Finite Group Databasebeta

← #120376

Subgroups — #120376

Order 512.

422 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17817C211yesmaximal, characteristic
2#17810C211yesmaximal, characteristic
3#8054C211yesmaximal, characteristic
4#7990C211yesmaximal, characteristic
5#7959C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1574—22no—
16#1573—22no—
17#1573C414yescomplemented
18#1572—22no—
19#1567—21no—
20#1566—21no—
21#1565—22no—
22#1298—22no—
23#1290—22no—
24#1286—22no—
25#1286—21no—
26#1286—21no—
27#1280—22no—
28C22×M16—21no—
29C22×C4⋊C4—21no—
30C22×C42C2311yescharacteristic
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432C2×C412yes—
38#432D412yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431C2×C412yes—
48#431D412yes—
49#431D412yes—
50#431Q812yes—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423D411yescharacteristic
57#423D411yescharacteristic
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63#407—21no—
64#407—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8D412yes—
68C4×M16—21no—
69C4×M16Q812yes—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#382—21no—
74#382—21no—
75#382—21no—
76#382—21no—
77C2×#95—21no—
78C2×#95Q812yes—
79#335—24no—
80#335—24no—
81#334—24no—
82#334—24no—
83C23×C4—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—81no—
93C2×M16—81no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—41no—
97C22×C8—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×Q811yescharacteristic
108C2×C42C2×Q811yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8C22⋊C412yes—
129C4⋊C8C22⋊C412yes—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8C4⋊C414yes—
140#97—22no—
141#97—22no—
142#97—22no—
143#97—41no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—41no—
150#95—22no—
151#95—22no—
152#95—41no—
153#95—41no—
154C24C22⋊Q811yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—81no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4C2×C4⋊C411yescharacteristic
166C22×C4C4×D411yescharacteristic
167C22×C4C4×Q811yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4C22⋊Q811yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4#12311yescharacteristic
177C22×C4#12511yescharacteristic
178C22×C4#12511yescharacteristic
179C22×C4#12511yescharacteristic
180C22×C4#12611yescharacteristic
181C22×C4C4⋊Q811yescharacteristic
182C22×C4C4⋊Q811yescharacteristic
183M16—41no—
184M16—41no—
185M16—41no—
186M16—41no—
187M16—82no—
188M16—81no—
189M16—81no—
190M16—82no—
191C2×C8—24no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C4⋊C4—82no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C42C2×C22⋊C411yescharacteristic
206C42#11711yescharacteristic
207C42C4×D411yescharacteristic
208C42C4×Q811yescharacteristic
209C42#12011yescharacteristic
210C42#12111yescharacteristic
211C42C22⋊Q811yescharacteristic
212C42C22⋊Q811yescharacteristic
213C42C22⋊Q811yescharacteristic
214C42C22⋊Q811yescharacteristic
215C42#12311yescharacteristic
216C42#12311yescharacteristic
217C42#12411yescharacteristic
218C42#12511yescharacteristic
219C42#12611yescharacteristic
220C42C4⋊Q811yescharacteristic
221C23—21no—
222C23—21no—
223C23—22no—
224C23—41no—
225C23#38911yescharacteristic
226C23#39611yescharacteristic
227C23#39811yescharacteristic
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—164no—
254C2×C4#38011yescharacteristic
255C2×C4#38611yescharacteristic
256C2×C4#38711yescharacteristic
257C2×C4#38711yescharacteristic
258C2×C4#38911yescharacteristic
259C2×C4#39111yescharacteristic
260C2×C4#39411yescharacteristic
261C2×C4#39511yescharacteristic
262C2×C4#39611yescharacteristic
263C2×C4#39811yescharacteristic
264C2×C4#39811yescharacteristic
265C2×C4#40011yescharacteristic
266C2×C4#47111yescharacteristic
267C2×C4#47111yescharacteristic
268C2×C4#47211yescharacteristic
269C2×C4#47311yescharacteristic
270C8—44no—
271C8—42no—
272C8—42no—
273C8—82no—
274C8—82no—
275C8—82no—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—42no—
285C22—42no—
286C22#144111yescharacteristic
287C22#195312yes—
288C22#206311yescharacteristic
289C22#206411yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—162no—
301C4#190711yescharacteristic
302C4#190811yescharacteristic
303C4#208911yescharacteristic
304C4#209011yescharacteristic
305C2—21no—
306C2—21no—
307C2—22no—
308C2—41no—
309C2#1121611yescharacteristic
310C2#1123411yescharacteristic
311C2#1125011yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
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
17. #1573C4semidirect

Maximal subgroups

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

subgroupindex
#178172
#178102
#80542
#79902
#79592
C2×#12862
#79142