3×#106C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#103C2211yescharacteristic, complemented
47C3×C22⋊C8C2211yescharacteristic
48#835C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#815C2211yescharacteristic, complemented
53Dic3⋊C8C2211yescharacteristic, complemented
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#809C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#555—31no—
61#546—31no—
62#489—31no—
63#485D311yescharacteristic, complemented
64#482—31no—
65#474—31no—
66#468—31no—
67#466—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#418—31no—
74#416—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67D411yescharacteristic
80#239—22no—
81#239—41no—
82C2×D12—22no—
83C2×D12—21no—
84C2×D12—21no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—41no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×Dic6—21no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42—21no—
105#217D411yescharacteristic
106#215—41no—
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)⋊C4—21no—
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4D411yescharacteristic
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121#208—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124D24—22no—
125D24—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×SD16—31no—
130C2×D8—31no—
131C2×D8—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#124—31no—
136#124—61no—
137#124—61no—
138#123—61no—
139C22⋊Q8C2×D311yescharacteristic
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—31no—
149#117—31no—
150#117—61no—
151#117C2×D311yescharacteristic
152C2×C22⋊C4—61no—
153#107—31no—
154#107C2×D311yescharacteristic, complemented
155#106C2×D311yescharacteristic, complemented
156C4⋊C8—31no—
157#103—31no—
158#103C2×D311yescharacteristic, complemented
159#103C2×D311yescharacteristic, complemented
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—81no—
177C22×D3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—41no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194D12—21no—
195D12—21no—
196D12—21no—
197D12—21no—
198D12—21no—
199D12—42no—
200D12—42no—
201D12—81no—
202D12—81no—
203D12#4112yes—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—42no—
207C4×D3—42no—
208C4×D3—81no—
209Dic6—81no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C24—61no—
215#41—62no—
216#41—121no—
217C2×Q8—61no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—62no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—121no—
230C22×C4—31no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235SD16—121no—
236D8—62no—
237D8—121no—
238M16—121no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—121no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4C22×D311yescharacteristic
249C4⋊C4C22×D311yescharacteristic
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—31no—
254C22⋊C4—31no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C42—21no—
264C42—31no—
265C42—61no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—22no—
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—41no—
275C2×D3—41no—
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—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4C2×#4111yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—21no—
291Dic3—21no—
292Dic3—21no—
293Dic3—41no—
294C23—31no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—122no—
299C23—122no—
300C23—241no—
301C23D3×D411yescharacteristic
302Q8—21no—
303Q8—21no—
304Q8—241no—
305D4—32no—
306D4—61no—
307D4—61no—
308D4—61no—
309D4—61no—
310D4—61no—
311D4—122no—
312D4—122no—
313D4—121no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—241no—
332C2×C4D3×D411yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—21no—
335C8—21no—
336C8—61no—
337C8—61no—
338C2×C3—41no—
339C2×C3#54611yescharacteristic
340C2×C3#57511yescharacteristic
341C2×C3#57611yescharacteristic
342D3—41no—
343D3—42no—
344D3—81no—
345D3—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—61no—
350C22—62no—
351C22—61no—
352C22—61no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—244no—
362C22C2×D3×D411yescharacteristic
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—61no—
369C4—61no—
370C4—61no—
371C4—121no—
372C4D3×#4111yescharacteristic
373C4#100911yescharacteristic
374C3#331911yescharacteristic, complemented, Sylow
375C2—41no—
376C2—121no—
377C2—122no—
378C2—241no—
379C2—241no—
380C2#565411yescharacteristic
381C2#576911yescharacteristic
382C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
hypercenter#485D3
hypocenterC3#3319
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #5630C2semidirect
2. #5532C2semidirect
3. C3×#485C2semidirect
4. #5049C2semidirect
5. #5001C2semidirect
6. #4890C2semidirect
7. #4883C2semidirect
8. #4869C2semidirect
9. #4866C2semidirect
10. #4819C2semidirect
11. #4810C2semidirect
12. #4804C2semidirect
13. #4794C2semidirect
14. #4732C2semidirect
15. #4728C2semidirect
22. C2×#215C22semidirect
28. C2×D3×C8C22semidirect
31. #895C22semidirect
34. #893C22semidirect
35. #892C22semidirect
36. #888C22semidirect
43. C3×#107C22semidirect
44. C3×#106C22semidirect
45. C3×#103C22semidirect
46. C3×#103C22semidirect
48. #835C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #815C22semidirect
53. Dic3⋊C8C22semidirect
58. #809C22semidirect
59. #807C22semidirect
63. #485D3semidirect
154. #107C2×D3semidirect
155. #106C2×D3semidirect
158. #103C2×D3semidirect
159. #103C2×D3semidirect
374. C3#3319semidirect

Maximal subgroups

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

subgroupindex
#56302
#55322
C3×#4852
#50492
#50012
#48902
#48832
#48692
#48662
#48192
#48102
#48042
#47942
#47322
#47282
#33193