Finite Group Databasebeta

← #124049

Subgroups — #124049

Order 512.

830 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 146 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
2#7906C212yesmaximal, complemented
3#7104C212yesmaximal, complemented
4#7103C212yesmaximal
5C22×C4×D4C411yescharacteristic
6C2×#431—22no—
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C22⋊C4—22no—
9C2×#376C2211yescharacteristic
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1439—22no—
13#1438C411yescharacteristic
14#1438C2211yescharacteristic
15#1280—22no—
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⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C22⋊C4—21no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#431—28no—
33#431—44no—
34C2×C4⋊C8—42no—
35C2×C22⋊C8—24no—
36C2×C4×C8—42no—
37C4×C4⋊C4—42no—
38C4×C22⋊C4—44no—
39C4×C22⋊C4—44no—
40#376—41no—
41#376—41no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95C2×C411yescharacteristic
45C2×#95C2×C411yescharacteristic
46C2×#95C2×C411yescharacteristic
47C2×#95D412yes—
48C2×#95C2311yescharacteristic
49C25—21no—
50C22×D4—21no—
51C22×D4—41no—
52C22×D4—41no—
53C22×D4—44no—
54C22×D4C2×C812yes—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—24no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—42no—
62C23×C4—44no—
63C23×C4—44no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C22×C8—44no—
70C4×D4—44no—
71C4×D4—44no—
72C4×D4—44no—
73C4×D4—44no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4C2×C812yes—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—44no—
87C2×C22⋊C4—44no—
88C2×C22⋊C4—44no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42—21no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42—42no—
98C2×C42—44no—
99C2×C42C2×C812yes—
100C2×C42M1612yes—
101C2×C42D814yes—
102C2×C42SD1614yes—
103C4⋊C8—48no—
104C22⋊C8—28no—
105C22⋊C8—44no—
106C4×C8—48no—
107#95—41no—
108#95—42no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—44no—
113#95—42no—
114#95—42no—
115C24—24no—
116C24—22no—
117C24—42no—
118C24—42no—
119C24—42no—
120C24—41no—
121C24—41no—
122C24—42no—
123C24—42no—
124C24C2×C22⋊C411yescharacteristic
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—48no—
130C2×D4—44no—
131C2×D4—44no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—416no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4C22⋊C812yes—
170C22×C4C22⋊C812yes—
171C22×C4#9911yescharacteristic
172C22×C4#9911yescharacteristic
173C22×C4#10012yes—
174C22×C4#10214yes—
175C22×C4#10214yes—
176C22×C4#10414yes—
177C22×C4C22×C811yescharacteristic
178C22×C4C2×M1611yescharacteristic
179C22×C4C2×D812yes—
180C22×C4C2×SD1612yes—
181C2×C8—48no—
182C2×C8—84no—
183C2×C8—84no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—48no—
187C22⋊C4—48no—
188C22⋊C4—416no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—24no—
196C42—24no—
197C42—24no—
198C42—24no—
199C42—84no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—24no—
204C23—44no—
205C23—44no—
206C23—44no—
207C23—42no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—42no—
213C23—44no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23—82no—
223C23—81no—
224C23C2×C22⋊C811yescharacteristic
225C23C2×#9911yescharacteristic
226C23C2×#10011yescharacteristic
227C23C2×#10212yes—
228C23C2×#10412yes—
229D4—44no—
230D4—44no—
231D4—48no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—88no—
271C2×C4—88no—
272C2×C4#32312yes—
273C2×C4#32312yes—
274C2×C4D4⋊C818yes—
275C2×C4#32714yes—
276C2×C4#32714yes—
277C2×C4#32914yes—
278C2×C4#33114yes—
279C8—88no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—48no—
289C22—44no—
290C22—84no—
291C22—84no—
292C22—84no—
293C22—82no—
294C22—82no—
295C22—84no—
296C22—84no—
297C22C2×#32311yescharacteristic
298C22C2×D4⋊C812yes—
299C22C2×#32712yes—
300C22C2×#32911yescharacteristic
301C22C2×#33111yescharacteristic
302C22#212012yes—
303C22#212611yescharacteristic
304C22#213211yescharacteristic
305C4—41no—
306C4—41no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—44no—
311C4—82no—
312C4—82no—
313C4—82no—
314C4#127014yes—
315C4#127014yes—
316C2—21no—
317C2—22no—
318C2—21no—
319C2—84no—
320C2C2×#127011yescharacteristic
321C2#1142511yescharacteristic
322C2#1143111yescharacteristic
323C2#1169112yes—
324C2#1169712yes—
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×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7906C2semidirect
3. #7104C2semidirect
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
#79062
#71042
#71032