Finite Group Databasebeta

← #120806

Subgroups — #120806

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21363C211yesmaximal, characteristic
2#8089C211yesmaximal, characteristic
3#8014C211yesmaximal, characteristic
4#7984C211yesmaximal, characteristic
5#7952C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18#1776—22no—
19#1302—22no—
20#1294—22no—
21#1289—21no—
22#1289—21no—
23#1285—22no—
24#1285C414yescomplemented
25#1282—22no—
26C2×#117—21no—
27C22×C4⋊C4—21no—
28C22×C42—21no—
29C22×C42C2311yescharacteristic
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432C2×C412yes—
40#432C2×C412yes—
41#432Q812yes—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46C2×#107—22no—
47C2×#106—22no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8Q812yes—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8D412yes—
56C2×#97—21no—
57C2×#97—21no—
58C2×#97D412yes—
59C2×C4×C8—21no—
60#384—21no—
61#383—21no—
62#382—21no—
63#382—21no—
64#381—21no—
65#380—21no—
66#380—21no—
67#380D411yescharacteristic
68#380D411yescharacteristic
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4C2×C412yes—
75C4×C4⋊C4D412yes—
76C4×C22⋊C4—21no—
77#376—21no—
78C2×#95—21no—
79#340—22no—
80#340—22no—
81C23×C4—41no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4#4111yescharacteristic
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89#117—81no—
90#117—81no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—41no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—81no—
107C2×C42—81no—
108C2×C42C22×C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120#107—44no—
121#106—44no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138#97—22no—
139#97—22no—
140#97—41no—
141#97—41no—
142#97—41no—
143#97C4⋊C414yes—
144C4×C8—41no—
145C4×C8—41no—
146#95—22no—
147#95—22no—
148#95—21no—
149#95—21no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155C24#12111yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—81no—
167C22×C4—82no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4#11711yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12311yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12411yescharacteristic
183C22×C4#12411yescharacteristic
184C22×C4#12511yescharacteristic
185C22×C4#12511yescharacteristic
186C22×C4#12611yescharacteristic
187C22×C4C4⋊Q811yescharacteristic
188C2×C8—24no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—24no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—82no—
212C42—82no—
213C42—82no—
214C42—82no—
215C42C2×C4⋊C411yescharacteristic
216C42#11711yescharacteristic
217C42C4×D411yescharacteristic
218C42C4×Q811yescharacteristic
219C42C22⋊Q811yescharacteristic
220C42C22⋊Q811yescharacteristic
221C42C22⋊Q811yescharacteristic
222C42#12311yescharacteristic
223C42#12411yescharacteristic
224C42#12411yescharacteristic
225C42#12511yescharacteristic
226C42#12511yescharacteristic
227C42#12511yescharacteristic
228C42#12611yescharacteristic
229C42#12611yescharacteristic
230C42#12611yescharacteristic
231C23—41no—
232C23—41no—
233C23#38811yescharacteristic
234C23#39411yescharacteristic
235C23#39811yescharacteristic
236C23#47111yescharacteristic
237C23#47111yescharacteristic
238C23#47211yescharacteristic
239C23#47311yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
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—22no—
253C2×C4—22no—
254C2×C4—22no—
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—44no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4#38211yescharacteristic
279C2×C4#38511yescharacteristic
280C2×C4#38711yescharacteristic
281C2×C4#38711yescharacteristic
282C2×C4#38811yescharacteristic
283C2×C4#38911yescharacteristic
284C2×C4#39311yescharacteristic
285C2×C4#39711yescharacteristic
286C2×C4#39811yescharacteristic
287C2×C4#40011yescharacteristic
288C2×C4#40011yescharacteristic
289C2×C4#40011yescharacteristic
290C8—44no—
291C8—82no—
292C8—82no—
293C8—82no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22#144211yescharacteristic
305C22#193011yescharacteristic
306C22#193111yescharacteristic
307C22#205111yescharacteristic
308C22#205211yescharacteristic
309C22#208911yescharacteristic
310C22#209011yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—84no—
323C4—161no—
324C4—162no—
325C4—161no—
326C4#203112yes—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2#1127711yescharacteristic
334C2#1128211yescharacteristic
335C2#1129011yescharacteristic
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
24. #1285C4semidirect

Maximal subgroups

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

subgroupindex
#213632
#80892
#80142
#79842
#79522
C2×#12852
#79282