Finite Group Databasebeta

← #130808

Subgroups — #130808

Order 512.

597 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 96 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#12870C212yesmaximal, complemented
3#11889C212yesmaximal, complemented
4#11466C211yesmaximal, characteristic
5#8608C211yesmaximal, 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×#427C2211yescharacteristic
18C22×#107C411yescharacteristic, complemented
19#2112C2212yes—
20#1807C2212yes—
21#1799—21no—
22#1386—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—
43C2×M32—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#427D414yes—
59C2×#107—21no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107C2×C411yescharacteristic
63C2×#107C2×C411yescharacteristic
64C2×#107C2×C411yescharacteristic
65C2×#106—21no—
66C2×#106—21no—
67C2×#106—22no—
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—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
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
137M32—44no—
138C2×C16—42no—
139#107—24no—
140#107—21no—
141#107—21no—
142#107—21no—
143#107C22⋊C412yes—
144#107C22⋊C412yes—
145#107C22⋊C412yes—
146#106—24no—
147#106—24no—
148#106—44no—
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—41no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4#10211yescharacteristic
183C22×C4#10211yescharacteristic
184C22×C4#10211yescharacteristic
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4#12012yes—
187C22×C4C2×SD1611yescharacteristic
188C22×C4AGL(1,ℤ/8)12yes—
189Q16—42no—
190Q16—42no—
191C2×C8—24no—
192C2×C8—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8C2×C22⋊C411yescharacteristic
202C2×C8C2×C22⋊C411yescharacteristic
203C2×C8#12012yes—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—44no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42—44no—
229C42—44no—
230C16—82no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23#36012yes—
240C23C2×#10211yescharacteristic
241C23#44712yes—
242Q8—42no—
243Q8—42no—
244Q8—82no—
245D4—44no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—88no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—162no—
274C2×C4—162no—
275C2×C4—162no—
276C2×C4#37911yescharacteristic
277C2×C4#41811yescharacteristic
278C2×C4#45012yes—
279C8—22no—
280C8—21no—
281C8—41no—
282C8—41no—
283C8#41012yes—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22#178611yescharacteristic
297C22C2×#36011yescharacteristic
298C22#214511yescharacteristic
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4—82no—
307C4—82no—
308C4—322no—
309C4#178211yescharacteristic
310C4#179611yescharacteristic
311C2—21no—
312C2—21no—
313C2—42no—
314C2—42no—
315C2#837611yescharacteristic
316C2#1273511yescharacteristic
317C2#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. #12870C2semidirect
3. #11889C2semidirect
9. C2×#499C4semidirect
13. C2×#467C4semidirect
15. C2×C4×Q16C4semidirect
18. C22×#107C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33362
#128702
#118892
#114662
#86082