Finite Group Databasebeta

← #106665

Subgroups — #106665

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2C2×#1822C211yesmaximal, characteristic
3#20688C211yesmaximal, characteristic
4#8108C212yesmaximal
5#7925C212yesmaximal
6C22×#117C2211yescharacteristic
7#2924—24no—
8#2924—24no—
9C2×#433—22no—
10C2×#432C412yescomplemented
11C2×#431C412yescomplemented
12C2×#424—21no—
13C2×#407C2211yescharacteristic
14C2×#407C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C2×#405—21no—
17#2280C2212yes—
18#1822—21no—
19#1822—21no—
20#1822—21no—
21#1822—21no—
22#1820—22no—
23#1819C414yes—
24C2×#343C412yescomplemented
25C2×#343C412yescomplemented
26#1740—21no—
27#1740—21no—
28#1305—24no—
29#1282—24no—
30C24×C4C2311yescharacteristic
31C22×M16—21no—
32C22×M16C2×C414yes—
33C23×C8—21no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117D412yes—
40C2×#117D411yescharacteristic
41C2×#117D411yescharacteristic
42C2×#117D411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117Q812yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C412yes—
48C22×C42C2×C412yes—
49#433—216no—
50#432—24no—
51#432—24no—
52#431—24no—
53#431—24no—
54#424—28no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57#407—22no—
58#407—22no—
59#407—22no—
60#407—22no—
61#407—22no—
62#407D414yes—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2×C411yescharacteristic
69C2×C22⋊C8C2×C411yescharacteristic
70C2×C22⋊C8C2×C412yes—
71C2×C22⋊C8D414yes—
72#405—28no—
73C2×#97—22no—
74C2×C4×C8—22no—
75#381—22no—
76#380—22no—
77#380—22no—
78#379C2×C414yes—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81#343—22no—
82#343—22no—
83#343—22no—
84#343—22no—
85#342—24no—
86#342—24no—
87C25C22×C411yescharacteristic
88C23×C4—44no—
89C23×C4C4212yes—
90C23×C4C4212yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×Q811yescharacteristic
99C23×C4#4112yes—
100C23×C4#4112yes—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—44no—
105C2×M16—48no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—24no—
110C22×C8—24no—
111C22×C8—44no—
112#117—22no—
113#117—22no—
114#117—24no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117—24no—
120#117—24no—
121#117—22no—
122#117—22no—
123#117—24no—
124#117—48no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C412yes—
135C2×C4⋊C4C4⋊C412yes—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C4⋊C412yes—
144C2×C22⋊C4C4⋊C412yes—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—42no—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C4⋊C412yes—
153C2×C42C4⋊C412yes—
154C4⋊C8—28no—
155C4⋊C8—28no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—22no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163C22⋊C8—42no—
164C22⋊C8—42no—
165C22⋊C8C22⋊C412yes—
166C22⋊C8C22⋊C412yes—
167C22⋊C8C22⋊C414yes—
168#97—28no—
169C4×C8—28no—
170#95—42no—
171#95—42no—
172C24—42no—
173C24—42no—
174C24—81no—
175C24C2×C4211yescharacteristic
176C24C2×C22⋊C411yescharacteristic
177C24C2×C4⋊C411yescharacteristic
178C24C4×D412yes—
179C24C4×D412yes—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—82no—
197C22×C4#9512yes—
198C22×C4#9512yes—
199C22×C4#9512yes—
200C22×C4#9512yes—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C2×C4⋊C411yescharacteristic
206C22×C4C2×C4⋊C411yescharacteristic
207C22×C4#11711yescharacteristic
208C22×C4#11711yescharacteristic
209C22×C4C4×D412yes—
210C22×C4C4×D412yes—
211C22×C4#12012yes—
212C22×C4#12012yes—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4C22⋊Q812yes—
216C22×C4C22⋊Q812yes—
217C22×C4#12312yes—
218C22×C4#12312yes—
219M16—48no—
220M16—48no—
221M16—84no—
222C2×C8—22no—
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—48no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—84no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—84no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—84no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—84no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—42no—
267C23—41no—
268C23—42no—
269C23—88no—
270C23C2×#9511yescharacteristic
271C23C4×C22⋊C411yescharacteristic
272C23C4×C22⋊C411yescharacteristic
273C23#38512yes—
274C23#38612yes—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4#37911yescharacteristic
308C2×C4#38011yescharacteristic
309C2×C4#38011yescharacteristic
310C2×C4#38111yescharacteristic
311C2×C4#38512yes—
312C2×C4#38612yes—
313C2×C4#41014yes—
314C8—42no—
315C8—42no—
316C8—44no—
317C8—84no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—22no—
331C22—44no—
332C22—44no—
333C22—82no—
334C22—84no—
335C22—84no—
336C22C22⋊#9511yescharacteristic
337C22#174012yes—
338C22#175712yes—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—164no—
349C4#189914yes—
350C2—22no—
351C2—22no—
352C2—42no—
353C2—41no—
354C2—42no—
355C2#1073611yescharacteristic
356C2#1076411yescharacteristic
357C2#1077111yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#432C4semidirect
11. C2×#431C4semidirect
24. C2×#343C4semidirect
25. C2×#343C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C2×#18222
#206882
#81082
#79252