Finite Group Databasebeta

← #241314

Subgroups — #241314

Order 512.

854 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 218 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#37623C211yesmaximal, characteristic
2#21987C212yesmaximal
3C2×#1845C211yesmaximal, characteristic
4#10579C214yesmaximal, complemented
5#10576C212yesmaximal, complemented
6#10573C212yesmaximal, complemented
7#8040C211yesmaximal, characteristic
8#7898C212yesmaximal
9C2×#432C2212yes—
10C22×#107—22no—
11C22×C4×C8C2211yescharacteristic
12#2843—22no—
13#2660—22no—
14#2647—22no—
15#2627—22no—
16#2571C2211yescharacteristic
17#2552C2212yes—
18#2551C2211yescharacteristic
19#2442—22no—
20#2440—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2283C2212yes—
24#1948C2214yes—
25#1947C2214yes—
26#1853—22no—
27#1852—22no—
28#1847—22no—
29#1845—22no—
30#1845C2212yes—
31#1845C2212yes—
32#1836—22no—
33#1296C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C23×C8—21no—
38C2×C4⋊Q8—22no—
39C2×#125—22no—
40C2×C22⋊Q8—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4D412yes—
45C22×C42C2311yescharacteristic
46#432—24no—
47#432Q818yescomplemented
48C2×#107—24no—
49C2×#107—24no—
50C2×#107—24no—
51C2×#107—48no—
52C2×#106—24no—
53C2×C4⋊C8C2314yes—
54C2×C22⋊C8—22no—
55C2×#97—22no—
56C2×C4×C8—21no—
57C2×C4×C8—22no—
58C2×C4×C8Q814yescomplemented
59C2×C4×C8C2312yes—
60#400—24no—
61#400—22no—
62#400—22no—
63#398—24no—
64#398—24no—
65#398—24no—
66#398—24no—
67#395—24no—
68#395—22no—
69#395—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#385—24no—
75#384—22no—
76#384—22no—
77#384—22no—
78#384—21no—
79#384D412yes—
80#384C2312yes—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#383—22no—
84#382—22no—
85#382D412yes—
86#382D412yes—
87#381—22no—
88#380—22no—
89#380D412yes—
90#380D412yes—
91C4×C4⋊C4C2312yes—
92C4×C4⋊C4C2312yes—
93C4×C22⋊C4—22no—
94#376—22no—
95#340—24no—
96#340—24no—
97C22×Q8—22no—
98C23×C4—22no—
99C23×C4—41no—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C22×C8—24no—
103C22×C8—24no—
104C22×C8—24no—
105C22×C8—22no—
106C22×C8—24no—
107C4⋊Q8—84no—
108#125—84no—
109C22⋊Q8—84no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—84no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×Q812yes—
140C2×C42C2×Q814yes—
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#107—48no—
144#107—48no—
145#107—84no—
146#106—84no—
147C4⋊C8—24no—
148C4⋊C8C2×Q818yes—
149C22⋊C8—28no—
150#97—28no—
151C4×C8—24no—
152C4×C8—22no—
153C4×C8C2×Q814yes—
154#95—22no—
155#95—22no—
156#95—24no—
157#95—24no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166C24#12011yescharacteristic
167C2×Q8—82no—
168C2×Q8—84no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4—82no—
184C22×C4#12012yes—
185C22×C4#12011yescharacteristic
186C22×C4C22⋊Q814yes—
187C22×C4C22⋊Q812yes—
188C22×C4C4⋊Q812yes—
189C22×C4C4⋊Q814yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D412yes—
192C2×C8—24no—
193C2×C8—216no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—28no—
197C2×C8—44no—
198C2×C8—44no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C42—44no—
222C42—44no—
223C42C22⋊Q812yes—
224C42C22⋊Q814yes—
225C42C4⋊Q812yes—
226C42C4⋊Q814yes—
227C42C22×Q811yescharacteristic
228C42C22×Q812yes—
229C42C2×#4111yescharacteristic
230C23—22no—
231C23—22no—
232C23#39512yes—
233C23#45312yes—
234C23#45612yes—
235C23C2×#12011yescharacteristic
236Q8—164no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—28no—
246C2×C4—28no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—84no—
264C2×C4—82no—
265C2×C4—164no—
266C2×C4—164no—
267C2×C4#39514yes—
268C2×C4#39512yes—
269C2×C4C2×C22⋊Q811yescharacteristic
270C2×C4C2×C22⋊Q812yes—
271C2×C4C2×C4⋊Q811yescharacteristic
272C2×C4C2×C4⋊Q812yes—
273C2×C4#57614yes—
274C2×C4#57814yes—
275C8—28no—
276C8—48no—
277C22—24no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22#206014yes—
282C22C2×#39511yescharacteristic
283C22#300812yes—
284C22#301112yes—
285C22C2×#45311yescharacteristic
286C22C2×#45611yescharacteristic
287C4—24no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—82no—
292C4—82no—
293C4—82no—
294C4—82no—
295C4—82no—
296C4—82no—
297C4—162no—
298C4#308314yes—
299C4#316414yes—
300C2—24no—
301C2#1868211yescharacteristic
302C2#1868511yescharacteristic
303C2#1869912yes—
304C2#2355512yes—
305C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
4. #10579C2semidirect
5. #10576C2semidirect
6. #10573C2semidirect
47. #432Q8semidirect
58. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#376232
#219872
C2×#18452
#105792
#105762
#105732
#80402
#78982