Finite Group Databasebeta

← #100008

Subgroups — #100008

Order 512.

402 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 84 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#23344C211yesmaximal, characteristic
2#22085C211yesmaximal, characteristic
3#9650C211yesmaximal, characteristic
4#7593C211yesmaximal, characteristic
5#7583C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C2×#484C411yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#1985—21no—
12#1943—21no—
13#1940—21no—
14#1935—21no—
15#1924—21no—
16#1924—21no—
17#1918—21no—
18#1917—21no—
19#1917—21no—
20#1872C411yescharacteristic
21#1865—21no—
22#1853C411yescharacteristic
23#1824C411yescharacteristic
24#1817C2211yescharacteristic
25#1817C2211yescharacteristic
26#1797C2211yescharacteristic
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29C2×#324—21no—
30C2×#324—21no—
31#1272—21no—
32#1271—21no—
33#1271—21no—
34#1271—21no—
35C2×C22⋊Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41#484—22no—
42#484—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×C8—21no—
61#391D411yescharacteristic
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389D411yescharacteristic
67#387—21no—
68#385—21no—
69#380C2×C411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4D411yescharacteristic
72C4×C22⋊C4—21no—
73C2×#95—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#336—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#336D411yescharacteristic
89#336D411yescharacteristic
90#324—42no—
91#324—42no—
92C22×Q8C22⋊C411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—41no—
105C22×C8—41no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C4×Q8—81no—
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—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—81no—
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4M1612yes—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—81no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141#107—24no—
142#106—81no—
143C4⋊C8—81no—
144C4⋊C8—81no—
145C4⋊C8—81no—
146#103—24no—
147#103—81no—
148C22⋊C8—21no—
149C22⋊C8—22no—
150C22⋊C8—21no—
151C22⋊C8—21no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—81no—
156C22⋊C8—81no—
157C22⋊C8M1612yes—
158#97—81no—
159#97—81no—
160#97—81no—
161C4×C8—81no—
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—22no—
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—81no—
198C22×C4—81no—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4C2×M1611yescharacteristic
207C22×C4#13111yescharacteristic
208C22×C4#13111yescharacteristic
209C2×C8—22no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
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—41no—
227C2×C8—41no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8—81no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—81no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#38611yescharacteristic
270C23#43511yescharacteristic
271C23#43611yescharacteristic
272Q8—42no—
273Q8—81no—
274C2×C4—21no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—161no—
319C2×C4—161no—
320C2×C4#40711yescharacteristic
321C2×C4#40811yescharacteristic
322C2×C4#43511yescharacteristic
323C2×C4#43611yescharacteristic
324C2×C4#45311yescharacteristic
325C2×C4#45411yescharacteristic
326C8—42no—
327C8—81no—
328C8—81no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—82no—
333C8—161no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#187011yescharacteristic
345C22#188111yescharacteristic
346C22#188311yescharacteristic
347C22#190211yescharacteristic
348C22#190511yescharacteristic
349C22#191311yescharacteristic
350C22#191511yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—82no—
365C4—161no—
366C2—41no—
367C2—41no—
368C2#895611yescharacteristic
369C2#909211yescharacteristic
370C2#911011yescharacteristic
371C2#933111yescharacteristic
372C2#933611yescharacteristic
373C2#933911yescharacteristic
374C2#934411yescharacteristic
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
#233442
#220852
#96502
#75932
#75832
#70332
#70212