Finite Group Databasebeta

← #99710

Subgroups — #99710

Order 512.

536 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 90 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#23451C211yesmaximal, characteristic
2#21909C211yesmaximal, characteristic
3#8314C212yesmaximal, complemented
4#7563C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C2×#533C2211yescharacteristic
7C2×#416—21no—
8#2297C2211yescharacteristic
9#2026—21no—
10#2025—21no—
11#2024C412yes—
12#2023C412yes—
13#1924—22no—
14#1923—22no—
15#1797C2212yes—
16#1780—21no—
17#1750—21no—
18C4×C22⋊C8C2212yes—
19#1735—21no—
20C2×#336C2211yescharacteristic
21C2×Q8⋊C8—22no—
22C2×Q8⋊C8—22no—
23#1272—21no—
24#1270—21no—
25C22×#41D411yescharacteristic
26C23×C8—21no—
27#533—44no—
28C2×C4⋊Q8C2×C412yes—
29C2×#125—21no—
30C2×C22⋊Q8—22no—
31C2×C4×Q8—22no—
32C2×#117—21no—
33C2×#117D411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×C4⋊C8D412yes—
36C2×C4⋊C8D412yes—
37#416—44no—
38C2×#103—22no—
39C2×#102—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—21no—
45C2×#97—21no—
46C2×C4×C8—22no—
47C2×C4×C8—21no—
48#391C2×C411yescharacteristic
49#391D411yescharacteristic
50#390C2×C411yescharacteristic
51#390D411yescharacteristic
52#386—22no—
53C4×C22⋊C4—22no—
54#340—22no—
55#340—22no—
56#336—22no—
57#336—22no—
58#336—21no—
59#336—22no—
60#336C2×C412yes—
61Q8⋊C8—44no—
62Q8⋊C8—44no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—42no—
68C22×Q8—22no—
69C22×Q8C22⋊C411yescharacteristic
70C22×D4—21no—
71C22×D4C22⋊C411yescharacteristic
72C23×C4—21no—
73C23×C4C2×D411yescharacteristic
74C23×C4#4112yes—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—42no—
83C4⋊Q8—44no—
84#125—44no—
85C22⋊Q8—44no—
86C22⋊Q8—44no—
87C4×Q8—24no—
88C4×Q8—24no—
89C4×Q8—42no—
90C4×Q8—42no—
91#117—24no—
92#117—42no—
93#117—41no—
94#117—41no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4C22⋊C411yescharacteristic
99C2×C4⋊C4C22⋊C411yescharacteristic
100C2×C22⋊C4—22no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—41no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—41no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C4⋊C8—42no—
111C4⋊C8—42no—
112#103—44no—
113#102—44no—
114C22⋊C8—24no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—81no—
119C22⋊C8—81no—
120C22⋊C8—82no—
121#97—81no—
122C4×C8—42no—
123C4×C8—42no—
124C4×C8—81no—
125#95—22no—
126#95—22no—
127C24—21no—
128C24#12311yescharacteristic
129#41—44no—
130#41—44no—
131#41—82no—
132#41—82no—
133C2×Q8—24no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—42no—
148C2×D4—81no—
149C2×D4—81no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C4×D412yes—
167C22×C4#12011yescharacteristic
168C22×C4#12112yes—
169C22×C4#13512yes—
170C22×C4#13712yes—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—82no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—82no—
197C22⋊C4—81no—
198C42—22no—
199C42—22no—
200C42—41no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42—44no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23—41no—
213C23—41no—
214C23#38611yescharacteristic
215C23#48512yes—
216Q8—44no—
217Q8—44no—
218Q8—42no—
219Q8—81no—
220Q8—82no—
221Q8—82no—
222D4—42no—
223D4—81no—
224D4—84no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4#41611yescharacteristic
251C2×C4#41911yescharacteristic
252C2×C4#44312yes—
253C2×C4#44412yes—
254C2×C4#45212yes—
255C2×C4#45511yescharacteristic
256C2×C4#45511yescharacteristic
257C2×C4B(4,𝔽2)12yes—
258C8—82no—
259C8—84no—
260C8—82no—
261C8—82no—
262C22—24no—
263C22—22no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—82no—
268C22—81no—
269C22—81no—
270C22—82no—
271C22—81no—
272C22#187311yescharacteristic
273C22#187612yes—
274C22#188112yes—
275C22#188511yescharacteristic
276C22#188911yescharacteristic
277C4—21no—
278C4—21no—
279C4—22no—
280C4—22no—
281C4—42no—
282C4—81no—
283C4—81no—
284C4—84no—
285C4—81no—
286C4#160612yes—
287C4#160612yes—
288C2—42no—
289C2—161no—
290C2#899412yes—
291C2#900511yescharacteristic
292C2#901611yescharacteristic
293C2#903712yes—
294C2#906711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8314C2semidirect
5. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#234512
#219092
#83142
#75632
#70292