Finite Group Databasebeta

← #120335

Subgroups — #120335

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21963C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#8105C211yesmaximal, characteristic
4#7984C211yesmaximal, characteristic
5#7960C211yesmaximal, characteristic
6#7954C211yesmaximal, characteristic
7#7945C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—22no—
16#1948—21no—
17#1947—21no—
18#1809—22no—
19#1573—22no—
20#1571C412yescomplemented
21#1570C412yescomplemented
22#1569—22no—
23#1305—22no—
24#1289—22no—
25#1286—22no—
26#1285—21no—
27#1285—21no—
28#1284—22no—
29C22×M16—21no—
30C2×#117—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432Q812yes—
40#431—21no—
41#431—21no—
42#431—21no—
43#431C2×C412yes—
44C2×#107—22no—
45C2×#106—22no—
46#423—21no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423C2×C412yes—
51#423D412yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2×C411yescharacteristic
54C2×C4⋊C8C2×C411yescharacteristic
55C2×C4⋊C8D412yes—
56#407—21no—
57#407D412yes—
58C2×C22⋊C8—21no—
59C4×M16—21no—
60C4×M16Q812yes—
61C2×#97—21no—
62#384—21no—
63#383—21no—
64#383—21no—
65#382—21no—
66#382—21no—
67#382—21no—
68#381—21no—
69#381—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380D412yes—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4D411yescharacteristic
78C4×C4⋊C4D411yescharacteristic
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81#376—21no—
82#376Q812yes—
83#340—22no—
84#340—22no—
85#335—24no—
86#334—24no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—82no—
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—21no—
98#117—82no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C42—22no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—41no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42C2×Q811yescharacteristic
118C2×C42C2×Q811yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124#107—82no—
125#106—82no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8C22⋊C414yes—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8C4⋊C414yes—
144#97—22no—
145#97—22no—
146#97—41no—
147#97—41no—
148C4×C8—22no—
149C4×C8—22no—
150#95—21no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161C24#12111yescharacteristic
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4C2×C4⋊C411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4C4×Q811yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4C22⋊Q811yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12511yescharacteristic
187C22×C4#12611yescharacteristic
188C22×C4C4⋊Q811yescharacteristic
189M16—82no—
190M16—82no—
191M16—82no—
192M16—82no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—81no—
201C2×C8—81no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C42—81no—
217C42—81no—
218C42—82no—
219C42—82no—
220C42C2×C22⋊C411yescharacteristic
221C42#11711yescharacteristic
222C42C4×D411yescharacteristic
223C42C4×Q811yescharacteristic
224C42#12111yescharacteristic
225C42C22⋊Q811yescharacteristic
226C42C22⋊Q811yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42C22⋊Q811yescharacteristic
229C42#12311yescharacteristic
230C42#12511yescharacteristic
231C42#12511yescharacteristic
232C42#12511yescharacteristic
233C42#12611yescharacteristic
234C42C4⋊Q811yescharacteristic
235C42C4⋊Q811yescharacteristic
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23#38911yescharacteristic
241C23#39411yescharacteristic
242C23#39611yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4#38011yescharacteristic
279C2×C4#38611yescharacteristic
280C2×C4#38711yescharacteristic
281C2×C4#38711yescharacteristic
282C2×C4#38911yescharacteristic
283C2×C4#39111yescharacteristic
284C2×C4#39511yescharacteristic
285C2×C4#39611yescharacteristic
286C2×C4#39811yescharacteristic
287C2×C4#39811yescharacteristic
288C2×C4#39811yescharacteristic
289C2×C4#40011yescharacteristic
290C8—44no—
291C8—44no—
292C8—82no—
293C8—81no—
294C8—81no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—81no—
304C22—81no—
305C22#144111yescharacteristic
306C22#204811yescharacteristic
307C22#204911yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—161no—
320C4—162no—
321C4—161no—
322C4—162no—
323C4#190411yescharacteristic
324C4#190511yescharacteristic
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2#1121311yescharacteristic
330C2#1123411yescharacteristic
331C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
20. #1571C4semidirect
21. #1570C4semidirect

Maximal subgroups

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

subgroupindex
#219632
#178222
#81052
#79842
#79602
#79542
#79452