Finite Group Databasebeta

← #120367

Subgroups — #120367

Order 512.

419 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17822C211yesmaximal, characteristic
2#17817C211yesmaximal, characteristic
3#8129C211yesmaximal, characteristic
4#8042C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6#7953C211yesmaximal, characteristic
7#7863C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2285C2211yescharacteristic
15#1574—22no—
16#1573—22no—
17#1573—22no—
18#1571—21no—
19#1570—21no—
20#1569—22no—
21#1567C412yescomplemented
22#1566C412yescomplemented
23#1309—22no—
24#1296—21no—
25#1296—21no—
26#1286—22no—
27#1285—22no—
28#1275—22no—
29C22×M16—21no—
30C22×M16—21no—
31C2×#117—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432D412yes—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431D412yes—
47#431D412yes—
48#431Q812yes—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423C2×C412yes—
57#423C2×C411yescharacteristic
58#423C2×C411yescharacteristic
59#423D412yes—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67#407—21no—
68#407—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C4×M16—21no—
72C4×M16Q812yes—
73C2×#97—21no—
74C2×C4×C8C2×C412yes—
75#384—21no—
76#382—21no—
77#380—21no—
78#380—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4Q812yes—
81#335—24no—
82#335—24no—
83#334—24no—
84#334—24no—
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—81no—
96C2×M16—81no—
97C2×M16—82no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102C22×C8—41no—
103#117—81no—
104#117—81no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C42—22no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42C2×Q811yescharacteristic
118C2×C42C2×Q811yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C4⋊C8—22no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C4⋊C8—21no—
127C4⋊C8—22no—
128C4⋊C8—21no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8C22⋊C412yes—
141C4⋊C8C22⋊C412yes—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#97—22no—
151#97—22no—
152#97—41no—
153#97—41no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8C4⋊C414yes—
158#95—22no—
159#95—41no—
160C24#12611yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—81no—
169C22×C4#11711yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4C4×Q811yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12411yescharacteristic
183C22×C4#12511yescharacteristic
184C22×C4#12511yescharacteristic
185C22×C4#12511yescharacteristic
186C22×C4#12611yescharacteristic
187M16—81no—
188M16—81no—
189M16—82no—
190M16—81no—
191M16—81no—
192M16—82no—
193M16—82no—
194M16—82no—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—82no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C22⋊C4—82no—
216C42—82no—
217C42C2×C22⋊C411yescharacteristic
218C42C2×C4⋊C411yescharacteristic
219C42C4×D411yescharacteristic
220C42C4×Q811yescharacteristic
221C42#12011yescharacteristic
222C42#12111yescharacteristic
223C42#12111yescharacteristic
224C42#12111yescharacteristic
225C42C22⋊Q811yescharacteristic
226C42C22⋊Q811yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42#12311yescharacteristic
229C42#12511yescharacteristic
230C42C4⋊Q811yescharacteristic
231C42C4⋊Q811yescharacteristic
232C42C4⋊Q811yescharacteristic
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23#38711yescharacteristic
238C23#39611yescharacteristic
239C23#40011yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4#38011yescharacteristic
269C2×C4#38611yescharacteristic
270C2×C4#38711yescharacteristic
271C2×C4#38911yescharacteristic
272C2×C4#38911yescharacteristic
273C2×C4#39111yescharacteristic
274C2×C4#39411yescharacteristic
275C2×C4#39511yescharacteristic
276C2×C4#39611yescharacteristic
277C2×C4#39811yescharacteristic
278C2×C4#39811yescharacteristic
279C2×C4#39811yescharacteristic
280C2×C4#47111yescharacteristic
281C2×C4#47111yescharacteristic
282C2×C4#47211yescharacteristic
283C2×C4#47311yescharacteristic
284C8—44no—
285C8—42no—
286C8—42no—
287C8—82no—
288C8—81no—
289C8—82no—
290C8—81no—
291C8—82no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—81no—
301C22—81no—
302C22#144111yescharacteristic
303C22#206311yescharacteristic
304C22#206411yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—162no—
315C4#190211yescharacteristic
316C4#190311yescharacteristic
317C4#195111yescharacteristic
318C4#195211yescharacteristic
319C4#205111yescharacteristic
320C4#205211yescharacteristic
321C2—21no—
322C2—21no—
323C2—41no—
324C2—41no—
325C2#1121411yescharacteristic
326C2#1124611yescharacteristic
327C2#1124911yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1567C4semidirect
22. #1566C4semidirect

Maximal subgroups

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

subgroupindex
#178222
#178172
#81292
#80422
#79572
#79532
#78632