Finite Group Databasebeta

← #126160

Subgroups — #126160

Order 512.

480 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 62 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#15778C211yesmaximal, characteristic
2#10568C211yesmaximal, characteristic
3#10567C211yesmaximal, characteristic
4#10407C211yesmaximal, characteristic
5#7123C211yesmaximal, characteristic
6#7122C211yesmaximal, characteristic
7#7122C211yesmaximal, characteristic
8#2659C2211yescharacteristic
9C2×#398—21no—
10C2×#389—21no—
11C2×#387—21no—
12#1872—21no—
13#1853—21no—
14#1852—21no—
15C2×#328—21no—
16#1444C2211yescharacteristic
17#1442C2211yescharacteristic
18#1442C2211yescharacteristic
19#1441—21no—
20#1441—21no—
21#1434—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
35C22×C42—21no—
36C2×#107—41no—
37C2×#106—41no—
38C2×#103—41no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42#398—41no—
43#398—44no—
44#395—41no—
45#389—44no—
46#387—42no—
47#387—42no—
48#380—21no—
49#380—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62C2×#95C2311yescharacteristic
63#340—41no—
64#340—41no—
65#340—41no—
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—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C2×C4⋊C4—24no—
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—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—81no—
109C2×C22⋊C4—41no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42—42no—
113#107—82no—
114#106—82no—
115#103—82no—
116C22⋊C8—22no—
117C22⋊C8—22no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95SD1612yes—
141#95Q1612yes—
142C24#12011yescharacteristic
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—81no—
146C2×Q8—81no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
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×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4C2×SD1611yescharacteristic
204C22×C4C2×Q1611yescharacteristic
205C22×C4#13511yescharacteristic
206C22×C4AGL(1,ℤ/8)11yescharacteristic
207C22×C4AGL(1,ℤ/8)11yescharacteristic
208C22×C4#13711yescharacteristic
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C4⋊C4—48no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—84no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—84no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—84no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23#44911yescharacteristic
245C23#45011yescharacteristic
246C23#45111yescharacteristic
247C23#45411yescharacteristic
248C23#45611yescharacteristic
249C23#45611yescharacteristic
250C23B(4,𝔽2)11yescharacteristic
251Q8—82no—
252Q8—161no—
253C2×C4—24no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
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—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C8—82no—
323C8—82no—
324C8—161no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22#159711yescharacteristic
338C22#160411yescharacteristic
339C22#160511yescharacteristic
340C22#163711yescharacteristic
341C22#164811yescharacteristic
342C22#165411yescharacteristic
343C22#166311yescharacteristic
344C22#219611yescharacteristic
345C22#219711yescharacteristic
346C22#220111yescharacteristic
347C22#220411yescharacteristic
348C4—44no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—82no—
356C4—82no—
357C4—82no—
358C4—82no—
359C4—161no—
360C4—161no—
361C4—161no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2#777211yescharacteristic
367C2#1213311yescharacteristic
368C2#1214711yescharacteristic
369C2#1224411yescharacteristic
370C2#1224611yescharacteristic
371C2#1225211yescharacteristic
372C2#1225811yescharacteristic
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
#157782
#105682
#105672
#104072
#71232
#71222