Finite Group Databasebeta

← #129496

Subgroups — #129496

Order 512.

958 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#436C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3#12599C211yesmaximal, characteristic, complemented
4C2×#1330C214yesmaximal, complemented
5C23×M16—21no—
6C22×C4×D4C411yescharacteristic
7C2×#436—28no—
8C2×#436—21no—
9C2×#436—21no—
10C2×#436C412yes—
11C2×#436C2212yes—
12C22×C4⋊C8C411yescharacteristic
13C22×C22⋊C8—21no—
14C22×C4×C8C2211yescharacteristic
15C2×C4⋊C16C2212yescomplemented
16C2×C22⋊C16—21no—
17C2×C22⋊C16—21no—
18C2×#346C2212yescomplemented
19#1374—21no—
20#1374—21no—
21#1330—28no—
22C23×D4C811yescharacteristic
23C24×C4—21no—
24C22×M16—28no—
25C22×M16—22no—
26C22×M16—22no—
27C22×M16—44no—
28C22×M16—42no—
29C22×M16—42no—
30C23×C8—21no—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×C4×D4—24no—
34C2×C4×D4—24no—
35C2×C4×D4—21no—
36C2×C4×D4—21no—
37C2×C4×D4C812yes—
38C2×C4×D4C2×C412yes—
39C22×C4⋊C4C811yescharacteristic
40C22×C22⋊C4—21no—
41C22×C42C2×C411yescharacteristic
42C22×C16—22no—
43C22×C16—22no—
44#436—28no—
45#436—28no—
46#436—24no—
47#436—24no—
48#436—24no—
49#436D418yes—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8C2×C412yes—
53C2×C22⋊C8—28no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C4×C8—24no—
57C2×C4×C8C2×C411yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C4⋊C16—24no—
62C22⋊C16—24no—
63C22⋊C16—42no—
64#346—24no—
65C25—21no—
66C22×D4—24no—
67C22×D4—21no—
68C22×D4—42no—
69C22×D4—42no—
70C22×D4C2×C812yes—
71C23×C4—24no—
72C23×C4—24no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C2×C811yescharacteristic
82C23×C4M1611yescharacteristic
83C2×M16—28no—
84C2×M16—28no—
85C2×M16—44no—
86C2×M16—416no—
87C2×M16—416no—
88C2×M16—44no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—44no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—41no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8C22⋊C411yescharacteristic
102C22×C8C22⋊C411yescharacteristic
103C22×C8D812yes—
104C22×C8SD1612yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4×D4—24no—
108C4×D4—28no—
109C4×D4—24no—
110C4×D4—24no—
111C4×D4—24no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4C22⋊C414yes—
115C4×D4M1614yes—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4C2×C812yes—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42C22⋊C411yescharacteristic
125C2×C42C2×C811yescharacteristic
126C2×C42M1611yescharacteristic
127C2×C42C22×C411yescharacteristic
128C2×C16—24no—
129C2×C16—42no—
130C2×C16—42no—
131C2×C16—42no—
132C2×C16—42no—
133C2×C16—42no—
134C4⋊C8—24no—
135C4⋊C8—22no—
136C4⋊C8C22⋊C414yes—
137C22⋊C8—28no—
138C22⋊C8—28no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8C2×D414yes—
143C24—24no—
144C24—22no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24C22⋊C811yescharacteristic
152C2×D4—24no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—416no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4C22⋊C814yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—48no—
176C22×C4—44no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C22⋊C811yescharacteristic
188C22×C4C22⋊C811yescharacteristic
189C22×C4C22⋊C811yescharacteristic
190C22×C4#10212yes—
191C22×C4#10412yes—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C22×C811yescharacteristic
194C22×C4C2×M1611yescharacteristic
195M16—416no—
196M16—48no—
197M16—48no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—21no—
205C2×C8—24no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—44no—
209C2×C8—48no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8#10212yes—
213C2×C8#12012yes—
214C2×C8#12012yes—
215C2×C8C2×D811yescharacteristic
216C2×C8C2×SD1611yescharacteristic
217C2×C8AGL(1,ℤ/8)12yes—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4C22⋊C814yes—
221C22⋊C4—28no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C42—21no—
225C42—22no—
226C42—21no—
227C42—22no—
228C42C2×C22⋊C412yes—
229C42C2×M1612yes—
230C16—44no—
231C16—82no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—24no—
237C23—48no—
238C23—48no—
239C23—48no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—84no—
247C23—84no—
248C23—84no—
249C23—84no—
250C23D4⋊C812yes—
251C23#34912yes—
252C23#35012yes—
253C23C2×C22⋊C811yescharacteristic
254D4—44no—
255D4—48no—
256D4—44no—
257C2×C4—28no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—44no—
275C2×C4—416no—
276C2×C4—48no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4D4⋊C812yes—
284C2×C4#34912yes—
285C2×C4#37911yescharacteristic
286C2×C4C2×C22⋊C812yes—
287C2×C4#40712yes—
288C2×C4#40711yescharacteristic
289C2×C4C2×#10211yescharacteristic
290C2×C4#41811yescharacteristic
291C2×C4C2×#10411yescharacteristic
292C2×C4#42111yescharacteristic
293C8—22no—
294C8—22no—
295C8—44no—
296C8—41no—
297C8#44712yes—
298C8#45012yes—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—816no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22#133014yescomplemented
314C22C2×D4⋊C811yescharacteristic
315C22#147811yescharacteristic
316C22#177911yescharacteristic
317C22C2×#34911yescharacteristic
318C22C2×#34911yescharacteristic
319C22C2×#35011yescharacteristic
320C22#211711yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—41no—
327C4#148611yescharacteristic
328C4#149011yescharacteristic
329C4#178611yescharacteristic
330C4#178911yescharacteristic
331C2—24no—
332C2—84no—
333C2#831011yescharacteristic
334C2C2×#133011yescharacteristic
335C2#1158711yescharacteristic
336C2#1163411yescharacteristic
337C2#1163611yescharacteristic
338C2#1268111yescharacteristic
339C2#1268311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. #12599C2semidirect
4. C2×#1330C2semidirect
15. C2×C4⋊C16C22semidirect
18. C2×#346C22semidirect
313. C22#1330semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
#127832
#125992
C2×#13302