Finite Group Databasebeta

← #130805

Subgroups — #130805

Order 512.

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

normal subgroups only

This group has 600 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×#3338C211yesmaximal, characteristic
2#12871C212yesmaximal, complemented
3#11896C212yesmaximal, complemented
4#11466C211yesmaximal, characteristic
5#8611C211yesmaximal, characteristic
6C2×#554—21no—
7#3338—24no—
8#3338—24no—
9C2×#500C411yescharacteristic
10C2×#484—21no—
11C2×#484—21no—
12C2×#477—21no—
13C2×#467C411yescharacteristic
14C2×#467C2211yescharacteristic
15C2×C4×Q16C411yescharacteristic
16C2×C8×D4C2211yescharacteristic
17C22×#107C411yescharacteristic
18C22×#107C2211yescharacteristic
19#2112C2212yes—
20#1810C2212yes—
21#1465—21no—
22#1387—24no—
23#1350—24no—
24#1322—21no—
25#1314—21no—
26#1293—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—
43C2×M32—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×#108—22no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—21no—
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—
72#407—22no—
73C2×C22⋊C8D411yescharacteristic
74C2×C22⋊C8D411yescharacteristic
75C2×C4×C8C2×C411yescharacteristic
76C22⋊C16—22no—
77C22⋊C16—22no—
78#341—22no—
79#324—41no—
80#323—41no—
81C22×Q8—21no—
82C22×Q8—41no—
83C22×D4SD1611yescharacteristic
84C23×C4—41no—
85C23×C4—41no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×Q16—21no—
89C2×Q16—42no—
90C2×Q16C22⋊C412yes—
91C2×M16—42no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22×C411yescharacteristic
99C22×C8C2×D412yes—
100C22×C8C2×D412yes—
101C22×C8C2×D411yescharacteristic
102C4⋊Q8—24no—
103C4⋊Q8—42no—
104#125—48no—
105C22⋊Q8—24no—
106C22⋊Q8—24no—
107C22⋊Q8—48no—
108C22⋊Q8—48no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C4×Q8—24no—
112C4×Q8—42no—
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—48no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4D811yescharacteristic
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4D811yescharacteristic
133C2×C22⋊C4SD1611yescharacteristic
134C2×C42—41no—
135C2×C42C22⋊C411yescharacteristic
136M32—44no—
137C2×C16—42no—
138#108—44no—
139#107—24no—
140#107—24no—
141#107—21no—
142#107—21no—
143#107—21no—
144#107—24no—
145#107C22⋊C412yes—
146#107C22⋊C412yes—
147#107C22⋊C412yes—
148C4⋊C8—22no—
149#103—24no—
150#103—24no—
151#103—24no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C4×C8—21no—
157C4×C8—21no—
158C24#10211yescharacteristic
159C24C2×SD1611yescharacteristic
160C2×Q8—22no—
161C2×Q8—44no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—162no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
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—44no—
181C22×C4—162no—
182C22×C4—162no—
183C22×C4#10211yescharacteristic
184C22×C4#10211yescharacteristic
185C22×C4#10211yescharacteristic
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4#12012yes—
188C22×C4C2×D811yescharacteristic
189C22×C4AGL(1,ℤ/8)12yes—
190Q16—42no—
191Q16—42no—
192M16—82no—
193M16—82no—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—82no—
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—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C42—21no—
226C42—21no—
227C42—44no—
228C16—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23#36212yes—
238C23C2×#10211yescharacteristic
239C23#45012yes—
240Q8—42no—
241Q8—42no—
242Q8—82no—
243Q8—164no—
244D4—28no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—164no—
269C2×C4—164no—
270C2×C4#36212yes—
271C2×C4#37911yescharacteristic
272C2×C4#41811yescharacteristic
273C2×C4#44712yes—
274C8—22no—
275C8—21no—
276C8—41no—
277C8—41no—
278C8—162no—
279C8#41012yes—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22#178611yescharacteristic
293C22C2×#36211yescharacteristic
294C22C2×#36211yescharacteristic
295C4—22no—
296C4—21no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4—82no—
303C4—82no—
304C4#178211yescharacteristic
305C4#179611yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#837611yescharacteristic
311C2#1273711yescharacteristic
312C2#1274311yescharacteristic
subgroupG/H
center, socleC22#1786
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4SD16
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. #12871C2semidirect
3. #11896C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33382
#128712
#118962
#114662
#86112