Finite Group Databasebeta

← #124757

Subgroups — #124757

Order 512.

462 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C4×#466C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3#7874C211yesmaximal, characteristic, complemented
4#7121C212yesmaximal, complemented
5#7112C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×C4×D8C411yescharacteristic
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10C4×#121—21no—
11C2×C4⋊C16—21no—
12C2×C22⋊C16—21no—
13#1970C2211yescharacteristic
14#1942C2211yescharacteristic
15#1934—21no—
16#1893—21no—
17#1808C411yescharacteristic
18C4×#107C411yescharacteristic
19C4×#102—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×D8C2×C411yescharacteristic
24C22×M16—41no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#121—21no—
28C2×C4×D4—21no—
29C2×C4×D4—41no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#466—22no—
33#466—22no—
34C4×D8—42no—
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38#423—24no—
39#423—42no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—41no—
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#390—21no—
51#389—21no—
52#388—41no—
53#386—41no—
54#380—21no—
55C4×C4⋊C4—21no—
56C4×C22⋊C4—41no—
57C4⋊C16—42no—
58C22⋊C16—42no—
59#340—21no—
60#340—21no—
61#340—21no—
62C22×D4—21no—
63C22×D4—41no—
64C23×C4—41no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×D8—21no—
69C2×D8—41no—
70C2×D8—41no—
71C2×D8M1612yescomplemented
72C2×M16—44no—
73C2×M16—82no—
74C2×M16—82no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—42no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22×C411yescharacteristic
87C22×C8C2×D411yescharacteristic
88#121—24no—
89#121—42no—
90C4×D4—42no—
91C4×D4—42no—
92C4×D4—84no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—41no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42D811yescharacteristic
107C2×C42SD1611yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C16—42no—
110C2×C16—42no—
111C2×C16—42no—
112#107—21no—
113#107—41no—
114#107—41no—
115#107M1612yescomplemented
116C4⋊C8—24no—
117C4⋊C8—42no—
118C4⋊C8—44no—
119#102—24no—
120#102—42no—
121#102—41no—
122#102—41no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8—41no—
126C4×C8—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130C24—41no—
131C24#10211yescharacteristic
132C2×D4—22no—
133C2×D4—44no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—82no—
140C2×D4—82no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4#10211yescharacteristic
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4C2×SD1611yescharacteristic
166D8—42no—
167D8—42no—
168M16—88no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—44no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8C2×M1611yescharacteristic
187C2×C8C2×M1611yescharacteristic
188C2×C8AGL(1,ℤ/8)11yescharacteristic
189C2×C8#13711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—82no—
197C22⋊C4—44no—
198C22⋊C4—82no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—22no—
204C42—22no—
205C42—21no—
206C42—21no—
207C42—82no—
208C42#9912yes—
209C42#10412yes—
210C16—82no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—42no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23—81no—
223C23—81no—
224C23#36012yes—
225C23C2×#10211yescharacteristic
226D4—42no—
227D4—42no—
228D4—88no—
229D4—81no—
230D4—81no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—164no—
258C2×C4#32712yes—
259C2×C4#32712yes—
260C2×C4#35912yes—
261C2×C4#36112yes—
262C2×C4#40711yescharacteristic
263C2×C4C2×#9911yescharacteristic
264C2×C4#41311yescharacteristic
265C2×C4#41711yescharacteristic
266C2×C4C2×#10411yescharacteristic
267C2×C4#42111yescharacteristic
268C2×C4#50911yescharacteristic
269C2×C4#51011yescharacteristic
270C8—21no—
271C8—21no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—84no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—161no—
288C22—161no—
289C22—161no—
290C22—161no—
291C22#148611yescharacteristic
292C22C2×#32711yescharacteristic
293C22#154711yescharacteristic
294C22#213911yescharacteristic
295C22C2×#35911yescharacteristic
296C22C2×#36011yescharacteristic
297C22C2×#36111yescharacteristic
298C4—21no—
299C4—22no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—82no—
304C4—161no—
305C4—161no—
306C4#134914yes—
307C4#146511yescharacteristic
308C4#148911yescharacteristic
309C4#151311yescharacteristic
310C4#152911yescharacteristic
311C2—42no—
312C2—162no—
313C2#761611yescharacteristic
314C2#1166011yescharacteristic
315C2#1166311yescharacteristic
316C2#1170411yescharacteristic
317C2C2×#134911yescharacteristic
318C2#1197711yescharacteristic
319C2#1198611yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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. #12783C2semidirect
3. #7874C2semidirect
4. #7121C2semidirect
5. #7112C2semidirect
71. C2×D8M16semidirect
115. #107M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
#127832
#78742
#71212
#71122