Finite Group Databasebeta

← #121349

Subgroups — #121349

Order 512.

596 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 154 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#21961C211yesmaximal, characteristic
2#21365C212yesmaximal
3#10835C211yesmaximal, characteristic
4#8034C212yesmaximal
5C2×#1290C211yesmaximal, characteristic
6C2×#432C2212yes—
7C22×C4⋊C8C2211yescharacteristic
8C22×#97C2211yescharacteristic
9#2285C2212yes—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1948—22no—
12#1947—22no—
13#1947—22no—
14#1947—22no—
15#1853—22no—
16#1852—22no—
17#1808—22no—
18#1776—24no—
19C2×#336—22no—
20#1295—24no—
21#1290C414yescomplemented
22C23×C8—21no—
23C23×C8D412yes—
24C2×#117—22no—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C42—21no—
28C22×C42C2311yescharacteristic
29#432—22no—
30#432—22no—
31#432C2×C414yes—
32#432Q814yes—
33C2×#107—24no—
34C2×#107—22no—
35C2×#107—22no—
36C2×#106—24no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2×C412yes—
41C2×C4⋊C8Q812yes—
42C2×C22⋊C8—22no—
43C2×#97—22no—
44C2×#97—22no—
45C2×#97—21no—
46C2×#97D414yes—
47C2×#97Q812yes—
48#384D412yes—
49#384D412yes—
50#382—22no—
51#380—22no—
52#380—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C2×#95—21no—
58#340—24no—
59#340—24no—
60#340—24no—
61#336—44no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4#4111yescharacteristic
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—24no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8C4⋊C414yes—
80#117—84no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—42no—
93C2×C42—42no—
94C2×C42—41no—
95C2×C42—42no—
96C2×C42—41no—
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D412yes—
99C2×C42C2×D412yes—
100C2×C42C2×Q812yes—
101C2×C42C2×Q811yescharacteristic
102C2×C42C2×Q811yescharacteristic
103C2×C42#4112yes—
104C2×C42#4111yescharacteristic
105#107—48no—
106#107—82no—
107#107—82no—
108#106—48no—
109C4⋊C8—24no—
110C4⋊C8—24no—
111C4⋊C8—24no—
112C4⋊C8—24no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115#97—24no—
116#97—24no—
117#97—42no—
118#97C4⋊C418yes—
119#95—42no—
120#95—42no—
121#95—42no—
122C24#12111yescharacteristic
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4C2×C4⋊C411yescharacteristic
144C22×C4C4×D411yescharacteristic
145C22×C4C4×Q811yescharacteristic
146C22×C4#12012yes—
147C22×C4#12011yescharacteristic
148C22×C4#12112yes—
149C22×C4C22⋊Q811yescharacteristic
150C22×C4C22⋊Q812yes—
151C22×C4#12311yescharacteristic
152C22×C4#12512yes—
153C22×C4#12511yescharacteristic
154C22×C4#12711yescharacteristic
155C22×C4C4⋊Q811yescharacteristic
156C22×C4C4⋊Q811yescharacteristic
157C2×C8—24no—
158C2×C8—28no—
159C2×C8—24no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—44no—
171C2×C8—82no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C22⋊C4—84no—
185C42—82no—
186C42—84no—
187C42—81no—
188C42—81no—
189C42C2×C4⋊C412yes—
190C42C4×D412yes—
191C42C22⋊Q812yes—
192C42C22⋊Q812yes—
193C42C22⋊Q812yes—
194C42#12312yes—
195C42C4⋊Q812yes—
196C42C4⋊Q812yes—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23#38911yescharacteristic
202C23#39211yescharacteristic
203C23#39811yescharacteristic
204C23#47212yes—
205C23#47312yes—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
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—41no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—81no—
233C2×C4—81no—
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#38411yescharacteristic
243C2×C4#38512yes—
244C2×C4#38511yescharacteristic
245C2×C4#38912yes—
246C2×C4#39512yes—
247C2×C4#39511yescharacteristic
248C2×C4#39812yes—
249C2×C4#39911yescharacteristic
250C2×C4#45311yescharacteristic
251C2×C4#45411yescharacteristic
252C2×C4#45511yescharacteristic
253C2×C4#45611yescharacteristic
254C8—48no—
255C8—44no—
256C8—44no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22#144411yescharacteristic
266C22#195212yes—
267C22#200611yescharacteristic
268C22#200711yescharacteristic
269C22#201011yescharacteristic
270C22#201011yescharacteristic
271C22#201811yescharacteristic
272C22#202011yescharacteristic
273C22#208912yes—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—21no—
279C4—21no—
280C4—82no—
281C4—82no—
282C4—162no—
283C4—162no—
284C4#185611yescharacteristic
285C4#185711yescharacteristic
286C4#206011yescharacteristic
287C4#206111yescharacteristic
288C2—22no—
289C2—22no—
290C2#1002611yescharacteristic
291C2#1002811yescharacteristic
292C2#1003011yescharacteristic
293C2#1003111yescharacteristic
294C2#1133111yescharacteristic
295C2#1133511yescharacteristic
296C2#1134211yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1290C4semidirect

Maximal subgroups

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

subgroupindex
#219612
#213652
#108352
#80342
C2×#12902