Finite Group Databasebeta

← #140603

Subgroups — #140603

Order 512.

698 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 142 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×#1954C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3C4×C22⋊C16C211yesmaximal, characteristic, complemented
4C2×#1314C214yesmaximal, complemented
5C22×C4×D4C411yescharacteristic
6C22×C4⋊C8C411yescharacteristic
7C22×C22⋊C8C412yes—
8C22×C22⋊C8C2212yes—
9C22×C4×C8C2211yescharacteristic
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×C22⋊C16C2212yes—
13C2×C4×C16—21no—
14#1954—21no—
15#1954—21no—
16#1954—21no—
17#1954—21no—
18#1374—21no—
19#1374—21no—
20#1314—28no—
21C23×D4C811yescharacteristic
22C24×C4C2×C412yes—
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C412yes—
25C23×C8C2311yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4—21no—
28C2×C4×D4—21no—
29C2×C4×D4—21no—
30C2×C4×D4—42no—
31C2×C4×D4—42no—
32C22×C4⋊C4C811yescharacteristic
33C22×C22⋊C4C812yes—
34C22×C42C2×C411yescharacteristic
35C22×C16—22no—
36C22×C16—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—42no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—44no—
47C2×C4×C8—42no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C4⋊C16—42no—
53C22⋊C16—22no—
54C22⋊C16—24no—
55C22⋊C16D414yescomplemented
56C4×C16—42no—
57C25C2×C812yes—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4—22no—
61C22×D4—42no—
62C22×D4—42no—
63C22×D4C1614yescomplemented
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—24no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C2×C811yescharacteristic
73C23×C4C2×C811yescharacteristic
74C23×C4C2×C812yes—
75C23×C4C22×C411yescharacteristic
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—24no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D412yes—
90C22×C8#4112yes—
91C4×D4—44no—
92C4×D4—44no—
93C4×D4—42no—
94C4×D4—42no—
95C4×D4—42no—
96C4×D4—42no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—42no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—44no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22⋊C411yescharacteristic
107C2×C42M1611yescharacteristic
108C2×C42M1611yescharacteristic
109C2×C16—24no—
110C2×C16—42no—
111C2×C16—42no—
112C2×C16—42no—
113C2×C16—42no—
114C2×C16—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C22⋊C8—44no—
118C22⋊C8—44no—
119C22⋊C8—44no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—41no—
125C4×C8—41no—
126C24—22no—
127C24—22no—
128C24—24no—
129C24—24no—
130C24—42no—
131C24—42no—
132C24—42no—
133C24—42no—
134C24C2×C1614yes—
135C24C22×C811yescharacteristic
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—48no—
141C2×D4—48no—
142C2×D4—42no—
143C2×D4—42no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—41no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4—84no—
170C22×C4C22⋊C811yescharacteristic
171C22×C4C22⋊C811yescharacteristic
172C22×C4C22⋊C811yescharacteristic
173C22×C4C22⋊C811yescharacteristic
174C22×C4C2×C1612yes—
175C22×C4M3212yes—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4C4×D412yes—
178C22×C4C2×M1611yescharacteristic
179C22×C4#13112yes—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—44no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—84no—
196C2×C8#10012yes—
197C2×C8#12112yes—
198C2×C8#12411yescharacteristic
199C2×C8#12711yescharacteristic
200C4⋊C4—42no—
201C4⋊C4—42no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C42—22no—
205C42—22no—
206C42—21no—
207C42—21no—
208C42#9912yes—
209C42#10012yes—
210C16—44no—
211C16—82no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—24no—
217C23—28no—
218C23—44no—
219C23—44no—
220C23—44no—
221C23—44no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—44no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—84no—
230C23—84no—
231C23—84no—
232C23—84no—
233C23C2×C22⋊C811yescharacteristic
234C23C8×D412yes—
235C23C22×C1611yescharacteristic
236C23C2×M3211yescharacteristic
237C23#50412yes—
238D4—48no—
239D4—48no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—88no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4#32312yes—
266C2×C4#32312yes—
267C2×C4C22⋊C1612yes—
268C2×C4C22⋊C1612yes—
269C2×C4#34912yes—
270C2×C4#39011yescharacteristic
271C2×C4#40811yescharacteristic
272C2×C4C2×#9911yescharacteristic
273C2×C4#41011yescharacteristic
274C2×C4C2×#10011yescharacteristic
275C2×C4C2×#10011yescharacteristic
276C2×C4#43612yes—
277C8—22no—
278C8—21no—
279C8—21no—
280C8—22no—
281C8—41no—
282C8—82no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—88no—
295C22—88no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22C2×#32311yescharacteristic
301C22#146011yescharacteristic
302C22#195411yescharacteristic
303C22C2×C22⋊C1611yescharacteristic
304C22#211311yescharacteristic
305C22C2×#34911yescharacteristic
306C22#211511yescharacteristic
307C22D4×C1612yes—
308C22#216912yes—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—41no—
313C4—82no—
314C4#131414yescomplemented
315C4#146911yescharacteristic
316C4#147111yescharacteristic
317C4#196411yescharacteristic
318C4#196511yescharacteristic
319C2—22no—
320C2—22no—
321C2—84no—
322C2#964911yescharacteristic
323C2C2×#131411yescharacteristic
324C2#1141911yescharacteristic
325C2#1148711yescharacteristic
326C2#1148911yescharacteristic
327C2#1318911yescharacteristic
328C2#1319411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. C4×C22⋊C16C2semidirect
4. C2×#1314C2semidirect
55. C22⋊C16D4semidirect
63. C22×D4C16semidirect
314. C4#1314semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
#127832
C4×C22⋊C162
C2×#13142