Finite Group Databasebeta

← #124980

Subgroups — #124980

Order 512.

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

normal subgroups only

This group has 666 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#14159C211yesmaximal, characteristic
2#8559C212yesmaximal
3#7132C212yesmaximal, complemented
4#7131C212yesmaximal, complemented
5C22×C22⋊Q8C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7C22×#95C411yescharacteristic
8#1797—22no—
9C2×#323—22no—
10#1446C411yescharacteristic
11#1446C2211yescharacteristic
12#1445C411yescharacteristic
13#1445C2211yescharacteristic
14#1436—22no—
15C22⋊#95—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C24×C4—21no—
19C24×C4C2×C411yescharacteristic
20C2×C22⋊Q8—22no—
21C2×C22⋊Q8—22no—
22C2×C22⋊Q8—41no—
23C2×C22⋊Q8—41no—
24C2×C22⋊Q8—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4D412yes—
29C22×C42D412yes—
30C2×C4⋊C8—42no—
31C2×C22⋊C8—24no—
32C4×C22⋊C4—41no—
33C4×C22⋊C4—41no—
34C2×#95—22no—
35C2×#95—21no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—42no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95C2311yescharacteristic
44#336—42no—
45#323—44no—
46C25C22⋊C411yescharacteristic
47C22×Q8—21no—
48C22×Q8—41no—
49C22×Q8—41no—
50C22×Q8—82no—
51C22×Q8M1612yes—
52C23×C4—22no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—44no—
65C23×C4—82no—
66C23×C4—82no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D412yes—
71C22×C8—44no—
72C22⋊Q8—44no—
73C22⋊Q8—44no—
74C22⋊Q8—44no—
75C22⋊Q8—44no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4M1612yes—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4SD1614yes—
90C2×C22⋊C4Q1614yes—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—82no—
94C4⋊C8—84no—
95C22⋊C8—44no—
96C22⋊C8—44no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—44no—
102#95—44no—
103#95—42no—
104#95—82no—
105#95—82no—
106C24—21no—
107C24—21no—
108C24—22no—
109C24—41no—
110C24—41no—
111C24—42no—
112C24#10012yes—
113C24#10314yes—
114C24C2×C22⋊C411yescharacteristic
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—82no—
118C2×Q8—82no—
119C2×Q8—84no—
120C2×Q8—84no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
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—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—48no—
152C22×C4—41no—
153C22×C4—41no—
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×C4—88no—
166C22×C4—88no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4#9911yescharacteristic
170C22×C4#9911yescharacteristic
171C22×C4#10314yes—
172C22×C4C2×M1611yescharacteristic
173C22×C4C2×M1611yescharacteristic
174C22×C4C2×SD1612yes—
175C22×C4C2×Q1612yes—
176C2×C8—84no—
177C2×C8—84no—
178C2×C8—84no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C22⋊C4—44no—
184C22⋊C4—44no—
185C22⋊C4—48no—
186C42—44no—
187C42—82no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—24no—
193C23—24no—
194C23—22no—
195C23—42no—
196C23—42no—
197C23—44no—
198C23—44no—
199C23—44no—
200C23—44no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23#33314yes—
207C23#40711yescharacteristic
208C23C2×#9911yescharacteristic
209C23C2×#10011yescharacteristic
210C23C2×#10312yes—
211C23#42112yes—
212Q8—84no—
213Q8—84no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
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—41no—
232C2×C4—41no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—816no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C8—164no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—24no—
273C22—44no—
274C22—44no—
275C22—48no—
276C22—44no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22#146211yescharacteristic
281C22#148712yes—
282C22#152012yes—
283C22#153311yescharacteristic
284C22C2×#33311yescharacteristic
285C22#212112yes—
286C22#212611yescharacteristic
287C22#213211yescharacteristic
288C4—42no—
289C4—42no—
290C4—44no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4—82no—
298C4—82no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2—44no—
303C2#764311yescharacteristic
304C2#1150211yescharacteristic
305C2#1150711yescharacteristic
306C2#1190212yes—
307C2#1190612yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×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
3. #7132C2semidirect
4. #7131C2semidirect

Maximal subgroups

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

subgroupindex
#141592
#85592
#71322
#71312