Finite Group Databasebeta

← #331295

Subgroups — #331295

Order 512.

773 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 154 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#29878C211yesmaximal, characteristic, complemented
2#29842C212yesmaximal, complemented
3#29807C212yesmaximal, complemented
4#29806C211yesmaximal, characteristic
5#10679C212yesmaximal, complemented
6#10671C212yesmaximal, complemented
7#10308C212yesmaximal, complemented
8#10305C212yesmaximal, complemented
9#7851C211yesmaximal, characteristic, complemented
10C2×#500C2211yescharacteristic
11C2×#493C2211yescharacteristic
12C2×#492C2212yes—
13C2×#486C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2857C2212yes—
18#2856C2212yes—
19#2087—22no—
20#2076—22no—
21#2067—24no—
22#2055—24no—
23#2054—22no—
24#2033—24no—
25#2000—24no—
26#1999—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1957C2212yes—
30#1957C2212yes—
31#1957C2212yes—
32#1956C2212yes—
33#1845C2212yes—
34#1834C2211yescharacteristic
35#1834C2212yes—
36#1834C2211yescharacteristic
37#1273C2212yes—
38#1273C2212yes—
39C22×SD16—24no—
40C22×D8—22no—
41C2×C4⋊Q8C2312yes—
42C2×#127C2312yes—
43C2×C22⋊Q8—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#121—22no—
47C22×C42C2311yescharacteristic
48#500—41no—
49#500—41no—
50#493—41no—
51#493—41no—
52#492—42no—
53#492—42no—
54#486—42no—
55#486—42no—
56#431—41no—
57#431—41no—
58#431—41no—
59#431—41no—
60#431—41no—
61#431—41no—
62C2×#107—22no—
63C2×#107—24no—
64C2×C4⋊C8D412yes—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8D412yes—
67C2×#103—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×C22⋊C8D412yes—
73C2×C22⋊C8D412yes—
74C2×C22⋊C8D412yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2312yes—
79#391—22no—
80#390—22no—
81#390—22no—
82#390—22no—
83#389—22no—
84#389—22no—
85#384C2312yes—
86#384C2312yes—
87#380—22no—
88#380—22no—
89#340—22no—
90#340—24no—
91#340—24no—
92#340—22no—
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×C8C2×D412yes—
108C22×C8C2×D414yes—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115#127—22no—
116#127—22no—
117#127—22no—
118#127—22no—
119C22⋊Q8—84no—
120#121—84no—
121#121—84no—
122#121—84no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—44no—
145#107—84no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149#103—84no—
150#102—48no—
151#102—84no—
152#102—84no—
153#102—84no—
154C22⋊C8—42no—
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—
165#95—42no—
166#95—42no—
167C24—42no—
168C24#12711yescharacteristic
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—84no—
186C2×D4—84no—
187C2×D4—84no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4#12711yescharacteristic
202C22×C4#12711yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207SD16—816no—
208D8—88no—
209C2×C8—24no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—84no—
230C22⋊C4—84no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42H(5,𝔽2)11yescharacteristic
241C42H(5,𝔽2)11yescharacteristic
242C42H(5,𝔽2)11yescharacteristic
243C422−1+411yescharacteristic
244C23—22no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23C2×#12711yescharacteristic
257Q8—84no—
258Q8—84no—
259D4—84no—
260D4—84no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—84no—
265D4—168no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—82no—
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#53411yescharacteristic
289C2×C4#53411yescharacteristic
290C2×C4#53411yescharacteristic
291C2×C4#53711yescharacteristic
292C2×C4#53711yescharacteristic
293C2×C4#53711yescharacteristic
294C2×C4#57614yes—
295C2×C4#57612yes—
296C2×C4#57714yes—
297C2×C4#57812yes—
298C8—44no—
299C8—42no—
300C8—42no—
301C8—44no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22#265611yescharacteristic
315C22#315412yes—
316C22#315511yescharacteristic
317C22#315512yes—
318C22#315611yescharacteristic
319C4—41no—
320C4—41no—
321C4—41no—
322C4—162no—
323C4—162no—
324C4—162no—
325C4#319912yes—
326C4#320012yes—
327C4#320312yes—
328C4#325912yes—
329C4#326012yes—
330C4#326212yes—
331C2—81no—
332C2—322no—
333C2#1973512yes—
334C2#1974012yes—
335C2#3025311yescharacteristic
336C2#3025411yescharacteristic
337C2#3025511yescharacteristic
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. #29878C2semidirect
2. #29842C2semidirect
3. #29807C2semidirect
5. #10679C2semidirect
6. #10671C2semidirect
7. #10308C2semidirect
8. #10305C2semidirect
9. #7851C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#298422
#298072
#298062
#106792
#106712
#103082
#103052
#78512