Finite Group Databasebeta

← #121219

Subgroups — #121219

Order 512.

367 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8128C211yesmaximal, characteristic
2#8093C211yesmaximal, characteristic
3#8079C211yesmaximal, characteristic
4#8076C211yesmaximal, characteristic
5#8065C211yesmaximal, characteristic
6#7899C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2283C2211yescharacteristic
15#1309—22no—
16#1303—21no—
17#1303—21no—
18#1301—22no—
19#1301—22no—
20#1299C414yescomplemented
21#1279—21no—
22#1279—21no—
23#1274—22no—
24C22×M16—21no—
25C22×M16—21no—
26C22×C42C2311yescharacteristic
27#432—21no—
28#432—21no—
29#432—21no—
30#432—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#431D412yes—
43#423—21no—
44#423—21no—
45#423—21no—
46#423—21no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423C2×C412yes—
51#423D412yes—
52#423Q812yes—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59#407—21no—
60#407—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C4×M16—21no—
64C4×M16C2×C412yes—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67#383—21no—
68#382—21no—
69#381—21no—
70#380—21no—
71C4×C4⋊C4D411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74#376—21no—
75C23×C4#4111yescharacteristic
76C23×C4#4111yescharacteristic
77C23×C4#4111yescharacteristic
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—81no—
85C2×M16—81no—
86C2×M16—81no—
87C2×M16—81no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C22⋊C4—41no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×Q811yescharacteristic
102C2×C42#4111yescharacteristic
103C2×C42#4111yescharacteristic
104C2×C42#4111yescharacteristic
105C2×C42#4111yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133#97—22no—
134#97—41no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—41no—
139C4×C8—41no—
140C4×C8—41no—
141#95—21no—
142#95—21no—
143#95—41no—
144#95—41no—
145#95—41no—
146C24#12611yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—81no—
155C22×C4#11711yescharacteristic
156C22×C4C4×D411yescharacteristic
157C22×C4C4×Q811yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4C22⋊Q811yescharacteristic
160C22×C4#12311yescharacteristic
161C22×C4#12311yescharacteristic
162C22×C4#12311yescharacteristic
163C22×C4#12311yescharacteristic
164C22×C4#12411yescharacteristic
165C22×C4#12511yescharacteristic
166C22×C4#12611yescharacteristic
167C22×C4#12611yescharacteristic
168C22×C4#12611yescharacteristic
169C22×C4#12611yescharacteristic
170C22×C4#12611yescharacteristic
171C22×C4#12611yescharacteristic
172C22×C4#12611yescharacteristic
173M16—82no—
174M16—81no—
175M16—81no—
176M16—82no—
177M16—82no—
178M16—82no—
179M16—82no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—81no—
190C2×C8—81no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C22⋊C4—82no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42#11711yescharacteristic
199C42#11711yescharacteristic
200C42C4×D411yescharacteristic
201C42C4×D411yescharacteristic
202C42#12111yescharacteristic
203C42#12111yescharacteristic
204C42C22⋊Q811yescharacteristic
205C42C22⋊Q811yescharacteristic
206C42#12311yescharacteristic
207C42#12311yescharacteristic
208C42#12411yescharacteristic
209C42#12411yescharacteristic
210C42#12511yescharacteristic
211C42#12511yescharacteristic
212C42#12611yescharacteristic
213C42#12611yescharacteristic
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23#38711yescharacteristic
219C23#39711yescharacteristic
220C23#40111yescharacteristic
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—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
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#38811yescharacteristic
253C2×C4#38811yescharacteristic
254C2×C4#38811yescharacteristic
255C2×C4#39611yescharacteristic
256C2×C4#39711yescharacteristic
257C2×C4#39711yescharacteristic
258C2×C4#40011yescharacteristic
259C2×C4#40011yescharacteristic
260C2×C4#40011yescharacteristic
261C2×C4#47111yescharacteristic
262C2×C4#47111yescharacteristic
263C2×C4#47211yescharacteristic
264C2×C4#47311yescharacteristic
265C8—82no—
266C8—82no—
267C8—82no—
268C8—82no—
269C8—82no—
270C8—82no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—81no—
280C22—81no—
281C22#144311yescharacteristic
282C22#207811yescharacteristic
283C22#207911yescharacteristic
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—161no—
294C4—161no—
295C4#193011yescharacteristic
296C4#193111yescharacteristic
297C4#206311yescharacteristic
298C4#206411yescharacteristic
299C4#208012yes—
300C2—21no—
301C2—21no—
302C2—41no—
303C2—41no—
304C2#1130311yescharacteristic
305C2#1130511yescharacteristic
306C2#1131311yescharacteristic
subgroupG/H
centerC2×C4#388
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1443
prime radicalC24#126
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
20. #1299C4semidirect

Maximal subgroups

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

subgroupindex
#81282
#80932
#80792
#80762
#80652
#78992
#78582