Finite Group Databasebeta

← #120913

Subgroups — #120913

Order 512.

423 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20825C211yesmaximal, characteristic
2#8110C211yesmaximal, characteristic
3#8073C211yesmaximal, characteristic
4#7994C211yesmaximal, characteristic
5#7973C211yesmaximal, characteristic
6#7951C211yesmaximal, characteristic
7#7887C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1846—21no—
16#1835—21no—
17#1809—22no—
18#1776—22no—
19#1305—22no—
20#1300—22no—
21#1291C414yescomplemented
22#1288—22no—
23#1285—22no—
24#1278—21no—
25#1278—21no—
26C22×M16—21no—
27C22×M16—21no—
28C22×C42C2311yescharacteristic
29#432—21no—
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432C2×C412yes—
36#432C2×C412yes—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41C2×#107—22no—
42C2×#106—22no—
43#423—21no—
44#423Q811yescharacteristic
45#423Q811yescharacteristic
46C2×C4⋊C8—21no—
47#407—21no—
48#407—21no—
49C2×C22⋊C8—21no—
50C4×M16—21no—
51C4×M16—21no—
52C4×M16—21no—
53C4×M16—21no—
54C4×M16—21no—
55C4×M16—21no—
56C4×M16—21no—
57C4×M16C2×C412yes—
58C4×M16D412yes—
59C4×M16Q812yes—
60C2×#97—21no—
61C2×#97—21no—
62C2×#97—21no—
63C2×C4×C8—21no—
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66#383—21no—
67#383—21no—
68#382—21no—
69#382—21no—
70#381—21no—
71#381—21no—
72#380—21no—
73#380D412yes—
74C4×C4⋊C4—21no—
75C4×C4⋊C4D412yes—
76C4×C22⋊C4—21no—
77C4×C22⋊C4—21no—
78#376—21no—
79#376—21no—
80#340—24no—
81C23×C4C2×D411yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—44no—
92C2×M16—41no—
93C2×M16—41no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—41no—
97C22×C8—41no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—41no—
102C2×C42—22no—
103C2×C42—41no—
104C2×C42C22×C411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×Q811yescharacteristic
109C2×C42C2×Q811yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116#107—44no—
117#106—44no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132#97—22no—
133#97—22no—
134#97—22no—
135#97—22no—
136#97—22no—
137#97—41no—
138#97—41no—
139#97—41no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—22no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—41no—
146C4×C8C4⋊C412yes—
147C4×C8C4⋊C412yes—
148#95—22no—
149#95—22no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24#12311yescharacteristic
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—81no—
163C22×C4#11711yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×Q811yescharacteristic
166C22×C4#12011yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4#12411yescharacteristic
174C22×C4#12411yescharacteristic
175C22×C4#12511yescharacteristic
176C22×C4#12511yescharacteristic
177C22×C4#12511yescharacteristic
178C22×C4#12611yescharacteristic
179C22×C4#12611yescharacteristic
180C22×C4#12611yescharacteristic
181M16—24no—
182M16—44no—
183M16—44no—
184M16—42no—
185M16—44no—
186M16—81no—
187M16—81no—
188M16—81no—
189M16—81no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—44no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—82no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C42—82no—
210C42—82no—
211C42C2×C4⋊C411yescharacteristic
212C42#11711yescharacteristic
213C42C4×D411yescharacteristic
214C42C4×D411yescharacteristic
215C42C22⋊Q811yescharacteristic
216C42C22⋊Q811yescharacteristic
217C42C22⋊Q811yescharacteristic
218C42#12311yescharacteristic
219C42#12311yescharacteristic
220C42#12311yescharacteristic
221C42#12411yescharacteristic
222C42#12411yescharacteristic
223C42#12511yescharacteristic
224C42#12511yescharacteristic
225C42#12611yescharacteristic
226C42C4⋊Q811yescharacteristic
227C23—22no—
228C23—41no—
229C23—41no—
230C23#38711yescharacteristic
231C23#39411yescharacteristic
232C23#40011yescharacteristic
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
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—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4#38211yescharacteristic
261C2×C4#38511yescharacteristic
262C2×C4#38711yescharacteristic
263C2×C4#38811yescharacteristic
264C2×C4#38811yescharacteristic
265C2×C4#38911yescharacteristic
266C2×C4#39311yescharacteristic
267C2×C4#39711yescharacteristic
268C2×C4#39811yescharacteristic
269C2×C4#39811yescharacteristic
270C2×C4#40011yescharacteristic
271C2×C4#40011yescharacteristic
272C8—24no—
273C8—42no—
274C8—44no—
275C8—82no—
276C8—82no—
277C8—81no—
278C8—81no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—82no—
288C22#144211yescharacteristic
289C22#205012yes—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—162no—
302C4—162no—
303C4#203111yescharacteristic
304C4#203111yescharacteristic
305C2—22no—
306C2—41no—
307C2—41no—
308C2#1126211yescharacteristic
309C2#1128411yescharacteristic
310C2#1129411yescharacteristic
subgroupG/H
center, socleC22#1442
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. #1291C4semidirect

Maximal subgroups

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

subgroupindex
#208252
#81102
#80732
#79942
#79732
#79512
#78872