Finite Group Databasebeta

← #238149

Subgroups — #238149

Order 512.

930 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 202 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#21840C212yesmaximal
3C2×#1956C211yesmaximal, characteristic
4#10355C214yesmaximal, complemented
5#10352C212yesmaximal, complemented
6#10349C212yesmaximal, complemented
7#8040C211yesmaximal, characteristic
8#7898C212yesmaximal
9C22×C4⋊Q8C2211yescharacteristic
10C2×#432C2212yes—
11C22×#103—22no—
12C22×C4×C8C2211yescharacteristic
13#2843—22no—
14#2660—22no—
15#2647—22no—
16#2627—22no—
17#2571C2211yescharacteristic
18#2552C2212yes—
19#2551C2211yescharacteristic
20#2442—22no—
21#2440—22no—
22C2×#384C2211yescharacteristic
23#2283C2212yes—
24#1993—22no—
25#1993—22no—
26#1987C2214yes—
27#1987C2214yes—
28#1956—22no—
29#1956C2212yes—
30#1956C2212yes—
31#1823—22no—
32#1823—22no—
33#1296C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C23×Q8—21no—
38C23×C8—21no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8Q814yescomplemented
45C2×C4⋊Q8C2312yes—
46C2×#125—22no—
47C2×C22⋊Q8—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D412yes—
51C22×C42C2311yescharacteristic
52#432—24no—
53#432—24no—
54C2×C4⋊C8C2314yes—
55C2×#103—24no—
56C2×#103—24no—
57C2×#103—48no—
58C2×C22⋊C8—22no—
59C2×#97—22no—
60C2×C4×C8—24no—
61C2×C4×C8—21no—
62C2×C4×C8C2312yes—
63#400—24no—
64#400—22no—
65#400—22no—
66#398—24no—
67#398—24no—
68#398—24no—
69#398—24no—
70#395—24no—
71#395—22no—
72#395—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#385—24no—
78#384—22no—
79#384—22no—
80#384D412yes—
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—
97#340—24no—
98#340—24no—
99C22×Q8—22no—
100C22×Q8—24no—
101C22×Q8—22no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—41no—
105C22×Q8—42no—
106C23×C4—22no—
107C23×C4—41no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C22×C8—24no—
111C22×C8—22no—
112C22×C8—24no—
113C22×C8—42no—
114C22×C8—42no—
115C4⋊Q8—28no—
116C4⋊Q8—24no—
117C4⋊Q8—24no—
118C4⋊Q8—22no—
119C4⋊Q8—42no—
120C4⋊Q8—44no—
121C4⋊Q8—84no—
122C4⋊Q8C2×Q814yes—
123#125—84no—
124C22⋊Q8—84no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—84no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42C2×D412yes—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×Q812yes—
153C2×C42#4114yes—
154C2×C42#4112yes—
155C2×C42C2411yescharacteristic
156C4⋊C8—24no—
157C4⋊C8—24no—
158#103—48no—
159#103—48no—
160C22⋊C8—44no—
161#97—44no—
162C4×C8—24no—
163C4×C8—42no—
164#95—22no—
165#95—22no—
166#95—24no—
167#95—24no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176C24#12011yescharacteristic
177C2×Q8—24no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—416no—
181C2×Q8—42no—
182C2×Q8—44no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×Q8—88no—
186C2×Q8—82no—
187C2×Q8—84no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—84no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4#12012yes—
205C22×C4#12011yescharacteristic
206C22×C4C22⋊Q814yes—
207C22×C4C22⋊Q812yes—
208C22×C4#12414yes—
209C22×C4#12412yes—
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D412yes—
212C2×C8—24no—
213C2×C8—42no—
214C2×C8—48no—
215C2×C8—42no—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—44no—
219C4⋊C4—28no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C42—44no—
240C42—44no—
241C42C22⋊Q812yes—
242C42C22⋊Q814yes—
243C42#12412yes—
244C42#12414yes—
245C42C22×Q811yescharacteristic
246C42C2×#4112yes—
247C42C2×#4111yescharacteristic
248C23—22no—
249C23—22no—
250C23#39312yes—
251C23#45312yes—
252C23#45412yes—
253C23C2×#12011yescharacteristic
254Q8—48no—
255Q8—84no—
256Q8—88no—
257Q8—164no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—88no—
288C2×C4—164no—
289C2×C4#39314yes—
290C2×C4#39312yes—
291C2×C4C2×C22⋊Q811yescharacteristic
292C2×C4C2×C22⋊Q812yes—
293C2×C4C2×#12411yescharacteristic
294C2×C4C2×#12412yes—
295C2×C4#57714yes—
296C2×C4#57814yes—
297C8—44no—
298C8—84no—
299C22—24no—
300C22—24no—
301C22—22no—
302C22—22no—
303C22#202714yes—
304C22C2×#39311yescharacteristic
305C22#300912yes—
306C22#301112yes—
307C22C2×#45311yescharacteristic
308C22C2×#45411yescharacteristic
309C4—24no—
310C4—21no—
311C4—21no—
312C4—22no—
313C4—82no—
314C4—82no—
315C4—84no—
316C4—82no—
317C4—82no—
318C4—82no—
319C4—82no—
320C4#308214yes—
321C4#313714yes—
322C2—24no—
323C2#1868311yescharacteristic
324C2#1868511yescharacteristic
325C2#1869512yes—
326C2#2345712yes—
327C2C2×#202711yescharacteristic
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. #10355C2semidirect
5. #10352C2semidirect
6. #10349C2semidirect
44. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#376232
#218402
C2×#19562
#103552
#103522
#103492
#80402
#78982