Finite Group Databasebeta

← #120361

Subgroups — #120361

Order 512.

430 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21962C211yesmaximal, characteristic
2#17807C211yesmaximal, characteristic
3#8110C211yesmaximal, characteristic
4#8094C211yesmaximal, characteristic
5#8082C211yesmaximal, characteristic
6#8070C211yesmaximal, characteristic
7#7953C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—22no—
16#1948—21no—
17#1947—21no—
18#1808—21no—
19#1807—21no—
20#1573—21no—
21#1573—21no—
22#1571C412yescomplemented
23#1570C412yescomplemented
24#1565—22no—
25#1305—21no—
26#1305—21no—
27#1303—22no—
28#1301—22no—
29#1300—22no—
30#1285—22no—
31C23×C8—21no—
32C2×#117—21no—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432D412yes—
40#432Q812yes—
41#432Q812yes—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431C2×C412yes—
50C2×#107—22no—
51C2×#106—22no—
52#423—21no—
53#423—21no—
54#423C2×C411yescharacteristic
55#423C2×C411yescharacteristic
56#423Q812yes—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2×C412yes—
61C2×C4⋊C8D412yes—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64#407—21no—
65#407—21no—
66#407—21no—
67#407—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72#384—21no—
73#383—21no—
74#382—21no—
75#382—21no—
76#381—21no—
77#380—21no—
78#380—21no—
79#380D411yescharacteristic
80#380D411yescharacteristic
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4D412yes—
84C4×C22⋊C4—21no—
85#376—21no—
86#340—22no—
87#340—22no—
88#335—22no—
89#335—22no—
90#334—22no—
91#334—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—22no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—41no—
99C2×M16—41no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—82no—
105#117—82no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—41no—
114C2×C42—22no—
115C2×C42—41no—
116C2×C42C22×C411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×Q811yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128#107—82no—
129#106—82no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—22no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8C22⋊C412yes—
144C4⋊C8C22⋊C412yes—
145C4⋊C8C4⋊C414yes—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154#97—22no—
155#97—22no—
156#97—22no—
157#97—41no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—41no—
162#95—21no—
163#95—21no—
164#95—22no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169C24#12311yescharacteristic
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4#11711yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4C4×Q811yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12411yescharacteristic
191C22×C4#12511yescharacteristic
192C22×C4#12511yescharacteristic
193C22×C4#12511yescharacteristic
194C22×C4#12611yescharacteristic
195C22×C4C4⋊Q811yescharacteristic
196M16—44no—
197M16—81no—
198M16—81no—
199M16—81no—
200M16—81no—
201M16—81no—
202M16—81no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—82no—
213C2×C8—82no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C42—82no—
226C42—82no—
227C42C2×C22⋊C411yescharacteristic
228C42C2×C4⋊C411yescharacteristic
229C42C4×D411yescharacteristic
230C42C4×Q811yescharacteristic
231C42#12111yescharacteristic
232C42#12111yescharacteristic
233C42C22⋊Q811yescharacteristic
234C42C22⋊Q811yescharacteristic
235C42C22⋊Q811yescharacteristic
236C42C22⋊Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42#12311yescharacteristic
239C42#12511yescharacteristic
240C42#12611yescharacteristic
241C42C4⋊Q811yescharacteristic
242C42C4⋊Q811yescharacteristic
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23#38711yescharacteristic
248C23#39411yescharacteristic
249C23#39811yescharacteristic
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4#38011yescharacteristic
283C2×C4#38611yescharacteristic
284C2×C4#38711yescharacteristic
285C2×C4#38911yescharacteristic
286C2×C4#38911yescharacteristic
287C2×C4#39111yescharacteristic
288C2×C4#39511yescharacteristic
289C2×C4#39611yescharacteristic
290C2×C4#39611yescharacteristic
291C2×C4#39811yescharacteristic
292C2×C4#39811yescharacteristic
293C2×C4#40011yescharacteristic
294C8—44no—
295C8—42no—
296C8—42no—
297C8—82no—
298C8—82no—
299C8—81no—
300C8—81no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—42no—
311C22#144111yescharacteristic
312C22#204811yescharacteristic
313C22#204911yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—161no—
325C4—162no—
326C4—161no—
327C4#181412yes—
328C4#190411yescharacteristic
329C4#190511yescharacteristic
330C2—21no—
331C2—21no—
332C2—41no—
333C2—41no—
334C2#1121311yescharacteristic
335C2#1124111yescharacteristic
336C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
22. #1571C4semidirect
23. #1570C4semidirect

Maximal subgroups

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

subgroupindex
#219622
#178072
#81102
#80942
#80822
#80702
#79532