Finite Group Databasebeta

← #104780

Subgroups — #104780

Order 512.

408 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 64 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#17055C211yesmaximal, characteristic
2#9889C211yesmaximal, characteristic
3#9888C211yesmaximal, characteristic
4#9683C211yesmaximal, characteristic
5#7036C211yesmaximal, characteristic
6#7036C211yesmaximal, characteristic
7#7036C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9#1749C2211yescharacteristic
10#1749C2211yescharacteristic
11#1749C2211yescharacteristic
12C2×#328—21no—
13C2×#328—21no—
14#1445—21no—
15#1444—21no—
16#1443—21no—
17#1442—21no—
18#1441—21no—
19#1441—21no—
20#1434C2211yescharacteristic
21#1434C2211yescharacteristic
22#1434C2211yescharacteristic
23#1294—21no—
24#1294—21no—
25#1294—21no—
26#1294—21no—
27C2×C4⋊Q8Q811yescharacteristic
28C2×C22⋊Q8—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C42C2311yescharacteristic
33C2×C22⋊C8—21no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36C2×#97—21no—
37C2×#97—21no—
38C2×#97—21no—
39#391—21no—
40#389—21no—
41#389—21no—
42#384Q811yescharacteristic
43#380—21no—
44#380—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×#95—21no—
55C2×#95—21no—
56C2×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#336—21no—
63#336—21no—
64#336—21no—
65#328—42no—
66#328—42no—
67C22×Q8—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C4⋊Q8—81no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
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—
101C2×C42—81no—
102C2×C42C2×Q811yescharacteristic
103C2×C42#4111yescharacteristic
104C2×C42#4111yescharacteristic
105C2×C42#4111yescharacteristic
106C22⋊C8—22no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112#97—81no—
113#97—81no—
114#97—81no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—41no—
128#95—41no—
129#95—81no—
130#95—81no—
131#95—81no—
132#95—81no—
133#95—81no—
134C24#12011yescharacteristic
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—81no—
138C2×Q8—81no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
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×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12411yescharacteristic
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#39311yescharacteristic
232C23#45311yescharacteristic
233C23#45411yescharacteristic
234C23#45511yescharacteristic
235C23#45511yescharacteristic
236C23#45611yescharacteristic
237C23#45611yescharacteristic
238Q8—161no—
239Q8—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
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—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C8—82no—
313C8—82no—
314C8—82no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22#165911yescharacteristic
328C22#166011yescharacteristic
329C22#166411yescharacteristic
330C22#166411yescharacteristic
331C22#202711yescharacteristic
332C22#202811yescharacteristic
333C22#202811yescharacteristic
334C22#203111yescharacteristic
335C22#203111yescharacteristic
336C22#203111yescharacteristic
337C22#203211yescharacteristic
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—82no—
345C4—82no—
346C4—82no—
347C4—161no—
348C4—161no—
349C4—161no—
350C4—161no—
351C4—161no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2#1035911yescharacteristic
357C2#1037811yescharacteristic
358C2#1038111yescharacteristic
359C2#1038111yescharacteristic
360C2#1038311yescharacteristic
361C2#1038411yescharacteristic
362C2#1038511yescharacteristic
subgroupG/H
center, socleC23#393
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#170552
#98892
#98882
#96832
#70362