Finite Group Databasebeta

← #304748

Subgroups — #304748

Order 512.

904 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 178 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
1D4×#125C211yesmaximal, characteristic
2#33195C211yesmaximal, characteristic
3#30228C211yesmaximal, characteristic, complemented
4#28755C211yesmaximal, characteristic, complemented
5#10356C214yesmaximal, complemented
6#9904C211yesmaximal, characteristic, complemented
7#9872C211yesmaximal, characteristic, complemented
8#8981C214yesmaximal
9#8174C211yesmaximal, characteristic
10C22×#125—21no—
11C2×#491—21no—
12C2×#474C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2851—21no—
16#2552C2212yes—
17#2506—22no—
18#2473—24no—
19#2442—22no—
20D4×C4⋊C4—22no—
21D4×C4⋊C4C2212yes—
22C4×#125C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#2099—24no—
25#2043—22no—
26#1990C2214yes—
27#1988C2212yes—
28#1981C2214yes—
29#1960—21no—
30#1947—22no—
31#1842C2211yescharacteristic
32#1802—22no—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×D4⋊C8C2212yes—
36#1296C2214yes—
37#1279C2214yes—
38#1279C2212yes—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2312yes—
42C2×#125—24no—
43C2×#125—22no—
44C2×#125—21no—
45C2×#125—44no—
46C2×#125—44no—
47C2×#125D411yescharacteristic
48C2×C4×D4—22no—
49C2×C4×D4D412yes—
50C2×C4×D4C2311yescharacteristic
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C42—21no—
54#491—44no—
55#474—28no—
56C2×#107—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—24no—
61C2×#103—24no—
62C2×#102—22no—
63C2×#97—21no—
64C2×C4×C8C2312yes—
65#400—24no—
66#398—24no—
67#398—24no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389D412yes—
73#389D414yes—
74#387—24no—
75#387—22no—
76#385—24no—
77#385—24no—
78#385—24no—
79#384—22no—
80#384—22no—
81#384—22no—
82#384C2312yes—
83#382—21no—
84#382D411yescharacteristic
85#380—24no—
86#380—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4C2312yes—
90C4×C4⋊C4C2311yescharacteristic
91C4×C22⋊C4—21no—
92C4×C22⋊C4—22no—
93C43C2311yescharacteristic
94#340—24no—
95#340—28no—
96#336—22no—
97#336—21no—
98D4⋊C8—24no—
99C22×Q8—22no—
100C22×D4C2×D411yescharacteristic
101C23×C4—21no—
102C23×C4—24no—
103C23×C4—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—82no—
112#125—22no—
113#125—44no—
114#125—416no—
115#125—44no—
116#125—42no—
117#125—42no—
118#125D814yes—
119C4×D4—24no—
120C4×D4—22no—
121C4×D4—42no—
122C4×D4—44no—
123C4×D4—44no—
124C4×D4#4114yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—416no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D414yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42C2×D412yes—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2411yescharacteristic
159#107—44no—
160C4⋊C8—22no—
161C4⋊C8—24no—
162C4⋊C8—22no—
163C4⋊C8—44no—
164#103—48no—
165#103—84no—
166#102—44no—
167#97—42no—
168C4×C8—22no—
169C4×C8—22no—
170#95—24no—
171#95—24no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—24no—
176#95—21no—
177#95—22no—
178C24—21no—
179C2×Q8—44no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—44no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4#12412yes—
186C2×D4#13714yes—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—48no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—82no—
202C22×C4—84no—
203C22×C4#12012yes—
204C22×C4#12012yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C2×C8—44no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—44no—
214C2×C8—42no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—28no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—44no—
222C4⋊C4—48no—
223C4⋊C4—48no—
224C4⋊C4—48no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—48no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4#12412yes—
238C4⋊C4C2×D814yes—
239C22⋊C4—24no—
240C22⋊C4—44no—
241C22⋊C4—48no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—42no—
246C42—41no—
247C42—41no—
248C42—42no—
249C42—42no—
250C42—44no—
251C42C2×D812yes—
252C42AGL(1,ℤ/8)12yes—
253C42C2×#4112yes—
254C422−1+412yes—
255C23—22no—
256C23—24no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23C2×#12011yescharacteristic
261Q8—84no—
262Q8—84no—
263D4—22no—
264D4—24no—
265D4#49114yescomplemented
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—88no—
290C2×C4—816no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4#44714yes—
295C2×C4C2×#12411yescharacteristic
296C2×C4#53611yescharacteristic
297C2×C4#54812yes—
298C2×C4#54812yes—
299C2×C4C22×D811yescharacteristic
300C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
301C2×C4C2×#13712yes—
302C2×C4#57512yes—
303C2×C4#57712yes—
304C8—82no—
305C8—82no—
306C8—82no—
307C22—24no—
308C22—42no—
309C22—42no—
310C22—44no—
311C22—44no—
312C22#243611yescharacteristic
313C22C2×#44711yescharacteristic
314C22#300712yes—
315C22#301011yescharacteristic
316C22#302212yes—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—44no—
321C4—42no—
322C4—82no—
323C4—84no—
324C4—82no—
325C4—162no—
326C4C2×#49111yescharacteristic
327C4#314211yescharacteristic
328C4#323612yes—
329C4#335812yes—
330C4#337312yes—
331C2—44no—
332C2#1879812yes—
333C2#1884512yes—
334C2#2591211yescharacteristic
335C2#2592711yescharacteristic
336C2#2593311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30228C2semidirect
4. #28755C2semidirect
5. #10356C2semidirect
6. #9904C2semidirect
7. #9872C2semidirect
265. D4#491semidirect

Maximal subgroups

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

subgroupindex
D4×#1252
#331952
#302282
#287552
#103562
#99042
#98722
#89812
#81742