Finite Group Databasebeta

← #99680

Subgroups — #99680

Order 512.

444 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 102 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#23801C211yesmaximal, characteristic
2#23249C211yesmaximal, characteristic
3#9868C212yesmaximal
4C4×#333C211yesmaximal, characteristic
5#7027C212yesmaximal
6C2×C4×Q16—21no—
7C2×C4×Q16—21no—
8C4×C4⋊Q8C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2095—21no—
11#2095C411yescharacteristic
12#2095C411yescharacteristic
13#2016—21no—
14#2016C411yescharacteristic
15#2016C411yescharacteristic
16#1985—21no—
17#1984—21no—
18#1940—22no—
19#1771—22no—
20#1767C2212yes—
21C4×C4⋊C8C2212yes—
22#1744C2211yescharacteristic
23C2×#333—21no—
24#1296—21no—
25#1279—22no—
26#1279—22no—
27C22×Q16—21no—
28C22×Q16—21no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8D411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8D411yescharacteristic
33C2×C4×Q8D411yescharacteristic
34C4×Q16—42no—
35C4×Q16—42no—
36C4×Q16—42no—
37C4×Q16—42no—
38C2×#107—21no—
39C2×#107—21no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D412yes—
44C2×C4⋊C8D412yes—
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#103—21no—
49C2×#97—22no—
50C2×#97—21no—
51C2×C4×C8—21no—
52C2×C4×C8—22no—
53C2×C4×C8—21no—
54#391—21no—
55#391C2×C411yescharacteristic
56#391C2×C411yescharacteristic
57#389—22no—
58#389—22no—
59#384C2×C411yescharacteristic
60#384D411yescharacteristic
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C43C2311yescharacteristic
64#336—22no—
65#336—22no—
66#336—21no—
67#336—22no—
68#336—22no—
69#336C2×C411yescharacteristic
70#336C2×C411yescharacteristic
71#333—24no—
72C22×Q8—21no—
73C22×Q8C22⋊C411yescharacteristic
74C22×Q8C22⋊C411yescharacteristic
75C2×Q16—42no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×Q16—42no—
79C2×Q16—42no—
80C2×Q16—42no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—42no—
86C4⋊Q8—22no—
87C4⋊Q8—41no—
88C4⋊Q8—42no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C22⋊C411yescharacteristic
106C2×C4⋊C4C22⋊C411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—41no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D412yes—
114C2×C42#4112yes—
115#107—42no—
116#107—42no—
117C4⋊C8—24no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—42no—
122C4⋊C8—81no—
123C4⋊C8—81no—
124C4⋊C8Q1614yescomplemented
125#103—42no—
126#103—42no—
127#103—42no—
128#103—42no—
129#97—42no—
130#97—42no—
131#97—81no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—42no—
135C4×C8—42no—
136C4×C8—81no—
137#95—22no—
138#95—22no—
139#95—42no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8#10412yes—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C4×D412yes—
162C22×C4#12011yescharacteristic
163C22×C4#12112yes—
164C22×C4#12311yescharacteristic
165Q16—48no—
166Q16—48no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—82no—
181C2×C8—82no—
182C4⋊C4—24no—
183C4⋊C4—44no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—44no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—41no—
200C42—42no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—81no—
206C42—81no—
207C42C2×Q1612yes—
208C42#13512yes—
209C42#13712yes—
210C42#13712yes—
211C23#38611yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—81no—
217Q8—81no—
218Q8—81no—
219Q8—81no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—81no—
243C2×C4C2×#10411yescharacteristic
244C2×C4#42111yescharacteristic
245C2×C4C4×Q1612yes—
246C2×C4#44112yes—
247C2×C4#45112yes—
248C2×C4#45212yes—
249C2×C4#45311yescharacteristic
250C2×C4#45311yescharacteristic
251C2×C4#46212yes—
252C2×C4#46412yes—
253C8—42no—
254C8—42no—
255C8—44no—
256C8—84no—
257C8—82no—
258C22#187111yescharacteristic
259C22#187812yes—
260C22#188112yes—
261C22#188711yescharacteristic
262C22#189111yescharacteristic
263C4—22no—
264C4—22no—
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—42no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—82no—
275C4—161no—
276C4#163212yes—
277C4#163311yescharacteristic
278C4#163411yescharacteristic
279C2#899012yes—
280C2#900711yescharacteristic
281C2#903111yescharacteristic
282C2#904812yes—
283C2#906611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socle, prime radicalC23#386
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
124. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#238012
#232492
#98682
C4×#3332
#70272