Finite Group Databasebeta

← #136639

Subgroups — #136639

Order 512.

466 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C4×#467C211yesmaximal, characteristic
2C2×#1566C211yesmaximal, characteristic
3#12599C211yesmaximal, characteristic
4C2×#1334C212yesmaximal, complemented
5#7178C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×C4×Q16C2211yescharacteristic
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×C22⋊C16—21no—
12C2×#346C2212yes—
13#1969C411yescharacteristic
14#1942C411yescharacteristic
15#1933—21no—
16#1892—21no—
17#1808C411yescharacteristic
18C4×#107C2211yescharacteristic
19C4×#103—21no—
20#1566—24no—
21C2×#334—21no—
22C2×#334C2212yes—
23#1374—21no—
24#1334—24no—
25C22×Q16C2×C411yescharacteristic
26C22×M16—41no—
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C22⋊Q8—21no—
30C2×C4×Q8—21no—
31C22×C42D411yescharacteristic
32C22×C16—22no—
33#467—22no—
34#467—22no—
35C4×Q16—22no—
36C4×Q16—22no—
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40#423—24no—
41#423—42no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—41no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C4×C8—22no—
48C2×C4×C8—22no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—21no—
54#389—21no—
55#387—41no—
56#385—41no—
57#380—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4—41no—
60C4×C22⋊C4—41no—
61C22⋊C16—24no—
62#346—22no—
63#346—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#334—24no—
68#334—22no—
69#334D414yes—
70C22×Q8—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4D811yescharacteristic
73C23×C4SD1611yescharacteristic
74C2×Q16—22no—
75C2×Q16—21no—
76C2×Q16—22no—
77C2×Q16M1612yescomplemented
78C2×M16—44no—
79C2×M16—82no—
80C2×M16—82no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—42no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22×C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22⋊Q8—24no—
94C22⋊Q8—42no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×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×C42—41no—
112C2×C42—41no—
113C2×C42C22⋊C411yescharacteristic
114C2×C42D811yescharacteristic
115C2×C42SD1611yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C16—24no—
118C2×C16—42no—
119C2×C16—42no—
120#107—21no—
121#107—22no—
122#107—22no—
123#107M1612yescomplemented
124C4⋊C8—24no—
125C4⋊C8—44no—
126C4⋊C8—42no—
127#103—24no—
128#103—42no—
129#103—42no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8C2×D412yes—
135C4×C8#4112yes—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141C24#10211yescharacteristic
142C2×Q8—22no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C2×D811yescharacteristic
171C22×C4C2×SD1611yescharacteristic
172Q16—24no—
173Q16—42no—
174M16—88no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—44no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8#10412yes—
189C2×C8C4×D412yes—
190C2×C8#12112yes—
191C2×C8C2×M1611yescharacteristic
192C2×C8C2×M1611yescharacteristic
193C2×C8#13112yes—
194C4⋊C4—22no—
195C4⋊C4—44no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—82no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C22⋊C4—44no—
205C22⋊C4—82no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—82no—
213C42—81no—
214C42—81no—
215C42#12011yescharacteristic
216C42#12311yescharacteristic
217C42AGL(1,ℤ/8)11yescharacteristic
218C42#13711yescharacteristic
219C16—44no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23#36012yes—
227C23C2×#10211yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—81no—
231Q8—81no—
232C2×C4—24no—
233C2×C4—24no—
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—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—81no—
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#36012yes—
264C2×C4#36212yes—
265C2×C4#38611yescharacteristic
266C2×C4#40711yescharacteristic
267C2×C4#40811yescharacteristic
268C2×C4#41711yescharacteristic
269C2×C4C2×#10411yescharacteristic
270C2×C4#42111yescharacteristic
271C2×C4#44711yescharacteristic
272C2×C4#44811yescharacteristic
273C2×C4#48011yescharacteristic
274C2×C4#48111yescharacteristic
275C8—22no—
276C8—22no—
277C8—41no—
278C8—84no—
279C8#43512yes—
280C8#43612yes—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22#148611yescharacteristic
289C22#149211yescharacteristic
290C22#187411yescharacteristic
291C22C2×#36011yescharacteristic
292C22C2×#36011yescharacteristic
293C22C2×#36211yescharacteristic
294C22#214811yescharacteristic
295C4—21no—
296C4—22no—
297C4—21no—
298C4—22no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—161no—
305C4—161no—
306C4#148911yescharacteristic
307C4#149511yescharacteristic
308C4#187011yescharacteristic
309C4#187111yescharacteristic
310C2—42no—
311C2#894411yescharacteristic
312C2#1166411yescharacteristic
313C2#1166611yescharacteristic
314C2#1167711yescharacteristic
315C2#1168111yescharacteristic
316C2#1300111yescharacteristic
317C2#1300311yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1334C2semidirect
5. #7178C2semidirect
77. C2×Q16M16semidirect
123. #107M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
C2×#15662
#125992
C2×#13342
#71782