Finite Group Databasebeta

← #136003

Subgroups — #136003

Order 512.

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

normal subgroups only

This group has 539 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#14501C211yesmaximal, characteristic
2C2×#1319C211yesmaximal, characteristic
3#11055C211yesmaximal, characteristic
4#8857C212yesmaximal
5#7251C212yesmaximal
6C2×#419—21no—
7#2316C2211yescharacteristic
8C2×#391—21no—
9C2×#391C2211yescharacteristic
10#1853—22no—
11C2×#343C2211yescharacteristic
12C2×#339—21no—
13C2×#324C2212yes—
14#1442—22no—
15#1441C2212yes—
16#1436—22no—
17#1434—21no—
18#1319C414yescomplemented
19#1294—22no—
20C23×Q8D411yescharacteristic
21C22×M16—21no—
22C2×#117—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4D412yes—
25C22×C4⋊C4C2311yescharacteristic
26C22×C42—21no—
27C22×C42D411yescharacteristic
28C2×#106—42no—
29#419—42no—
30#419—42no—
31C2×#103—42no—
32C2×C22⋊C8—22no—
33#391—22no—
34#391—42no—
35#391—42no—
36#391C2×C412yes—
37#391Q812yes—
38#389—42no—
39#380—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95D412yes—
48#343—22no—
49#340—42no—
50#339—42no—
51#324—22no—
52#324C2×C414yes—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—21no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8C4⋊C412yes—
61C22×Q8SD1612yes—
62C22×Q8SD1612yes—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C2×M16—22no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—41no—
76C22×C8—42no—
77#117—41no—
78#117—41no—
79#117—41no—
80#117—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4C22×C411yescharacteristic
92C2×C4⋊C4C2×Q811yescharacteristic
93C2×C4⋊C4#4112yes—
94C2×C22⋊C4—41no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—42no—
100C2×C42—81no—
101C2×C42—81no—
102C2×C42C4⋊C412yes—
103#106—84no—
104#103—48no—
105C22⋊C8—24no—
106C22⋊C8—42no—
107#95—24no—
108#95—24no—
109#95—42no—
110#95—44no—
111#95—44no—
112#95—44no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—81no—
117#95—81no—
118C24#12011yescharacteristic
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—81no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×Q8—81no—
133C2×Q8—81no—
134C2×Q8#10614yescomplemented
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C2×C4⋊C411yescharacteristic
171C22×C4C4×D412yes—
172C22×C4C22⋊Q812yes—
173C22×C4#12311yescharacteristic
174C22×C4C2×SD1611yescharacteristic
175C22×C4C2×SD1611yescharacteristic
176C22×C4AGL(1,ℤ/8)11yescharacteristic
177C22×C4#13711yescharacteristic
178M16—42no—
179M16—42no—
180M16—81no—
181M16—81no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—82no—
187C2×C8—82no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—42no—
201C42—81no—
202C42—81no—
203C42—81no—
204C42—84no—
205C23—22no—
206C23—22no—
207C23#38511yescharacteristic
208C23#44811yescharacteristic
209C23#45011yescharacteristic
210C23#45311yescharacteristic
211C23#45511yescharacteristic
212C23B(4,𝔽2)11yescharacteristic
213C23#45811yescharacteristic
214Q8—44no—
215Q8—82no—
216Q8—82no—
217Q8—82no—
218C2×C4—21no—
219C2×C4—21no—
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—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4C2×#10611yescharacteristic
267C2×C4#42811yescharacteristic
268C2×C4#48111yescharacteristic
269C2×C4#48311yescharacteristic
270C8—82no—
271C8—84no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22#159611yescharacteristic
281C22#160111yescharacteristic
282C22#160811yescharacteristic
283C22#161311yescharacteristic
284C22#185311yescharacteristic
285C22#185611yescharacteristic
286C22#186711yescharacteristic
287C22#219712yes—
288C22#220312yes—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—44no—
294C4—42no—
295C4—42no—
296C4—82no—
297C4—84no—
298C4—162no—
299C4—162no—
300C2—21no—
301C2—21no—
302C2—21no—
303C2—21no—
304C2#885411yescharacteristic
305C2#1212812yes—
306C2#1221012yes—
307C2#1296511yescharacteristic
308C2#1297111yescharacteristic
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. #1319C4semidirect
134. C2×Q8#106semidirect

Maximal subgroups

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

subgroupindex
#145012
C2×#13192
#110552
#88572
#72512