Finite Group Databasebeta

← #101710

Subgroups — #101710

Order 512.

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

normal subgroups only

This group has 362 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#29553C211yesmaximal, characteristic
2#21344C211yesmaximal, characteristic
3#9890C211yesmaximal, characteristic
4#7662C211yesmaximal, characteristic
5#7615C211yesmaximal, characteristic
6#7030C211yesmaximal, characteristic
7#7027C211yesmaximal, characteristic
8#2862C2211yescharacteristic
9#2295C2211yescharacteristic
10#2057C411yescharacteristic
11#2043C411yescharacteristic
12#2024C411yescharacteristic
13#2016C411yescharacteristic
14#1985—21no—
15#1969—21no—
16#1940—21no—
17#1933—21no—
18#1928—21no—
19#1916—22no—
20#1878—21no—
21#1831C2211yescharacteristic
22#1768—21no—
23#1767C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25#1746—21no—
26#1744C2211yescharacteristic
27C4×#97C2211yescharacteristic
28#1295—21no—
29#1295—21no—
30#1272—21no—
31#1272—22no—
32#1271—21no—
33C22×Q16—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×#107—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×#103—21no—
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#97—21no—
47C2×#97—21no—
48C2×#97D411yescharacteristic
49C2×#97D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53#391—21no—
54#391—21no—
55#391—21no—
56#391—21no—
57#391C2×C411yescharacteristic
58#391D411yescharacteristic
59#389C2×C411yescharacteristic
60#389D411yescharacteristic
61#387—21no—
62#387—21no—
63#382—21no—
64#382—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4D411yescharacteristic
67#376—21no—
68C43C2311yescharacteristic
69#340—21no—
70#340—21no—
71#336—22no—
72#336—22no—
73#336—22no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336C2×C411yescharacteristic
79#336C2×C411yescharacteristic
80#336D412yes—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8C22⋊C411yescharacteristic
84C2×Q16—42no—
85C2×Q16—42no—
86C2×Q16—81no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—41no—
93C22×C8—41no—
94C4⋊Q8—81no—
95C4⋊Q8—81no—
96C4×Q8—41no—
97C4×Q8—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C22⋊C411yescharacteristic
106C2×C4⋊C4C22⋊C411yescharacteristic
107C2×C4⋊C4C22⋊C411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117#107—81no—
118C4⋊C8—22no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—44no—
123C4⋊C8—81no—
124#103—42no—
125#103—42no—
126#103—81no—
127#103—81no—
128#97—21no—
129#97—21no—
130#97—21no—
131#97—41no—
132#97—41no—
133#97—41no—
134#97—41no—
135#97Q1612yescomplemented
136C4×C8—42no—
137C4×C8—42no—
138C4×C8—41no—
139C4×C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—81no—
164C2×Q8—81no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12311yescharacteristic
178Q16—84no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—41no—
221C42—41no—
222C42—41no—
223C42—42no—
224C42—42no—
225C42C2×Q1611yescharacteristic
226C42#13511yescharacteristic
227C42#13711yescharacteristic
228C42#13711yescharacteristic
229C23#38611yescharacteristic
230Q8—42no—
231Q8—81no—
232Q8—81no—
233Q8—81no—
234Q8—81no—
235Q8—81no—
236Q8—161no—
237Q8—161no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4#42111yescharacteristic
272C2×C4#42111yescharacteristic
273C2×C4C4×Q1611yescharacteristic
274C2×C4#44111yescharacteristic
275C2×C4#44311yescharacteristic
276C2×C4#44411yescharacteristic
277C2×C4#45111yescharacteristic
278C2×C4#45211yescharacteristic
279C2×C4#46211yescharacteristic
280C2×C4#46411yescharacteristic
281C2×C4#47111yescharacteristic
282C2×C4#47311yescharacteristic
283C8—42no—
284C8—42no—
285C8—42no—
286C8—82no—
287C8—81no—
288C8—82no—
289C8—161no—
290C22#187611yescharacteristic
291C22#187811yescharacteristic
292C22#188411yescharacteristic
293C22#189711yescharacteristic
294C22#190611yescharacteristic
295C22#190811yescharacteristic
296C22#191411yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—161no—
315C4—161no—
316C2#905611yescharacteristic
317C2#921411yescharacteristic
318C2#923611yescharacteristic
319C2#926511yescharacteristic
320C2#926811yescharacteristic
321C2#938911yescharacteristic
322C2#943111yescharacteristic
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

Complements

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

NH ≅ G/Ntype
135. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#295532
#213442
#98902
#76622
#76152
#70302
#70272