Finite Group Databasebeta

← #100232

Subgroups — #100232

Order 512.

418 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 86 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#22325C211yesmaximal, characteristic
2#21579C211yesmaximal, characteristic
3#8707C211yesmaximal, characteristic
4#8701C211yesmaximal, characteristic
5#7600C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C2×#483C411yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1985—21no—
12#1928—21no—
13#1924—21no—
14#1924—21no—
15#1919—21no—
16#1917—21no—
17#1916—21no—
18#1872C411yescharacteristic
19#1852C411yescharacteristic
20#1823C411yescharacteristic
21#1817C2211yescharacteristic
22#1817C2211yescharacteristic
23#1797C2211yescharacteristic
24#1776—21no—
25#1771—21no—
26#1763—21no—
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1279—21no—
32#1279—21no—
33#1272—21no—
34#1271—21no—
35C2×C22⋊Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41#483—22no—
42#483—22no—
43C2×#107—21no—
44C2×#106—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8C2×C411yescharacteristic
54C2×C22⋊C8C2×C411yescharacteristic
55C2×C22⋊C8D411yescharacteristic
56C2×C22⋊C8D411yescharacteristic
57C2×#97—21no—
58C2×#97—21no—
59C2×#97—21no—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8D411yescharacteristic
62#391D411yescharacteristic
63#389D411yescharacteristic
64#387—21no—
65#385—21no—
66#380C2×C411yescharacteristic
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74C2×#95—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90C22×Q8C22⋊C411yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—41no—
103C22×C8—41no—
104C22⋊Q8—21no—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C4×Q8—41no—
109C4×Q8—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4M1612yes—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—82no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138#107—42no—
139#106—24no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—42no—
143C4⋊C8—81no—
144#103—24no—
145#103—42no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148C22⋊C8—21no—
149C22⋊C8—42no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—81no—
155C22⋊C8M1612yes—
156#97—41no—
157#97—41no—
158#97—42no—
159#97—81no—
160C4×C8—41no—
161C4×C8—41no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—41no—
168#95—41no—
169C24C4×D411yescharacteristic
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
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—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4C2×M1611yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4#13111yescharacteristic
207C22×C4#13111yescharacteristic
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—81no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—81no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—82no—
263C42—82no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#38611yescharacteristic
271C23#43511yescharacteristic
272C23#43611yescharacteristic
273Q8—42no—
274Q8—81no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4#40711yescharacteristic
320C2×C4#40811yescharacteristic
321C2×C4#43511yescharacteristic
322C2×C4#43611yescharacteristic
323C2×C4#45811yescharacteristic
324C2×C4#45811yescharacteristic
325C2×C4#47111yescharacteristic
326C2×C4#47311yescharacteristic
327C8—42no—
328C8—42no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—81no—
333C8—161no—
334C8—161no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22#187011yescharacteristic
346C22#189011yescharacteristic
347C22#189111yescharacteristic
348C22#190311yescharacteristic
349C22#190511yescharacteristic
350C22#190611yescharacteristic
351C22#190811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—82no—
366C4—82no—
367C4—81no—
368C2—41no—
369C2—41no—
370C2#896811yescharacteristic
371C2#909411yescharacteristic
372C2#909811yescharacteristic
373C2#952111yescharacteristic
374C2#952611yescharacteristic
375C2#952811yescharacteristic
376C2#953211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
#223252
#215792
#87072
#87012
#76002
#70332
#70212