Finite Group Databasebeta

← #608530

Subgroups — #608530

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15210C212yesmaximal
2#15208C216yesmaximal
3#15198C211yesmaximal, characteristic
4#15189C211yesmaximal, characteristic
5#15187C213yesmaximal
6#15186C213yesmaximal
7#13855C212yesmaximal
8#13846C216yesmaximal
9C4×#391C213yesmaximal
10C4×#389C213yesmaximal
11C4×#376C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#391C2213yes—
14C2×#391C2213yes—
15C2×#389C2213yes—
16C2×#389C2213yes—
17C2×C4×C4⋊C4C2216yes—
18C2×C4×C4⋊C4C2213yes—
19C2×#376C2213yes—
20C2×#376C2211yescharacteristic
21C2×C43C2213yes—
22#1445C2216yes—
23#1445C2216yes—
24#1443C2216yes—
25#1443C2216yes—
26#1443C2216yes—
27#1443C2216yes—
28#1443C2212yes—
29#1443C2212yes—
30#1442C2216yes—
31#1442C2216yes—
32#1441C2216yes—
33#1441C2216yes—
34#1436C2216yes—
35#1436C2216yes—
36#1436C2216yes—
37#1436C2216yes—
38C4⋊#95C2213yes—
39C4⋊#95C2213yes—
40C4⋊#95C2213yes—
41C4⋊#95C2213yes—
42C4×#95C2216yes—
43C4×#95C2216yes—
44C4×#95C2216yes—
45C4×#95C2213yes—
46C4×#95C2211yescharacteristic
47C4×#95C2213yes—
48C4×#95C2211yescharacteristic
49C23×Q8C2311yescharacteristic
50C2×C4×Q8—212no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2313yes—
54C22×C4⋊C4C2316yes—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2313yes—
57C22×C42C2313yes—
58C22×C42C2313yes—
59C22×C42C2312yes—
60C22×C42C2316yes—
61C22×C42C2313yes—
62C22×C42C2313yes—
63C22×C42C2313yes—
64C22×C42C2313yes—
65C22×C42C2311yescharacteristic
66#391—212no—
67#391—212no—
68#389—212no—
69#389—212no—
70C4×C4⋊C4—224no—
71C4×C4⋊C4—212no—
72#376—212no—
73#376Q818yescomplemented
74C2×#95C2313yes—
75C2×#95C2313yes—
76C2×#95C2313yes—
77C2×#95C2313yes—
78C2×#95C2316yes—
79C2×#95C2316yes—
80C2×#95C2316yes—
81C2×#95C2316yes—
82C2×#95C2316yes—
83C2×#95C2316yes—
84C2×#95C2316yes—
85C2×#95C2316yes—
86C2×#95C2316yes—
87C2×#95C2316yes—
88C2×#95C2316yes—
89C2×#95C2316yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2313yes—
94C2×#95C2311yescharacteristic
95C43—212no—
96C22×Q8—26no—
97C22×Q8—26no—
98C22×Q8#4114yes—
99C23×C4C2416yes—
100C23×C4C2412yes—
101C23×C4C2413yes—
102C23×C4C2413yes—
103C23×C4C2416yes—
104C23×C4C2413yes—
105C23×C4C2413yes—
106C23×C4C2412yes—
107C23×C4C2411yescharacteristic
108C23×C4C2411yescharacteristic
109C23×C4C2411yescharacteristic
110C4×Q8—416no—
111C2×C4⋊C4—26no—
112C2×C4⋊C4—26no—
113C2×C4⋊C4—212no—
114C2×C4⋊C4—26no—
115C2×C4⋊C4—212no—
116C2×C4⋊C4—212no—
117C2×C4⋊C4—26no—
118C2×C4⋊C4—212no—
119C2×C4⋊C4—212no—
120C2×C4⋊C4—26no—
121C2×C4⋊C4—26no—
122C2×C4⋊C4—46no—
123C2×C4⋊C4—46no—
124C2×C4⋊C4#41112yes—
125C2×C42—26no—
126C2×C42—26no—
127C2×C42—26no—
128C2×C42—26no—
129C2×C42—212no—
130C2×C42—26no—
131C2×C42—26no—
132C2×C42—26no—
133C2×C42—26no—
134C2×C42—212no—
135C2×C42—212no—
136C2×C42—212no—
137C2×C42—26no—
138C2×C42—26no—
139C2×C42—26no—
140C2×C42—26no—
141C2×C42—43no—
142C2×C42—43no—
143C2×C42—43no—
144C2×C42—43no—
145C2×C42C2×Q8112yes—
146C2×C42#41112yes—
147C2×C42#41112yes—
148#95—26no—
149#95—26no—
150#95—26no—
151#95—26no—
152#95—212no—
153#95—212no—
154#95—212no—
155#95—212no—
156#95—212no—
157#95—212no—
158#95—212no—
159#95—212no—
160#95—212no—
161#95—46no—
162#95—46no—
163#95—46no—
164#95—46no—
