Finite Group Databasebeta

← #101500

Subgroups — #101500

Order 512.

359 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 82 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#29278C211yesmaximal, characteristic
2#22381C211yesmaximal, characteristic
3#9677C211yesmaximal, characteristic
4#8582C211yesmaximal, characteristic
5#7622C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C4×C4⋊Q8C2211yescharacteristic
9C4×#125C2211yescharacteristic
10#2100C411yescharacteristic
11#2100C411yescharacteristic
12#2058C411yescharacteristic
13#2057C411yescharacteristic
14#1989—21no—
15#1949—21no—
16#1940—21no—
17#1919—21no—
18#1918—21no—
19#1917—21no—
20#1917—21no—
21#1917—21no—
22#1916—21no—
23#1839C2211yescharacteristic
24#1831C2211yescharacteristic
25#1771—21no—
26#1767C2211yescharacteristic
27#1767C2211yescharacteristic
28C4×C4⋊C8C2211yescharacteristic
29C2×#330—21no—
30C2×Q8⋊C8—21no—
31C2×Q8⋊C8—21no—
32#1295—21no—
33#1271—21no—
34#1271—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×#125D411yescharacteristic
37C2×C4×Q8—21no—
38C2×#107—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×#97—21no—
47C2×#97—21no—
48C2×#97—21no—
49C2×#97—21no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#391—21no—
53#389—21no—
54#389—21no—
55#389C2×C411yescharacteristic
56#389C2×C411yescharacteristic
57#387—21no—
58#387—21no—
59#384C2×C411yescharacteristic
60#382D411yescharacteristic
61C4×C4⋊C4—21no—
62C4×C4⋊C4—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4D411yescharacteristic
66C4×C4⋊C4D411yescharacteristic
67C43C2311yescharacteristic
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#330—42no—
89Q8⋊C8—42no—
90Q8⋊C8—42no—
91C22×Q8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—42no—
101#125—21no—
102#125—21no—
103#125—21no—
104#125—21no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
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—41no—
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—41no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133#107—42no—
134#106—42no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—81no—
144#97—42no—
145#97—41no—
146#97—41no—
147#97—41no—
148#97—41no—
149#97—81no—
150C4×C8—41no—
151C4×C8—41no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—41no—
159#95—41no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12311yescharacteristic
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—81no—
239C42—81no—
240C42AGL(1,ℤ/8)11yescharacteristic
241C42#13711yescharacteristic
242C42#13711yescharacteristic
243C42#13711yescharacteristic
244C23#38611yescharacteristic
245Q8—42no—
246Q8—42no—
247Q8—81no—
248Q8—81no—
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—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
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—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4#41711yescharacteristic
284C2×C4#41911yescharacteristic
285C2×C4#44311yescharacteristic
286C2×C4#44411yescharacteristic
287C2×C4#45311yescharacteristic
288C2×C4#45411yescharacteristic
289C2×C4#46811yescharacteristic
290C2×C4#46811yescharacteristic
291C2×C4#47011yescharacteristic
292C2×C4#47011yescharacteristic
293C8—42no—
294C8—81no—
295C8—81no—
296C8—81no—
297C8—81no—
298C8—81no—
299C8—82no—
300C8—82no—
301C22#187611yescharacteristic
302C22#188311yescharacteristic
303C22#189111yescharacteristic
304C22#189411yescharacteristic
305C22#190211yescharacteristic
306C22#190611yescharacteristic
307C22#191211yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—161no—
326C2#905211yescharacteristic
327C2#920911yescharacteristic
328C2#928611yescharacteristic
329C2#932111yescharacteristic
330C2#938211yescharacteristic
331C2#948811yescharacteristic
332C2#953111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#292782
#223812
#96772
#85822
#76222
#70412
#70222