Finite Group Databasebeta

← #136815

Subgroups — #136815

Order 512.

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

normal subgroups only

This group has 568 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
1#33489C211yesmaximal, characteristic, complemented
2#21439C211yesmaximal, characteristic
3#12681C212yesmaximal, complemented
4#11466C211yesmaximal, characteristic, complemented
5#7201C212yesmaximal, complemented
6C23×M16C2211yescharacteristic
7C2×#435—21no—
8C2×C8×D4C2211yescharacteristic
9C4×#120C411yescharacteristic
10#2169—24no—
11#2162—21no—
12C2×#349—24no—
13#2112—21no—
14#2112C2212yes—
15#2112C2212yes—
16#1870—21no—
17#1869C411yescharacteristic
18#1851—21no—
19#1850C411yescharacteristic
20#1779C411yescharacteristic
21#1753—21no—
22#1752C2211yescharacteristic
23#1322—21no—
24#1314—21no—
25C24×C4C2×C411yescharacteristic
26C22×M16—22no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—22no—
30C22×M16—44no—
31C23×C8C2×C411yescharacteristic
32C23×C8D411yescharacteristic
33C23×C8C2311yescharacteristic
34C2×#120C811yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×M32—22no—
38C2×M32—22no—
39C2×M32D414yescomplemented
40#435—44no—
41C8×D4—22no—
42C8×D4—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×#97—21no—
52C2×C4×C8D411yescharacteristic
53#386—21no—
54#386C811yescharacteristic
55#385—21no—
56#385C811yescharacteristic
57#379C811yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C411yescharacteristic
60C4⋊C16—22no—
61#349—48no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64C22⋊C16—22no—
65C22⋊C16—22no—
66C22⋊C16—22no—
67C4×C16—22no—
68#336—21no—
69#336—21no—
70#336—21no—
71C25C2×C811yescharacteristic
72C22×D4—21no—
73C22×D4C2×C811yescharacteristic
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C2×C811yescharacteristic
82C23×C4C22×C411yescharacteristic
83C2×M16—28no—
84C2×M16—24no—
85C2×M16—42no—
86C2×M16—44no—
87C2×M16—42no—
88C2×M16—44no—
89C2×M16—42no—
90C2×M16—44no—
91C2×M16—41no—
92C2×M16—42no—
93C2×M16—41no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C2×D412yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8#4112yes—
108#120—42no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—42no—
112C4×D4—42no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4M1611yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4C2×C811yescharacteristic
120C2×C22⋊C4C2×C811yescharacteristic
121C2×C22⋊C4C2×C811yescharacteristic
122C2×C22⋊C4M1611yescharacteristic
123C2×C42—21no—
124C2×C42C22⋊C411yescharacteristic
125M32—44no—
126M32—44no—
127M32—82no—
128C2×C16—24no—
129C2×C16—42no—
130C4⋊C8—22no—
131C4⋊C8—42no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—42no—
139C4×C8—21no—
140C4×C8—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24—22no—
146C24—42no—
147C24—41no—
148C24—41no—
149C24—81no—
150C24C22⋊C811yescharacteristic
151C24C22×C811yescharacteristic
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—81no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4C22⋊C811yescharacteristic
185C22×C4C22⋊C811yescharacteristic
186C22×C4C22⋊C811yescharacteristic
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C4×D412yes—
189C22×C4C2×M1611yescharacteristic
190C22×C4#13112yes—
191M16—48no—
192M16—44no—
193M16—44no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—81no—
212C2×C8#12011yescharacteristic
213C2×C8#12112yes—
214C2×C8#12311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—42no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—81no—
225C42—21no—
226C42—21no—
227C42—41no—
228C42—41no—
229C16—44no—
230C16—82no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—24no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—82no—
250C23—82no—
251C23—84no—
252C23—84no—
253C23C2×C22⋊C811yescharacteristic
254C23C8×D412yes—
255D4—44no—
256D4—84no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4#34912yescomplemented
290C2×C4#38611yescharacteristic
291C2×C4#40811yescharacteristic
292C2×C4#43512yes—
293C8—22no—
294C8—21no—
295C8—42no—
296C8—41no—
297C8—41no—
298C8—81no—
299C8B(4,𝔽2)12yes—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—84no—
318C22—84no—
319C22—81no—
320C22—81no—
321C22#186911yescharacteristic
322C22C2×#34911yescharacteristic
323C22#211511yescharacteristic
324C4—21no—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4#188911yescharacteristic
333C4#189111yescharacteristic
334C2—21no—
335C2—21no—
336C2—42no—
337C2—81no—
338C2—82no—
339C2#896511yescharacteristic
340C2#1298511yescharacteristic
341C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC2×#120C8
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33489C2semidirect
3. #12681C2semidirect
4. #11466C2semidirect
5. #7201C2semidirect
39. C2×M32D4semidirect
289. C2×C4#349semidirect

Maximal subgroups

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

subgroupindex
#334892
#214392
#126812
#114662
#72012