Finite Group Databasebeta

← #122115

Subgroups — #122115

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35344C211yesmaximal, characteristic
2#8096C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7976C211yesmaximal, characteristic
5#7945C211yesmaximal, characteristic
6#7940C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2864—21no—
12#2862—21no—
13#2628—22no—
14#2607—22no—
15#2607—22no—
16#2605—22no—
17#2605C414yescomplemented
18#2331—22no—
19#2283C2211yescharacteristic
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1303—22no—
24#1289—22no—
25#1288—22no—
26#1284—22no—
27#1284—22no—
28#1276—21no—
29#1276—21no—
30C2×C4⋊Q8—22no—
31C2×#117—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432Q812yescomplemented
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431Q812yescomplemented
45#423—21no—
46#423—21no—
47C2×C4⋊C8Q812yescomplemented
48#407—21no—
49#407—21no—
50C2×C22⋊C8—21no—
51C4×M16—21no—
52C4×M16—21no—
53C2×C4×C8—21no—
54#400—24no—
55#399—24no—
56#395—24no—
57#393—24no—
58#391—24no—
59#391—24no—
60#391—24no—
61#391—24no—
62#389—22no—
63#389—22no—
64#387—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#383—21no—
69#383C2×C412yes—
70#383D412yes—
71#382—22no—
72#382—21no—
73#382—21no—
74#382—21no—
75#381—21no—
76#381—21no—
77#381D412yes—
78#380—21no—
79#380—21no—
80#380D412yes—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2×C412yes—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4D412yes—
91#376—21no—
92#376—21no—
93#376C2×C412yes—
94#376Q812yescomplemented
95C43D411yescharacteristic
96C43D411yescharacteristic
97C22×Q8—24no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—41no—
102C2×M16—41no—
103C22×C8—22no—
104C4⋊Q8—48no—
105#117—44no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—41no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—22no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42C22⋊C412yes—
125C2×C42C22⋊C412yes—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×Q811yescharacteristic
131C2×C42C2×Q811yescharacteristic
132C2×C42C2×Q811yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#97—22no—
151#97—41no—
152C4×C8—22no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—41no—
171#95—41no—
172#95C22⋊C414yes—
173#95C22⋊C414yes—
174C24#12011yescharacteristic
175C2×Q8—48no—
176C2×Q8—84no—
177C2×Q8—84no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4C4⋊Q811yescharacteristic
202C22×C4C4⋊Q811yescharacteristic
203C22×C4C4⋊Q811yescharacteristic
204M16—82no—
205M16—82no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—82no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C22⋊C4—44no—
218C22⋊C4—82no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—24no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42—82no—
226C42—82no—
227C42—81no—
228C42—81no—
229C42—82no—
230C42C4×Q811yescharacteristic
231C42C4×Q811yescharacteristic
232C42C4×Q811yescharacteristic
233C42C4×Q811yescharacteristic
234C42C22⋊Q811yescharacteristic
235C42C22⋊Q811yescharacteristic
236C42C22⋊Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42#12411yescharacteristic
241C42#12411yescharacteristic
242C42#12411yescharacteristic
243C42C4⋊Q811yescharacteristic
244C42C4⋊Q811yescharacteristic
245C42C4⋊Q811yescharacteristic
246C23—22no—
247C23—41no—
248C23—41no—
249C23#37911yescharacteristic
250C23#39311yescharacteristic
251C23#39511yescharacteristic
252Q8—168no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
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—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#39111yescharacteristic
287C2×C4#39111yescharacteristic
288C2×C4#39111yescharacteristic
289C2×C4#39111yescharacteristic
290C2×C4#39111yescharacteristic
291C2×C4#39111yescharacteristic
292C2×C4#39311yescharacteristic
293C2×C4#39311yescharacteristic
294C2×C4#39311yescharacteristic
295C2×C4#39511yescharacteristic
296C2×C4#39511yescharacteristic
297C2×C4#39511yescharacteristic
298C8—82no—
299C8—82no—
300C8—82no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—82no—
310C22#144611yescharacteristic
311C22#179412yes—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—44no—
324C4—82no—
325C4—84no—
326C4—84no—
327C4—162no—
328C4#203112yes—
329C2—22no—
330C2—41no—
331C2—41no—
332C2#1138711yescharacteristic
333C2#1138911yescharacteristic
334C2#1139311yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
17. #2605C4semidirect
36. #432Q8semidirect
44. #431Q8semidirect
47. C2×C4⋊C8Q8semidirect
94. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#353442
#80962
#79802
#79762
#79452
#79402
#78692