Finite Group Databasebeta

← #143772

Subgroups — #143772

Order 512.

392 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×#2227C211yesmaximal, characteristic
2#28265C211yesmaximal, characteristic
3C2×#1676C211yesmaximal, characteristic
4#13282C211yesmaximal, characteristic
5C2×Q8⋊C16C211yesmaximal, characteristic
6C2×#1332C211yesmaximal, characteristic
7#7274C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#477—21no—
10C2×C8×Q8C2211yescharacteristic
11C2×C4×Q16C2211yescharacteristic
12#2582—21no—
13#2227—22no—
14#2227—22no—
15C2×#366—21no—
16C2×C4⋊C16C2211yescharacteristic
17C2×#358—21no—
18C2×C4×C16C2211yescharacteristic
19#2068—22no—
20#1969—21no—
21#1942—21no—
22#1845C2211yescharacteristic
23#1803—21no—
24#1676—22no—
25#1676—22no—
26C2×#335C2211yescharacteristic
27C2×#333—21no—
28#1380—22no—
29Q8⋊C16—22no—
30#1332—22no—
31C22×Q16D411yescharacteristic
32C2×C4⋊Q8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8D411yescharacteristic
35C22×C16—21no—
36C22×C16—21no—
37#500—21no—
38#500—21no—
39#500—21no—
40#500D412yes—
41#477—24no—
42#477—42no—
43C8×Q8—21no—
44C8×Q8—21no—
45C8×Q8—21no—
46C8×Q8—21no—
47C4×Q16—21no—
48C4×Q16—21no—
49C4×Q16—21no—
50C4×Q16—21no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—41no—
55C2×#107D411yescharacteristic
56C2×#107D412yes—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—21no—
61C2×#103—22no—
62C2×#103—41no—
63C2×C4×C8—21no—
64C2×C4×C8C2311yescharacteristic
65#398—42no—
66#391—22no—
67#391—41no—
68#389—22no—
69#389—41no—
70#384—21no—
71#366—24no—
72C4⋊C16—21no—
73C4⋊C16D412yes—
74#358—24no—
75C4×C16—21no—
76C4×C16—21no—
77#340—22no—
78#335—21no—
79#335D412yes—
80#333—24no—
81C22×Q8—21no—
82C22×Q8—41no—
83C22×Q8D811yescharacteristic
84C2×Q16—21no—
85C2×Q16—21no—
86C2×Q16—21no—
87C2×Q16—21no—
88C2×Q16—21no—
89C2×Q16Q1612yes—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C4⋊Q8—22no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4D811yescharacteristic
116C2×C42—21no—
117C2×C42—41no—
118C2×C42C2×D411yescharacteristic
119C2×C16—22no—
120C2×C16—22no—
121C2×C16—41no—
122C2×C16—41no—
123C2×C16—41no—
124C2×C16—41no—
125#107—22no—
126#107—21no—
127#107—22no—
128#107—21no—
129#107—44no—
130#107—41no—
131#107—41no—
132#107—41no—
133#107—41no—
134#107—42no—
135C4⋊C8—21no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141#103—44no—
142#103—41no—
143#103—41no—
144#103—82no—
145C4×C8—41no—
146C4×C8—41no—
147C4×C8C2×D411yescharacteristic
148C4×C8C2×D411yescharacteristic
149C4×C8C2×D411yescharacteristic
150C4×C8#4111yescharacteristic
151#95—42no—
152#95—42no—
153#95—42no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—81no—
167C2×Q8—161no—
168C2×Q8—161no—
169C2×Q8Q3212yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—81no—
176C22×C4#12011yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4AGL(1,ℤ/8)11yescharacteristic
179Q16—22no—
180Q16—22no—
181Q16—41no—
182Q16—41no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8#12011yescharacteristic
192C2×C8#12111yescharacteristic
193C2×C8#12111yescharacteristic
194C2×C8#12711yescharacteristic
195C2×C8C2×Q1611yescharacteristic
196C2×C8#13511yescharacteristic
197C2×C8#13711yescharacteristic
198C2×C8#13711yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—161no—
216C4⋊C4—162no—
217C42—41no—
218C42—41no—
219C42—81no—
220C42—81no—
221C42#12011yescharacteristic
222C42#12111yescharacteristic
223C42#13511yescharacteristic
224C42AGL(1,ℤ/8)11yescharacteristic
225C16—42no—
226C16—42no—
227C23#44711yescharacteristic
228Q8—22no—
229Q8—22no—
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235Q8—81no—
236Q8—162no—
237Q8—162no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—161no—
257C2×C4—161no—
258C2×C4#39211yescharacteristic
259C2×C4#44711yescharacteristic
260C2×C4#44911yescharacteristic
261C2×C4#44911yescharacteristic
262C2×C4#45111yescharacteristic
263C2×C4#45211yescharacteristic
264C2×C4#45511yescharacteristic
265C2×C4#45611yescharacteristic
266C2×C4#46611yescharacteristic
267C2×C4#46911yescharacteristic
268C2×C4C2×Q3211yescharacteristic
269C2×C4#50811yescharacteristic
270C2×C4#51011yescharacteristic
271C2×C4#51011yescharacteristic
272C8—21no—
273C8—21no—
274C8—82no—
275C8—81no—
276C8#45111yescharacteristic
277C8#45211yescharacteristic
278C8#46711yescharacteristic
279C8#47011yescharacteristic
280C22#164111yescharacteristic
281C22#164911yescharacteristic
282C22#199911yescharacteristic
283C22#218811yescharacteristic
284C22#218911yescharacteristic
285C22#219211yescharacteristic
286C22#219311yescharacteristic
287C4—21no—
288C4—21no—
289C4—42no—
290C4—82no—
291C4—81no—
292C4—162no—
293C4—161no—
294C4—162no—
295C4#164411yescharacteristic
296C4#165211yescharacteristic
297C4#200111yescharacteristic
298C4#200711yescharacteristic
299C4#218811yescharacteristic
300C4#218911yescharacteristic
301C4#221611yescharacteristic
302C4#221911yescharacteristic
303C2#1023311yescharacteristic
304C2#1238511yescharacteristic
305C2#1238611yescharacteristic
306C2#1240911yescharacteristic
307C2#1241011yescharacteristic
308C2#1330611yescharacteristic
309C2#1330711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×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
C2×#22272
#282652
C2×#16762
#132822
C2×Q8⋊C162
C2×#13322
#72742