Finite Group Databasebeta

← #130800

Subgroups — #130800

Order 512.

620 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 110 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×#3338C211yesmaximal, characteristic
2#12867C212yesmaximal, complemented
3#11887C212yesmaximal, complemented
4#11465C211yesmaximal, characteristic
5#8609C211yesmaximal, characteristic
6C2×#554—21no—
7#3338—24no—
8#3338—24no—
9C2×#500C411yescharacteristic, complemented
10C2×#484—21no—
11C2×#484—21no—
12C2×#477—21no—
13C2×#467C411yescharacteristic, complemented
14C2×#467C2211yescharacteristic
15C2×C4×Q16C411yescharacteristic, complemented
16C2×C8×D4C2211yescharacteristic
17C22×#107C411yescharacteristic, complemented
18C22×#107C2211yescharacteristic
19C2×C22⋊C16C2212yes—
20#1808C2212yes—
21#1799—21no—
22#1380—24no—
23#1348—24no—
24#1322—21no—
25#1314—21no—
26#1289—21no—
27C22×Q16C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2311yescharacteristic
30#554—28no—
31#554—44no—
32C2×C4⋊Q8—21no—
33C2×#125—41no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8—41no—
37C2×C22⋊Q8—41no—
38C2×C4×Q8—21no—
39C2×C4×D4—41no—
40C2×C4×D4D411yescharacteristic
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C16—22no—
44#500—22no—
45#500—22no—
46#484—28no—
47#484—28no—
48#477—28no—
49#467—22no—
50#467—22no—
51#467—22no—
52#467D414yes—
53C4×Q16—22no—
54C4×Q16—22no—
55C8×D4—22no—
56C8×D4—22no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107C2×C411yescharacteristic
65C2×#107C2×C411yescharacteristic
66C2×#107C2×C411yescharacteristic
67C2×#107D414yes—
68C2×C4⋊C8D411yescharacteristic
69C2×#103—21no—
70C2×#103—21no—
71C2×#103—21no—
72C2×#99—41no—
73C2×C22⋊C8D411yescharacteristic
74C2×C22⋊C8D411yescharacteristic
75C2×C4×C8C2×C411yescharacteristic
76#380—22no—
77C22⋊C16—22no—
78C22⋊C16—22no—
79#342—41no—
80#340—22no—
81C22×Q8—21no—
82C22×Q8—41no—
83C22×D4D811yescharacteristic
84C23×C4—41no—
85C23×C4—41no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×Q16—21no—
89C2×Q16—42no—
90C2×Q16C22⋊C412yes—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22×C411yescharacteristic
98C22×C8C2×D412yes—
99C22×C8C2×D412yes—
100C22×C8C2×D411yescharacteristic
101C4⋊Q8—24no—
102C4⋊Q8—42no—
103#125—48no—
104C22⋊Q8—24no—
105C22⋊Q8—24no—
106C22⋊Q8—48no—
107C22⋊Q8—48no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C4×Q8—24no—
111C4×Q8—42no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4—48no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4SD1611yescharacteristic
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4D811yescharacteristic
134C2×C22⋊C4SD1611yescharacteristic
135C2×C42—41no—
136C2×C42C22⋊C411yescharacteristic
137C2×C16—44no—
138C2×C16—42no—
139#107—24no—
140#107—24no—
141#107—21no—
142#107—21no—
143#107—21no—
144#107—24no—
145#107—44no—
146#107C22⋊C412yes—
147#107C22⋊C412yes—
148#107C22⋊C412yes—
149C4⋊C8—22no—
150#103—24no—
151#103—24no—
152#103—24no—
153#99—82no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C4×C8—21no—
157C4×C8—21no—
158#95—42no—
159C24#10211yescharacteristic
160C24C2×D811yescharacteristic
161C2×Q8—22no—
162C2×Q8—44no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—162no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4SD3212yes—
170C2×D4Q3212yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—82no—
184C22×C4—162no—
185C22×C4—162no—
186C22×C4#10211yescharacteristic
187C22×C4#10211yescharacteristic
188C22×C4#10211yescharacteristic
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4#12012yes—
191C22×C4C2×SD1611yescharacteristic
192C22×C4AGL(1,ℤ/8)12yes—
193Q16—42no—
194Q16—42no—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8C2×C22⋊C411yescharacteristic
206C2×C8C2×C22⋊C411yescharacteristic
207C2×C8#12012yes—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—44no—
232C16—82no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23#35812yes—
240C23C2×#10211yescharacteristic
241C23#44712yes—
242C23C2×SD3211yescharacteristic
243C23C2×Q3211yescharacteristic
244C23#51012yes—
245Q8—42no—
246Q8—42no—
247Q8—82no—
248Q8—164no—
249D4—24no—
250D4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—164no—
275C2×C4—164no—
276C2×C4—162no—
277C2×C4—162no—
278C2×C4—162no—
279C2×C4#35812yes—
280C2×C4#37911yescharacteristic
281C2×C4#41811yescharacteristic
282C2×C4#45012yes—
283C8—22no—
284C8—21no—
285C8—41no—
286C8—41no—
287C8#41012yes—
288C22—21no—
289C22—22no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22#178611yescharacteristic
301C22C2×#35811yescharacteristic
302C22#214111yescharacteristic
303C22#218512yes—
304C22#218812yes—
305C4—21no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—82no—
311C4—82no—
312C4—82no—
313C4—82no—
314C4—322no—
315C4#178211yescharacteristic
316C4#179611yescharacteristic
317C2—22no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—42no—
322C2#837611yescharacteristic
323C2#1273211yescharacteristic
324C2#1274111yescharacteristic
subgroupG/H
center, socleC22#1786
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12867C2semidirect
3. #11887C2semidirect
9. C2×#500C4semidirect
13. C2×#467C4semidirect
15. C2×C4×Q16C4semidirect
17. C22×#107C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33382
#128672
#118872
#114652
#86092