Finite Group Databasebeta

← #136035

Subgroups — #136035

Order 512.

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

normal subgroups only

This group has 485 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#14677C211yesmaximal, characteristic
2C2×#1324C211yesmaximal, characteristic
3#11211C211yesmaximal, characteristic
4#8856C212yesmaximal
5#7251C212yesmaximal
6#2316C2211yescharacteristic
7C2×#382—21no—
8C2×#382C2211yescharacteristic
9#1852—22no—
10C2×#343C2211yescharacteristic
11#1736—21no—
12C2×#339—21no—
13C2×#324C2212yes—
14#1444—22no—
15#1442—22no—
16#1441—21no—
17#1441C2212yes—
18#1324C414yescomplemented
19#1294—22no—
20C22×M16—21no—
21C2×#117—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D412yes—
27C22×C4⋊C4C2311yescharacteristic
28C22×C42D411yescharacteristic
29C2×#107—42no—
30C2×C22⋊C8—22no—
31#389—41no—
32#389—41no—
33#382—22no—
34#382—42no—
35#382—42no—
36#382C2×C412yes—
37#382Q812yes—
38#380—22no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95D412yes—
46#343—22no—
47#340—42no—
48#340—41no—
49#340—41no—
50#339—42no—
51#324—22no—
52#324C2×C414yes—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4C2×D411yescharacteristic
58C23×C4C2×D412yes—
59C2×M16—22no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C22×C8—41no—
66C22×C8—42no—
67#117—41no—
68#117—41no—
69#117—41no—
70#117—41no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—82no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—82no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4C4⋊C412yes—
91C2×C4⋊C4C22×C411yescharacteristic
92C2×C4⋊C4C2×Q811yescharacteristic
93C2×C4⋊C4#4112yes—
94C2×C22⋊C4—41no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—41no—
99C2×C42C4⋊C412yes—
100C2×C42D812yes—
101C2×C42Q1612yes—
102#107—84no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105#95—24no—
106#95—24no—
107#95—42no—
108#95—44no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—82no—
113#95—82no—
114#95—81no—
115C24#12011yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
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—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4C2×C4⋊C411yescharacteristic
153C22×C4C4×D412yes—
154C22×C4C22⋊Q812yes—
155C22×C4#12311yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C2×Q1611yescharacteristic
158C22×C4AGL(1,ℤ/8)11yescharacteristic
159C22×C4#13711yescharacteristic
160M16—42no—
161M16—42no—
162M16—81no—
163M16—81no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—82no—
169C2×C8—82no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C4⋊C4—84no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—81no—
192C42—81no—
193C42#10714yescomplemented
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23#38511yescharacteristic
199C23#44711yescharacteristic
200C23#45111yescharacteristic
201C23#45411yescharacteristic
202C23#45611yescharacteristic
203C23B(4,𝔽2)11yescharacteristic
204C23#45811yescharacteristic
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
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—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4C2×#10711yescharacteristic
257C2×C4#42811yescharacteristic
258C2×C4#48011yescharacteristic
259C2×C4#48411yescharacteristic
260C8—82no—
261C8—84no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22#159811yescharacteristic
275C22#160511yescharacteristic
276C22#161011yescharacteristic
277C22#161711yescharacteristic
278C22#185211yescharacteristic
279C22#185711yescharacteristic
280C22#186711yescharacteristic
281C22#219911yescharacteristic
282C22#220111yescharacteristic
283C22#220211yescharacteristic
284C22#220311yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—44no—
290C4—42no—
291C4—42no—
292C4—82no—
293C4—162no—
294C4—161no—
295C4—162no—
296C4—161no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2—21no—
301C2#885611yescharacteristic
302C2#1215611yescharacteristic
303C2#1215711yescharacteristic
304C2#1226011yescharacteristic
305C2#1226111yescharacteristic
306C2#1296711yescharacteristic
307C2#1297011yescharacteristic
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. #1324C4semidirect
193. C42#107semidirect

Maximal subgroups

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

subgroupindex
#146772
C2×#13242
#112112
#88562
#72512