Finite Group Databasebeta

← #126147

Subgroups — #126147

Order 512.

421 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 60 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#17221C211yesmaximal, characteristic
2#10570C211yesmaximal, characteristic
3#10406C211yesmaximal, characteristic
4#10342C211yesmaximal, characteristic
5#7123C211yesmaximal, characteristic
6#7123C211yesmaximal, characteristic
7#7104C211yesmaximal, characteristic
8#2659C2211yescharacteristic
9C2×#398—21no—
10#1872—21no—
11#1853—21no—
12#1823—21no—
13C2×#328—21no—
14#1445C2211yescharacteristic
15#1444—21no—
16#1444—21no—
17#1444—21no—
18#1444C2211yescharacteristic
19#1444C2211yescharacteristic
20#1443—21no—
21#1443—21no—
22#1294—21no—
23#1294—21no—
24#1280C2211yescharacteristic
25#1280C2211yescharacteristic
26#1280C2211yescharacteristic
27C2×C22⋊Q8—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C2×#106—41no—
36C2×#103—41no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40#398—41no—
41#398—42no—
42#398—42no—
43#395—41no—
44#380—21no—
45#380—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95C2311yescharacteristic
60#340—41no—
61#340—41no—
62#340—41no—
63#340—41no—
64#328—42no—
65C22×Q8—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—41no—
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—81no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4SD1612yes—
110C2×C22⋊C4—41no—
111#106—82no—
112#103—82no—
113C22⋊C8—22no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119#95—24no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—41no—
125#95—41no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—81no—
136#95—81no—
137#95—81no—
138#95—81no—
139C24#12011yescharacteristic
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—81no—
143C2×Q8—81no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4C2×SD1611yescharacteristic
201C22×C4#13511yescharacteristic
202C22×C4#13511yescharacteristic
203C22×C4AGL(1,ℤ/8)11yescharacteristic
204C22×C4AGL(1,ℤ/8)11yescharacteristic
205C22×C4#13711yescharacteristic
206C2×C8—42no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#44911yescharacteristic
242C23#45011yescharacteristic
243C23#45211yescharacteristic
244C23#45411yescharacteristic
245C23#45511yescharacteristic
246C23#45611yescharacteristic
247C23B(4,𝔽2)11yescharacteristic
248Q8—82no—
249Q8—161no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C8—82no—
320C8—161no—
321C8—161no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22#159711yescharacteristic
335C22#160111yescharacteristic
336C22#160311yescharacteristic
337C22#163511yescharacteristic
338C22#164011yescharacteristic
339C22#165411yescharacteristic
340C22#166411yescharacteristic
341C22#220011yescharacteristic
342C22#220011yescharacteristic
343C22#220111yescharacteristic
344C22#220511yescharacteristic
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—82no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4—161no—
356C4—161no—
357C4—161no—
358C4—161no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2#777011yescharacteristic
364C2#1213811yescharacteristic
365C2#1214611yescharacteristic
366C2#1220611yescharacteristic
367C2#1221311yescharacteristic
368C2#1223211yescharacteristic
369C2#1224011yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutator, Frattini, cosocleC2×#95C23
prime radicalC24#120
LCSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#172212
#105702
#104062
#103422
#71232
#71042