Finite Group Databasebeta

← #120859

Subgroups — #120859

Order 512.

421 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20819C211yesmaximal, characteristic
2#8109C211yesmaximal, characteristic
3#8087C211yesmaximal, characteristic
4#8074C211yesmaximal, characteristic
5#7984C211yesmaximal, characteristic
6#7954C211yesmaximal, characteristic
7#7945C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1847—21no—
16#1836—21no—
17#1809—22no—
18C4×#107—21no—
19C4×#106—21no—
20#1305—22no—
21#1302—21no—
22#1302—21no—
23#1300C414yescomplemented
24#1289—22no—
25#1285—21no—
26#1285—21no—
27#1284—22no—
28C22×M16—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#432—21no—
38#432C2×C412yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43C2×#107—22no—
44C2×#106—22no—
45#423—21no—
46#423—21no—
47C2×C4⋊C8Q812yes—
48#407—21no—
49#407D412yes—
50C2×C22⋊C8—21no—
51C4×M16—21no—
52C4×M16—21no—
53C4×M16—21no—
54C4×M16—21no—
55C4×M16C2×C412yes—
56C4×M16Q812yes—
57C2×#97—21no—
58C2×#97D411yescharacteristic
59C2×#97D411yescharacteristic
60C2×C4×C8—21no—
61#384—21no—
62#383—21no—
63#383—21no—
64#382—21no—
65#382—21no—
66#382—21no—
67#381—21no—
68#381—21no—
69#380—21no—
70#380—21no—
71#380—21no—
72#380D412yes—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4D411yescharacteristic
77C4×C4⋊C4D411yescharacteristic
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80#376—21no—
81#376C2×C412yes—
82#340—22no—
83#340—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—44no—
91C2×M16—41no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95#117—82no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C42—22no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—41no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42C2×Q811yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121#107—44no—
122#106—44no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135#97—22no—
136#97—21no—
137#97—21no—
138#97—41no—
139#97—41no—
140#97—41no—
141#97C4⋊C412yes—
142#97C4⋊C412yes—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—41no—
146C4×C8—41no—
147#95—22no—
148#95—22no—
149#95—21no—
150#95—21no—
151#95—22no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158C24#12111yescharacteristic
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4#11711yescharacteristic
168C22×C4C4×D411yescharacteristic
169C22×C4C4×D411yescharacteristic
170C22×C4#12011yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4C22⋊Q811yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12311yescharacteristic
177C22×C4#12311yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12411yescharacteristic
180C22×C4#12411yescharacteristic
181C22×C4#12511yescharacteristic
182C22×C4#12511yescharacteristic
183C22×C4#12611yescharacteristic
184C22×C4C4⋊Q811yescharacteristic
185M16—44no—
186M16—44no—
187M16—82no—
188M16—82no—
189C2×C8—24no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—82no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C22⋊C4—81no—
205C22⋊C4—81no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C42—81no—
209C42—81no—
210C42—82no—
211C42—82no—
212C42C2×C4⋊C411yescharacteristic
213C42#11711yescharacteristic
214C42C4×D411yescharacteristic
215C42C4×Q811yescharacteristic
216C42C22⋊Q811yescharacteristic
217C42C22⋊Q811yescharacteristic
218C42C22⋊Q811yescharacteristic
219C42#12311yescharacteristic
220C42#12411yescharacteristic
221C42#12411yescharacteristic
222C42#12511yescharacteristic
223C42#12511yescharacteristic
224C42#12511yescharacteristic
225C42#12611yescharacteristic
226C42#12611yescharacteristic
227C42#12611yescharacteristic
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23#38811yescharacteristic
233C23#39411yescharacteristic
234C23#39811yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4#38211yescharacteristic
268C2×C4#38511yescharacteristic
269C2×C4#38711yescharacteristic
270C2×C4#38711yescharacteristic
271C2×C4#38811yescharacteristic
272C2×C4#38911yescharacteristic
273C2×C4#39311yescharacteristic
274C2×C4#39711yescharacteristic
275C2×C4#39811yescharacteristic
276C2×C4#40011yescharacteristic
277C2×C4#40011yescharacteristic
278C2×C4#40011yescharacteristic
279C8—24no—
280C8—42no—
281C8—82no—
282C8—82no—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—81no—
293C22—81no—
294C22#144211yescharacteristic
295C22#204811yescharacteristic
296C22#204911yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—161no—
309C4—162no—
310C4—161no—
311C4—162no—
312C4#203112yes—
313C2—21no—
314C2—21no—
315C2—41no—
316C2—41no—
317C2#1126211yescharacteristic
318C2#1127411yescharacteristic
319C2#1127711yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
23. #1300C4semidirect

Maximal subgroups

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

subgroupindex
#208192
#81092
#80872
#80742
#79842
#79542
#79452