Finite Group Databasebeta

← #124741

Subgroups — #124741

Order 512.

419 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 84 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×#469C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic, complemented
3#7862C211yesmaximal, characteristic, complemented
4#7127C212yesmaximal, complemented
5#7110C212yesmaximal, complemented
6C2×#469C411yescharacteristic
7C2×C4×D8C411yescharacteristic
8#2920—21no—
9C2×C4×M16C2211yescharacteristic
10C4×#121—21no—
11#2150—21no—
12C2×#349—21no—
13#1972C2211yescharacteristic
14#1944C2211yescharacteristic
15#1936—21no—
16#1895—21no—
17#1809C411yescharacteristic
18C4×#106C411yescharacteristic
19C4×#102—21no—
20#1377C2212yes—
21#1275—21no—
22#1275C2212yes—
23C22×D8C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16D411yescharacteristic
26C2×#121—21no—
27C2×C4×D4—21no—
28C2×C4×D4—41no—
29C22×C42D411yescharacteristic
30C2×M32—22no—
31#469—22no—
32#469—22no—
33C4×D8—41no—
34C4×D8—41no—
35#432—41no—
36#432—41no—
37#431—22no—
38#431—22no—
39C2×#107C2×C411yescharacteristic
40C2×#106C2×C411yescharacteristic
41C2×#106C2×C411yescharacteristic
42#423—41no—
43C2×#102—21no—
44C2×#102—21no—
45C2×#102—21no—
46#407—41no—
47C4×M16—22no—
48C4×M16—41no—
49C4×M16—41no—
50C2×#97D411yescharacteristic
51C2×#97C2311yescharacteristic
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#390—21no—
55#389—21no—
56#388—41no—
57#386—41no—
58#380—21no—
59C4×C4⋊C4—21no—
60C4×C22⋊C4—41no—
61C4⋊C16—41no—
62C4⋊C16—41no—
63#349—42no—
64#340—21no—
65#340—21no—
66#340—21no—
67C22×D4—21no—
68C22×D4—41no—
69C23×C4—41no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4D811yescharacteristic
72C23×C4SD1611yescharacteristic
73C2×D8—21no—
74C2×D8—41no—
75C2×D8—41no—
76C2×D8M1612yescomplemented
77C2×M16—21no—
78C2×M16—21no—
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—42no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D411yescharacteristic
90#121—24no—
91#121—42no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—84no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—41no—
111C2×C42C22⋊C411yescharacteristic
112C2×C42D811yescharacteristic
113C2×C42SD1611yescharacteristic
114C2×C42C2×D411yescharacteristic
115M32—42no—
116M32—42no—
117C2×C16—42no—
118#107—41no—
119#106—21no—
120#106—41no—
121#106M1612yescomplemented
122C4⋊C8—42no—
123C4⋊C8—42no—
124#102—24no—
125#102—42no—
126#102—41no—
127#102—41no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#97—41no—
131#97—41no—
132#97—42no—
133C4×C8—24no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138#95—41no—
139#95—41no—
140#95—41no—
141C24—41no—
142C24#10211yescharacteristic
143C2×D4—22no—
144C2×D4—44no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—82no—
151C2×D4—82no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4#10211yescharacteristic
172C22×C4#10211yescharacteristic
173C22×C4#10211yescharacteristic
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×D811yescharacteristic
176C22×C4C2×SD1611yescharacteristic
177D8—42no—
178D8—42no—
179M16—42no—
180M16—42no—
181M16—42no—
182M16—42no—
183M16—84no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—44no—
195C2×C8C2×M1611yescharacteristic
196C2×C8C2×M1611yescharacteristic
197C2×C8AGL(1,ℤ/8)11yescharacteristic
198C2×C8#13711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C22⋊C4—44no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—82no—
219C42#9911yescharacteristic
220C42#9911yescharacteristic
221C42#10411yescharacteristic
222C42#10411yescharacteristic
223C16—82no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23C2×#10211yescharacteristic
237D4—42no—
238D4—42no—
239D4—88no—
240D4—81no—
241D4—81no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—162no—
269C2×C4—162no—
270C2×C4#32711yescharacteristic
271C2×C4#32711yescharacteristic
272C2×C4#32711yescharacteristic
273C2×C4#32711yescharacteristic
274C2×C4#40711yescharacteristic
275C2×C4C2×#9911yescharacteristic
276C2×C4#41311yescharacteristic
277C2×C4#41711yescharacteristic
278C2×C4C2×#10411yescharacteristic
279C2×C4#42111yescharacteristic
280C8—21no—
281C8—21no—
282C8—21no—
283C8—21no—
284C8—41no—
285C8—41no—
286C8—84no—
287C22—21no—
288C22—21no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—161no—
298C22—161no—
299C22—161no—
300C22—161no—
301C22#148611yescharacteristic
302C22C2×#32711yescharacteristic
303C22#154711yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—82no—
311C4—161no—
312C4—161no—
313C4#133912yes—
314C4#146511yescharacteristic
315C4#148911yescharacteristic
316C4#151311yescharacteristic
317C4#152911yescharacteristic
318C2—21no—
319C2—21no—
320C2—81no—
321C2—162no—
322C2#761611yescharacteristic
323C2C2×#133911yescharacteristic
324C2#1168811yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#469C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12785C2semidirect
3. #7862C2semidirect
4. #7127C2semidirect
5. #7110C2semidirect
76. C2×D8M16semidirect
121. #106M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4692
#127852
#78622
#71272
#71102