Finite Group Databasebeta

← #120209

Subgroups — #120209

Order 512.

597 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 126 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#21939C211yesmaximal, characteristic
2#21931C211yesmaximal, characteristic
3#8102C212yesmaximal
4#7982C211yesmaximal, characteristic
5#7982C212yesmaximal
6C22×C22⋊C8C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2280C2212yes—
9#2280C2212yes—
10#2280C2211yescharacteristic
11#1854—24no—
12#1850—21no—
13#1850—21no—
14#1850C412yescomplemented
15#1850C412yescomplemented
16#1808—21no—
17#1807—21no—
18#1798—22no—
19#1304—24no—
20#1289—22no—
21#1289—24no—
22C24×C4C2311yescharacteristic
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#107—22no—
31C2×#106—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C412yes—
41C2×C22⋊C8D412yes—
42#381—21no—
43#381—22no—
44#381Q814yes—
45#380—21no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380D414yes—
50#380Q812yes—
51#379—22no—
52#379—22no—
53#379D412yes—
54C4×C22⋊C4—21no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59C4×C22⋊C4C2×C412yes—
60#340—22no—
61#340—22no—
62#336—22no—
63#336—22no—
64C25C2×D411yescharacteristic
65C23×C4—81no—
66C23×C4—81no—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D412yes—
71C23×C4C2×Q812yes—
72C23×C4C2×Q811yescharacteristic
73C23×C4#4112yes—
74C23×C4#4111yescharacteristic
75C23×C4#4111yescharacteristic
76C23×C4#4112yes—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—81no—
86C22×C8—81no—
87C22×C8—81no—
88C22×C8—81no—
89C22×C8C22⋊C412yes—
90C22×C8C22⋊C412yes—
91#117—81no—
92#117—81no—
93#117—82no—
94#117—82no—
95#117—82no—
96#117—82no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4C4⋊C414yes—
109C2×C42—22no—
110C2×C42—42no—
111C2×C42—42no—
112#107—82no—
113#106—82no—
114C4⋊C8—82no—
115C4⋊C8—82no—
116C22⋊C8—24no—
117C22⋊C8—24no—
118C22⋊C8—41no—
119C22⋊C8—42no—
120C22⋊C8—41no—
121#95—22no—
122#95—22no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—42no—
127C24—42no—
128C24—42no—
129C24—81no—
130C24C2×C4⋊C411yescharacteristic
131C24#12111yescharacteristic
132C24#12111yescharacteristic
133C24#12112yes—
134C24C22⋊Q812yes—
135C22×C4—24no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—81no—
149C22×C4—82no—
150C22×C4—81no—
151C22×C4—82no—
152C22×C4—81no—
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4#11711yescharacteristic
155C22×C4C4×D412yes—
156C22×C4C4×Q812yes—
157C22×C4#12011yescharacteristic
158C22×C4#12112yes—
159C22×C4C22⋊Q812yes—
160C22×C4C22⋊Q812yes—
161C22×C4C22⋊Q811yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4#12311yescharacteristic
164C22×C4#12312yes—
165C22×C4#12411yescharacteristic
166C22×C4#12512yes—
167C22×C4#12512yes—
168C22×C4#12612yes—
169C22×C4C4⋊Q811yescharacteristic
170C22×C4C4⋊Q812yes—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—81no—
182C2×C8—81no—
183C2×C8—82no—
184C2×C8—82no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—82no—
193C22⋊C4—84no—
194C22⋊C4—84no—
195C22⋊C4—84no—
196C22⋊C4—84no—
197C22⋊C4—84no—
198C22⋊C4—84no—
199C42—42no—
200C42—42no—
201C42—84no—
202C42—84no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23—88no—
219C23#38011yescharacteristic
220C23#38912yes—
221C23#39411yescharacteristic
222C23#39612yes—
223C23#39811yescharacteristic
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4#38611yescharacteristic
257C2×C4#38712yes—
258C2×C4#39111yescharacteristic
259C2×C4#39511yescharacteristic
260C2×C4#39812yes—
261C2×C4#40011yescharacteristic
262C8—42no—
263C8—42no—
264C8—81no—
265C8—81no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—22no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—21no—
279C22—44no—
280C22—44no—
281C22—82no—
282C22—84no—
283C22—84no—
284C22#144111yescharacteristic
285C22#181212yes—
286C22#205312yes—
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—21no—
292C4—84no—
293C4—164no—
294C4—164no—
295C4#191311yescharacteristic
296C4#191311yescharacteristic
297C2—22no—
298C2—22no—
299C2—42no—
300C2—42no—
301C2—41no—
302C2#1120511yescharacteristic
303C2#1122911yescharacteristic
304C2#1123611yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1850C4semidirect
15. #1850C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#219312
#81022
#79822