Finite Group Databasebeta

← #136032

Subgroups — #136032

Order 512.

597 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 94 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#14151C211yesmaximal, characteristic
2C2×#1325C211yesmaximal, characteristic
3#11210C211yesmaximal, characteristic
4#8856C212yesmaximal
5#7251C212yesmaximal
6#2316C2211yescharacteristic
7C2×#389—22no—
8C2×#384C2211yescharacteristic
9C2×C4×C4⋊C4—21no—
10#1852—22no—
11C2×#343C2211yescharacteristic
12#1736—21no—
13C2×#339—21no—
14C2×#324C2212yes—
15#1441—21no—
16#1441C2212yes—
17C4⋊#95—22no—
18#1325C414yescomplemented
19#1294—22no—
20C22×M16—21no—
21C2×#117—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D412yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C42—22no—
30C22×C42D411yescharacteristic
31C2×#107—42no—
32C2×C22⋊C8—22no—
33#389—41no—
34#389—41no—
35#389—48no—
36#384—22no—
37#384C2×C412yes—
38#384Q812yes—
39#380—22no—
40C4×C4⋊C4—42no—
41C4×C4⋊C4—42no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95D412yes—
47#343—22no—
48#340—42no—
49#340—41no—
50#340—41no—
51#339—42no—
52#324—22no—
53#324C2×C414yes—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×D412yes—
60C2×M16—22no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C22×C8—41no—
67C22×C8—42no—
68#117—41no—
69#117—41no—
70#117—41no—
71#117—41no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4C4⋊C412yes—
96C2×C4⋊C4C22×C411yescharacteristic
97C2×C4⋊C4C2×Q811yescharacteristic
98C2×C4⋊C4#4112yes—
99C2×C22⋊C4—41no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—41no—
104C2×C42—44no—
105C2×C42—44no—
106C2×C42—44no—
107C2×C42C4⋊C412yes—
108C2×C42D812yes—
109C2×C42Q1612yes—
110#107—84no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113#95—42no—
114#95—44no—
115#95—44no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—81no—
120C24#12011yescharacteristic
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
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—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4C2×C4⋊C411yescharacteristic
158C22×C4C4×D412yes—
159C22×C4C22⋊Q812yes—
160C22×C4#12311yescharacteristic
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—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—48no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4—48no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—82no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—88no—
199C42—81no—
200C42—81no—
201C42#10714yescomplemented
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23#38511yescharacteristic
207C23#44711yescharacteristic
208C23#45111yescharacteristic
209C23#45411yescharacteristic
210C23#45611yescharacteristic
211C23B(4,𝔽2)11yescharacteristic
212C23#45811yescharacteristic
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
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×C4C2×#10711yescharacteristic
265C2×C4#42811yescharacteristic
266C2×C4#48011yescharacteristic
267C2×C4#48411yescharacteristic
268C8—82no—
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#159811yescharacteristic
283C22#160511yescharacteristic
284C22#161011yescharacteristic
285C22#161711yescharacteristic
286C22#185211yescharacteristic
287C22#185711yescharacteristic
288C22#186711yescharacteristic
289C22#219611yescharacteristic
290C22#219811yescharacteristic
291C22#220411yescharacteristic
292C22#220511yescharacteristic
293C4—28no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—42no—
299C4—42no—
300C4—84no—
301C4—84no—
302C4—82no—
303C4—161no—
304C4—161no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2#885611yescharacteristic
310C2#1215211yescharacteristic
311C2#1215311yescharacteristic
312C2#1225611yescharacteristic
313C2#1225711yescharacteristic
314C2#1296411yescharacteristic
315C2#1297311yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C2×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
18. #1325C4semidirect
201. C42#107semidirect

Maximal subgroups

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

subgroupindex
#141512
C2×#13252
#112102
#88562
#72512