Finite Group Databasebeta

← #100006

Subgroups — #100006

Order 512.

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

normal subgroups only

This group has 392 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#22084C211yesmaximal, characteristic
3#9879C211yesmaximal, characteristic
4#7593C211yesmaximal, characteristic
5#7583C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C2×#483C411yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#1985—21no—
12#1944—21no—
13#1940—21no—
14#1935—21no—
15#1924—21no—
16#1924—21no—
17#1920—21no—
18#1917—21no—
19#1917—21no—
20#1872C411yescharacteristic
21#1863—21no—
22#1852C411yescharacteristic
23#1823C411yescharacteristic
24#1817C2211yescharacteristic
25#1817C2211yescharacteristic
26#1797C2211yescharacteristic
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29#1280—21no—
30#1280—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#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×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
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—81no—
109C2×C4⋊C4—21no—
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—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4M1612yes—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—81no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136#107—81no—
137#106—24no—
138C4⋊C8—81no—
139C4⋊C8—81no—
140C4⋊C8—81no—
141#103—24no—
142#103—81no—
143C22⋊C8—21no—
144C22⋊C8—22no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—81no—
151C22⋊C8—81no—
152C22⋊C8M1612yes—
153#97—81no—
154#97—81no—
155#97—81no—
156C4×C8—81no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—41no—
163#95—41no—
164C24C4×D411yescharacteristic
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—41no—
168C2×Q8—41no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
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—42no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4C2×M1611yescharacteristic
198C22×C4C2×M1611yescharacteristic
199C22×C4#13111yescharacteristic
200C22×C4#13111yescharacteristic
201C2×C8—22no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
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—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—81no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—41no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23#38611yescharacteristic
260C23#43511yescharacteristic
261C23#43611yescharacteristic
262Q8—42no—
263Q8—81no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—162no—
307C2×C4#40711yescharacteristic
308C2×C4#40811yescharacteristic
309C2×C4#43511yescharacteristic
310C2×C4#43611yescharacteristic
311C2×C4#45511yescharacteristic
312C2×C4#45611yescharacteristic
313C8—42no—
314C8—81no—
315C8—81no—
316C8—81no—
317C8—81no—
318C8—81no—
319C8—82no—
320C8—161no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#187011yescharacteristic
332C22#188111yescharacteristic
333C22#188211yescharacteristic
334C22#190311yescharacteristic
335C22#190511yescharacteristic
336C22#191311yescharacteristic
337C22#191511yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—82no—
350C4—81no—
351C4—81no—
352C4—161no—
353C2—41no—
354C2—41no—
355C2#895511yescharacteristic
356C2#909411yescharacteristic
357C2#911011yescharacteristic
358C2#933211yescharacteristic
359C2#933611yescharacteristic
360C2#933811yescharacteristic
361C2#934211yescharacteristic
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
#220842
#98792
#75932
#75832
#70332
#70212