Finite Group Databasebeta

← #120362

Subgroups — #120362

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic
2#17807C211yesmaximal, characteristic
3#8109C211yesmaximal, characteristic
4#8078C211yesmaximal, characteristic
5#8061C211yesmaximal, characteristic
6#8045C211yesmaximal, characteristic
7#7961C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2285C2211yescharacteristic
15#1950—22no—
16#1950—22no—
17#1810—22no—
18#1573—22no—
19#1571C412yescomplemented
20#1570C412yescomplemented
21#1565—22no—
22#1305—22no—
23#1301—21no—
24#1301—21no—
25#1299—22no—
26#1296—22no—
27#1286—21no—
28#1286—21no—
29C22×M16—21no—
30C23×C8—21no—
31C2×#117—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432Q812yes—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431C2×C412yes—
49C2×#108—24no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423C2×C411yescharacteristic
56#423C2×C411yescharacteristic
57#423D411yescharacteristic
58#423D411yescharacteristic
59#423Q812yes—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2×C412yes—
66C2×C4⋊C8D412yes—
67C2×C4⋊C8D412yes—
68#407—21no—
69#407—21no—
70#407—21no—
71#407D412yes—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16D412yes—
76#384—21no—
77#382—21no—
78#380—21no—
79#380Q812yes—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82#341—24no—
83#335—24no—
84#334—24no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—22no—
95C2×M16—41no—
96C2×M16—81no—
97C2×M16—81no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—82no—
103#117—81no—
104#117—81no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C22⋊C4—22no—
109C2×C42—41no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
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
122#108—84no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8C22⋊C414yes—
141C4⋊C8C4⋊C412yes—
142C4⋊C8C4⋊C412yes—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#97—22no—
152#97—22no—
153#97—41no—
154#97—41no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—41no—
158C4×C8—41no—
159#95—22no—
160#95—41no—
161C24C22⋊Q811yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—81no—
170C22×C4#11711yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4C4×Q811yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12511yescharacteristic
183C22×C4#12611yescharacteristic
184C22×C4#12611yescharacteristic
185C22×C4C4⋊Q811yescharacteristic
186C22×C4C4⋊Q811yescharacteristic
187C22×C4C4⋊Q811yescharacteristic
188M16—41no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—42no—
193M16—42no—
194M16—42no—
195M16—42no—
196M16—82no—
197M16—81no—
198M16—81no—
199M16—82no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—24no—
203C2×C8—44no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—81no—
212C2×C8—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—82no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C22⋊C4—82no—
219C42—82no—
220C42C2×C22⋊C411yescharacteristic
221C42C2×C4⋊C411yescharacteristic
222C42C4×D411yescharacteristic
223C42C4×Q811yescharacteristic
224C42#12111yescharacteristic
225C42#12111yescharacteristic
226C42#12111yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42C22⋊Q811yescharacteristic
229C42C22⋊Q811yescharacteristic
230C42#12311yescharacteristic
231C42#12311yescharacteristic
232C42#12411yescharacteristic
233C42#12511yescharacteristic
234C42#12511yescharacteristic
235C42#12511yescharacteristic
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23#38711yescharacteristic
241C23#39511yescharacteristic
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—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#38011yescharacteristic
272C2×C4#38611yescharacteristic
273C2×C4#38711yescharacteristic
274C2×C4#38911yescharacteristic
275C2×C4#38911yescharacteristic
276C2×C4#39111yescharacteristic
277C2×C4#39411yescharacteristic
278C2×C4#39611yescharacteristic
279C2×C4#39811yescharacteristic
280C2×C4#39811yescharacteristic
281C2×C4#39811yescharacteristic
282C2×C4#40011yescharacteristic
283C2×C4#47111yescharacteristic
284C2×C4#47111yescharacteristic
285C2×C4#47211yescharacteristic
286C2×C4#47311yescharacteristic
287C8—42no—
288C8—42no—
289C8—44no—
290C8—82no—
291C8—82no—
292C8—82no—
293C8—81no—
294C8—81no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—42no—
305C22#144111yescharacteristic
306C22#206311yescharacteristic
307C22#206411yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—162no—
318C4#181511yescharacteristic
319C4#181611yescharacteristic
320C4#190411yescharacteristic
321C4#190511yescharacteristic
322C4#206311yescharacteristic
323C4#206411yescharacteristic
324C2—21no—
325C2—21no—
326C2—41no—
327C2—41no—
328C2#1121311yescharacteristic
329C2#1124211yescharacteristic
330C2#1125411yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
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. #1571C4semidirect
20. #1570C4semidirect

Maximal subgroups

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

subgroupindex
#219692
#178072
#81092
#80782
#80612
#80452
#79612