Finite Group Databasebeta

← #100048

Subgroups — #100048

Order 512.

379 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 98 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#22373C211yesmaximal, characteristic
2#22250C211yesmaximal, characteristic
3#9673C211yesmaximal, characteristic
4#7604C211yesmaximal, characteristic
5#7585C211yesmaximal, characteristic
6#7023C211yesmaximal, characteristic
7#7023C211yesmaximal, characteristic
8C2×#501C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#125C2211yescharacteristic
11#1993C411yescharacteristic
12#1992C411yescharacteristic
13#1939—21no—
14#1921—21no—
15#1919—21no—
16#1919—21no—
17#1918—21no—
18#1916—21no—
19#1916—21no—
20#1916—21no—
21#1916—21no—
22#1839C2211yescharacteristic
23#1839C2211yescharacteristic
24#1836C411yescharacteristic
25C8×C4⋊C4—21no—
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1745C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1271—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×#125C2×C411yescharacteristic
37C2×C4×Q8—21no—
38#501—22no—
39#501C814yescomplemented
40C2×#107—21no—
41C2×#106—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×#97C2×C411yescharacteristic
47C2×#97C2×C411yescharacteristic
48C2×C4×C8—21no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#391—21no—
54#389—21no—
55#389—21no—
56#389D411yescharacteristic
57#389D411yescharacteristic
58#387—21no—
59#387—21no—
60#384C2×C411yescharacteristic
61#382C2×C411yescharacteristic
62C4×C4⋊C4—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4D411yescharacteristic
67C4×C4⋊C4D411yescharacteristic
68C43C2311yescharacteristic
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90Q8⋊C8—42no—
91Q8⋊C8—42no—
92C22×Q8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—41no—
100C22×C8—41no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—21no—
104C4⋊Q8C2×C812yes—
105#125—21no—
106#125—21no—
107#125C2×C812yes—
108#125M1612yes—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42C22×C411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137#107—24no—
138#106—24no—
139C4⋊C8—22no—
140C4⋊C8—41no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143C4⋊C8—81no—
144#97—21no—
145#97—21no—
146#97—21no—
147#97C2×C812yes—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8—42no—
151C4×C8—81no—
152C4×C8—81no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—41no—
160#95—41no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4#12011yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12311yescharacteristic
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4C22⋊C812yes—
226C4⋊C4C22⋊C812yes—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—81no—
240C42—81no—
241C42C22×C811yescharacteristic
242C42C2×M1611yescharacteristic
243C42#13111yescharacteristic
244C42#13111yescharacteristic
245C23#38611yescharacteristic
246Q8—42no—
247Q8—42no—
248Q8—81no—
249Q8—81no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4C2×C22⋊C811yescharacteristic
285C2×C4#40811yescharacteristic
286C2×C4C8×D411yescharacteristic
287C2×C4C8×D411yescharacteristic
288C2×C4#43511yescharacteristic
289C2×C4#43511yescharacteristic
290C2×C4#45311yescharacteristic
291C2×C4#45411yescharacteristic
292C2×C4#47111yescharacteristic
293C2×C4#47311yescharacteristic
294C8—42no—
295C8—42no—
296C8—81no—
297C8—81no—
298C8—81no—
299C8—81no—
300C8—82no—
301C8—161no—
302C22#186911yescharacteristic
303C22#188111yescharacteristic
304C22#188311yescharacteristic
305C22#190611yescharacteristic
306C22#190811yescharacteristic
307C22#190911yescharacteristic
308C22#191211yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—41no—
321C4—41no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—82no—
326C4—161no—
327C2#895411yescharacteristic
328C2#909611yescharacteristic
329C2#910011yescharacteristic
330C2#936911yescharacteristic
331C2#937411yescharacteristic
332C2#937711yescharacteristic
333C2#938211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
39. #501C8semidirect

Maximal subgroups

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

subgroupindex
#223732
#222502
#96732
#76042
#75852
#70232