Finite Group Databasebeta

← #120512

Subgroups — #120512

Order 512.

746 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 134 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
1#21948C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3#10737C212yesmaximal
4#8051C212yesmaximal
5#7982C211yesmaximal, characteristic
6C23×M16C2211yescharacteristic
7C22×C4⋊C8—21no—
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10#2280C2212yes—
11C2×C4×C22⋊C4C2211yescharacteristic
12#1855—24no—
13#1851—22no—
14#1851C414yescomplemented
15#1810—21no—
16#1810—21no—
17C2×#344—24no—
18C2×#336—21no—
19#1297—24no—
20#1289—22no—
21C24×C4C2311yescharacteristic
22C22×M16—21no—
23C22×M16—22no—
24C22×M16—22no—
25C22×M16—44no—
26C22×M16C2×C412yes—
27C22×M16D414yes—
28C23×C8—21no—
29C23×C8D411yescharacteristic
30C23×C8D411yescharacteristic
31C2×#117—22no—
32C22×C22⋊C4—21no—
33C22×C42—21no—
34C2×#108—22no—
35C2×#108—22no—
36C2×C4⋊C8—24no—
37C2×C4⋊C8—44no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8C2×C412yes—
47C2×C22⋊C8D412yes—
48#381Q814yes—
49#380—22no—
50#380—22no—
51#379—22no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4C2×C412yes—
57C4×C22⋊C4D412yes—
58C2×#95Q811yescharacteristic
59C2×#95Q811yescharacteristic
60#344—48no—
61#341—22no—
62#341—22no—
63#336—24no—
64C25#4111yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×Q811yescharacteristic
75C23×C4C2×Q812yes—
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4111yescharacteristic
79C2×M16—28no—
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—44no—
84C2×M16—44no—
85C2×M16—48no—
86C2×M16—44no—
87C2×M16—42no—
88C2×M16—44no—
89C2×M16—44no—
90C22×C8—24no—
91C22×C8—22no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—24no—
97C22×C8—42no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8C22⋊C412yes—
103C22×C8C22⋊C412yes—
104#117—44no—
105#117—44no—
106C2×C4⋊C4—42no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118#108—82no—
119#108—82no—
120C4⋊C8—48no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127#95—21no—
128#95—21no—
129#95—24no—
130#95—24no—
131#95C4⋊C412yes—
132#95C4⋊C412yes—
133C24—42no—
134C24—42no—
135C24—41no—
136C24—41no—
137C24#11711yescharacteristic
138C24#12111yescharacteristic
139C24C22⋊Q811yescharacteristic
140C24C22⋊Q812yes—
141C24#12612yes—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C2×C4⋊C411yescharacteristic
168C22×C4C4×D412yes—
169C22×C4C4×Q812yes—
170C22×C4#12011yescharacteristic
171C22×C4#12112yes—
172C22×C4#12112yes—
173C22×C4#12111yescharacteristic
174C22×C4C22⋊Q812yes—
175C22×C4C22⋊Q812yes—
176C22×C4C22⋊Q811yescharacteristic
177C22×C4#12311yescharacteristic
178C22×C4#12312yes—
179C22×C4#12411yescharacteristic
180C22×C4#12512yes—
181C22×C4#12512yes—
182C22×C4C4⋊Q812yes—
183C22×C4C4⋊Q811yescharacteristic
184M16—28no—
185M16—48no—
186M16—44no—
187M16—44no—
188M16—44no—
189M16—84no—
190C2×C8—24no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—44no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—48no—
202C2×C8—44no—
203C2×C8—81no—
204C2×C8—81no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—48no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—84no—
213C42—42no—
214C42—42no—
215C42—44no—
216C42—84no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—24no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—24no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—84no—
236C23—84no—
237C23#38011yescharacteristic
238C23#38712yes—
239C23#39511yescharacteristic
240C23#39612yes—
241C23#40011yescharacteristic
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4#38611yescharacteristic
275C2×C4#38912yes—
276C2×C4#39111yescharacteristic
277C2×C4#39411yescharacteristic
278C2×C4#39812yes—
279C2×C4#39811yescharacteristic
280C2×C4B(4,𝔽2)12yes—
281C2×C4#45812yes—
282C8—44no—
283C8—42no—
284C8—42no—
285C8—84no—
286C22—24no—
287C22—24no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—84no—
305C22—84no—
306C22#144111yescharacteristic
307C22#181411yescharacteristic
308C22#181411yescharacteristic
309C22#206212yes—
310C22#210212yes—
311C4—22no—
312C4—21no—
313C4—22no—
314C4—21no—
315C4—44no—
316C4—82no—
317C4—164no—
318C4#188911yescharacteristic
319C4#189011yescharacteristic
320C4#204412yes—
321C2—21no—
322C2—21no—
323C2—22no—
324C2—22no—
325C2—42no—
326C2—42no—
327C2#1122211yescharacteristic
328C2#1124011yescharacteristic
329C2#1124411yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1441
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1851C4semidirect

Maximal subgroups

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

subgroupindex
#219482
C2×#18512
#107372
#80512
#79822