Finite Group Databasebeta

← #126205

Subgroups — #126205

Order 512.

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

normal subgroups only

This group has 389 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#17298C211yesmaximal, characteristic
2#10569C211yesmaximal, characteristic
3#10407C211yesmaximal, characteristic
4#10343C211yesmaximal, characteristic
5#7132C211yesmaximal, characteristic
6#7122C211yesmaximal, characteristic
7#7122C211yesmaximal, characteristic
8#2659C2211yescharacteristic
9#1872—21no—
10#1852—21no—
11#1824—21no—
12C2×#328—21no—
13#1445C2211yescharacteristic
14#1444—21no—
15#1444—21no—
16#1443—21no—
17#1443—21no—
18#1442—21no—
19#1442C2211yescharacteristic
20#1442C2211yescharacteristic
21#1441—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⋊C4D411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C2×#107—41no—
34C2×#103—41no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38#398—41no—
39#395—41no—
40#380—21no—
41#380—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—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×#95D411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95C2311yescharacteristic
58#340—41no—
59#340—41no—
60#340—41no—
61#340—41no—
62#328—42no—
63C22×Q8—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—41no—
76C22⋊Q8—41no—
77C22⋊Q8—41no—
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C22⋊C4—41no—
101#107—82no—
102#103—82no—
103C22⋊C8—22no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106C22⋊C8—41no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—41no—
113#95—41no—
114#95—42no—
115#95—42no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—81no—
123#95—81no—
124#95—81no—
125#95—81no—
126#95—81no—
127#95—81no—
128#95—81no—
129#95—81no—
130#95Q1612yes—
131C24#12011yescharacteristic
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—81no—
135C2×Q8—81no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
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×C4C2×Q1611yescharacteristic
193C22×C4#13511yescharacteristic
194C22×C4#13511yescharacteristic
195C22×C4AGL(1,ℤ/8)11yescharacteristic
196C22×C4#13711yescharacteristic
197C22×C4#13711yescharacteristic
198C2×C8—42no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23#44911yescharacteristic
230C23#45111yescharacteristic
231C23#45211yescharacteristic
232C23#45411yescharacteristic
233C23#45511yescharacteristic
234C23#45611yescharacteristic
235C23B(4,𝔽2)11yescharacteristic
236Q8—82no—
237Q8—161no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
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—
307C8—82no—
308C8—161no—
309C8—161no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22#159711yescharacteristic
323C22#160511yescharacteristic
324C22#160611yescharacteristic
325C22#163611yescharacteristic
326C22#164411yescharacteristic
327C22#165011yescharacteristic
328C22#166411yescharacteristic
329C22#219711yescharacteristic
330C22#220011yescharacteristic
331C22#220111yescharacteristic
332C22#220211yescharacteristic
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—82no—
340C4—82no—
341C4—161no—
342C4—161no—
343C4—161no—
344C4—161no—
345C4—161no—
346C4—161no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2—21no—
351C2#777711yescharacteristic
352C2#1213811yescharacteristic
353C2#1214111yescharacteristic
354C2#1225411yescharacteristic
355C2#1226111yescharacteristic
356C2#1226911yescharacteristic
357C2#1227411yescharacteristic
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
#172982
#105692
#104072
#103432
#71322
#71222