Finite Group Databasebeta

← #82576

Subgroups — #82576

Order 384.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#520C211yesmaximal, characteristic, complemented
2#5534C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic
5#5030C211yesmaximal, characteristic, complemented
6#5028C211yesmaximal, characteristic, complemented
7C4×#215C211yesmaximal, characteristic, complemented
8#5020C211yesmaximal, characteristic, complemented
9C4×#214C211yesmaximal, characteristic, complemented
10#5016C211yesmaximal, characteristic, complemented
11#5014C211yesmaximal, characteristic, complemented
12#5008C211yesmaximal, characteristic, complemented
13#5000C211yesmaximal, characteristic, complemented
14#4971C211yesmaximal, characteristic, complemented
15#4968C211yesmaximal, characteristic, complemented
16#3125—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20C3×C4×D4—21no—
21C3×C4×D4—21no—
22C3×C4×D4C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24C3×#117—21no—
25C3×#117C2211yescharacteristic
26C2×C3×C4⋊C4—21no—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#215C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#924C2211yescharacteristic
34#898—21no—
35#896—21no—
36#885—21no—
37#883—21no—
38#875C2211yescharacteristic
39C4×D12C2211yescharacteristic
40#869C2211yescharacteristic
41C4×Dic6C2211yescharacteristic
42#835C2211yescharacteristic
43#835C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#830C2211yescharacteristic
47#830C2211yescharacteristic
48#810C2211yescharacteristic
49#810C2211yescharacteristic
50#809C2211yescharacteristic
51#809C2211yescharacteristic
52#808C2211yescharacteristic
53#808C2211yescharacteristic
54#808C2211yescharacteristic
55#808C2211yescharacteristic
56C4⋊(C3⋊C8)C2211yescharacteristic
57C4⋊(C3⋊C8)C2211yescharacteristic
58#803C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#542—31no—
61#520D311yescharacteristic, complemented
62#483—31no—
63#481—31no—
64#476—31no—
65#475—31no—
66#464—31no—
67#463—31no—
68#452—31no—
69#449—31no—
70#442—31no—
71#441—31no—
72C4×SD16—31no—
73C4×SD16—31no—
74#433—31no—
75C3×#41—22no—
76C3×#41—22no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×D12C2311yescharacteristic
87C2×C4×D3—21no—
88C2×Dic6—21no—
89C2×Dic6C2311yescharacteristic
90C3×C4⋊C4—21no—
91C3×C4⋊C4—21no—
92C3×C4⋊C4—21no—
93C3×C4⋊C4D411yescharacteristic
94C3×C4⋊C4D411yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C22⋊C4—21no—
99C3×C22⋊C4D411yescharacteristic
100C3×C22⋊C4D411yescharacteristic
101C3×C42—21no—
102C3×C42—21no—
103C3×C42C2311yescharacteristic
104C3×C42C2311yescharacteristic
105#217—21no—
106#216—22no—
107#215—22no—
108#214—22no—
109#213—22no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119C4×Dic3—21no—
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×#41C2×D311yescharacteristic
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×D8—31no—
129C4⋊Q8—31no—
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121—31no—
137C4×Q8—31no—
138C4×Q8C2×D311yescharacteristic
139C4×Q8C2×D311yescharacteristic
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—31no—
143C4×D4C2×D311yescharacteristic
144C4×D4C2×D311yescharacteristic
145#117—21no—
146#117C2×D311yescharacteristic
147C2×C4⋊C4—21no—
148C2×C4⋊C4C2×D311yescharacteristic
149#106—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C22×D3—21no—
170C3×Q8—21no—
171C3×Q8#4112yes—
172C3×D4—24no—
173C3×D4—21no—
174C3×D4#4112yes—
175C2×C3×C4—24no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—81no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194D12—21no—
195D12—21no—
196C4×D3—81no—
197Dic6—21no—
198Dic6—21no—
199Dic6—81no—
200C3⋊C8—21no—
201C3⋊C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204#41—22no—
205#41—22no—
206C2×Q8—31no—
207C2×Q8—61no—
208C2×Q8C22×D311yescharacteristic
209C2×D4—21no—
210C2×D4—31no—
211C2×D4—61no—
212C2×D4C22×D311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219Q16—62no—
220SD16—62no—
221SD16—62no—
222D8—62no—
223C2×C8—31no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—31no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4#6711yescharacteristic
236C4⋊C4#6711yescharacteristic
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4#6711yescharacteristic
247C22⋊C4#6711yescharacteristic
248C42—21no—
249C42—21no—
250C42—61no—
251C42C22×D311yescharacteristic
252C42C22×D311yescharacteristic
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3C22×D411yescharacteristic
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—21no—
264C3×C4—22no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4C2×#4111yescharacteristic
272C3×C4C2×#4111yescharacteristic
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276C23—21no—
277C23—61no—
278C23C2×#6711yescharacteristic
279Q8—21no—
280Q8—61no—
281Q8—61no—
282Q8—241no—
283Q8#23512yes—
284D4—24no—
285D4—21no—
286D4—61no—
287D4—61no—
288D4—241no—
289D4#23512yes—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—241no—
303C2×C4C2×#6711yescharacteristic
304C2×C4C2×#6711yescharacteristic
305C2×C4C2×#6711yescharacteristic
306C2×C4C2×#6711yescharacteristic
307C2×C4C2×#6711yescharacteristic
308C2×C4C23×D311yescharacteristic
309C8—61no—
310C8—61no—
311C8—61no—
312C8—61no—
313C2×C3—41no—
314C2×C3—42no—
315C2×C3#52511yescharacteristic
316C2×C3#57711yescharacteristic
317C2×C3#57711yescharacteristic
318D3—81no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22C22×#6711yescharacteristic
329C4—21no—
330C4—22no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4C2×#23511yescharacteristic
341C4C2×#23511yescharacteristic
342C3#312511yescharacteristic, complemented, Sylow
343C2—41no—
344C2—42no—
345C2—241no—
346C2#579111yescharacteristic
347C2#583611yescharacteristic
348C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#520C2
socleC22×C3C22×D4
prime radical#5238C2
hypercenter#520D3
hypocenterC3#3125
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C3×#520C2semidirect
2. #5534C2semidirect
3. #5239C2semidirect
5. #5030C2semidirect
6. #5028C2semidirect
7. C4×#215C2semidirect
8. #5020C2semidirect
9. C4×#214C2semidirect
10. #5016C2semidirect
11. #5014C2semidirect
12. #5008C2semidirect
13. #5000C2semidirect
14. #4971C2semidirect
15. #4968C2semidirect
61. #520D3semidirect
342. C3#3125semidirect

Maximal subgroups

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

subgroupindex
C3×#5202
#55342
#52392
#52382
#50302
#50282
C4×#2152
#50202
C4×#2142
#50162
#50142
#50082
#50002
#49712
#49682
#31253