Finite Group Databasebeta

← #120398

Subgroups — #120398

Order 512.

447 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#21363C211yesmaximal, characteristic
2#17821C211yesmaximal, characteristic
3#8106C211yesmaximal, characteristic
4#8098C211yesmaximal, characteristic
5#7997C211yesmaximal, characteristic
6#7976C211yesmaximal, characteristic
7#7950C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18#1776—22no—
19#1573—22no—
20#1572—22no—
21#1569C414yescomplemented
22#1305—21no—
23#1305—21no—
24#1303—22no—
25#1291—22no—
26#1288—22no—
27#1285—22no—
28C22×C4⋊C4—21no—
29C22×C42—21no—
30C22×C42C2311yescharacteristic
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432C2×C412yes—
40#432D412yes—
41#432D412yes—
42#432Q812yes—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431Q812yes—
50#431Q812yes—
51C2×#107—22no—
52C2×#106—22no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423C2×C412yes—
57C2×C4⋊C8D412yes—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62C2×C22⋊C8—21no—
63C4×M16—21no—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16C2×C412yes—
67C2×#97—21no—
68#383—21no—
69#382—21no—
70#381—21no—
71#380D411yescharacteristic
72#380D411yescharacteristic
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4D412yes—
78C4×C22⋊C4—21no—
79#376—21no—
80C2×#95—21no—
81#340—22no—
82#340—22no—
83#335—24no—
84#334—24no—
85C23×C4—41no—
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—41no—
92C2×M16—41no—
93C22×C8—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—41no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—22no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—82no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×Q811yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117#107—44no—
118#106—44no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8C22⋊C414yes—
129C22⋊C8—22no—
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—
139#97—22no—
140#97—22no—
141#97—41no—
142#97—41no—
143#97—41no—
144#97C4⋊C414yes—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—41no—
149#95—21no—
150#95—21no—
151#95—22no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24#12311yescharacteristic
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—81no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4#11711yescharacteristic
168C22×C4C4×D411yescharacteristic
169C22×C4C4×Q811yescharacteristic
170C22×C4#12011yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12411yescharacteristic
180C22×C4#12511yescharacteristic
181C22×C4#12511yescharacteristic
182C22×C4#12511yescharacteristic
183C22×C4#12611yescharacteristic
184C22×C4C4⋊Q811yescharacteristic
185M16—82no—
186M16—81no—
187M16—81no—
188M16—82no—
189M16—82no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C4⋊C4—24no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C42—82no—
205C42—82no—
206C42—82no—
207C42C2×C22⋊C411yescharacteristic
208C42C2×C4⋊C411yescharacteristic
209C42C4×D411yescharacteristic
210C42C4×Q811yescharacteristic
211C42#12111yescharacteristic
212C42#12111yescharacteristic
213C42C22⋊Q811yescharacteristic
214C42C22⋊Q811yescharacteristic
215C42C22⋊Q811yescharacteristic
216C42C22⋊Q811yescharacteristic
217C42C22⋊Q811yescharacteristic
218C42#12311yescharacteristic
219C42#12511yescharacteristic
220C42#12611yescharacteristic
221C42C4⋊Q811yescharacteristic
222C42C4⋊Q811yescharacteristic
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23#38711yescharacteristic
228C23#39411yescharacteristic
229C23#39811yescharacteristic
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
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—44no—
251C2×C4—82no—
252C2×C4—88no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4#38011yescharacteristic
261C2×C4#38611yescharacteristic
262C2×C4#38711yescharacteristic
263C2×C4#38911yescharacteristic
264C2×C4#38911yescharacteristic
265C2×C4#39111yescharacteristic
266C2×C4#39511yescharacteristic
267C2×C4#39611yescharacteristic
268C2×C4#39611yescharacteristic
269C2×C4#39811yescharacteristic
270C2×C4#39811yescharacteristic
271C2×C4#40011yescharacteristic
272C8—44no—
273C8—44no—
274C8—82no—
275C8—82no—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22#144111yescharacteristic
288C22#204811yescharacteristic
289C22#204911yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—84no—
302C4—161no—
303C4—161no—
304C4#190612yes—
305C2—21no—
306C2—21no—
307C2—41no—
308C2—41no—
309C2#1121511yescharacteristic
310C2#1124511yescharacteristic
311C2#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
21. #1569C4semidirect

Maximal subgroups

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

subgroupindex
#213632
#178212
#81062
#80982
#79972
#79762
#79502