Finite Group Databasebeta

← #120555

Subgroups — #120555

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21963C211yesmaximal, characteristic
3#8121C211yesmaximal, characteristic
4#8094C211yesmaximal, characteristic
5#7977C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7951C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1950—22no—
16#1949—22no—
17#1948—21no—
18#1947—21no—
19#1855—22no—
20#1849C412yescomplemented
21#1838C412yescomplemented
22#1809—22no—
23#1307—22no—
24#1303—22no—
25#1288—22no—
26#1285—22no—
27#1285—21no—
28#1285—21no—
29C22×M16D412yes—
30C2×#117—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432D412yes—
42#432Q812yes—
43#432Q812yes—
44#431—21no—
45#431—21no—
46#431—21no—
47#431Q812yes—
48C2×#108—24no—
49C2×#107—22no—
50C2×#106—22no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423C2×C411yescharacteristic
55#423C2×C411yescharacteristic
56#423C2×C412yes—
57#423D412yes—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62#407—21no—
63#407—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C4×M16—21no—
67C4×M16—21no—
68C2×#97—21no—
69C2×#97C2×C412yes—
70#384—21no—
71#383—21no—
72#382—21no—
73#382—21no—
74#381—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D411yescharacteristic
80C4×C4⋊C4D411yescharacteristic
81C4×C4⋊C4D412yes—
82C4×C22⋊C4—21no—
83#376—21no—
84#341—24no—
85#340—22no—
86#340—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—41no—
94C2×M16—82no—
95C2×M16C22⋊C414yes—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—41no—
99#117—82no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C22×C411yescharacteristic
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
121C2×C42#4111yescharacteristic
122#108—84no—
123#107—82no—
124#106—82no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147#97—22no—
148#97—22no—
149#97—41no—
150#97—41no—
151#97—41no—
152#97C4⋊C414yes—
153C4×C8—22no—
154C4×C8—41no—
155#95—21no—
156#95—21no—
157#95—22no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162C24#12411yescharacteristic
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×Q811yescharacteristic
173C22×C4C4×Q811yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12511yescharacteristic
185C22×C4#12511yescharacteristic
186C22×C4#12611yescharacteristic
187C22×C4#12611yescharacteristic
188C22×C4C4⋊Q811yescharacteristic
189M16—44no—
190M16—42no—
191M16—42no—
192M16—82no—
193M16—81no—
194M16—81no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—82no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C42—81no—
217C42—81no—
218C42—82no—
219C42C2×C4⋊C411yescharacteristic
220C42#11711yescharacteristic
221C42C4×D411yescharacteristic
222C42C4×D411yescharacteristic
223C42#12111yescharacteristic
224C42#12111yescharacteristic
225C42#12111yescharacteristic
226C42C22⋊Q811yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42C22⋊Q811yescharacteristic
229C42C22⋊Q811yescharacteristic
230C42C22⋊Q811yescharacteristic
231C42#12511yescharacteristic
232C42#12511yescharacteristic
233C42C4⋊Q811yescharacteristic
234C42C4⋊Q811yescharacteristic
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#39111yescharacteristic
240C23#39411yescharacteristic
241C23#40011yescharacteristic
242C2×C4—21no—
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—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4#38011yescharacteristic
275C2×C4#38611yescharacteristic
276C2×C4#38711yescharacteristic
277C2×C4#38711yescharacteristic
278C2×C4#38911yescharacteristic
279C2×C4#38911yescharacteristic
280C2×C4#39511yescharacteristic
281C2×C4#39611yescharacteristic
282C2×C4#39611yescharacteristic
283C2×C4#39811yescharacteristic
284C2×C4#39811yescharacteristic
285C2×C4#39811yescharacteristic
286C8—44no—
287C8—44no—
288C8—82no—
289C8—82no—
290C8—81no—
291C8—81no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—81no—
301C22—81no—
302C22#144111yescharacteristic
303C22#204811yescharacteristic
304C22#204911yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—161no—
317C4—162no—
318C4—161no—
319C4#191111yescharacteristic
320C4#191211yescharacteristic
321C2—21no—
322C2—21no—
323C2—41no—
324C2—41no—
325C2#1122411yescharacteristic
326C2#1125111yescharacteristic
327C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
20. #1849C4semidirect
21. #1838C4semidirect

Maximal subgroups

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

subgroupindex
#219942
#219632
#81212
#80942
#79772
#79522
#79512