Finite Group Databasebeta

← #121501

Subgroups — #121501

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3#17806C211yesmaximal, characteristic
4#8070C211yesmaximal, characteristic
5#7966C211yesmaximal, characteristic
6#7928C211yesmaximal, characteristic
7#7899C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2283C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17#1849—21no—
18#1838—21no—
19#1574—21no—
20#1574—21no—
21#1574—22no—
22#1573—21no—
23#1573—21no—
24#1571—21no—
25#1570—21no—
26#1565—22no—
27#1564—22no—
28#1300—22no—
29#1287C414yescomplemented
30#1282—22no—
31#1279—22no—
32C22×M16—21no—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432C2×C412yes—
39#432C2×C412yes—
40#432Q812yes—
41#432Q812yes—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431—21no—
51#431—21no—
52#431C2×C412yes—
53#431Q812yes—
54C2×#108—24no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423D411yescharacteristic
60#423D411yescharacteristic
61#423D412yes—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68#407—21no—
69#407—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8D412yes—
73C4×M16—21no—
74C4×M16D412yes—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8D412yes—
78#383—21no—
79#382—21no—
80#381—21no—
81#380Q812yes—
82C4×C4⋊C4—21no—
83C4×C22⋊C4—21no—
84#376—21no—
85#341—24no—
86#335—22no—
87#335—22no—
88#335—24no—
89#334—22no—
90#334—22no—
91#334—24no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×Q811yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—41no—
100C2×M16—82no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106C2×C4⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C42—41no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×Q811yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42C2×Q811yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121#108—84no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—21no—
126C4⋊C8—22no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—22no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8C4⋊C414yes—
150#97—22no—
151#97—22no—
152#97—41no—
153#97C4⋊C414yes—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8C4⋊C414yes—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164C24C22⋊Q811yescharacteristic
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C2×C4⋊C411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4C4×Q811yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4#12311yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12511yescharacteristic
183C22×C4#12711yescharacteristic
184C22×C4C4⋊Q811yescharacteristic
185C22×C4C4⋊Q811yescharacteristic
186C22×C4C4⋊Q811yescharacteristic
187C22×C4C4⋊Q811yescharacteristic
188M16—44no—
189M16—42no—
190M16—44no—
191M16—42no—
192M16—82no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—44no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—82no—
205C2×C8—81no—
206C2×C8—81no—
207C4⋊C4—44no—
208C22⋊C4—44no—
209C42—81no—
210C42—81no—
211C42C2×C4⋊C411yescharacteristic
212C42C2×C4⋊C411yescharacteristic
213C42C4×D411yescharacteristic
214C42C4×D411yescharacteristic
215C42#12011yescharacteristic
216C42#12011yescharacteristic
217C42#12111yescharacteristic
218C42#12111yescharacteristic
219C42C22⋊Q811yescharacteristic
220C42C22⋊Q811yescharacteristic
221C42C22⋊Q811yescharacteristic
222C42C22⋊Q811yescharacteristic
223C42#12511yescharacteristic
224C42#12511yescharacteristic
225C42C4⋊Q811yescharacteristic
226C42C4⋊Q811yescharacteristic
227C23—22no—
228C23—41no—
229C23—41no—
230C23#38911yescharacteristic
231C23#39511yescharacteristic
232C23#39911yescharacteristic
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#38411yescharacteristic
258C2×C4#38511yescharacteristic
259C2×C4#38511yescharacteristic
260C2×C4#38511yescharacteristic
261C2×C4#38911yescharacteristic
262C2×C4#38911yescharacteristic
263C2×C4#39211yescharacteristic
264C2×C4#39511yescharacteristic
265C2×C4#39511yescharacteristic
266C2×C4#39811yescharacteristic
267C2×C4#39811yescharacteristic
268C2×C4#39811yescharacteristic
269C8—44no—
270C8—44no—
271C8—44no—
272C8—81no—
273C8—82no—
274C8—81no—
275C8—81no—
276C8—81no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—82no—
286C22#144411yescharacteristic
287C22#209812yes—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—162no—
299C4#195312yes—
300C4#200811yescharacteristic
301C4#200911yescharacteristic
302C2—22no—
303C2—41no—
304C2—41no—
305C2#1133211yescharacteristic
306C2#1133311yescharacteristic
307C2#1133611yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1444
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
29. #1287C4semidirect

Maximal subgroups

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

subgroupindex
#219942
#178092
#178062
#80702
#79662
#79282
#78992