Finite Group Databasebeta

← #99102

Subgroups — #99102

Order 512.

1060 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13756C211yesmaximal, characteristic
2#10712C212yesmaximal
3#7025C214yesmaximal, complemented
4C22×#117C411yescharacteristic
5C22×#117C411yescharacteristic
6C23×C42C2211yescharacteristic
7C22×C22⋊C8—22no—
8C2×#336—22no—
9C2×#336C2214yes—
10C4⋊#95C412yes—
11C4⋊#95C2212yes—
12C22⋊#95—22no—
13C4×#95—22no—
14C24×C4C2×C411yescharacteristic
15C24×C4D412yes—
16C23×C8—24no—
17C2×#117—22no—
18C2×#117—24no—
19C2×#117—42no—
20C2×#117—42no—
21C2×#117—42no—
22C2×#117—42no—
23C2×#117C814yescomplemented
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—21no—
27C22×C22⋊C4—21no—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42—24no—
31C22×C42—42no—
32C22×C42—42no—
33C22×C42C2×C411yescharacteristic
34C22×C42D412yes—
35C22×C42C2311yescharacteristic
36C2×C22⋊C8—28no—
37C2×C22⋊C8—44no—
38C2×C22⋊C8—44no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43#336—28no—
44#336—44no—
45C25C22⋊C411yescharacteristic
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—24no—
53C23×C4—24no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—24no—
59C23×C4—24no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×C812yes—
65C23×C4M1612yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C22×C8—28no—
69C22×C8—44no—
70C22×C8—44no—
71C22×C8—44no—
72C22×C8—44no—
73C22×C8—44no—
74C22×C8—44no—
75#117—416no—
76#117—48no—
77#117—48no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C2×C814yes—
85C2×C4⋊C4M1614yes—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—24no—
93C2×C42—24no—
94C2×C42—24no—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—42no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—416no—
106C2×C42—42no—
107C2×C42#4114yes—
108C22⋊C8—416no—
109#95—44no—
110#95—44no—
111#95—44no—
112#95—42no—
113#95—42no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24C22⋊C812yes—
123C24#9912yes—
124C24#10012yes—
125C24C2×C22⋊C411yescharacteristic
126C22×C4—28no—
127C22×C4—28no—
128C22×C4—28no—
129C22×C4—28no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—48no—
161C22×C4—48no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C22⋊C812yes—
179C22×C4#9912yes—
180C22×C4#10112yes—
181C22×C4#10414yes—
182C22×C4#11712yes—
183C22×C4#12314yes—
184C22×C4C22×C811yescharacteristic
185C22×C4C2×M1612yes—
186C22×C4C2×M1611yescharacteristic
187C22×C4#13514yes—
188C22×C4AGL(1,ℤ/8)12yes—
189C22×C4#13712yes—
190C2×C8—48no—
191C2×C8—48no—
192C2×C8—48no—
193C2×C8—84no—
194C2×C8—84no—
195C2×C8—84no—
196C2×C8—84no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C42—44no—
204C42—48no—
205C42—44no—
206C42—48no—
207C42—48no—
208C23—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23#32314yes—
231C23#38111yescharacteristic
232C23C2×C22⋊C811yescharacteristic
233C23#40711yescharacteristic
234C23C2×#9911yescharacteristic
235C23C2×#9911yescharacteristic
236C23C2×#10011yescharacteristic
237C23C2×#10111yescharacteristic
238C23#41612yes—
239C23#41712yes—
240C23C2×#10412yes—
241C23#42112yes—
242C2×C4—216no—
243C2×C4—216no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—416no—
270C2×C4—416no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4#32414yes—
290C2×C4#43112yes—
291C2×C4#43212yes—
292C2×C4#48214yes—
293C2×C4#48514yes—
294C8—88no—
295C22—28no—
296C22—24no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22C2×#32311yescharacteristic
307C22C2×#32411yescharacteristic
308C22#146211yescharacteristic
309C22#146311yescharacteristic
310C22#147612yes—
311C22#147712yes—
312C22#148812yes—
313C22#148912yes—
314C22#150112yes—
315C22#150412yes—
316C22#151312yes—
317C22#151612yes—
318C22#152611yescharacteristic
319C22#153012yes—
320C22#153711yescharacteristic
321C22#154411yescharacteristic
322C22#155112yes—
323C22#156111yescharacteristic
324C22#181711yescharacteristic
325C22#181911yescharacteristic
326C22#182111yescharacteristic
327C22#182512yes—
328C22#182612yes—
329C4—24no—
330C4—24no—
331C4—24no—
332C4—22no—
333C4—22no—
334C4—44no—
335C4—84no—
336C4—84no—
337C2—44no—
338C2#756411yescharacteristic
339C2#757812yes—
340C2#759811yescharacteristic
341C2#761212yes—
342C2#763211yescharacteristic
343C2#764611yescharacteristic
344C2#869811yescharacteristic
345C2#870812yes—
346C2#871512yes—
347C2#872211yescharacteristic
348C2#872911yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7025C2semidirect
23. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
#137562
#107122
#70252