Finite Group Databasebeta

← #145804

Subgroups — #145804

Order 512.

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

normal subgroups only

This group has 523 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#17080C211yesmaximal, characteristic
2#11296C211yesmaximal, characteristic
3#8856C212yesmaximal
4#7251C212yesmaximal
5#7189C211yesmaximal, characteristic
6#2316C2211yescharacteristic
7C2×#391—21no—
8#1852—22no—
9C2×#343C2211yescharacteristic
10#1736—21no—
11C2×#339—21no—
12C2×#324C2212yes—
13#1446—22no—
14#1444—22no—
15#1441—21no—
16#1441C2212yes—
17C4⋊#95C2211yescharacteristic
18#1294—22no—
19C23×Q8—21no—
20C22×M16—21no—
21C2×#117—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4D411yescharacteristic
25C22×C4⋊C4D412yes—
26C22×C4⋊C4C2311yescharacteristic
27C22×C42D411yescharacteristic
28C2×#107—42no—
29C2×C22⋊C8—22no—
30#391—42no—
31#391—42no—
32#389—41no—
33#389—41no—
34#380—22no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95D412yes—
42#343—21no—
43#343Q812yescomplemented
44#340—42no—
45#340—41no—
46#340—41no—
47#339—42no—
48#324—22no—
49#324Q814yescomplemented
50C22×Q8—24no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8—81no—
54C22×Q8—81no—
55C22×Q8—81no—
56C22×Q8—81no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D412yes—
63C2×M16—22no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C22×C8—41no—
70C22×C8—42no—
71#117—41no—
72#117—41no—
73#117—41no—
74#117—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—82no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4Q1612yes—
92C2×C4⋊C4C2×Q812yes—
93C2×C4⋊C4C2×Q811yescharacteristic
94C2×C4⋊C4#4111yescharacteristic
95C2×C22⋊C4—41no—
96C2×C42—21no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42D812yes—
101#107—84no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104#95—42no—
105#95—44no—
106#95—44no—
107#95—42no—
108#95—42no—
109#95—42no—
110#95—82no—
111#95—82no—
112#95—81no—
113C24#12011yescharacteristic
114C2×Q8—48no—
115C2×Q8—84no—
116C2×Q8—84no—
117C2×Q8—84no—
118C2×Q8—84no—
119C2×Q8—84no—
120C2×Q8—84no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4C22⋊Q811yescharacteristic
158C22×C4C22⋊Q812yes—
159C22×C4C4⋊Q811yescharacteristic
160C22×C4C4⋊Q812yes—
161C22×C4C2×D811yescharacteristic
162C22×C4C2×Q1611yescharacteristic
163C22×C4AGL(1,ℤ/8)11yescharacteristic
164C22×C4#13711yescharacteristic
165M16—42no—
166M16—42no—
167M16—81no—
168M16—81no—
169C2×C8—44no—
170C2×C8—42no—
171C2×C8—81no—
172C2×C8—81no—
173C2×C8—82no—
174C2×C8—82no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—44no—
184C4⋊C4—81no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—84no—
189C4⋊C4—84no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C42—42no—
193C42—81no—
194C42—81no—
195C42—81no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23#39511yescharacteristic
201C23#44711yescharacteristic
202C23#45111yescharacteristic
203C23#45411yescharacteristic
204C23#45511yescharacteristic
205C23B(4,𝔽2)11yescharacteristic
206C23#45811yescharacteristic
207Q8—816no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#47411yescharacteristic
260C2×C4#47711yescharacteristic
261C2×C4#50011yescharacteristic
262C2×C4#50111yescharacteristic
263C8—82no—
264C8—84no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22#159811yescharacteristic
278C22#160211yescharacteristic
279C22#160711yescharacteristic
280C22#161711yescharacteristic
281C22#205711yescharacteristic
282C22#206111yescharacteristic
283C22#206211yescharacteristic
284C22#219811yescharacteristic
285C22#219911yescharacteristic
286C22#220211yescharacteristic
287C22#220511yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—44no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—82no—
297C4—161no—
298C4—161no—
299C4—162no—
300C2—21no—
301C2—21no—
302C2—21no—
303C2—21no—
304C2#1056911yescharacteristic
305C2#1215311yescharacteristic
306C2#1215611yescharacteristic
307C2#1222111yescharacteristic
308C2#1222411yescharacteristic
309C2#1346311yescharacteristic
310C2#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
43. #343Q8semidirect
49. #324Q8semidirect

Maximal subgroups

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

subgroupindex
#170802
#112962
#88562
#72512
#71892