Finite Group Databasebeta

← #100706

Subgroups — #100706

Order 512.

353 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 80 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#29987C211yesmaximal, characteristic
2#22838C211yesmaximal, characteristic
3#7720C211yesmaximal, characteristic, complemented
4#7683C211yesmaximal, characteristic, complemented
5#7580C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8#2859C2211yescharacteristic
9#2293C2211yescharacteristic
10#2100C411yescharacteristic
11#2099C411yescharacteristic
12#2094C411yescharacteristic
13#2093C411yescharacteristic
14#2059—21no—
15#2025—21no—
16#1955—21no—
17#1949—21no—
18#1924—21no—
19#1923—21no—
20#1839C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1818—21no—
24#1804—21no—
25#1771—21no—
26#1767C2211yescharacteristic
27#1750—21no—
28#1744C2211yescharacteristic
29#1271—21no—
30#1271—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C2×#125—21no—
36C2×C4×D4D411yescharacteristic
37C2×#117—21no—
38C2×#107—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×#97—21no—
50C2×#97—21no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53#390C2×C411yescharacteristic
54#390D411yescharacteristic
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389C2×C411yescharacteristic
60#389D411yescharacteristic
61#388—21no—
62#388—21no—
63#384C2×C411yescharacteristic
64#384C2×C411yescharacteristic
65#382—21no—
66C4×C4⋊C4D411yescharacteristic
67C4×C22⋊C4—21no—
68C43C2311yescharacteristic
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#336—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#336C2×C411yescharacteristic
83#336C2×C411yescharacteristic
84#336D411yescharacteristic
85#336D411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88C22×D4C22⋊C411yescharacteristic
89C23×C4—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—41no—
97C22×C8—41no—
98#125—81no—
99#125—81no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104#117—81no—
105#117—81no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42C22×C411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132#107—81no—
133#106—81no—
134C4⋊C8—42no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139#102—81no—
140#102—81no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143C22⋊C8—81no—
144C22⋊C8—81no—
145#97—41no—
146#97—41no—
147#97—81no—
148C4×C8—22no—
149C4×C8—41no—
150C4×C8—81no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—41no—
155#95—41no—
156C24—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
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×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—42no—
183C2×C8—41no—
184C2×C8—41no—
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—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—21no—
228C42—21no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42#13511yescharacteristic
237C42#13511yescharacteristic
238C42#13711yescharacteristic
239C42#13711yescharacteristic
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#38611yescharacteristic
248D4—42no—
249D4—81no—
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—21no—
266C2×C4—21no—
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—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4#41611yescharacteristic
287C2×C4#41911yescharacteristic
288C2×C4#44311yescharacteristic
289C2×C4#44411yescharacteristic
290C2×C4#45211yescharacteristic
291C2×C4#45211yescharacteristic
292C2×C4#48511yescharacteristic
293C2×C4#48511yescharacteristic
294C8—42no—
295C8—81no—
296C8—81no—
297C8—81no—
298C8—81no—
299C8—81no—
300C8—82no—
301C8—161no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#187311yescharacteristic
310C22#187611yescharacteristic
311C22#188111yescharacteristic
312C22#190411yescharacteristic
313C22#190611yescharacteristic
314C22#191011yescharacteristic
315C22#191511yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—161no—
331C4—161no—
332C2—161no—
333C2#899411yescharacteristic
334C2#917711yescharacteristic
335C2#919411yescharacteristic
336C2#926611yescharacteristic
337C2#928111yescharacteristic
338C2#933511yescharacteristic
339C2#937011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7720C2semidirect
4. #7683C2semidirect
6. #7041C2semidirect
7. #7018C2semidirect

Maximal subgroups

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

subgroupindex
#299872
#228382
#77202
#76832
#75802
#70412
#70182