Finite Group Databasebeta

← #96926

Subgroups — #96926

Order 512.

980 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×#2861C211yesmaximal, characteristic
2#10795C212yesmaximal, complemented
3C2×#1272C212yesmaximal, complemented
4C2×#1270C212yesmaximal
5C22×#125C411yescharacteristic
6C22×#124C411yescharacteristic
7#2861—24no—
8#2861—24no—
9C2×#391C411yescharacteristic
10C2×#391C2211yescharacteristic
11C2×#390C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#388—22no—
14C2×#388—22no—
15C2×C43C2211yescharacteristic
16C2×#336—22no—
17C2×#336—22no—
18C2×#336C2214yes—
19#1272—24no—
20#1270—24no—
21C23×Q8C2×C411yescharacteristic
22C23×D4C2×C411yescharacteristic
23C23×C8—24no—
24C2×#125—22no—
25C2×#125—22no—
26C2×#125—42no—
27C2×#125—42no—
28C2×#124—22no—
29C2×#124—22no—
30C2×#124—42no—
31C2×#124—42no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45#391—22no—
46#391—22no—
47#391—22no—
48#391D414yes—
49#390—22no—
50#390—22no—
51#390—22no—
52#390D414yes—
53#388—216no—
54#388—216no—
55C2×#95—22no—
56C2×#95—22no—
57C43—22no—
58C43—22no—
59#336—28no—
60#336—24no—
61#336—24no—
62#336—44no—
63C25—21no—
64C22×Q8—21no—
65C22×Q8—24no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—42no—
69C22×Q8C22⋊C412yes—
70C22×D4—21no—
71C22×D4—24no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—42no—
75C22×D4C22⋊C412yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C412yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D412yes—
84C22×C8—28no—
85C22×C8—44no—
86C22×C8—44no—
87C22×C8—44no—
88C22×C8—44no—
89C22×C8—44no—
90C22×C8—44no—
91#125—48no—
92#125—48no—
93#124—48no—
94#124—48no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C22⋊C412yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—24no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—24no—
132C2×C42—24no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139#95—28no—
140#95—28no—
141C24—22no—
142C24—42no—
143C24—42no—
144C24—41no—
145C24—41no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24C2×C22⋊C411yescharacteristic
151C2×Q8—24no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—84no—
158C2×Q8—84no—
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—84no—
166C2×D4—84no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4#10012yes—
204C22×C4#10012yes—
205C22×C4#10414yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4#12014yes—
209C22×C4#13114yes—
210C22×C4#13514yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13712yes—
213C2×C8—48no—
214C2×C8—48no—
215C2×C8—48no—
216C2×C8—84no—
217C2×C8—84no—
218C2×C8—84no—
219C2×C8—84no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C22⋊C4—416no—
227C22⋊C4—48no—
228C22⋊C4—48no—
229C22⋊C4—48no—
230C22⋊C4—48no—
231C42—24no—
232C42—24no—
233C42—42no—
234C42—42no—
235C42—48no—
236C42—42no—
237C42—42no—
238C42—44no—
239C42—44no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—44no—
244C23—44no—
245C23—81no—
246C23—81no—
247C23—82no—
248C23—81no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—81no—
260C23—82no—
261C23#37911yescharacteristic
262C23#40812yes—
263C23#41012yes—
264C23C2×#10011yescharacteristic
265C23C2×#10011yescharacteristic
266C23#41612yes—
267C23#41712yes—
268C23C2×#10412yes—
269C23#42112yes—
270Q8—84no—
271Q8—84no—
272D4—84no—
273D4—84no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4#40812yes—
321C2×C4#40812yes—
322C2×C4#44914yes—
323C2×C4#45214yes—
324C8—88no—
325C22—82no—
326C22—84no—
327C22—84no—
328C22—82no—
329C22—82no—
330C22—162no—
331C22—162no—
332C22—162no—
333C22—162no—
334C22#146912yes—
335C22#146912yes—
336C22#148212yes—
337C22#148312yes—
338C22#149412yes—
339C22#149512yes—
340C22#150012yes—
341C22#150912yes—
342C22#151212yes—
343C22#152112yes—
344C22#152511yescharacteristic
345C22#153012yes—
346C22#153611yescharacteristic
347C22#154311yescharacteristic
348C22#155112yes—
349C22#156011yescharacteristic
350C22#178011yescharacteristic
351C22#178211yescharacteristic
352C22#178311yescharacteristic
353C22#178812yes—
354C22#178912yes—
355C4—22no—
356C4—22no—
357C4—24no—
358C4—24no—
359C4—44no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—82no—
365C2—162no—
366C2#765612yes—
367C2#767612yes—
368C2#769612yes—
369C2#771612yes—
370C2#830511yescharacteristic
371C2#831512yes—
372C2#833012yes—
373C2#833711yescharacteristic
374C2#834411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10795C2semidirect
3. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#107952
C2×#12722
C2×#12702