165#95—46no—
166#95—46no—
167#95C2×Q8112yes—
168#95C2×Q814yes—
169C24C2511yescharacteristic
170C2×Q8—224no—
171C2×Q8—44no—
172C2×Q8—412no—
173C22×C4—21no—
174C22×C4—23no—
175C22×C4—23no—
176C22×C4—23no—
177C22×C4—23no—
178C22×C4—23no—
179C22×C4—23no—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—23no—
183C22×C4—23no—
184C22×C4—23no—
185C22×C4—26no—
186C22×C4—26no—
187C22×C4—26no—
188C22×C4—26no—
189C22×C4—23no—
190C22×C4—23no—
191C22×C4—23no—
192C22×C4—23no—
193C22×C4—23no—
194C22×C4—23no—
195C22×C4—26no—
196C22×C4—26no—
197C22×C4—23no—
198C22×C4—23no—
199C22×C4—23no—
200C22×C4—23no—
201C22×C4—23no—
202C22×C4—23no—
203C22×C4—26no—
204C22×C4—26no—
205C22×C4—23no—
206C22×C4—23no—
207C22×C4—26no—
208C22×C4—26no—
209C22×C4—26no—
210C22×C4—26no—
211C22×C4—26no—
212C22×C4—26no—
213C22×C4—22no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—26no—
217C22×C4C22×Q816yes—
218C22×C4C22×Q816yes—
219C22×C4C22×Q812yes—
220C22×C4C2×#4114yes—
221C22×C4C2×#4116yes—
222C22×C4C2×#4116yes—
223C22×C4C2×#41112yes—
224C22×C4C2×#4116yes—
225C22×C4C2×#4116yes—
226C22×C4C2×#4116yes—
227C22×C4C2×#4112yes—
228C22×C4C2×#4116yes—
229C22×C4C2×#4116yes—
230C4⋊C4—224no—
231C4⋊C4—424no—
232C4⋊C4—424no—
233C4⋊C4—48no—
234C4⋊C4—44no—
235C4⋊C4—412no—
236C42—224no—
237C42—224no—
238C42—412no—
239C42—412no—
240C42—48no—
241C42—412no—
242C42—412no—
243C42—424no—
244C42#126116yes—
245C23C23×Q811yescharacteristic
246C23C22×#4113yes—
247C23C22×#4113yes—
248C23C22×#4111yescharacteristic
249C23C22×#4113yes—
250C23#58511yescharacteristic
251C23#58513yes—
252Q8—416no—
253C2×C4—26no—
254C2×C4—26no—
255C2×C4—26no—
256C2×C4—26no—
257C2×C4—212no—
258C2×C4—212no—
259C2×C4—26no—
260C2×C4—26no—
261C2×C4—26no—
262C2×C4—26no—
263C2×C4—212no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—26no—
270C2×C4—26no—
271C2×C4—42no—
272C2×C4—46no—
273C2×C4—43no—
274C2×C4—43no—
275C2×C4—43no—
276C2×C4—43no—
277C2×C4—46no—
278C2×C4—43no—
279C2×C4—43no—
280C2×C4—43no—
281C2×C4—43no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—43no—
285C2×C4—43no—
286C2×C4—43no—
287C2×C4—43no—
288C2×C4—46no—
289C2×C4—43no—
290C2×C4—43no—
291C2×C4—43no—
292C2×C4—43no—
293C2×C4—42no—
294C2×C4—46no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—46no—
300C2×C4—46no—
301C2×C4—46no—
302C2×C4—46no—
303C2×C4C2×#12614yes—
304C2×C4C2×#12614yes—
305C2×C4C2×#12614yes—
306C2×C4#529112yes—
307C2×C4#529112yes—
308C2×C4#529112yes—
309C2×C4#533112yes—
310C22C22×#12611yescharacteristic
311C22C2×#52913yes—
312C22C2×#52913yes—
313C22C2×#52913yes—
314C22C2×#53313yes—
315C22#345513yes—
316C22#345513yes—
317C22#347513yes—
318C22#347513yes—
319C22Q8×#4113yes—
320C22Q8×#4111yescharacteristic
321C22#348013yes—
322C22#348013yes—
323C4—24no—
324C4—46no—
325C4—412no—
326C4—46no—
327C4—46no—
328C4—46no—
329C4—44no—
330C4—82no—
331C4—86no—
332C4#240118yes—
333C4#286318yes—
334C2C2×#240111yescharacteristic
335C2#3463911yescharacteristic
336C2#3491313yes—
337C2#3491513yes—
338C2#3491613yes—
339C2#3491613yes—
340C2C2×#286311yescharacteristic
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
73. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#152102
#152082
#151982
#151892
#151872
#151862
#138552
#138462
C4×#3912
C4×#3892
C4×#3762