Finite Group Databasebeta

← #134308

Subgroups — #134308

Order 512.

578 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 122 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×#382C211yesmaximal, characteristic
2C2×#1324C212yesmaximal
3#8560C212yesmaximal
4#7189C212yesmaximal
5C2×#382C411yescharacteristic
6C2×#382C2211yescharacteristic
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C43C411yescharacteristic
10#1797—22no—
11C2×#324C2214yes—
12C4⋊#95C411yescharacteristic
13C4⋊#95C2211yescharacteristic
14#1434—22no—
15C4×#95—22no—
16#1324—24no—
17#1280—22no—
18C22×C4⋊C4C2×C411yescharacteristic
19C22×C4⋊C4C2×C411yescharacteristic
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4C2311yescharacteristic
22C22×C42—21no—
23C22×C42—21no—
24C22×C42—21no—
25C22×C42C2×C411yescharacteristic
26C22×C42C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29C2×C4⋊C8—42no—
30C2×C22⋊C8—24no—
31#382—22no—
32#382—22no—
33#382C814yescomplemented
34#382D412yes—
35#382Q812yes—
36C4×C4⋊C4—41no—
37C4×C4⋊C4—41no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95D412yes—
46C43—22no—
47C43—22no—
48#336—42no—
49#324—24no—
50#324—24no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D412yes—
60C22×C8—44no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4C4⋊C412yes—
72C2×C4⋊C4C2×C812yes—
73C2×C4⋊C4C2×C812yes—
74C2×C4⋊C4M1612yes—
75C2×C4⋊C4C2×D411yescharacteristic
76C2×C4⋊C4C2×Q811yescharacteristic
77C2×C4⋊C4#4112yes—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—22no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—24no—
87C2×C42—24no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—22no—
91C2×C42—41no—
92C2×C42—41no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42—42no—
96C2×C42—42no—
97C2×C42—44no—
98C2×C42—44no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42C4⋊C412yes—
102C2×C42C2×C812yes—
103C2×C42M1612yes—
104C2×C42M1612yes—
105C4⋊C8—84no—
106C22⋊C8—44no—
107C22⋊C8—44no—
108#95—28no—
109#95—24no—
110#95—24no—
111#95—42no—
112#95—44no—
113#95—44no—
114#95—42no—
115C24C2×C22⋊C411yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊C814yes—
161C22×C4#9911yescharacteristic
162C22×C4#9911yescharacteristic
163C22×C4#10112yes—
164C22×C4C2×C4⋊C411yescharacteristic
165C22×C4#11711yescharacteristic
166C22×C4#12112yes—
167C22×C4C22⋊Q812yes—
168C22×C4C22×C811yescharacteristic
169C22×C4C2×M1611yescharacteristic
170C22×C4C2×M1611yescharacteristic
171C22×C4C2×M1611yescharacteristic
172C2×C8—84no—
173C2×C8—84no—
174C2×C8—84no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—42no—
192C42—48no—
193C42—42no—
194C42—48no—
195C42—44no—
196C42—82no—
197C42—82no—
198C42—82no—
199C42C4⋊C814yescomplemented
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23#38011yescharacteristic
205C23C2×C22⋊C811yescharacteristic
206C23#40711yescharacteristic
207C23C2×#9911yescharacteristic
208C23#41011yescharacteristic
209C23C2×#10111yescharacteristic
210C23#41311yescharacteristic
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#35412yes—
271C2×C4#35512yes—
272C2×C4C2×C4⋊C811yescharacteristic
273C2×C4#42311yescharacteristic
274C2×C4#43111yescharacteristic
275C2×C4#43211yescharacteristic
276C8—164no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22#145911yescharacteristic
290C22#146111yescharacteristic
291C22#146311yescharacteristic
292C22#146411yescharacteristic
293C22#179711yescharacteristic
294C22#179911yescharacteristic
295C22#180111yescharacteristic
296C22C2×#35411yescharacteristic
297C22#212611yescharacteristic
298C22C2×#35511yescharacteristic
299C22#213211yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—44no—
307C4—44no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—82no—
313C4—82no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—21no—
318C2#856011yescharacteristic
319C2#1147611yescharacteristic
320C2#1147711yescharacteristic
321C2#1151111yescharacteristic
322C2#1151311yescharacteristic
323C2#1279811yescharacteristic
324C2#1280111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
33. #382C8semidirect
199. C42C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3822
C2×#13242
#85602
#71892