Finite Group Databasebeta

← #110233

Subgroups — #110233

Order 512.

702 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 166 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#20534C211yesmaximal, characteristic
2C2×#1534C212yesmaximal
3#14160C211yesmaximal, characteristic
4C2×#1285C212yesmaximal
5#7912C211yesmaximal, characteristic
6C22×#125C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1771—24no—
12#1749—21no—
13C4×C22⋊C8—21no—
14#1534—24no—
15#1534C418yescomplemented
16C2×#330—24no—
17#1441—22no—
18#1436—21no—
19#1436—21no—
20C4⋊#95C412yes—
21C4⋊#95C412yes—
22#1285—24no—
23#1280—22no—
24C2×#125—22no—
25C2×#125—22no—
26C2×#125—22no—
27C2×#125—22no—
28C2×#125—22no—
29C2×#125C2×C414yes—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C412yes—
36C22×C42C2×C412yes—
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432C2×C414yes—
46#432C2×C414yes—
47#432D414yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×#97—22no—
53C2×#97—22no—
54C2×#97D414yes—
55C2×C4×C8—22no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D412yes—
66#336—22no—
67#336—22no—
68#330—216no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C412yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C22×C8—24no—
78C22×C8—42no—
79#125—24no—
80#125—24no—
81#125—24no—
82#125—24no—
83#125—42no—
84#125—42no—
85#125—42no—
86#125—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C4212yes—
108C2×C4⋊C4C4212yes—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—24no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42C22×C412yes—
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42#4112yes—
122C2×C42#4112yes—
123C2×C42#4112yes—
124C4⋊C8—24no—
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—42no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132#97—24no—
133#97—24no—
134#97—42no—
135#97—82no—
136#97C22⋊C418yes—
137C4×C8—82no—
138#95—24no—
139#95—44no—
140#95—44no—
141#95—42no—
142#95—82no—
143#95—82no—
144C24C2×C22⋊C411yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4#10114yes—
177C22×C4C2×C4211yescharacteristic
178C22×C4#11711yescharacteristic
179C22×C4C4×D412yes—
180C22×C4C4×D412yes—
181C22×C4C4×D412yes—
182C22×C4C4×D412yes—
183C22×C4#12011yescharacteristic
184C22×C4#12112yes—
185C22×C4#12112yes—
186C22×C4#12311yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4#12711yescharacteristic
189C2×C8—28no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—84no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—48no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C42—42no—
206C42—42no—
207C42—81no—
208C42—81no—
209C42—84no—
210C42C2×C22⋊C412yes—
211C42#11712yes—
212C42C4×D412yes—
213C42C4×D412yes—
214C42#12112yes—
215C42#12312yes—
216C42#12412yes—
217C42#12612yes—
218C23—22no—
219C23—21no—
220C23—21no—
221C23C4×C22⋊C411yescharacteristic
222C23#38611yescharacteristic
223C23#39011yescharacteristic
224C23C2×#10112yes—
225C23#41312yes—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C4×C22⋊C412yes—
269C2×C4#38612yes—
270C2×C4#38812yes—
271C2×C4#38812yes—
272C2×C4#38812yes—
273C2×C4#39012yes—
274C8—48no—
275C8—84no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22#143711yescharacteristic
285C22C4×#10112yes—
286C22#188412yes—
287C22#190112yes—
288C22#191111yescharacteristic
289C22#191211yescharacteristic
290C22#196611yescharacteristic
291C22#196711yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—22no—
297C4—22no—
298C4—44no—
299C4—84no—
300C4—82no—
301C4—82no—
302C4—82no—
303C2—21no—
304C2—21no—
305C2—22no—
306C2#944912yes—
307C2#945012yes—
308C2#1094911yescharacteristic
309C2#1095911yescharacteristic
310C2#1096311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1534C4semidirect

Maximal subgroups

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

subgroupindex
#205342
C2×#15342
#141602
C2×#12852
#79122