Finite Group Databasebeta

← #124141

Subgroups — #124141

Order 512.

726 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 98 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#14290C211yesmaximal, characteristic
2C4×#323C212yesmaximal, complemented
3#7132C212yesmaximal, complemented
4#7103C212yesmaximal, complemented
5C22×#121C411yescharacteristic
6C2×#381C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8C4×C22⋊C8—22no—
9C2×#323—22no—
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1438—22no—
13#1438C411yescharacteristic
14#1438C2211yescharacteristic
15#1437—22no—
16#1280C2214yes—
17C23×D4—21no—
18C23×D4C2×C411yescharacteristic
19C24×C4C2×C411yescharacteristic
20C2×#121—22no—
21C2×#121—42no—
22C2×#121—42no—
23C2×#121C814yescomplemented
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C22⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4D412yes—
30C22×C42D412yes—
31C2×C22⋊C8—24no—
32C2×C4×C8—42no—
33#381—42no—
34C4×C22⋊C4—42no—
35C2×#95—21no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95C2×C411yescharacteristic
39C2×#95C2×C411yescharacteristic
40C2×#95C2×C411yescharacteristic
41C2×#95C2311yescharacteristic
42#336—42no—
43#323—28no—
44C25—21no—
45C25C22⋊C411yescharacteristic
46C22×D4—24no—
47C22×D4—21no—
48C22×D4—41no—
49C22×D4—41no—
50C22×D4—42no—
51C22×D4—42no—
52C22×D4—82no—
53C22×D4—82no—
54C22×D4—82no—
55C22×D4C2×C812yes—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—82no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×C812yes—
65C23×C4M1612yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C22×C8—44no—
69#121—48no—
70#121—48no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4C2×C812yes—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—82no—
85C2×C22⋊C4—82no—
86C2×C22⋊C4—82no—
87C2×C22⋊C4—82no—
88C2×C22⋊C4—82no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—82no—
92C22⋊C8—28no—
93C22⋊C8—44no—
94C4×C8—84no—
95#95—41no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—42no—
100#95—82no—
101#95—82no—
102C24—21no—
103C24—21no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—42no—
108C24—41no—
109C24—41no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24—42no—
114C24—41no—
115C24—41no—
116C24—42no—
117C24—42no—
118C24C22⋊C812yes—
119C24C2×C22⋊C411yescharacteristic
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—48no—
123C2×D4—82no—
124C2×D4—88no—
125C2×D4—84no—
126C2×D4—84no—
127C2×D4—82no—
128C2×D4—88no—
129C2×D4—84no—
130C2×D4—84no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—84no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4C22⋊C812yes—
166C22×C4#9912yes—
167C22×C4#10012yes—
168C22×C4#10414yes—
169C22×C4C22×C811yescharacteristic
170C22×C4C2×M1611yescharacteristic
171C22×C4#13514yes—
172C2×C8—48no—
173C2×C8—84no—
174C2×C8—84no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—88no—
180C22⋊C4—88no—
181C22⋊C4—84no—
182C42—44no—
183C42—82no—
184C23—22no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—24no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—44no—
196C23—41no—
197C23—42no—
198C23—41no—
199C23—41no—
200C23—42no—
201C23—41no—
202C23—42no—
203C23—44no—
204C23—82no—
205C23—82no—
206C23—82no—
207C23—82no—
208C23—82no—
209C23—82no—
210C23—82no—
211C23—82no—
212C23—82no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—81no—
219C23—82no—
220C23—81no—
221C23—81no—
222C23—82no—
223C23—81no—
224C23C2×C22⋊C811yescharacteristic
225C23C2×#9911yescharacteristic
226C23C2×#10011yescharacteristic
227C23#41612yes—
228C23C2×#10412yes—
229D4—84no—
230D4—84no—
231D4—816no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4#32314yes—
270C8—88no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—84no—
283C22—82no—
284C22—82no—
285C22—84no—
286C22—84no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—162no—
293C22—162no—
294C22—162no—
295C22—162no—
296C22C2×#32311yescharacteristic
297C22#147612yes—
298C22#150112yes—
299C22#152511yescharacteristic
300C22#154311yescharacteristic
301C22#212112yes—
302C22#212811yescharacteristic
303C22#213411yescharacteristic
304C4—24no—
305C4—41no—
306C4—41no—
307C4—44no—
308C4—44no—
309C4—82no—
310C4—161no—
311C4—161no—
312C2—21no—
313C2—21no—
314C2—22no—
315C2—82no—
316C2—162no—
317C2#756311yescharacteristic
318C2#1142611yescharacteristic
319C2#1143311yescharacteristic
320C2#1172912yes—
321C2#1173712yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#323C2semidirect
3. #7132C2semidirect
4. #7103C2semidirect
23. C2×#121C8semidirect

Maximal subgroups

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

subgroupindex
#142902
C4×#3232
#71322
#71032