Finite Group Databasebeta

← #99938

Subgroups — #99938

Order 512.

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

normal subgroups only

This group has 395 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#22221C211yesmaximal, characteristic
3#9873C211yesmaximal, characteristic
4#7591C211yesmaximal, characteristic
5#7591C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic
7#7014C211yesmaximal, characteristic
8C2×#487C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#1990C411yescharacteristic
12#1984—22no—
13#1956C411yescharacteristic
14#1939—21no—
15#1928—22no—
16#1916—22no—
17#1863—22no—
18#1845C411yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21C8×C4⋊C4—21no—
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25#1279—21no—
26#1279—21no—
27#1271—22no—
28#1271—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8—21no—
33#487—22no—
34#487C814yescomplemented
35C2×#107—22no—
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#391—21no—
49#389—21no—
50#389—21no—
51#389—21no—
52#389D412yes—
53#384C2×C411yescharacteristic
54#384C2×C411yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4D412yes—
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×Q8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8—41no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8C2×C812yes—
88C4⋊Q8C2×C812yes—
89C4⋊Q8M1612yes—
90C4×Q8—82no—
91C4×Q8—82no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C22⋊C412yes—
103C2×C4⋊C4C22⋊C412yes—
104C2×C42—22no—
105C2×C42—41no—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113#107—82no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—82no—
119#103—28no—
120#103—82no—
121C4×C8—22no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—41no—
129C4×C8—82no—
130C4×C8C2×C812yes—
131C4×C8Q1612yes—
132#95—22no—
133#95—22no—
134#95—41no—
135#95—41no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—41no—
142C2×Q8—81no—
143C2×Q8—81no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D411yescharacteristic
150C22×C4C4×D411yescharacteristic
151C22×C4#12011yescharacteristic
152C22×C4#12111yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4#12311yescharacteristic
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—81no—
172C2×C8—81no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—81no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—82no—
193C4⋊C4C22⋊C814yes—
194C42—21no—
195C42—21no—
196C42—41no—
197C42—41no—
198C42—41no—
199C42—41no—
200C42—42no—
201C42—42no—
202C42—82no—
203C42C22×C811yescharacteristic
204C42C2×M1611yescharacteristic
205C42#13111yescharacteristic
206C42#13111yescharacteristic
207C42C2×Q1611yescharacteristic
208C42#13511yescharacteristic
209C42#13711yescharacteristic
210C42#13711yescharacteristic
211C23#38611yescharacteristic
212Q8—44no—
213Q8—82no—
214Q8—82no—
215Q8—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4C2×C22⋊C811yescharacteristic
243C2×C4#40811yescharacteristic
244C2×C4C8×D411yescharacteristic
245C2×C4C8×D411yescharacteristic
246C2×C4#43511yescharacteristic
247C2×C4#43511yescharacteristic
248C2×C4C4×Q1611yescharacteristic
249C2×C4#44111yescharacteristic
250C2×C4#44311yescharacteristic
251C2×C4#44411yescharacteristic
252C2×C4#45111yescharacteristic
253C2×C4#45211yescharacteristic
254C2×C4#46211yescharacteristic
255C2×C4#46411yescharacteristic
256C8—42no—
257C8—42no—
258C8—82no—
259C8—81no—
260C8—81no—
261C8—82no—
262C8—161no—
263C22#186911yescharacteristic
264C22#187611yescharacteristic
265C22#187811yescharacteristic
266C22#190211yescharacteristic
267C22#190411yescharacteristic
268C22#191011yescharacteristic
269C22#191211yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—44no—
281C4—82no—
282C4—82no—
283C4—161no—
284C4C8×Q1611yescharacteristic
285C4#157911yescharacteristic
286C4#158411yescharacteristic
287C4#158711yescharacteristic
288C2#894611yescharacteristic
289C2#908711yescharacteristic
290C2#910211yescharacteristic
291C2#928111yescharacteristic
292C2#928611yescharacteristic
293C2#929011yescharacteristic
294C2#929811yescharacteristic
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
34. #487C8semidirect

Maximal subgroups

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

subgroupindex
#223072
#222212
#98732
#75912
#70182
#70142