Finite Group Databasebeta

← #124122

Subgroups — #124122

Order 512.

764 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 146 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#14281C211yesmaximal, characteristic
2C2×#1280C212yesmaximal
3#7132C212yesmaximal, complemented
4#7131C212yesmaximal, complemented
5C22×C22⋊Q8C411yescharacteristic
6C22×C22⋊C8—22no—
7C2×#380C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1446C411yescharacteristic
10#1446C2211yescharacteristic
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1441—22no—
14C22⋊#95—22no—
15#1280—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C23×C8—42no—
21C2×C22⋊Q8—22no—
22C2×C22⋊Q8—42no—
23C2×C22⋊Q8—42no—
24C2×C22⋊Q8C814yescomplemented
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C2×C22⋊C8—28no—
31C2×C22⋊C8—24no—
32C2×C22⋊C8—42no—
33C2×C22⋊C8—44no—
34#380—42no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—42no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95D412yes—
44C2×#95C2311yescharacteristic
45C25C22⋊C411yescharacteristic
46C22×Q8—21no—
47C22×Q8—41no—
48C22×Q8—41no—
49C22×Q8—82no—
50C22×Q8C2×C812yescomplemented
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—24no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—82no—
61C23×C4—82no—
62C23×C4—82no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C412yes—
65C23×C4C2×C812yes—
66C23×C4M1612yes—
67C23×C4SD1614yes—
68C23×C4Q1614yes—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D412yes—
71C22×C8—48no—
72C22×C8—44no—
73C22×C8—84no—
74C22×C8—84no—
75C22⋊Q8—48no—
76C22⋊Q8—48no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—82no—
85C2×C4⋊C4C2×C812yescomplemented
86C2×C22⋊C4—24no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—44no—
91C22⋊C8—28no—
92C22⋊C8—48no—
93C22⋊C8—44no—
94#95—41no—
95#95—42no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—82no—
100#95—82no—
101#95—82no—
102#95—82no—
103C24—42no—
104C24—41no—
105C24—41no—
106C24C22⋊C812yes—
107C24#9912yes—
108C24#10012yes—
109C24#10314yes—
110C24#10414yes—
111C24C2×C22⋊C411yescharacteristic
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—82no—
115C2×Q8—82no—
116C2×Q8—84no—
117C2×Q8—84no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—41no—
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—84no—
161C22×C4—84no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—84no—
167C22×C4—88no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4C22⋊C812yes—
171C22×C4#10314yes—
172C22×C4C22×C811yescharacteristic
173C22×C4C2×M1611yescharacteristic
174C22×C4C2×SD1612yes—
175C22×C4C2×Q1612yes—
176C2×C8—48no—
177C2×C8—816no—
178C2×C8—84no—
179C2×C8—84no—
180C4⋊C4—42no—
181C4⋊C4—48no—
182C4⋊C4—42no—
183C4⋊C4—84no—
184C22⋊C4—48no—
185C22⋊C4—84no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—44no—
196C23—44no—
197C23—42no—
198C23—42no—
199C23—44no—
200C23—44no—
201C23#32314yes—
202C23Q8⋊C818yes—
203C23#32814yes—
204C23#32814yes—
205C23#32914yes—
206C23#33314yes—
207C23C2×C22⋊C811yescharacteristic
208C23C2×#9911yescharacteristic
209C23C2×#10011yescharacteristic
210C23C2×#10312yes—
211C23C2×#10412yes—
212Q8—84no—
213Q8—84no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C8—88no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—24no—
275C22—24no—
276C22—22no—
277C22—48no—
278C22—44no—
279C22—44no—
280C22#127218yes—
281C22C2×#32311yescharacteristic
282C22C2×Q8⋊C812yes—
283C22C2×#32812yes—
284C22C2×#32911yescharacteristic
285C22C2×#33311yescharacteristic
286C22#212112yes—
287C22#212711yescharacteristic
288C22#213311yescharacteristic
289C4—42no—
290C4—41no—
291C4—41no—
292C4—42no—
293C4—44no—
294C4—42no—
295C4—82no—
296C4—82no—
297C4—161no—
298C4—161no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2—28no—
303C2C2×#127211yescharacteristic
304C2#1142611yescharacteristic
305C2#1143211yescharacteristic
306C2#1171612yes—
307C2#1172212yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7132C2semidirect
4. #7131C2semidirect
24. C2×C22⋊Q8C8semidirect
50. C22×Q8C2×C8semidirect
85. C2×C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#142812
C2×#12802
#71322
#71312