Finite Group Databasebeta

← #80164

Subgroups — #80164

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5806C211yesmaximal, characteristic
2#5639C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5215C211yesmaximal, characteristic, complemented
5#5174C211yesmaximal, characteristic, complemented
6#5173C211yesmaximal, characteristic, complemented
7#5166C211yesmaximal, characteristic, complemented
8Dic3×SD16C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#4777C211yesmaximal, characteristic, complemented
11#4771C211yesmaximal, characteristic, complemented
12#4768C211yesmaximal, characteristic, complemented
13#4766C211yesmaximal, characteristic, complemented
14#4729C211yesmaximal, characteristic, complemented
15#4723C211yesmaximal, characteristic, complemented
16#3323—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#953—21no—
23C2×#217—21no—
24#946C2211yescharacteristic, complemented
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27#939—21no—
28#938C2211yescharacteristic
29#936C2211yescharacteristic
30#935—21no—
31D4×Dic3C2211yescharacteristic
32C4×#67C2211yescharacteristic
33#926C2211yescharacteristic
34#924C2211yescharacteristic
35C2×#208C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37#892—21no—
38D3×C4⋊C4C2211yescharacteristic, complemented
39#890C2211yescharacteristic, complemented
40#889—21no—
41#883—21no—
42#879—21no—
43C3×#102C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#818C2211yescharacteristic, complemented
53#817C2211yescharacteristic, complemented
54#816C2211yescharacteristic, complemented
55#816C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#552—31no—
61#546—31no—
62#488—31no—
63#485—31no—
64#481—31no—
65#478—31no—
66#470—31no—
67#468—31no—
68#464—31no—
69#452—31no—
70#450D311yescharacteristic, complemented
71C4×SD16—31no—
72#436—31no—
73#418—31no—
74#417—31no—
75C24×C3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—41no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
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×Dic6C2311yescharacteristic
93C3×SD16—22no—
94C3×SD16—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217D411yescharacteristic
107#216—41no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123#208—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C22×D4C2×D311yescharacteristic
129C2×Q16—31no—
130C2×SD16C2×D311yescharacteristic
131C2×SD16C2×D311yescharacteristic
132C2×M16—31no—
133C2×M16—31no—
134#126—61no—
135#125—31no—
136#125—61no—
137#124—31no—
138#123—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8C2×D311yescharacteristic
143#121—31no—
144#121—61no—
145#120—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149#117—31no—
150#117—61no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—61no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—81no—
175C23×C3C2×D411yescharacteristic
176C22×D3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—42no—
183C3×D4#4112yes—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—41no—
191#67—42no—
192#67—81no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—42no—
198C2×Dic3—81no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C4×D3—21no—
203C4×D3—21no—
204C4×D3—42no—
205Dic6—21no—
206Dic6—21no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C24—21no—
212#41—62no—
213#41—121no—
214C2×Q8—31no—
215C2×Q8C22×D311yescharacteristic
216C2×D4—22no—
217C2×D4—21no—
218C2×D4—31no—
219C2×D4—41no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4C22×D311yescharacteristic
223C2×D4C22×D311yescharacteristic
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229Q16—121no—
230SD16—22no—
231SD16—41no—
232M16—121no—
233M16—121no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4C22×D311yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—31no—
252C22⋊C4—31no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C42—31no—
261C42—61no—
262C42—61no—
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—84no—
273C22×C3C22×D411yescharacteristic
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—21no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#4111yescharacteristic
281C3×C4H(5,𝔽2)11yescharacteristic
282Dic3—21no—
283Dic3—21no—
284Dic3—21no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289C23—21no—
290C23—21no—
291C23—31no—
292C23—42no—
293C23—42no—
294C23—81no—
295C23D3×D411yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—61no—
299Q8—61no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—121no—
305D4—122no—
306D4—241no—
307D4#23712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—241no—
324C2×C4D3×D411yescharacteristic
325C2×C4C23×D311yescharacteristic
326C8—21no—
327C8—21no—
328C8—61no—
329C8—61no—
330C2×C3—41no—
331C2×C3—42no—
332C2×C3—81no—
333C2×C3#54611yescharacteristic
334C2×C3#57511yescharacteristic
335C2×C3#57711yescharacteristic
336D3—41no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—61no—
347C22—61no—
348C22—61no—
349C22—84no—
350C22C2×D3×D411yescharacteristic
351C4—21no—
352C4—41no—
353C4—41no—
354C4—61no—
355C4—61no—
356C4—61no—
357C4—121no—
358C4—121no—
359C4—121no—
360C4—121no—
361C4C2×#23711yescharacteristic
362C4#100411yescharacteristic
363C3#332311yescharacteristic, complemented, Sylow
364C2—41no—
365C2—42no—
366C2—81no—
367C2—121no—
368C2#559111yescharacteristic
369C2#576211yescharacteristic
370C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radical#5806C2
hypercenter#450D3
hypocenterC3#3323
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #5639C2semidirect
3. C3×#450C2semidirect
4. #5215C2semidirect
5. #5174C2semidirect
6. #5173C2semidirect
7. #5166C2semidirect
8. Dic3×SD16C2semidirect
9. #5052C2semidirect
10. #4777C2semidirect
11. #4771C2semidirect
12. #4768C2semidirect
13. #4766C2semidirect
14. #4729C2semidirect
15. #4723C2semidirect
24. #946C22semidirect
25. Q8×Dic3C22semidirect
26. C2×#216C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
38. D3×C4⋊C4C22semidirect
39. #890C22semidirect
46. #835C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #818C22semidirect
53. #817C22semidirect
54. #816C22semidirect
55. #816C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #810C22semidirect
59. #808C22semidirect
70. #450D3semidirect
363. C3#3323semidirect

Maximal subgroups

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

subgroupindex
#58062
#56392
C3×#4502
#52152
#51742
#51732
#51662
Dic3×SD162
#50522
#47772
#47712
#47682
#47662
#47292
#47232
#33233