Finite Group Databasebeta

← #331201

Subgroups — #331201

Order 512.

804 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 162 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#33042C212yesmaximal, complemented
2#29878C211yesmaximal, characteristic, complemented
3#29807C212yesmaximal, complemented
4#29806C211yesmaximal, characteristic
5#10682C212yesmaximal, complemented
6#10678C212yesmaximal, complemented
7#10308C212yesmaximal, complemented
8#10305C212yesmaximal, complemented
9#7849C211yesmaximal, characteristic, complemented
10C2×#500C2211yescharacteristic
11C2×#493C2211yescharacteristic
12C2×#492C2212yes—
13C2×#486C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857C2212yes—
18#2856C2212yes—
19#2087—22no—
20#2076—22no—
21#2055—24no—
22#2054—22no—
23#2000—24no—
24#1999—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1957C2212yes—
28#1957C2212yes—
29#1957C2212yes—
30#1956C2212yes—
31#1934—24no—
32#1893—24no—
33#1834C2212yes—
34#1834C2211yescharacteristic
35#1834C2212yes—
36#1834C2211yescharacteristic
37#1273C2212yes—
38#1273C2212yes—
39C22×SD16—24no—
40C22×D8—22no—
41C23×C8D412yes—
42C2×C4⋊Q8C2312yes—
43C2×#127C2312yes—
44C2×C22⋊Q8—22no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—22no—
48C22×C42C2311yescharacteristic
49#500—41no—
50#500—41no—
51#493—41no—
52#493—41no—
53#492—42no—
54#492—42no—
55#486—42no—
56#486—42no—
57#431—41no—
58#431—41no—
59#431—41no—
60#431—41no—
61C2×#107—22no—
62C2×C4⋊C8D412yes—
63C2×C4⋊C8D412yes—
64C2×#103—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×C22⋊C8D412yes—
70C2×C22⋊C8D412yes—
71C2×C4×C8—41no—
72C2×C4×C8—41no—
73C2×C4×C8D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#390—22no—
80#390—22no—
81#390—22no—
82#389—22no—
83#389—22no—
84#384C2312yes—
85#384C2312yes—
86#380—22no—
87#380—22no—
88#340—22no—
89#340—24no—
90#340—22no—
91#340—24no—
92#340—24no—
93#340—24no—
94C22×Q8—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—42no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×SD16—48no—
103C2×SD16—84no—
104C2×SD16—84no—
105C2×D8—44no—
106C2×D8—84no—
107C22×C8—24no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D414yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118#127—22no—
119#127—22no—
120#127—22no—
121#127—22no—
122C22⋊Q8—84no—
123#121—48no—
124#121—84no—
125#121—84no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2411yescharacteristic
147#107—44no—
148C4⋊C8—42no—
149C4⋊C8—42no—
150#103—84no—
151#102—48no—
152#102—48no—
153#102—84no—
154#102—84no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—42no—
166#95—42no—
167#95—42no—
168C24—42no—
169C24#12711yescharacteristic
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—84no—
187C2×D4—84no—
188C2×D4—84no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4#12711yescharacteristic
202C22×C4#12711yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4#13514yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208SD16—816no—
209D8—88no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—48no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42H(5,𝔽2)11yescharacteristic
243C42H(5,𝔽2)11yescharacteristic
244C42H(5,𝔽2)11yescharacteristic
245C422−1+411yescharacteristic
246C23—21no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23#49512yes—
258C23C2×#12711yescharacteristic
259Q8—84no—
260Q8—84no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—84no—
265D4—84no—
266D4—84no—
267D4—168no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—82no—
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#49512yes—
290C2×C4#53411yescharacteristic
291C2×C4#53411yescharacteristic
292C2×C4#53411yescharacteristic
293C2×C4#53711yescharacteristic
294C2×C4#53711yescharacteristic
295C2×C4#53711yescharacteristic
296C2×C4C2×#13512yes—
297C2×C4#57512yes—
298C2×C4#57614yes—
299C2×C4#57714yes—
300C8—44no—
301C8—44no—
302C8—42no—
303C8—42no—
304C22—22no—
305C22—24no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22—162no—
315C22—162no—
316C22#265611yescharacteristic
317C22C2×#49511yescharacteristic
318C22#315011yescharacteristic
319C22#315412yes—
320C22#315512yes—
321C4—22no—
322C4—41no—
323C4—41no—
324C4—162no—
325C4—162no—
326C4—162no—
327C4#319112yes—
328C4#319912yes—
329C4#320012yes—
330C4#325012yes—
331C4#325912yes—
332C4#326012yes—
333C2—42no—
334C2—322no—
335C2#1965812yes—
336C2#1970812yes—
337C2#3024111yescharacteristic
338C2#3025311yescharacteristic
339C2#3025411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#29806C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33042C2semidirect
2. #29878C2semidirect
3. #29807C2semidirect
5. #10682C2semidirect
6. #10678C2semidirect
7. #10308C2semidirect
8. #10305C2semidirect
9. #7849C2semidirect

Maximal subgroups

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

subgroupindex
#330422
#298782
#298072
#298062
#106822
#106782
#103082
#103052
#78492