Finite Group Databasebeta

← #124138

Subgroups — #124138

Order 512.

794 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 106 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
1#13847C211yesmaximal, characteristic
2C4×#324C212yesmaximal, complemented
3#7132C212yesmaximal, complemented
4#7103C212yesmaximal
5C22×C4×D4C411yescharacteristic
6C2×C4×C4⋊C4C2211yescharacteristic
7C2×C4×C22⋊C4—22no—
8C2×#376C411yescharacteristic
9C4×C22⋊C8—22no—
10C2×#324—22no—
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1439—22no—
14#1438C411yescharacteristic
15#1438C2211yescharacteristic
16#1280C2214yes—
17C23×D4C2×C411yescharacteristic
18C24×C4—21no—
19C2×C4×D4—22no—
20C2×C4×D4—41no—
21C2×C4×D4—41no—
22C2×C4×D4—42no—
23C2×C4×D4C814yescomplemented
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D412yes—
26C22×C22⋊C4—21no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32C2×C22⋊C8—24no—
33C2×C4×C8—42no—
34C4×C4⋊C4—41no—
35C4×C4⋊C4—41no—
36C4×C22⋊C4—48no—
37#376—42no—
38C2×#95—21no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95C2×C411yescharacteristic
44C2×#95C2311yescharacteristic
45#336—42no—
46#324—28no—
47C25—21no—
48C22×D4—21no—
49C22×D4—41no—
50C22×D4—41no—
51C22×D4—44no—
52C22×D4C2×C812yes—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—24no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—42no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D412yes—
67C22×C8—44no—
68C4×D4—44no—
69C4×D4—44no—
70C4×D4—44no—
71C4×D4—44no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4C2×C812yes—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—44no—
87C2×C22⋊C4—44no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—22no—
91C2×C42—24no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—42no—
95C2×C42—42no—
96C2×C42—44no—
97C2×C42C2×C812yes—
98C2×C42M1612yes—
99C22⋊C8—28no—
100C22⋊C8—44no—
101C4×C8—84no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—42no—
107#95—44no—
108#95—44no—
109C24—24no—
110C24—22no—
111C24—42no—
112C24—42no—
113C24—42no—
114C24—41no—
115C24—41no—
116C24—42no—
117C24—42no—
118C24C2×C22⋊C411yescharacteristic
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—48no—
124C2×D4—44no—
125C2×D4—44no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—416no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4—88no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4C22⋊C812yes—
167C22×C4C22⋊C812yes—
168C22×C4#9911yescharacteristic
169C22×C4#9911yescharacteristic
170C22×C4#10012yes—
171C22×C4#10414yes—
172C22×C4C22×C811yescharacteristic
173C22×C4C2×M1611yescharacteristic
174C22×C4#13514yes—
175C2×C8—48no—
176C2×C8—84no—
177C2×C8—84no—
178C4⋊C4—48no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C22⋊C4—48no—
182C22⋊C4—416no—
183C22⋊C4—416no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—48no—
189C42—44no—
190C42—82no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—24no—
195C23—44no—
196C23—44no—
197C23—44no—
198C23—44no—
199C23—42no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—42no—
204C23—44no—
205C23—82no—
206C23—82no—
207C23—82no—
208C23—82no—
209C23—82no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—82no—
214C23—81no—
215C23C2×C22⋊C811yescharacteristic
216C23C2×#9911yescharacteristic
217C23C2×#10011yescharacteristic
218C23#41612yes—
219C23C2×#10412yes—
220D4—44no—
221D4—44no—
222D4—48no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—88no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—88no—
265C2×C4#32312yes—
266C2×C4#32312yes—
267C2×C4#35312yes—
268C2×C4#35512yes—
269C8—88no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—48no—
279C22—44no—
280C22—84no—
281C22—84no—
282C22—84no—
283C22—82no—
284C22—82no—
285C22—84no—
286C22—84no—
287C22C2×#32311yescharacteristic
288C22#147612yes—
289C22#150112yes—
290C22#152511yescharacteristic
291C22#154311yescharacteristic
292C22#212012yes—
293C22C2×#35311yescharacteristic
294C22C2×#35511yescharacteristic
295C4—22no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—44no—
300C4—44no—
301C4—82no—
302C4—82no—
303C4—82no—
304C4#131814yes—
305C2—21no—
306C2—21no—
307C2—22no—
308C2—84no—
309C2#756311yescharacteristic
310C2#1142511yescharacteristic
311C2C2×#131811yescharacteristic
312C2#1172612yes—
313C2#1173412yes—
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#324C2semidirect
3. #7132C2semidirect
23. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
#138472
C4×#3242
#71322
#71032