Finite Group Databasebeta

← #78763

Subgroups — #78763

Order 384.

410 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#5806C211yesmaximal, characteristic
2#5511C211yesmaximal, characteristic, complemented
3C2×C3×#102C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5116C211yesmaximal, characteristic, complemented
6#5111C211yesmaximal, characteristic, complemented
7#4970C211yesmaximal, characteristic, complemented
8#4783C211yesmaximal, characteristic, complemented
9#4780C211yesmaximal, characteristic, complemented
10#4774C211yesmaximal, characteristic, complemented
11#4771C211yesmaximal, characteristic, complemented
12#4767C211yesmaximal, characteristic, complemented
13#4765C211yesmaximal, characteristic, complemented
14#4759C211yesmaximal, characteristic, complemented
15#4757C211yesmaximal, characteristic, complemented
16#3314—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#953—21no—
22C2×#217—21no—
23#939—21no—
24#938C2211yescharacteristic
25#936C2211yescharacteristic
26#935—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C4×#67—21no—
31C4×#67C2211yescharacteristic
32#924—21no—
33#924C2211yescharacteristic
34#924C2211yescharacteristic
35C2×Dic12C2211yescharacteristic, complemented
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37#897C2211yescharacteristic, complemented
38#896—21no—
39D3×C4⋊C4C2211yescharacteristic, complemented
40#889—21no—
41#888C2211yescharacteristic, complemented
42#887C2211yescharacteristic, complemented
43C3×#102C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic, complemented
51#818C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#554—31no—
61#546—31no—
62#489—31no—
63#484—31no—
64#480—31no—
65#474—31no—
66#468—31no—
67#467—31no—
68#464—31no—
69#452—31no—
70#450—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74C2×#102D311yescharacteristic, complemented
75C24×C3—21no—
76C2×C3×D4—22no—
77C2×C3×D4—21no—
78C2×C3×D4—41no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67D411yescharacteristic
86C22×Dic3—21no—
87C22×Dic3—21no—
88#237—22no—
89#237—41no—
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C2×Dic6C2311yescharacteristic
95C2×C3×C8—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101#217—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217D411yescharacteristic
108#214—22no—
109#214—41no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C22×D4C2×D311yescharacteristic
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135#125—61no—
136#124—31no—
137#123—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—61no—
145#120—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4C2×D311yescharacteristic
152C2×C22⋊C4—61no—
153#107—31no—
154#107—31no—
155#107—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102C2×D311yescharacteristic
164#102C2×D311yescharacteristic
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—81no—
175C23×C3C2×D411yescharacteristic
176C22×D3C2×D411yescharacteristic
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—42no—
181C3×D4#4112yes—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—81no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—22no—
190#67—42no—
191#67—81no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—42no—
197C2×Dic3—81no—
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C4×D3—21no—
202C4×D3—21no—
203C4×D3—81no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—21no—
208Dic6—81no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C24—21no—
214#41—62no—
215#41—121no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—61no—
219C2×D4—22no—
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—41no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4C22×D311yescharacteristic
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4C22×D311yescharacteristic
233Q16—121no—
234SD16—62no—
235SD16—121no—
236M16—121no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—41no—
240C2×C8C22×D311yescharacteristic
241C2×C8C22×D311yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4C22×D311yescharacteristic
253C4⋊C4C22×D311yescharacteristic
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—
263C42—31no—
264C42—61no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—84no—
274C22×C3#13712yes—
275C22×C3C22×D411yescharacteristic
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—21no—
285Dic3—21no—
286Dic3—21no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—21no—
293C23—31no—
294C23—42no—
295C23—42no—
296C23—81no—
297C23D3×D411yescharacteristic
298Q8—61no—
299Q8—61no—
300Q8—61no—
301Q8—61no—
302Q8—241no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—62no—
308D4—122no—
309D4—241no—
310D4#23512yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—81no—
325C2×C4—122no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4D3×D411yescharacteristic
329C2×C4C23×D311yescharacteristic
330C8—21no—
331C8—21no—
332C8—61no—
333C8—61no—
334C2×C3—22no—
335C2×C3—42no—
336C2×C3—81no—
337C2×C3#54611yescharacteristic
338C2×C3C2×#13711yescharacteristic
339C2×C3#57611yescharacteristic
340D3—41no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—61no—
350C22—61no—
351C22—61no—
352C22—84no—
353C22#91512yes—
354C22C2×D3×D411yescharacteristic
355C4—21no—
356C4—41no—
357C4—41no—
358C4—61no—
359C4—61no—
360C4—61no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4C2×#23511yescharacteristic
366C4#100411yescharacteristic
367C3#331411yescharacteristic, complemented, Sylow
368C2—22no—
369C2—42no—
370C2—81no—
371C2—121no—
372C2#549011yescharacteristic
373C2#575711yescharacteristic
374C2C2×#91511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#102C2
socleC22×C3C22×D4
prime radical#5806C2
hypercenterC2×#102D3
hypocenterC3#3314
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #5511C2semidirect
3. C2×C3×#102C2semidirect
4. #5218C2semidirect
5. #5116C2semidirect
6. #5111C2semidirect
7. #4970C2semidirect
8. #4783C2semidirect
9. #4780C2semidirect
10. #4774C2semidirect
11. #4771C2semidirect
12. #4767C2semidirect
13. #4765C2semidirect
14. #4759C2semidirect
15. #4757C2semidirect
35. C2×Dic12C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
37. #897C22semidirect
39. D3×C4⋊C4C22semidirect
41. #888C22semidirect
42. #887C22semidirect
50. (C2×D3)⋊C8C22semidirect
51. #818C22semidirect
52. #816C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. #810C22semidirect
57. #810C22semidirect
58. #807C22semidirect
59. #807C22semidirect
74. C2×#102D3semidirect
367. C3#3314semidirect

Maximal subgroups

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

subgroupindex
#58062
#55112
C2×C3×#1022
#52182
#51162
#51112
#49702
#47832
#47802
#47742
#47712
#47672
#47652
#47592
#47572
#33143