Finite Group Databasebeta

← #99937

Subgroups — #99937

Order 512.

411 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 112 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#22307C211yesmaximal, characteristic
2#22270C211yesmaximal, characteristic
3#9873C211yesmaximal, characteristic
4#7591C211yesmaximal, characteristic
5#7590C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic
7#7014C211yesmaximal, characteristic
8C2×#494C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#1984—22no—
12#1956C411yescharacteristic
13#1956C411yescharacteristic
14#1939—21no—
15#1916—22no—
16#1878—22no—
17#1878—22no—
18#1834C411yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21C8×C4⋊C4—21no—
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25#1279—21no—
26#1279—21no—
27#1272—22no—
28#1271—22no—
29C22×Q16—22no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C4×Q8—21no—
33C2×C4×Q8D412yes—
34#494—22no—
35#494C814yescomplemented
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×#103—22no—
40C2×#103—22no—
41C2×C4×C8—21no—
42C2×C4×C8—22no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—21no—
48#391D412yes—
49#389—21no—
50#389—21no—
51#389—21no—
52#389—21no—
53#384C2×C411yescharacteristic
54#384C2×C411yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C43C2311yescharacteristic
60#340—22no—
61#336—22no—
62#336—22no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336D412yes—
73C22×Q8—21no—
74C22×Q8C22⋊C412yes—
75C2×Q16—28no—
76C2×Q16—82no—
77C2×Q16—82no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8C2×C812yes—
91C4⋊Q8C2×C812yes—
92C4⋊Q8M1612yes—
93C4×Q8—82no—
94C4×Q8—82no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C22⋊C412yes—
107C2×C42—22no—
108C2×C42—41no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—82no—
121#103—82no—
122#103—82no—
123C4×C8—22no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—41no—
131C4×C8—82no—
132C4×C8C2×C812yes—
133C4×C8Q1612yescomplemented
134#95—22no—
135#95—22no—
136#95—41no—
137#95—41no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—41no—
145C2×Q8C22⋊C814yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4C4×D411yescharacteristic
152C22×C4C4×D411yescharacteristic
153C22×C4#12011yescharacteristic
154C22×C4#12111yescharacteristic
155C22×C4#12111yescharacteristic
156C22×C4#12311yescharacteristic
157Q16—88no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8—81no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—81no—
194C4⋊C4—82no—
195C4⋊C4—81no—
196C4⋊C4—82no—
197C42—21no—
198C42—21no—
199C42—41no—
200C42—41no—
201C42—41no—
202C42—41no—
203C42—42no—
204C42—42no—
205C42—82no—
206C42C22×C811yescharacteristic
207C42C2×M1611yescharacteristic
208C42#13111yescharacteristic
209C42#13111yescharacteristic
210C42C2×Q1611yescharacteristic
211C42#13511yescharacteristic
212C42#13711yescharacteristic
213C42#13711yescharacteristic
214C23#38611yescharacteristic
215Q8—44no—
216Q8—44no—
217Q8—82no—
218Q8—82no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4C2×C22⋊C811yescharacteristic
246C2×C4#40811yescharacteristic
247C2×C4C8×D411yescharacteristic
248C2×C4C8×D411yescharacteristic
249C2×C4#43511yescharacteristic
250C2×C4#43511yescharacteristic
251C2×C4C4×Q1611yescharacteristic
252C2×C4#44111yescharacteristic
253C2×C4#44311yescharacteristic
254C2×C4#44411yescharacteristic
255C2×C4#45111yescharacteristic
256C2×C4#45211yescharacteristic
257C2×C4#46211yescharacteristic
258C2×C4#46411yescharacteristic
259C8—42no—
260C8—42no—
261C8—82no—
262C8—81no—
263C8—81no—
264C8—82no—
265C8—161no—
266C22#186911yescharacteristic
267C22#187611yescharacteristic
268C22#187811yescharacteristic
269C22#190211yescharacteristic
270C22#190511yescharacteristic
271C22#191011yescharacteristic
272C22#191111yescharacteristic
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—41no—
282C4—41no—
283C4—44no—
284C4—82no—
285C4—82no—
286C4—161no—
287C4C8×Q1611yescharacteristic
288C4#157911yescharacteristic
289C4#158411yescharacteristic
290C4#158711yescharacteristic
291C2#894611yescharacteristic
292C2#908811yescharacteristic
293C2#910111yescharacteristic
294C2#928211yescharacteristic
295C2#928511yescharacteristic
296C2#929011yescharacteristic
297C2#929711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #494C8semidirect
133. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#223072
#222702
#98732
#75912
#75902
#70182
#70142