Finite Group Databasebeta

← #120339

Subgroups — #120339

Order 512.

441 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21959C211yesmaximal, characteristic
2#17807C211yesmaximal, characteristic
3#8105C211yesmaximal, characteristic
4#8094C211yesmaximal, characteristic
5#8082C211yesmaximal, characteristic
6#7961C211yesmaximal, characteristic
7#7953C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15#1948—22no—
16#1948—22no—
17#1807—22no—
18#1573—22no—
19#1571C412yescomplemented
20#1570C412yescomplemented
21#1565—22no—
22#1305—22no—
23#1303—22no—
24#1301—22no—
25#1286—22no—
26#1285—22no—
27C23×C8—21no—
28C2×#117—21no—
29C2×#117—21no—
30C22×C42C2311yescharacteristic
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432Q812yes—
38#432Q812yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431C2×C412yes—
46#431Q812yes—
47C2×#106—24no—
48#423—21no—
49#423—21no—
50#423C2×C411yescharacteristic
51#423C2×C411yescharacteristic
52#423D412yes—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8C2×C412yes—
57C2×C4⋊C8D412yes—
58C2×C4⋊C8D412yes—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63C4×M16—21no—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16—21no—
67#384—21no—
68#384—21no—
69#382—21no—
70#382—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380D412yes—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4D412yes—
79#340—24no—
80#335—24no—
81#334—24no—
82C23×C4C2×D411yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—22no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—41no—
89C2×M16—41no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—82no—
94#117—82no—
95#117—82no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—41no—
104C2×C42—22no—
105C2×C42—41no—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42C2×Q811yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118#106—84no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—22no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8C22⋊C414yes—
132C4⋊C8C4⋊C414yes—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#97—22no—
142#97—22no—
143#97—41no—
144#97—41no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—41no—
148C4×C8—41no—
149#95—22no—
150#95—22no—
151#95—41no—
152#95—41no—
153C24#12311yescharacteristic
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—81no—
161C22×C4#11711yescharacteristic
162C22×C4C4×D411yescharacteristic
163C22×C4C4×Q811yescharacteristic
164C22×C4#12011yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4C22⋊Q811yescharacteristic
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4#12411yescharacteristic
174C22×C4#12511yescharacteristic
175C22×C4#12511yescharacteristic
176C22×C4#12511yescharacteristic
177C22×C4#12611yescharacteristic
178C22×C4C4⋊Q811yescharacteristic
179M16—42no—
180M16—42no—
181M16—81no—
182M16—81no—
183M16—82no—
184M16—81no—
185M16—81no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—44no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—82no—
194C2×C8—82no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C22⋊C4—82no—
204C22⋊C4—82no—
205C42—82no—
206C42—82no—
207C42C2×C22⋊C411yescharacteristic
208C42C2×C4⋊C411yescharacteristic
209C42C4×D411yescharacteristic
210C42C4×Q811yescharacteristic
211C42#12111yescharacteristic
212C42#12111yescharacteristic
213C42C22⋊Q811yescharacteristic
214C42C22⋊Q811yescharacteristic
215C42C22⋊Q811yescharacteristic
216C42C22⋊Q811yescharacteristic
217C42C22⋊Q811yescharacteristic
218C42#12311yescharacteristic
219C42#12511yescharacteristic
220C42#12611yescharacteristic
221C42C4⋊Q811yescharacteristic
222C42C4⋊Q811yescharacteristic
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23#38711yescharacteristic
228C23#39411yescharacteristic
229C23#39811yescharacteristic
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#38011yescharacteristic
258C2×C4#38611yescharacteristic
259C2×C4#38711yescharacteristic
260C2×C4#38911yescharacteristic
261C2×C4#38911yescharacteristic
262C2×C4#39111yescharacteristic
263C2×C4#39511yescharacteristic
264C2×C4#39611yescharacteristic
265C2×C4#39611yescharacteristic
266C2×C4#39811yescharacteristic
267C2×C4#39811yescharacteristic
268C2×C4#40011yescharacteristic
269C2×C4#47112yes—
270C2×C4#47112yes—
271C8—44no—
272C8—44no—
273C8—82no—
274C8—82no—
275C8—81no—
276C8—81no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22#144111yescharacteristic
288C22#204611yescharacteristic
289C22#204611yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—162no—
300C4—162no—
301C4#181512yes—
302C4#190411yescharacteristic
303C4#190511yescharacteristic
304C4#209012yes—
305C2—21no—
306C2—21no—
307C2—41no—
308C2—41no—
309C2#1121311yescharacteristic
310C2#1123811yescharacteristic
311C2#1125111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
#219592
#178072
#81052
#80942
#80822
#79612
#79532