Finite Group Databasebeta

← #121052

Subgroups — #121052

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21364C211yesmaximal, characteristic
2#8121C211yesmaximal, characteristic
3#8097C211yesmaximal, characteristic
4#8058C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6#7953C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18#1776—22no—
19#1307C414yescomplemented
20#1303—21no—
21#1303—21no—
22#1298—22no—
23#1286—22no—
24#1285—22no—
25#1276—22no—
26C22×M16—21no—
27C2×#117—21no—
28C2×#117—21no—
29C22×C42C2311yescharacteristic
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432C2×C412yes—
36#432C2×C412yes—
37#432Q812yes—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431D412yes—
46C2×#107—22no—
47C2×#106—22no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423C2×C412yes—
53#423Q811yescharacteristic
54#423Q811yescharacteristic
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⋊C8D412yes—
63C4×M16—21no—
64C4×M16—21no—
65C2×#97—21no—
66C2×C4×C8—21no—
67#384—21no—
68#382—21no—
69#380—21no—
70#380—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4D412yes—
74C4×C4⋊C4D411yescharacteristic
75C4×C4⋊C4D411yescharacteristic
76C4×C22⋊C4—21no—
77C4×C22⋊C4—21no—
78#376—21no—
79C43—21no—
80#340—22no—
81#340—22no—
82C23×C4C2×D411yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—22no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—81no—
91C2×M16—81no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95#117—42no—
96#117—42no—
97#117—81no—
98#117—81no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42C22×C411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×Q811yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123#107—44no—
124#106—44no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—22no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8C4⋊C414yes—
147#97—22no—
148#97—22no—
149#97—22no—
150#97—41no—
151C4×C8—22no—
152C4×C8—41no—
153C4×C8—41no—
154C4×C8—41no—
155#95—22no—
156#95—21no—
157#95—21no—
158#95—41no—
159#95—41no—
160C24#12411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4C2×C4⋊C411yescharacteristic
172C22×C4#11711yescharacteristic
173C22×C4#11711yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4#12311yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12411yescharacteristic
183C22×C4#12511yescharacteristic
184C22×C4#12511yescharacteristic
185C22×C4#12511yescharacteristic
186C22×C4#12511yescharacteristic
187C22×C4#12611yescharacteristic
188C22×C4#12611yescharacteristic
189M16—82no—
190M16—81no—
191M16—81no—
192M16—81no—
193M16—81no—
194M16—82no—
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—82no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C22⋊C4—44no—
216C22⋊C4—82no—
217C42—44no—
218C42—41no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—82no—
223C42—82no—
224C42C4×D411yescharacteristic
225C42C4×D411yescharacteristic
226C42C4×D411yescharacteristic
227C42C4×Q811yescharacteristic
228C42C22⋊Q811yescharacteristic
229C42C22⋊Q811yescharacteristic
230C42C22⋊Q811yescharacteristic
231C42C22⋊Q811yescharacteristic
232C42#12311yescharacteristic
233C42#12311yescharacteristic
234C42#12411yescharacteristic
235C42#12411yescharacteristic
236C42#12511yescharacteristic
237C42#12611yescharacteristic
238C42#12611yescharacteristic
239C42C4⋊Q811yescharacteristic
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23#38211yescharacteristic
245C23#39411yescharacteristic
246C23#40011yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#38511yescharacteristic
282C2×C4#38711yescharacteristic
283C2×C4#38711yescharacteristic
284C2×C4#38811yescharacteristic
285C2×C4#38811yescharacteristic
286C2×C4#38911yescharacteristic
287C2×C4#39311yescharacteristic
288C2×C4#39711yescharacteristic
289C2×C4#39811yescharacteristic
290C2×C4#39811yescharacteristic
291C2×C4#40011yescharacteristic
292C2×C4#40011yescharacteristic
293C2×C4#47111yescharacteristic
294C2×C4#47111yescharacteristic
295C2×C4#47211yescharacteristic
296C2×C4#47311yescharacteristic
297C8—44no—
298C8—82no—
299C8—82no—
300C8—82no—
301C8—81no—
302C8—81no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—81no—
312C22—81no—
313C22#144211yescharacteristic
314C22#205111yescharacteristic
315C22#205211yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—82no—
327C4—162no—
328C4#193011yescharacteristic
329C4#193111yescharacteristic
330C4#203111yescharacteristic
331C4#203111yescharacteristic
332C4#208911yescharacteristic
333C4#209011yescharacteristic
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2#1126311yescharacteristic
339C2#1128411yescharacteristic
340C2#1129011yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
19. #1307C4semidirect

Maximal subgroups

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

subgroupindex
#213642
#81212
#80972
#80582
#79572
#79532
#78692