Finite Group Databasebeta

← #119548

Subgroups — #119548

Order 512.

738 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 118 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#21798C211yesmaximal, characteristic
2#17393C211yesmaximal, characteristic
3#14288C211yesmaximal, characteristic
4#7975C212yesmaximal, complemented
5#7950C212yesmaximal, complemented
6C22×#124C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9C2×#383C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1967—24no—
12#1784—21no—
13#1783—21no—
14#1469—24no—
15#1462—22no—
16#1446—21no—
17#1439—22no—
18#1438—21no—
19#1437C414yes—
20#1288—24no—
21#1285—24no—
22C23×Q8D411yescharacteristic
23C23×D4D411yescharacteristic
24C2×#124—22no—
25C2×#124—22no—
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124—22no—
30C22×C4⋊C4—21no—
31C22×C22⋊C4—24no—
32C22×C22⋊C4C2×C412yes—
33C22×C22⋊C4D412yes—
34C22×C42C2×C412yes—
35C22×C42C2311yescharacteristic
36#432—22no—
37#432—22no—
38#432—22no—
39#432D414yes—
40#431—22no—
41#431—22no—
42#431—22no—
43#431—22no—
44#423—22no—
45#423—22no—
46C2×#101—24no—
47C2×#100—24no—
48#407—22no—
49#407—22no—
50C4×M16—22no—
51C4×M16D414yes—
52#383—22no—
53#383—22no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—24no—
60C2×#95C2×C412yes—
61#323—28no—
62C25C22⋊C412yes—
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—21no—
67C22×Q8—82no—
68C22×D4—21no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—82no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4C22⋊C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C2×M16—24no—
80C2×M16—42no—
81#124—42no—
82#124—42no—
83#124—42no—
84#124—42no—
85#124—42no—
86#124—42no—
87#124—42no—
88#124—42no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—48no—
101C2×C22⋊C4—84no—
102C2×C22⋊C4—84no—
103C2×C42—24no—
104C2×C42—82no—
105C2×C42C2×D412yes—
106C2×C42C2×D412yes—
107C2×C42#4112yes—
108C2×C42#4112yes—
109C2×C42#4112yes—
110C2×C42#4112yes—
111C4⋊C8—24no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#101—84no—
116#100—84no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#97—24no—
122#97—24no—
123C4×C8—24no—
124C4×C8—42no—
125#95—24no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—84no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—42no—
135C24—44no—
136C24—42no—
137C24—44no—
138C24C2×C22⋊C411yescharacteristic
139C2×Q8—44no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—82no—
143C2×Q8—84no—
144C2×Q8—84no—
145C2×D4—44no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—82no—
149C2×D4—84no—
150C2×D4—84no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4#11711yescharacteristic
170C22×C4#11711yescharacteristic
171C22×C4C4×D412yes—
172C22×C4C4×D412yes—
173C22×C4#12011yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12112yes—
176C22×C4#12112yes—
177C22×C4#12312yes—
178C22×C4#12311yescharacteristic
179C22×C4#12312yes—
180C22×C4#12311yescharacteristic
181M16—48no—
182M16—84no—
183C2×C8—44no—
184C2×C8—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—816no—
192C42—84no—
193C42#12112yes—
194C42#12112yes—
195C42#12312yes—
196C42#12312yes—
197C42#12412yes—
198C42#12412yes—
199C42#12612yes—
200C42#12612yes—
201C23—22no—
202C23—22no—
203C23—21no—
204C23—21no—
205C23—44no—
206C23—44no—
207C23—44no—
208C23—42no—
209C23—44no—
210C23—42no—
211C23—42no—
212C23—84no—
213C23—84no—
214C23—88no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23#38111yescharacteristic
220C23#38611yescharacteristic
221C23#38611yescharacteristic
222Q8—84no—
223Q8—84no—
224D4—84no—
225D4—84no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—48no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—84no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#38812yes—
254C2×C4#38812yes—
255C2×C4#39412yes—
256C2×C4#39412yes—
257C2×C4#39712yes—
258C2×C4#39712yes—
259C8—84no—
260C8—84no—
261C22—24no—
262C22—24no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—88no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—162no—
274C22—164no—
275C22—162no—
276C22#144011yescharacteristic
277C22#182712yes—
278C22#191112yes—
279C22#191411yescharacteristic
280C22#191511yescharacteristic
281C4—22no—
282C4—22no—
283C4—22no—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—84no—
288C4—82no—
289C4—162no—
290C2—21no—
291C2—22no—
292C2—22no—
293C2—21no—
294C2—162no—
295C2#1114911yescharacteristic
296C2#1115511yescharacteristic
297C2#1117311yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7975C2semidirect
5. #7950C2semidirect

Maximal subgroups

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

subgroupindex
#217982
#173932
#142882
#79752
#79502