Finite Group Databasebeta

← #99934

Subgroups — #99934

Order 512.

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

normal subgroups only

This group has 387 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#22248C211yesmaximal, characteristic
3#9873C211yesmaximal, characteristic
4#7591C211yesmaximal, characteristic
5#7590C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic
7#7014C211yesmaximal, characteristic
8C2×#500C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#1990C411yescharacteristic
12#1990C411yescharacteristic
13#1984—22no—
14#1939—21no—
15#1921—22no—
16#1921—22no—
17#1916—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—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8—21no—
33#500—22no—
34#500C814yescomplemented
35C2×#107—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×C4×C8—21no—
40C2×C4×C8—22no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#391—21no—
46#391—21no—
47#389—21no—
48#389—21no—
49#389—21no—
50#389D412yes—
51#384C2×C411yescharacteristic
52#384C2×C411yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4D412yes—
57C43C2311yescharacteristic
58#340—22no—
59#340—22no—
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—28no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—82no—
119C4×C8—22no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—41no—
127C4×C8—82no—
128C4×C8C2×C812yes—
129C4×C8Q1612yes—
130#95—22no—
131#95—22no—
132#95—41no—
133#95—41no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—81no—
139C2×Q8—81no—
140C2×Q8—81no—
141C2×Q8—81no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C4×D411yescharacteristic
148C22×C4C4×D411yescharacteristic
149C22×C4#12011yescharacteristic
150C22×C4#12111yescharacteristic
151C22×C4#12111yescharacteristic
152C22×C4#12311yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—42no—
169C2×C8—81no—
170C2×C8—81no—
171C2×C8—81no—
172C2×C8—81no—
173C2×C8—81no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—81no—
188C4⋊C4—82no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4C22⋊C814yes—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42—42no—
199C42—42no—
200C42—82no—
201C42C22×C811yescharacteristic
202C42C2×M1611yescharacteristic
203C42#13111yescharacteristic
204C42#13111yescharacteristic
205C42C2×Q1611yescharacteristic
206C42#13511yescharacteristic
207C42#13711yescharacteristic
208C42#13711yescharacteristic
209C23#38611yescharacteristic
210Q8—82no—
211Q8—82no—
212Q8—82no—
213Q8—82no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4C2×C22⋊C811yescharacteristic
241C2×C4#40811yescharacteristic
242C2×C4C8×D411yescharacteristic
243C2×C4C8×D411yescharacteristic
244C2×C4#43511yescharacteristic
245C2×C4#43511yescharacteristic
246C2×C4C4×Q1611yescharacteristic
247C2×C4#44111yescharacteristic
248C2×C4#44311yescharacteristic
249C2×C4#44411yescharacteristic
250C2×C4#45111yescharacteristic
251C2×C4#45211yescharacteristic
252C2×C4#46211yescharacteristic
253C2×C4#46411yescharacteristic
254C8—42no—
255C8—42no—
256C8—82no—
257C8—81no—
258C8—81no—
259C8—82no—
260C8—161no—
261C22#186911yescharacteristic
262C22#187611yescharacteristic
263C22#187811yescharacteristic
264C22#190311yescharacteristic
265C22#190411yescharacteristic
266C22#190911yescharacteristic
267C22#191211yescharacteristic
268C4—21no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—41no—
277C4—41no—
278C4—44no—
279C4—82no—
280C4—82no—
281C4—161no—
282C4C8×Q1611yescharacteristic
283C4#157911yescharacteristic
284C4#158411yescharacteristic
285C4#158711yescharacteristic
286C2#894611yescharacteristic
287C2#908911yescharacteristic
288C2#910011yescharacteristic
289C2#927911yescharacteristic
290C2#928411yescharacteristic
291C2#928811yescharacteristic
292C2#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. #500C8semidirect

Maximal subgroups

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

subgroupindex
#223072
#222482
#98732
#75912
#75902
#70182
#70142