Finite Group Databasebeta

← #124590

Subgroups — #124590

Order 512.

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

normal subgroups only

This group has 412 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
1C8×#121C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic, complemented
3#12783C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C2×C8×D4—21no—
7C2×C8×D4C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C4×#121C411yescharacteristic
10C2×C4⋊C16—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16—21no—
13#1954C2211yescharacteristic
14#1939C2211yescharacteristic
15#1923—21no—
16#1869—21no—
17#1797C411yescharacteristic
18C8×C4⋊C4C411yescharacteristic
19C8×C22⋊C4—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121C811yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C16—22no—
31C22×C16—22no—
32C8×D4—42no—
33C8×D4—44no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#390C811yescharacteristic
48#389C811yescharacteristic
49#388—21no—
50#386—21no—
51#380C811yescharacteristic
52C4×C4⋊C4C2×C411yescharacteristic
53C4×C22⋊C4—21no—
54C4⋊C16—42no—
55C22⋊C16—42no—
56C22⋊C16—42no—
57#336—21no—
58#336—21no—
59#336—21no—
60C22×D4—21no—
61C22×D4C2×C811yescharacteristic
62C23×C4—21no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×C811yescharacteristic
65C23×C4M1611yescharacteristic
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82#121—22no—
83#121—22no—
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—42no—
87C4×D4—42no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2×C811yescharacteristic
90C2×C4⋊C4C2×C811yescharacteristic
91C2×C4⋊C4C2×C811yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C2×C811yescharacteristic
102C2×C42M1611yescharacteristic
103C2×C42C22×C411yescharacteristic
104C2×C16—42no—
105C2×C16—42no—
106C2×C16—42no—
107C2×C16—42no—
108C2×C16—42no—
109C2×C16—42no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—41no—
116C22⋊C8—42no—
117C22⋊C8—41no—
118C4×C8—22no—
119C4×C8—22no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8—42no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C22⋊C811yescharacteristic
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4M3212yes—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4C22⋊C811yescharacteristic
158C22×C4C22⋊C811yescharacteristic
159C22×C4C22⋊C811yescharacteristic
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C22×C811yescharacteristic
162C22×C4C2×M1611yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—44no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—84no—
182C2×C8#9912yes—
183C2×C8AGL(1,ℤ/8)11yescharacteristic
184C2×C8AGL(1,ℤ/8)12yes—
185C2×C8#13711yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4M3212yes—
191C22⋊C4—24no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—41no—
201C42—41no—
202C42—41no—
203C42—41no—
204C16—82no—
205C16—82no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23#35012yes—
220C23C2×C22⋊C811yescharacteristic
221D4—42no—
222D4—42no—
223D4—48no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4#35012yes—
258C2×C4C2×#9911yescharacteristic
259C2×C4#41311yescharacteristic
260C2×C4#41711yescharacteristic
261C2×C4#41811yescharacteristic
262C2×C4#42111yescharacteristic
263C2×C4#42111yescharacteristic
264C2×C4C2×M3211yescharacteristic
265C2×C4C2×M3211yescharacteristic
266C8—22no—
267C8—22no—
268C8—21no—
269C8—21no—
270C8—41no—
271C8—81no—
272C8—81no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22#145911yescharacteristic
288C22#147711yescharacteristic
289C22#147811yescharacteristic
290C22#211211yescharacteristic
291C22#211511yescharacteristic
292C22C2×#35011yescharacteristic
293C22C2×#35011yescharacteristic
294C4—21no—
295C4—21no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4#151011yescharacteristic
304C4#151311yescharacteristic
305C4#153511yescharacteristic
306C4#155711yescharacteristic
307C2—42no—
308C2—82no—
309C2#759911yescharacteristic
310C2#1146611yescharacteristic
311C2#1146811yescharacteristic
312C2#1162311yescharacteristic
313C2#1162511yescharacteristic
314C2#1163111yescharacteristic
315C2#1163311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12844C2semidirect
3. #12783C2semidirect
4. #7118C2semidirect
5. #7102C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1212
#128442
#127832
#71182
#71022