Finite Group Databasebeta

← #100115

Subgroups — #100115

Order 512.

522 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 100 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#23269C211yesmaximal, characteristic, complemented
2#21701C211yesmaximal, characteristic
3#7601C211yesmaximal, characteristic, complemented
4#7587C212yesmaximal, complemented
5#7030C212yesmaximal, complemented
6C2×#496C411yescharacteristic
7C2×#442—21no—
8C2×#440—21no—
9C4×#127C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#1984—21no—
12#1970—21no—
13#1968—21no—
14#1961C411yescharacteristic
15#1959C411yescharacteristic
16#1923—22no—
17#1869—21no—
18#1847C411yescharacteristic
19#1831C2212yes—
20C8×C22⋊C4—22no—
21#1744C2212yes—
22C4×#97C2211yescharacteristic
23#1272—21no—
24#1270—21no—
25#1270—22no—
26#1270—22no—
27C22×SD16—21no—
28C22×D8—21no—
29C23×C8—21no—
30C2×#127C2×C411yescharacteristic
31C2×#124C2×C411yescharacteristic
32C2×C4×Q8D411yescharacteristic
33C2×C4×D4—21no—
34C2×C4×D4—22no—
35C2×C4×D4D411yescharacteristic
36#496—22no—
37#496C814yescomplemented
38#442—42no—
39#442—42no—
40#440—42no—
41#440—42no—
42C2×#107—21no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—22no—
46C2×#103—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—21no—
52C2×#97—21no—
53C2×#97C2×C411yescharacteristic
54C2×#97C2×C411yescharacteristic
55C2×C4×C8—22no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391D411yescharacteristic
59#390—21no—
60#390—22no—
61#390D411yescharacteristic
62#388—22no—
63#384C2×C411yescharacteristic
64#382C2×C411yescharacteristic
65C4×C22⋊C4—22no—
66C43C2311yescharacteristic
67#336—22no—
68#336—22no—
69#336—22no—
70#336—22no—
71#336D412yes—
72#336D412yes—
73C22×Q8C22⋊C411yescharacteristic
74C22×D4—21no—
75C22×D4—22no—
76C22×D4C22⋊C411yescharacteristic
77C23×C4—21no—
78C23×C4—41no—
79C2×SD16—24no—
80C2×SD16—42no—
81C2×SD16—42no—
82C2×D8—24no—
83C2×D8—42no—
84C2×D8—42no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—44no—
90C22×C8—42no—
91C22×C8—81no—
92#127—21no—
93#127—22no—
94#127C2×C812yescomplemented
95#124—22no—
96#124C2×C812yes—
97#124M1612yes—
98C4×Q8—41no—
99C4×Q8—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—82no—
107C4×D4—82no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C4⋊C4C22⋊C411yescharacteristic
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C42—21no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42#4112yes—
124#107—42no—
125#106—42no—
126C4⋊C8—82no—
127C4⋊C8—81no—
128#103—42no—
129#102—42no—
130#102—42no—
131#102—42no—
132C22⋊C8—44no—
133C22⋊C8—81no—
134#97—22no—
135#97—22no—
136#97—21no—
137#97—22no—
138#97C2×C812yescomplemented
139C4×C8—42no—
140C4×C8—44no—
141C4×C8—81no—
142#95—22no—
143#95—22no—
144C24—21no—
145C24—41no—
146C2×Q8—21no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8C22⋊C812yes—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—24no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4C22⋊C812yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—84no—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C4×D412yes—
177C22×C4#12011yescharacteristic
178C22×C4#12112yes—
179C22×C4#12311yescharacteristic
180SD16—48no—
181D8—48no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—84no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—21no—
214C42—21no—
215C42—22no—
216C42—22no—
217C42—41no—
218C42—42no—
219C42—41no—
220C42—41no—
221C42C22×C811yescharacteristic
222C42C2×M1611yescharacteristic
223C42#13112yes—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—81no—
231C23—81no—
232C23—82no—
233C23—82no—
234C23#38611yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—81no—
241D4—84no—
242D4—81no—
243D4—81no—
244D4—84no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—162no—
267C2×C4—162no—
268C2×C4C2×C22⋊C811yescharacteristic
269C2×C4#40811yescharacteristic
270C2×C4C8×D412yes—
271C2×C4#43512yes—
272C2×C4#45311yescharacteristic
273C2×C4#45311yescharacteristic
274C2×C4#47112yes—
275C2×C4#47212yes—
276C8—24no—
277C8—42no—
278C8—44no—
279C8—82no—
280C8—162no—
281C22—44no—
282C22—42no—
283C22—82no—
284C22—81no—
285C22—81no—
286C22—84no—
287C22—82no—
288C22—162no—
289C22—161no—
290C22—161no—
291C22#186911yescharacteristic
292C22#188411yescharacteristic
293C22#188711yescharacteristic
294C22#190612yes—
295C22#190712yes—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—161no—
309C2—82no—
310C2—162no—
311C2#895911yescharacteristic
312C2#909512yes—
313C2#943212yes—
314C2#943711yescharacteristic
315C2#944111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC2×#496C4
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
1. #23269C2semidirect
3. #7601C2semidirect
4. #7587C2semidirect
5. #7030C2semidirect
37. #496C8semidirect
94. #127C2×C8semidirect
138. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#232692
#217012
#76012
#75872
#70302