Finite Group Databasebeta

← #138282

Subgroups — #138282

Order 512.

470 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 102 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#33130C211yesmaximal, characteristic
2C2×C4×#108C211yesmaximal, characteristic
3#12844C211yesmaximal, characteristic
4C2×#1350C212yesmaximal
5#7259C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×#108—21no—
8C2×C4×M16—21no—
9C22×C4×C8C2211yescharacteristic
10#2855—21no—
11C2×C22⋊C16C2212yes—
12#1969C411yescharacteristic
13#1969C411yescharacteristic
14#1942C411yescharacteristic
15#1942C411yescharacteristic
16#1933—21no—
17#1933—21no—
18#1892—21no—
19#1808C2211yescharacteristic
20C4×#108—24no—
21C2×#341C2212yes—
22#1374—21no—
23#1374—21no—
24#1350—24no—
25C22×Q16C2×C411yescharacteristic
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×C22⋊Q8—21no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—22no—
33#467—22no—
34C2×#108—24no—
35C2×#108—24no—
36C2×#108—24no—
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C4×M16—24no—
44C4×M16—42no—
45C2×#97—41no—
46C2×C4×C8—22no—
47C2×C4×C8—22no—
48C2×C4×C8—41no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#391—21no—
54#391—21no—
55#389—21no—
56#389—21no—
57#387—41no—
58#387—41no—
59#385—41no—
60#381—41no—
61#380—21no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64#341—22no—
65#341D414yes—
66#340—21no—
67#340—21no—
68#340—21no—
69C22×Q8—21no—
70C23×C4SD1611yescharacteristic
71C23×C4SD1611yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×Q16—22no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—42no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22×C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D412yes—
94C22×C8#4112yes—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98C22⋊Q8—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113C2×C42D811yescharacteristic
114C2×C42D811yescharacteristic
115C2×C16—42no—
116C2×C16—42no—
117C2×C16—42no—
118#108—216no—
119#107—22no—
120#107—41no—
121#107—41no—
122#103—42no—
123#103—42no—
124#103—42no—
125#97—44no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—44no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138C24C2×SD1611yescharacteristic
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4#10211yescharacteristic
162C22×C4#10211yescharacteristic
163C22×C4#10211yescharacteristic
164C22×C4#10211yescharacteristic
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4#12011yescharacteristic
167C22×C4#12311yescharacteristic
168C22×C4C2×D811yescharacteristic
169C22×C4AGL(1,ℤ/8)11yescharacteristic
170C22×C4#13711yescharacteristic
171Q16—82no—
172M16—28no—
173M16—44no—
174M16—44no—
175M16—44no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8#10412yes—
192C2×C8#10412yes—
193C2×C8C4×D412yes—
194C2×C8#12112yes—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C16—82no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23C2×#10211yescharacteristic
224C23#44811yescharacteristic
225C23#48111yescharacteristic
226Q8—81no—
227Q8—81no—
228Q8—81no—
229Q8—81no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
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—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4#35912yes—
267C2×C4#36212yes—
268C2×C4#38611yescharacteristic
269C2×C4#41011yescharacteristic
270C2×C4#41011yescharacteristic
271C2×C4#41711yescharacteristic
272C2×C4C2×#10411yescharacteristic
273C2×C4C2×#10411yescharacteristic
274C2×C4#44711yescharacteristic
275C2×C4#48011yescharacteristic
276C8—22no—
277C8—22no—
278C8—22no—
279C8—22no—
280C8—48no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#150211yescharacteristic
292C22#150211yescharacteristic
293C22#187411yescharacteristic
294C22C2×#35911yescharacteristic
295C22#214311yescharacteristic
296C22C2×#36211yescharacteristic
297C22#214811yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—161no—
305C4—161no—
306C4—161no—
307C4—161no—
308C4#150911yescharacteristic
309C4#150911yescharacteristic
310C4#189611yescharacteristic
311C4#189911yescharacteristic
312C2—41no—
313C2—41no—
314C2#914811yescharacteristic
315C2#1174811yescharacteristic
316C2#1174911yescharacteristic
317C2#1175011yescharacteristic
318C2#1175211yescharacteristic
319C2#1299011yescharacteristic
320C2#1300311yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#331302
C2×C4×#1082
#128442
C2×#13502
#72592