Finite Group Databasebeta

← #130801

Subgroups — #130801

Order 512.

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

normal subgroups only

This group has 601 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
1C2×#3336C211yesmaximal, characteristic
2#12866C212yesmaximal, complemented
3#11887C212yesmaximal, complemented
4#11465C211yesmaximal, characteristic
5#8609C211yesmaximal, characteristic
6C2×#552—21no—
7#3336—24no—
8#3336—24no—
9C2×#499C411yescharacteristic, complemented
10C2×#485—21no—
11C2×#483—21no—
12C2×#479—21no—
13C2×#467C411yescharacteristic, complemented
14C2×#467C2211yescharacteristic
15C2×C4×Q16C411yescharacteristic, complemented
16C2×C8×D4C2211yescharacteristic
17C2×#427C411yescharacteristic, 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#552—28no—
31#552—44no—
32C2×#126—41no—
33C2×#125—21no—
34C2×#125—41no—
35C2×#123—41no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×D4D411yescharacteristic
40C2×#117—21no—
41C2×#117—41no—
42C22×C4⋊C4—21no—
43C22×C16—22no—
44#499—22no—
45#499—22no—
46#485—28no—
47#483—28no—
48#479—28no—
49#467—22no—
50#467—22no—
51#467—22no—
52#467D414yes—
53C4×Q16—22no—
54C4×Q16—22no—
55C8×D4—22no—
56C8×D4—22no—
57#427—22no—
58#427—22no—
59C2×#107—21no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107C2×C411yescharacteristic
63C2×#107C2×C411yescharacteristic
64C2×#107C2×C411yescharacteristic
65C2×#107D414yes—
66C2×#106—21no—
67C2×#106—21no—
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×D4D811yescharacteristic
83C23×C4—41no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×Q16—21no—
87C2×Q16—42no—
88C2×Q16C22⋊C412yes—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22×C411yescharacteristic
96C22×C8C2×D412yes—
97C22×C8C2×D412yes—
98C22×C8C2×D411yescharacteristic
99#126—48no—
100#125—24no—
101#125—42no—
102#125—48no—
103#123—48no—
104C22⋊Q8—24no—
105C22⋊Q8—24no—
106C22⋊Q8—42no—
107C22⋊Q8—42no—
108C4×Q8—24no—
109C4×Q8—42no—
110C4×D4—22no—
111C4×D4—22no—
112#117—24no—
113#117—48no—
114#117—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4SD1611yescharacteristic
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4D811yescharacteristic
133C2×C22⋊C4SD1611yescharacteristic
134C2×C42—41no—
135C2×C42—41no—
136C2×C42C22⋊C411yescharacteristic
137C2×C16—44no—
138C2×C16—42no—
139#107—21no—
140#107—21no—
141#107—21no—
142#107—24no—
143#107—44no—
144#107C22⋊C412yes—
145#107C22⋊C412yes—
146#107C22⋊C412yes—
147#106—24no—
148#106—24no—
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—42no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—44no—
180C22×C4—82no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4#10211yescharacteristic
185C22×C4#10211yescharacteristic
186C22×C4#10211yescharacteristic
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#12012yes—
189C22×C4C2×SD1611yescharacteristic
190C22×C4AGL(1,ℤ/8)12yes—
191Q16—42no—
192Q16—42no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8C2×C22⋊C411yescharacteristic
204C2×C8C2×C22⋊C411yescharacteristic
205C2×C8#12012yes—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—44no—
231C42—44no—
232C16—82no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C2×#10211yescharacteristic
241C23#44712yes—
242C23#50812yes—
243C23#50911yescharacteristic
244C23#51011yescharacteristic
245Q8—42no—
246Q8—42no—
247Q8—82no—
248D4—42no—
249D4—42no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
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—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—162no—
279C2×C4—162no—
280C2×C4—162no—
281C2×C4#37911yescharacteristic
282C2×C4#41811yescharacteristic
283C2×C4#45012yes—
284C8—22no—
285C8—21no—
286C8—41no—
287C8—41no—
288C8#41012yes—
289C22—21no—
290C22—22no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#178611yescharacteristic
302C22#213811yescharacteristic
303C22#213911yescharacteristic
304C22#218612yes—
305C22#218912yes—
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—82no—
314C4—82no—
315C4—322no—
316C4#178211yescharacteristic
317C4#179611yescharacteristic
318C2—22no—
319C2—21no—
320C2—21no—
321C2—22no—
322C2—81no—
323C2#837611yescharacteristic
324C2#1273311yescharacteristic
325C2#1274211yescharacteristic
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. #12866C2semidirect
3. #11887C2semidirect
9. C2×#499C4semidirect
13. C2×#467C4semidirect
15. C2×C4×Q16C4semidirect
17. C2×#427C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33362
#128662
#118872
#114652
#86092