Finite Group Databasebeta

← #120050

Subgroups — #120050

Order 512.

776 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 142 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#21934C211yesmaximal, characteristic
2#14295C211yesmaximal, characteristic
3C4×#342C211yesmaximal, characteristic
4#8019C212yesmaximal
5C2×#1284C212yesmaximal
6C2×#407C2212yes—
7C2×#381C2211yescharacteristic
8C2×#380C2211yescharacteristic
9C2×C4×C22⋊C4C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1851—24no—
12#1797—22no—
13C2×#342—22no—
14#1444C412yescomplemented
15#1442C412yescomplemented
16#1439—22no—
17#1436—22no—
18#1294—24no—
19#1284—24no—
20C24×C4C2311yescharacteristic
21C22×M16—22no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4C2×C412yes—
24C22×C22⋊C4—22no—
25C22×C22⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4D412yes—
28C22×C42—22no—
29C22×C42D412yes—
30C2×C4⋊C8—24no—
31#407—22no—
32#407—22no—
33#407—22no—
34#407—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8D414yes—
37#381—21no—
38#381—21no—
39#381D412yes—
40#381Q812yes—
41#380—22no—
42#380—22no—
43C4×C22⋊C4—22no—
44C4×C22⋊C4—22no—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4Q814yes—
49C2×#95—21no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C412yes—
58#342—28no—
59#336—24no—
60C25C2×D411yescharacteristic
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—42no—
64C23×C4—82no—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C412yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×Q811yescharacteristic
72C23×C4C2×Q812yes—
73C23×C4#4112yes—
74C23×C4#4111yescharacteristic
75C23×C4#4111yescharacteristic
76C23×C4#4112yes—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—84no—
80C22×C8—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—82no—
84C2×C4⋊C4—82no—
85C2×C4⋊C4—82no—
86C2×C22⋊C4—28no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—44no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—82no—
97C2×C22⋊C4—81no—
98C2×C22⋊C4—81no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—24no—
102C2×C42—42no—
103C2×C42—82no—
104C2×C42—84no—
105C4⋊C8—84no—
106C22⋊C8—24no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#95—22no—
111#95—22no—
112#95—22no—
113#95—22no—
114#95—22no—
115#95—24no—
116#95—44no—
117#95—41no—
118#95—42no—
119#95—41no—
120#95—82no—
121#95—82no—
122#95C4⋊C414yes—
123C24—22no—
124C24—22no—
125C24—41no—
126C24—42no—
127C24—41no—
128C24C2×C22⋊C411yescharacteristic
129C24#12112yes—
130C24#12111yescharacteristic
131C24C22⋊Q811yescharacteristic
132C24C22⋊Q812yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—48no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—81no—
163C22×C4—82no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—82no—
167C22×C4—81no—
168C22×C4—84no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4#10414yes—
172C22×C4C2×C4⋊C411yescharacteristic
173C22×C4#11711yescharacteristic
174C22×C4C4×D412yes—
175C22×C4C4×Q812yes—
176C22×C4#12011yescharacteristic
177C22×C4#12112yes—
178C22×C4#12111yescharacteristic
179C22×C4C22⋊Q812yes—
180C22×C4C22⋊Q812yes—
181C22×C4C22⋊Q811yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12312yes—
184C22×C4#12411yescharacteristic
185C22×C4#12512yes—
186C22×C4#12512yes—
187C22×C4#12612yes—
188C22×C4C4⋊Q812yes—
189C22×C4C4⋊Q811yescharacteristic
190M16—84no—
191M16—84no—
192C2×C8—48no—
193C2×C8—44no—
194C4⋊C4—84no—
195C4⋊C4—88no—
196C22⋊C4—28no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—84no—
203C42—44no—
204C42—82no—
205C42—84no—
206C42—84no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—22no—
213C23—24no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23#38011yescharacteristic
229C23#38611yescharacteristic
230C23#39111yescharacteristic
231C23#39612yes—
232C23#39812yes—
233C23C2×#10412yes—
234C23#42112yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—88no—
273C2×C4—88no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#38712yes—
283C2×C4#38912yes—
284C2×C4#39411yescharacteristic
285C2×C4#39511yescharacteristic
286C2×C4#39811yescharacteristic
287C2×C4#40011yescharacteristic
288C8—84no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—44no—
304C22—44no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22#144111yescharacteristic
310C22#180612yes—
311C22#187112yes—
312C22#190911yescharacteristic
313C22#191011yescharacteristic
314C22#191312yes—
315C22#199411yescharacteristic
316C22#199511yescharacteristic
317C4—21no—
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—44no—
324C4—84no—
325C4—88no—
326C4—82no—
327C4—162no—
328C2—22no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2—42no—
333C2#918312yes—
334C2#918612yes—
335C2#1120511yescharacteristic
336C2#1120911yescharacteristic
337C2#1122511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1444C4semidirect
15. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#219342
#142952
C4×#3422
#80192
C2×#12842