Finite Group Databasebeta

← #101760

Subgroups — #101760

Order 512.

636 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 104 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#28304C211yesmaximal, characteristic
2#13826C211yesmaximal, characteristic
3#9908C212yesmaximal
4#7643C211yesmaximal, characteristic
5#7028C212yesmaximal
6C2×#484—21no—
7C2×#451—21no—
8#2857C2211yescharacteristic
9C2×#391—21no—
10C2×#391C2211yescharacteristic
11#2095C412yescomplemented
12#2016C412yescomplemented
13#1969—22no—
14#1797C2212yes—
15C4×C22⋊C8C2211yescharacteristic
16#1436—22no—
17#1436—21no—
18C4×#95—22no—
19C4×#95C2212yes—
20#1294—22no—
21#1294—22no—
22#1272—22no—
23C23×Q8D411yescharacteristic
24C22×Q16—22no—
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×C22⋊Q8—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42—22no—
32C22×C42C2311yescharacteristic
33#484—42no—
34#484—42no—
35#451—42no—
36#451—42no—
37C2×#107—22no—
38C2×C4⋊C8—22no—
39C2×#103—22no—
40C2×#103—22no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—22no—
43C2×C4×C8—21no—
44#391—28no—
45#391—22no—
46#391—21no—
47#391C2×C412yes—
48#391D412yes—
49#389—22no—
50#384C2×C411yescharacteristic
51#380—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95D412yes—
58C2×#95D412yes—
59#336—22no—
60#336C2×C412yes—
61C22×Q8—24no—
62C22×Q8—21no—
63C22×Q8—21no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—81no—
67C22×Q8C22⋊C412yes—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C2×Q16—44no—
75C2×Q16—44no—
76C2×Q16—82no—
77C22×C8—22no—
78C22×C8—42no—
79C4⋊Q8—81no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4C22⋊C412yes—
95C2×C22⋊C4—41no—
96C2×C42—24no—
97C2×C42—44no—
98C2×C42—44no—
99C2×C42—42no—
100C2×C42—42no—
101C2×C42—81no—
102C2×C42—81no—
103C2×C42—81no—
104C2×C42—82no—
105C2×C42—81no—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109#107—44no—
110C4⋊C8—82no—
111#103—44no—
112#103—44no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C4×C8—81no—
118#95—28no—
119#95—28no—
120#95—24no—
121#95—24no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—44no—
126#95—42no—
127#95—81no—
128#95Q1614yescomplemented
129C24#12011yescharacteristic
130C2×Q8—24no—
131C2×Q8—22no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8—81no—
140C2×Q8—84no—
141C2×Q8—84no—
142C2×Q8#10414yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D412yes—
179C22×C4#12112yes—
180C22×C4#12311yescharacteristic
181C22×C4C2×Q1612yes—
182C22×C4#13512yes—
183C22×C4#13712yes—
184C22×C4#13712yes—
185Q16—88no—
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8—82no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—81no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—84no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—41no—
204C42—42no—
205C42—41no—
206C42—84no—
207C42—84no—
208C42—84no—
209C42—84no—
210C23—21no—
211C23—22no—
212C23—21no—
213C23#38611yescharacteristic
214C23#45112yes—
215C23#45212yes—
216C23#45311yescharacteristic
217C23#45311yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—82no—
221Q8—82no—
222Q8—84no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—28no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4C2×#10412yes—
270C2×C4C4×Q1612yes—
271C2×C4#44112yes—
272C2×C4#46212yes—
273C2×C4#46412yes—
274C8—44no—
275C8—162no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22#163212yes—
285C22#163311yescharacteristic
286C22#163411yescharacteristic
287C22#187812yes—
288C22#188711yescharacteristic
289C22#189611yescharacteristic
290C22#190612yes—
291C22#191311yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—22no—
297C4—48no—
298C4—41no—
299C4—44no—
300C4—84no—
301C4—82no—
302C4—161no—
303C4—161no—
304C2—22no—
305C2—21no—
306C2—21no—
307C2#906611yescharacteristic
308C2#923212yes—
309C2#927612yes—
310C2#940911yescharacteristic
311C2#944111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2095C4semidirect
12. #2016C4semidirect
128. #95Q16semidirect

Maximal subgroups

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

subgroupindex
#283042
#138262
#99082
#76432
#70282