Finite Group Databasebeta

← #129868

Subgroups — #129868

Order 512.

766 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 104 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×#2634C211yesmaximal, characteristic
2C2×#1353C212yesmaximal
3C2×#1346C212yesmaximal
4#10763C211yesmaximal, characteristic
5#8303C211yesmaximal, characteristic
6C22×#125C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×#101—21no—
9#2634—24no—
10#2634—24no—
11C2×#400—21no—
12C2×#400C411yescharacteristic
13C2×#400C2211yescharacteristic
14C2×#398C411yescharacteristic
15C2×#398C2211yescharacteristic
16C2×#395—21no—
17C2×#387—21no—
18C2×#387—21no—
19C2×#383C411yescharacteristic
20#1817—21no—
21C2×#343C2212yes—
22#1728—21no—
23C2×#324—21no—
24C2×#324C2212yes—
25#1353—24no—
26#1346—24no—
27C23×Q8D411yescharacteristic
28C22×M16—22no—
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—42no—
32C2×#125—42no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—42no—
36C2×C4×Q8—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C2×#101—24no—
50C2×#101—42no—
51C2×#101—42no—
52C2×C22⋊C8—22no—
53#400—28no—
54#400—22no—
55#400—22no—
56#400—22no—
57#400D414yes—
58#398—22no—
59#398—22no—
60#398—22no—
61#398D414yes—
62#395—28no—
63#387—28no—
64#387—28no—
65#383—22no—
66#383—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95C2×C411yescharacteristic
76C2×#95C2×C411yescharacteristic
77#343—22no—
78#343—22no—
79#336—42no—
80#336—42no—
81#324—22no—
82#324—22no—
83#324—42no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—21no—
89C22×Q8—21no—
90C22×Q8—41no—
91C22×Q8—41no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×M16—24no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C22×C8—42no—
110C22×C8—42no—
111#125—48no—
112#125—48no—
113C4×Q8—48no—
114C4×Q8—48no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—41no—
156C2×C42—41no—
157#101—48no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—21no—
166#95—21no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95C22⋊C412yes—
171#95C22⋊C412yes—
172C24C2×C22⋊C411yescharacteristic
173C2×Q8—24no—
174C2×Q8—24no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4#9912yes—
234C22×C4#9912yes—
235C22×C4#10412yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4#12012yes—
239C22×C4#12012yes—
240C22×C4#13512yes—
241C22×C4AGL(1,ℤ/8)11yescharacteristic
242C22×C4#13711yescharacteristic
243M16—44no—
244M16—44no—
245M16—82no—
246M16—82no—
247C2×C8—44no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—84no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—48no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—48no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—48no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C42—24no—
267C42—24no—
268C42—44no—
269C42—44no—
270C23—22no—
271C23—22no—
272C23#37911yescharacteristic
273C23C2×#9911yescharacteristic
274C23C2×#9911yescharacteristic
275C23#41611yescharacteristic
276C23#41711yescharacteristic
277C23C2×#10411yescharacteristic
278C23#42111yescharacteristic
279Q8—48no—
280Q8—48no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4#44912yes—
343C2×C4#45212yes—
344C8—84no—
345C8—162no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—22no—
354C22#150111yescharacteristic
355C22#150411yescharacteristic
356C22#151311yescharacteristic
357C22#151611yescharacteristic
358C22#178111yescharacteristic
359C22#178811yescharacteristic
360C22#178911yescharacteristic
361C22#212312yes—
362C22#213312yes—
363C4—22no—
364C4—22no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—41no—
369C4—41no—
370C4—42no—
371C4—82no—
372C4—82no—
373C4—82no—
374C4—81no—
375C4—82no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2#833111yescharacteristic
381C2#1174012yes—
382C2#1183512yes—
383C2#1269211yescharacteristic
384C2#1270211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26342
C2×#13532
C2×#13462
#107632
#83032