Finite Group Databasebeta

← #121194

Subgroups — #121194

Order 512.

393 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8097C211yesmaximal, characteristic
2#8065C211yesmaximal, characteristic
3#7958C211yesmaximal, characteristic
4#7940C211yesmaximal, characteristic
5#7924C211yesmaximal, characteristic
6#7899C211yesmaximal, characteristic
7#7886C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1303—22no—
16#1299—22no—
17#1286C412yescomplemented
18#1286C412yescomplemented
19#1284—22no—
20#1282—22no—
21#1279—22no—
22#1278—22no—
23C22×M16—21no—
24C2×#117—21no—
25C22×C42C2311yescharacteristic
26#432—21no—
27#432—21no—
28#432—21no—
29#432Q812yes—
30#431—21no—
31#431—21no—
32#431—21no—
33#431—21no—
34#431—21no—
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431C2×C412yes—
42#423—21no—
43#423—21no—
44#423—21no—
45#423—21no—
46#423C2×C412yes—
47#423D412yes—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52#407—21no—
53#407—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C4×M16—21no—
57C4×M16—21no—
58C2×C4×C8—21no—
59C2×C4×C8D412yes—
60#383—21no—
61#382C2×C411yescharacteristic
62#382C2×C411yescharacteristic
63#381—21no—
64#380—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4D412yes—
71#376—21no—
72#376—21no—
73C43—21no—
74C23×C4C2×D411yescharacteristic
75C23×C4#4111yescharacteristic
76C23×C4#4111yescharacteristic
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—82no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85#117—24no—
86#117—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—41no—
92C2×C42—22no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42—41no—
96C2×C42C22×C411yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×Q811yescharacteristic
100C2×C42#4111yescharacteristic
101C2×C42#4111yescharacteristic
102C2×C42#4111yescharacteristic
103C2×C42#4111yescharacteristic
104C2×C42#4111yescharacteristic
105C2×C42#4111yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128#97—22no—
129#97—41no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8—41no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—22no—
141#95—41no—
142#95—41no—
143#95—41no—
144C24#12311yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—81no—
153C22×C4#11711yescharacteristic
154C22×C4C4×D411yescharacteristic
155C22×C4C4×D411yescharacteristic
156C22×C4#12111yescharacteristic
157C22×C4#12111yescharacteristic
158C22×C4C22⋊Q811yescharacteristic
159C22×C4#12311yescharacteristic
160C22×C4#12311yescharacteristic
161C22×C4#12411yescharacteristic
162C22×C4#12411yescharacteristic
163C22×C4#12411yescharacteristic
164C22×C4#12511yescharacteristic
165C22×C4#12511yescharacteristic
166C22×C4#12611yescharacteristic
167C22×C4#12611yescharacteristic
168C22×C4#12611yescharacteristic
169C22×C4#12611yescharacteristic
170C22×C4#12611yescharacteristic
171M16—42no—
172M16—42no—
173M16—82no—
174M16—82no—
175M16—82no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C22⋊C4—82no—
190C42—44no—
191C42—81no—
192C42—81no—
193C42—81no—
194C42—81no—
195C42—82no—
196C42#11711yescharacteristic
197C42#11711yescharacteristic
198C42C4×D411yescharacteristic
199C42C4×Q811yescharacteristic
200C42#12111yescharacteristic
201C42C22⋊Q811yescharacteristic
202C42C22⋊Q811yescharacteristic
203C42#12311yescharacteristic
204C42#12311yescharacteristic
205C42#12311yescharacteristic
206C42#12511yescharacteristic
207C42#12611yescharacteristic
208C42#12611yescharacteristic
209C42#12611yescharacteristic
210C42#12611yescharacteristic
211C42#12611yescharacteristic
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216C23#38811yescharacteristic
217C23#39711yescharacteristic
218C23#40011yescharacteristic
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4#38311yescharacteristic
250C2×C4#38711yescharacteristic
251C2×C4#38711yescharacteristic
252C2×C4#38711yescharacteristic
253C2×C4#38811yescharacteristic
254C2×C4#38811yescharacteristic
255C2×C4#39611yescharacteristic
256C2×C4#39711yescharacteristic
257C2×C4#39711yescharacteristic
258C2×C4#40011yescharacteristic
259C2×C4#40011yescharacteristic
260C2×C4#40111yescharacteristic
261C2×C4#44312yes—
262C2×C4#44412yes—
263C8—82no—
264C8—82no—
265C8—82no—
266C8—82no—
267C8—82no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—81no—
277C22—81no—
278C22#144311yescharacteristic
279C22#193711yescharacteristic
280C22#193811yescharacteristic
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—84no—
291C4—161no—
292C4—161no—
293C4#192212yes—
294C4#192912yes—
295C4#208111yescharacteristic
296C4#208211yescharacteristic
297C2—21no—
298C2—21no—
299C2—41no—
300C2—41no—
301C2#1130111yescharacteristic
302C2#1130711yescharacteristic
303C2#1131711yescharacteristic
subgroupG/H
centerC2×C4#397
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1443
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1286C4semidirect
18. #1286C4semidirect

Maximal subgroups

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

subgroupindex
#80972
#80652
#79582
#79402
#79242
#78992
#78862