Finite Group Databasebeta

← #120363

Subgroups — #120363

Order 512.

430 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17817C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3#8129C211yesmaximal, characteristic
4#8095C211yesmaximal, characteristic
5#8094C211yesmaximal, characteristic
6#8041C211yesmaximal, characteristic
7#8004C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2285C2211yescharacteristic
15#1574—22no—
16#1574—22no—
17#1573—22no—
18#1571C412yescomplemented
19#1570C412yescomplemented
20#1567—21no—
21#1566—21no—
22#1565—22no—
23#1309—21no—
24#1309—21no—
25#1303—22no—
26#1303—21no—
27#1303—21no—
28#1296—22no—
29#1292—21no—
30#1292—21no—
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C2×#117—21no—
35C22×C42C2311yescharacteristic
36#432—21no—
37#432—21no—
38#432—21no—
39#432Q812yes—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431C2×C412yes—
46#431D412yes—
47#431D412yes—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423C2×C412yes—
59#423D411yescharacteristic
60#423D411yescharacteristic
61#423D411yescharacteristic
62#423D411yescharacteristic
63#423Q812yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2×C411yescharacteristic
70C2×C4⋊C8C2×C411yescharacteristic
71C2×C4⋊C8D412yes—
72#407—21no—
73#407—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C2×C4×C8—21no—
78#384—21no—
79#382Q812yes—
80#380—21no—
81#380—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84#335—24no—
85#335—24no—
86#334—24no—
87#334—24no—
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—81no—
102C2×M16—81no—
103C2×M16—82no—
104C2×M16—82no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—41no—
109C22×C8—41no—
110#117—81no—
111#117—81no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C42—41no—
117C2×C42C22×C411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42C2×Q811yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128C2×C42#4111yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—22no—
133C4⋊C8—21no—
134C4⋊C8—22no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8C22⋊C412yes—
150C4⋊C8C22⋊C412yes—
151C4⋊C8C4⋊C412yes—
152C4⋊C8C4⋊C412yes—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159#97—22no—
160#97—41no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—41no—
165#95—22no—
166#95—22no—
167C24#12611yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4#11711yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×Q811yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12411yescharacteristic
191C22×C4#12511yescharacteristic
192C22×C4#12511yescharacteristic
193C22×C4#12511yescharacteristic
194C22×C4#12611yescharacteristic
195M16—41no—
196M16—41no—
197M16—41no—
198M16—41no—
199M16—82no—
200M16—82no—
201M16—82no—
202M16—81no—
203M16—81no—
204M16—81no—
205M16—81no—
206M16—82no—
207M16—82no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—82no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C22⋊C4—82no—
228C42—81no—
229C42—81no—
230C42C2×C22⋊C411yescharacteristic
231C42C2×C4⋊C411yescharacteristic
232C42C4×D411yescharacteristic
233C42C4×Q811yescharacteristic
234C42#12011yescharacteristic
235C42#12111yescharacteristic
236C42#12111yescharacteristic
237C42#12111yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42C22⋊Q811yescharacteristic
241C42#12311yescharacteristic
242C42#12511yescharacteristic
243C42C4⋊Q811yescharacteristic
244C42C4⋊Q811yescharacteristic
245C42C4⋊Q811yescharacteristic
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23#38711yescharacteristic
251C23#39611yescharacteristic
252C23#40011yescharacteristic
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—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4#38011yescharacteristic
285C2×C4#38611yescharacteristic
286C2×C4#38711yescharacteristic
287C2×C4#38911yescharacteristic
288C2×C4#38911yescharacteristic
289C2×C4#39111yescharacteristic
290C2×C4#39411yescharacteristic
291C2×C4#39511yescharacteristic
292C2×C4#39611yescharacteristic
293C2×C4#39811yescharacteristic
294C2×C4#39811yescharacteristic
295C2×C4#39811yescharacteristic
296C2×C4#47111yescharacteristic
297C2×C4#47111yescharacteristic
298C2×C4#47211yescharacteristic
299C2×C4#47311yescharacteristic
300C8—42no—
301C8—42no—
302C8—42no—
303C8—42no—
304C8—82no—
305C8—81no—
306C8—81no—
307C8—82no—
308C8—82no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—81no—
318C22—81no—
319C22#144111yescharacteristic
320C22#206311yescharacteristic
321C22#206411yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—162no—
331C4#181511yescharacteristic
332C4#181611yescharacteristic
333C4#190211yescharacteristic
334C4#190311yescharacteristic
335C4#204811yescharacteristic
336C4#204911yescharacteristic
337C4#206311yescharacteristic
338C4#206411yescharacteristic
339C2—21no—
340C2—21no—
341C2—41no—
342C2—41no—
343C2#1121411yescharacteristic
344C2#1124211yescharacteristic
345C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#126
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
18. #1571C4semidirect
19. #1570C4semidirect

Maximal subgroups

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

subgroupindex
#178172
#178092
#81292
#80952
#80942
#80412
#80042