Finite Group Databasebeta

← #137233

Subgroups — #137233

Order 512.

506 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×#2075C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic
3#12871C212yesmaximal, complemented
4#11418C211yesmaximal, characteristic
5#7264C212yesmaximal, complemented
6C22×#107C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8C2211yescharacteristic
9C2×#380—21no—
10#2112C2212yes—
11#2075—21no—
12#2075—21no—
13#2075C412yes—
14#2075C412yes—
15#1855—24no—
16#1810—21no—
17#1810C2212yes—
18C2×#340—21no—
19C2×#340C2211yescharacteristic
20#1387—24no—
21#1314—21no—
22#1314—21no—
23C24×C4D411yescharacteristic
24C22×M16—22no—
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4—41no—
30C2×M32—22no—
31C2×#108—22no—
32C2×#108D414yes—
33C2×#107—22no—
34C2×#107—22no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37#407—22no—
38#407—22no—
39#407—41no—
40#407—41no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—41no—
44C2×C22⋊C8—41no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C22⋊C8D411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49#380—22no—
50#380—22no—
51#380—41no—
52#380—41no—
53C2×#95—41no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56#341—22no—
57#341—22no—
58#340—22no—
59#340—22no—
60#340—21no—
61#340C2×C412yes—
62C25SD1611yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4D811yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4SD1611yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—82no—
80C2×M16—82no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D412yes—
90C22×C8C2×D411yescharacteristic
91C22×C8#4112yes—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—84no—
105M32—44no—
106C2×C16—42no—
107#108—44no—
108#108—82no—
109#107—24no—
110#107—42no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#95—42no—
120#95—42no—
121C24—22no—
122C24—21no—
123C24—21no—
124C24—41no—
125C24—42no—
126C24—42no—
127C24#10211yescharacteristic
128C24#10211yescharacteristic
129C24C2×SD1611yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—82no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4#10211yescharacteristic
165C22×C4#10211yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4#12011yescharacteristic
168C22×C4#12311yescharacteristic
169C22×C4C2×D811yescharacteristic
170C22×C4AGL(1,ℤ/8)11yescharacteristic
171C22×C4#13711yescharacteristic
172M16—44no—
173M16—44no—
174M16—84no—
175C2×C8—24no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—44no—
185C2×C8—81no—
186C2×C8—84no—
187C2×C8C4×D412yes—
188C2×C8#12112yes—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—82no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C22⋊C4—84no—
197C22⋊C4—84no—
198C16—82no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—48no—
215C23—42no—
216C23—42no—
217C23—81no—
218C23—81no—
219C23—84no—
220C23#36112yes—
221C23C2×#10211yescharacteristic
222C23#44811yescharacteristic
223C23#48111yescharacteristic
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—164no—
254C2×C4—162no—
255C2×C4—162no—
256C2×C4#38611yescharacteristic
257C2×C4#41711yescharacteristic
258C2×C4#44711yescharacteristic
259C2×C4#45512yes—
260C2×C4#45812yes—
261C2×C4#48011yescharacteristic
262C8—22no—
263C8—22no—
264C8—41no—
265C8—41no—
266C8—84no—
267C22—24no—
268C22—24no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22—84no—
285C22—84no—
286C22—81no—
287C22#160812yes—
288C22#187411yescharacteristic
289C22C2×#36111yescharacteristic
290C22#214811yescharacteristic
291C22#223912yes—
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—162no—
297C4—162no—
298C4#160712yes—
299C4#188511yescharacteristic
300C4#189011yescharacteristic
301C2—21no—
302C2—22no—
303C2—21no—
304C2—22no—
305C2—42no—
306C2—81no—
307C2#902511yescharacteristic
308C2#1300411yescharacteristic
309C2#1300911yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12871C2semidirect
5. #7264C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
#219482
#128712
#114182
#72642