Finite Group Databasebeta

← #129302

Subgroups — #129302

Order 512.

812 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2920C211yesmaximal, characteristic
2#12849C212yesmaximal, complemented
3C2×#1327C212yesmaximal, complemented
4C2×#1320C212yesmaximal
5C23×C42C411yescharacteristic
6#2920—24no—
7#2920—24no—
8C2×#432—22no—
9C2×#431—22no—
10C2×#423C411yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#407C411yescharacteristic
13C2×#407C2211yescharacteristic
14C2×C4×M16C411yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C2×#349C2214yes—
17#1376—22no—
18#1375—22no—
19#1327—24no—
20#1320—24no—
21C24×C4—21no—
22C24×C4C2×C411yescharacteristic
23C22×M16C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16C2×C411yescharacteristic
26C22×M16C2311yescharacteristic
27C22×C42—22no—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42—22no—
31C22×C42—21no—
32C22×C42—21no—
33C22×C42—21no—
34C22×C42—22no—
35C22×C42C814yes—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C2×M32—24no—
39#432—216no—
40#431—216no—
41#423—22no—
42#423—22no—
43#423—22no—
44#423D414yes—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47#407—22no—
48#407—22no—
49#407—22no—
50#407D414yes—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C4×M16—22no—
54C4×M16—22no—
55C4×M16—22no—
56C4×M16—22no—
57C2×#97—21no—
58C2×#97D412yes—
59C2×C4×C8—21no—
60C2×C4×C8D412yes—
61#349—24no—
62#349—24no—
63C25M1611yescharacteristic
64C23×C4—21no—
65C23×C4—24no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—24no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—48no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4C22⋊C412yes—
82C23×C4C2×C812yes—
83C23×C4M1611yescharacteristic
84C23×C4C22×C411yescharacteristic
85C2×M16—21no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16C22⋊C412yes—
98C2×M16C22⋊C412yes—
99C2×M16C2×D412yes—
100C2×M16C2×D412yes—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8C22⋊C412yes—
104C22×C8C2×D412yes—
105C2×C42—24no—
106C2×C42—24no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—24no—
111C2×C42—21no—
112C2×C42—24no—
113C2×C42—24no—
114C2×C42—24no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42C22⋊C411yescharacteristic
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C2×C812yes—
129C2×C42C2×C812yes—
130C2×C42M1611yescharacteristic
131C2×C42M1611yescharacteristic
132C2×C42M1612yes—
133M32—44no—
134M32—44no—
135C2×C16—44no—
136C4⋊C8—28no—
137C4⋊C8—28no—
138C22⋊C8—28no—
139C22⋊C8—28no—
140#97—24no—
141#97—24no—
142C4×C8—24no—
143C4×C8—24no—
144C24—21no—
145C24—21no—
146C24—44no—
147C24—41no—
148C24—41no—
149C24C22⋊C812yes—
150C24C2×M1611yescharacteristic
151C22×C4—24no—
152C22×C4—28no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—416no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—416no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4C22⋊C812yes—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C22×C811yescharacteristic
202C22×C4C2×M1611yescharacteristic
203C22×C4C2×M1611yescharacteristic
204M16—42no—
205M16—42no—
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—42no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8#12014yes—
224C42—24no—
225C42—24no—
226C42—24no—
227C42—24no—
228C42—21no—
229C42—21no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—21no—
234C42—22no—
235C42—21no—
236C42C22⋊C812yes—
237C42C22⋊C812yes—
238C42C22⋊C812yes—
239C42C22⋊C812yes—
240C16—84no—
241C23—24no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—44no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—44no—
250C23—42no—
251C23—44no—
252C23—48no—
253C23—44no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊C811yescharacteristic
257C23#40711yescharacteristic
258C23#40711yescharacteristic
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—216no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4#37911yescharacteristic
297C2×C4C2×C22⋊C811yescharacteristic
298C2×C4C2×C22⋊C811yescharacteristic
299C2×C4#40711yescharacteristic
300C2×C4#41011yescharacteristic
301C2×C4#41011yescharacteristic
302C2×C4#41311yescharacteristic
303C2×C4#41311yescharacteristic
304C8—24no—
305C8—42no—
306C8—42no—
307C8—42no—
308C22—22no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—44no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—48no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22#146411yescharacteristic
324C22#146411yescharacteristic
325C22#177911yescharacteristic
326C4—24no—
327C4—24no—
328C4—21no—
329C4—21no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4#145811yescharacteristic
334C4#145811yescharacteristic
335C4#178211yescharacteristic
336C4#178511yescharacteristic
337C2—21no—
338C2—21no—
339C2—44no—
340C2—41no—
341C2—41no—
342C2#830011yescharacteristic
343C2#1152112yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1464
prime radicalC25M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12849C2semidirect
3. C2×#1327C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#128492
C2×#13272
C2×#13202