Finite Group Databasebeta

← #237839

Subgroups — #237839

Order 512.

1308 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 234 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#35181C211yesmaximal, characteristic, complemented
2#21803C212yesmaximal
3#14154C211yesmaximal, characteristic
4#10344C218yesmaximal, complemented
5#8014C212yesmaximal, complemented
6#7908C211yesmaximal, characteristic, complemented
7C22×#127C2211yescharacteristic
8C2×#431C2212yes—
9#2851—22no—
10#2646C2214yes—
11#2571—22no—
12#2570—24no—
13#2552—22no—
14#2317—22no—
15C2×#390—22no—
16C2×#390—22no—
17#2285C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1982—24no—
21#1957—22no—
22#1957—22no—
23#1786C2218yes—
24#1438C2214yes—
25#1294C2218yes—
26#1280C2214yes—
27C23×D4—21no—
28C23×D4—21no—
29C23×D4C2312yes—
30C2×#127—22no—
31C2×#127—22no—
32C2×#127—24no—
33C2×#127—22no—
34C2×#127Q814yes—
35C2×#125—22no—
36C2×#117—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D412yes—
39C22×C22⋊C4—28no—
40C22×C42—21no—
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43#431—24no—
44#431—24no—
45C2×C4⋊C8—22no—
46C2×#102—28no—
47C2×#102—28no—
48C2×C22⋊C8C2314yes—
49C2×C4×C8—22no—
50#400—28no—
51#398—28no—
52#390—48no—
53#390—48no—
54#389—24no—
55#389—24no—
56#389—24no—
57#387—28no—
58#385—28no—
59#384—24no—
60#384D412yes—
61#384D412yes—
62#382—24no—
63#382—22no—
64#382—22no—
65#380—22no—
66#380C2314yes—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4D414yes—
72C2×#95C2312yes—
73C2×#95C2312yes—
74C25—22no—
75C22×D4—24no—
76C22×D4—24no—
77C22×D4—24no—
78C22×D4—24no—
79C22×D4—22no—
80C22×D4—44no—
81C22×D4—44no—
82C22×D4—41no—
83C22×D4—42no—
84C22×D4—41no—
85C22×D4—42no—
86C22×D4—42no—
87C22×D4—44no—
88C22×D4—44no—
89C22×D4—84no—
90C22×D4C2×Q814yes—
91C23×C4—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D414yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—24no—
97#127—28no—
98#127—28no—
99#127—44no—
100#127—44no—
101#125—84no—
102#117—44no—
103#117—44no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×D414yes—
118C2×C4⋊C4C2×D414yes—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—416no—
121C2×C22⋊C4—416no—
122C2×C22⋊C4—88no—
123C2×C42—24no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—44no—
127C2×C42—42no—
128C2×C42—44no—
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×Q812yes—
132C2×C42#4114yes—
133C2×C42#4112yes—
134C4⋊C8—44no—
135#102—88no—
136#102—88no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C4×C8—44no—
140#95—28no—
141#95—28no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146C24—24no—
147C24—44no—
148C24—48no—
149C24—44no—
150C24—42no—
151C24—42no—
152C24—44no—
153C24—44no—
154C24C22×D411yescharacteristic
155C2×D4—28no—
156C2×D4—48no—
157C2×D4—48no—
158C2×D4—416no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—48no—
162C2×D4—44no—
163C2×D4—416no—
164C2×D4—82no—
165C2×D4—82no—
166C2×D4—88no—
167C2×D4—84no—
168C2×D4—88no—
169C2×D4—84no—
170C2×D4—84no—
171C2×D4—84no—
172C2×D4—82no—
173C2×D4—82no—
174C2×D4—84no—
175C2×D4—84no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4#12011yescharacteristic
192C22×C4#12012yes—
193C22×C4#12011yescharacteristic
194C22×C4C22⋊Q812yes—
195C22×C4C22⋊Q812yes—
196C22×C4#12412yes—
197C22×C4#12412yes—
198C22×C4#13514yes—
199C22×C4AGL(1,ℤ/8)14yes—
200C22×C4C22×D412yes—
201C22×C4C22×Q811yescharacteristic
202C22×C4C2×#4112yes—
203C22×C4C2×#4111yescharacteristic
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—88no—
217C4⋊C4—88no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—832no—
221C42—48no—
222C42—44no—
223C42—82no—
224C42—84no—
225C42C22⋊Q814yes—
226C42C22⋊Q814yes—
227C42#12414yes—
228C42#12414yes—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—48no—
233C23—48no—
234C23—48no—
235C23—44no—
236C23—88no—
237C23—84no—
238C23—84no—
239C23—88no—
240C23—88no—
241C23—88no—
242C23—82no—
243C23—82no—
244C23—84no—
245C23—82no—
246C23—82no—
247C23—84no—
248C23C2×#12011yescharacteristic
249C23C2×C22⋊Q811yescharacteristic
250C23C2×#12411yescharacteristic
251C23C2×#13512yes—
252C23C2×AGL(1,ℤ/8)12yes—
253D4—416no—
254D4—88no—
255D4—88no—
256D4—816no—
257D4—88no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—82no—
282C2×C4—84no—
283C2×C4#39312yes—
284C2×C4#39314yes—
285C2×C4#39312yes—
286C2×C4#44914yes—
287C2×C4#44914yes—
288C2×C4#45312yes—
289C2×C4#45412yes—
290C2×C4#47818yes—
291C2×C4#48814yes—
292C2×C4#49014yes—
293C2×C4C2×C22⋊Q812yes—
294C2×C4C2×#12412yes—
295C8—84no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—816no—
305C22—84no—
306C22—84no—
307C22—84no—
308C22—164no—
309C22—164no—
310C22—164no—
311C22—164no—
312C22C2×#39311yescharacteristic
313C22C2×#44912yes—
314C22C2×#45311yescharacteristic
315C22C2×#45411yescharacteristic
316C22#302712yes—
317C22C2×#47812yes—
318C22C2×#48811yescharacteristic
319C22C2×#49011yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—84no—
324C4—84no—
325C4—84no—
326C4—84no—
327C4#202518yes—
328C4#202714yes—
329C2—21no—
330C2—22no—
331C2—21no—
332C2—164no—
333C2#1839112yes—
334C2#1840512yes—
335C2C2×#202511yescharacteristic
336C2C2×#202711yescharacteristic
337C2#2346911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
4. #10344C2semidirect
5. #8014C2semidirect
6. #7908C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#218032
#141542
#103442
#80142
#79082