Finite Group Databasebeta

← #145777

Subgroups — #145777

Order 512.

583 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 80 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#17072C211yesmaximal, characteristic
2#11342C211yesmaximal, characteristic
3#8856C212yesmaximal
4#7251C212yesmaximal
5#7189C211yesmaximal, characteristic
6C2×#428—21no—
7C2×#395—22no—
8C2×#395—21no—
9#2316C2211yescharacteristic
10#1852—22no—
11C2×#343C2211yescharacteristic
12C2×#339—21no—
13C2×#324C2212yes—
14#1441—22no—
15#1441C2212yes—
16C4⋊#95—21no—
17C4⋊#95C2211yescharacteristic
18#1294—22no—
19C23×Q8—21no—
20C22×M16—21no—
21C2×#117—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D412yes—
27C22×C4⋊C4C2311yescharacteristic
28C22×C42D411yescharacteristic
29#428—42no—
30#428—42no—
31C2×#107—41no—
32C2×#107—42no—
33C2×#106—41no—
34C2×C22⋊C8—22no—
35#395—48no—
36#395—42no—
37#395—42no—
38#389—41no—
39#389—41no—
40#380—22no—
41C2×#95—22no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95D412yes—
47#343—21no—
48#343Q812yescomplemented
49#340—42no—
50#339—42no—
51#324—22no—
52#324Q814yescomplemented
53C22×Q8—44no—
54C22×Q8—42no—
55C22×Q8—82no—
56C22×Q8—81no—
57C22×Q8—81no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D412yes—
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C22×C8—41no—
71C22×C8—42no—
72#117—41no—
73#117—41no—
74#117—41no—
75#117—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4D812yes—
97C2×C4⋊C4C2×Q812yes—
98C2×C4⋊C4C2×Q811yescharacteristic
99C2×C4⋊C4#4111yescharacteristic
100C2×C22⋊C4—41no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42Q1612yes—
106#107—44no—
107#107—84no—
108#106—44no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111#95—42no—
112#95—44no—
113#95—44no—
114#95—44no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—42no—
119C24#12011yescharacteristic
120C2×Q8—88no—
121C2×Q8—84no—
122C2×Q8—88no—
123C2×Q8—84no—
124C2×Q8—84no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C22⋊Q811yescharacteristic
162C22×C4C22⋊Q812yes—
163C22×C4C4⋊Q811yescharacteristic
164C22×C4C4⋊Q812yes—
165C22×C4C2×D811yescharacteristic
166C22×C4C2×Q1611yescharacteristic
167C22×C4AGL(1,ℤ/8)11yescharacteristic
168C22×C4#13711yescharacteristic
169M16—24no—
170M16—42no—
171M16—42no—
172M16—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—82no—
178C2×C8—82no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4—88no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C42—42no—
198C42—81no—
199C42—81no—
200C42—81no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23#39511yescharacteristic
206C23#44711yescharacteristic
207C23#45111yescharacteristic
208C23#45311yescharacteristic
209C23#45611yescharacteristic
210C23B(4,𝔽2)11yescharacteristic
211C23#45811yescharacteristic
212Q8—816no—
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—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4#47411yescharacteristic
265C2×C4#47711yescharacteristic
266C2×C4#50011yescharacteristic
267C2×C4#50111yescharacteristic
268C8—44no—
269C8—84no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#159511yescharacteristic
283C22#160511yescharacteristic
284C22#161011yescharacteristic
285C22#161411yescharacteristic
286C22#205711yescharacteristic
287C22#206011yescharacteristic
288C22#206211yescharacteristic
289C22#219611yescharacteristic
290C22#220111yescharacteristic
291C22#220211yescharacteristic
292C22#220511yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—44no—
298C4—42no—
299C4—42no—
300C4—84no—
301C4—82no—
302C4—84no—
303C4—161no—
304C4—161no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2#1056711yescharacteristic
310C2#1209611yescharacteristic
311C2#1210111yescharacteristic
312C2#1225611yescharacteristic
313C2#1226111yescharacteristic
314C2#1346111yescharacteristic
315C2#1346411yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutator, Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
48. #343Q8semidirect
52. #324Q8semidirect

Maximal subgroups

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

subgroupindex
#170722
#113422
#88562
#72512
#71892