Finite Group Databasebeta

← #100111

Subgroups — #100111

Order 512.

428 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 100 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#23269C211yesmaximal, characteristic
2#21655C211yesmaximal, characteristic
3#7601C211yesmaximal, characteristic
4#7590C211yesmaximal, characteristic, complemented
5#7588C211yesmaximal, characteristic, complemented
6#7030C211yesmaximal, characteristic, complemented
7#7030C211yesmaximal, characteristic, complemented
8C2×#489C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#1993C411yescharacteristic
12#1984—21no—
13#1959C411yescharacteristic
14#1934—21no—
15#1933—21no—
16#1923—21no—
17#1923—21no—
18#1869—21no—
19#1836C411yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1763—21no—
23#1762—21no—
24C8×C22⋊C4—21no—
25C8×C22⋊C4—21no—
26#1744C2211yescharacteristic
27#1744C2211yescharacteristic
28C4×#97C2211yescharacteristic
29#1272—21no—
30#1272—21no—
31#1272—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C23×C8—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×#124C2×C411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41#489—22no—
42#489C814yescomplemented
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×#97—21no—
54C2×#97C2×C411yescharacteristic
55C2×#97C2×C411yescharacteristic
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#391—21no—
61#391D411yescharacteristic
62#390D411yescharacteristic
63#389—21no—
64#389—21no—
65#388—21no—
66#388—21no—
67#384C2×C411yescharacteristic
68#382C2×C411yescharacteristic
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C43C2311yescharacteristic
74#340—21no—
75#340—21no—
76#340—21no—
77#340—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#336D411yescharacteristic
87#336D411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8—21no—
91C22×Q8C22⋊C411yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C23×C4—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—81no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8C2×C812yescomplemented
109#124—21no—
110#124—21no—
111#124C2×C812yes—
112#124M1612yes—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—81no—
116C4×Q8—81no—
117C4×D4—41no—
118C4×D4—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C42—21no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142C4⋊C8—81no—
143C4⋊C8—81no—
144C4⋊C8—81no—
145#103—24no—
146#103—42no—
147#102—24no—
148#102—42no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—81no—
152#97—21no—
153#97—21no—
154#97—21no—
155#97—41no—
156#97—41no—
157#97C2×C812yescomplemented
158C4×C8—42no—
159C4×C8—42no—
160C4×C8—42no—
161C4×C8—81no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—41no—
167#95—41no—
168C24—21no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8C22⋊C812yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4C22⋊C812yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12311yescharacteristic
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—84no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—81no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—82no—
258C42C22×C811yescharacteristic
259C42C2×M1611yescharacteristic
260C42#13111yescharacteristic
261C42#13111yescharacteristic
262C23—22no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#38611yescharacteristic
270Q8—42no—
271Q8—81no—
272Q8—82no—
273Q8—82no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×C22⋊C811yescharacteristic
311C2×C4#40811yescharacteristic
312C2×C4C8×D411yescharacteristic
313C2×C4C8×D411yescharacteristic
314C2×C4#43511yescharacteristic
315C2×C4#43511yescharacteristic
316C2×C4#45511yescharacteristic
317C2×C4#45511yescharacteristic
318C2×C4#47111yescharacteristic
319C2×C4#47111yescharacteristic
320C2×C4#47211yescharacteristic
321C2×C4#47311yescharacteristic
322C8—42no—
323C8—42no—
324C8—42no—
325C8—42no—
326C8—81no—
327C8—81no—
328C8—161no—
329C8—161no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22#186911yescharacteristic
338C22#188411yescharacteristic
339C22#188511yescharacteristic
340C22#190611yescharacteristic
341C22#190611yescharacteristic
342C22#190711yescharacteristic
343C22#190811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—42no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—161no—
362C2—82no—
363C2#895711yescharacteristic
364C2#909511yescharacteristic
365C2#909611yescharacteristic
366C2#943111yescharacteristic
367C2#943211yescharacteristic
368C2#943311yescharacteristic
369C2#943911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
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
4. #7590C2semidirect
5. #7588C2semidirect
6. #7030C2semidirect
7. #7030C2semidirect
42. #489C8semidirect
108. C4⋊Q8C2×C8semidirect
157. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#232692
#216552
#76012
#75902
#75882
#70302