Finite Group Databasebeta

← #102303

Subgroups — #102303

Order 512.

578 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 126 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#33172C211yesmaximal, characteristic
2C8×#95C211yesmaximal, characteristic
3#9909C212yesmaximal
4#8596C211yesmaximal, characteristic
5#7668C212yesmaximal
6C2×#494C411yescharacteristic, complemented
7C2×#467—21no—
8C2×#467—21no—
9C22×C4×C8—21no—
10C22×C4×C8C2211yescharacteristic
11#2857C2212yes—
12#1969—21no—
13#1969—21no—
14#1956C412yescomplemented
15#1845C411yescharacteristic, complemented
16#1797C2212yes—
17C2×#336—21no—
18C2×#336C2211yescharacteristic
19C2×#335—21no—
20C4×#95C2211yescharacteristic
21#1294—24no—
22#1277—21no—
23#1271—24no—
24C22×Q16—21no—
25C22×Q16—21no—
26C23×C8—21no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4⋊Q8C2×C412yes—
30C2×C22⋊Q8—22no—
31C22×C42—21no—
32C22×C42C2311yescharacteristic
33#494—42no—
34#467—42no—
35#467—42no—
36#467—42no—
37#467—42no—
38C2×#107—21no—
39C2×#107—21no—
40C2×C4⋊C8—22no—
41C2×#103—22no—
42C2×#103—22no—
43C2×C22⋊C8—22no—
44C2×C4×C8—24no—
45C2×C4×C8—42no—
46C2×C4×C8—42no—
47C2×C4×C8—42no—
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#391—22no—
53#389—22no—
54#389—22no—
55#384C2×C412yes—
56#380—22no—
57#380—22no—
58C2×#95—21no—
59C2×#95D412yes—
60#336—24no—
61#336—22no—
62#336—21no—
63#336D412yes—
64#335—42no—
65C22×Q8—22no—
66C23×C4—22no—
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4#4111yescharacteristic
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16—42no—
74C2×Q16—42no—
75C2×Q16—42no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—42no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8D812yes—
95C22×C8SD1612yes—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42C22×C411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121#107—42no—
122#107—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125#103—44no—
126#103—44no—
127C22⋊C8—44no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8—48no—
133#95—24no—
134#95—22no—
135#95—42no—
136#95—42no—
137#95Q1614yescomplemented
138C24#12111yescharacteristic
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×D411yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12411yescharacteristic
168C22×C4#12711yescharacteristic
169C22×C4C2×D811yescharacteristic
170C22×C4C2×SD1611yescharacteristic
171C22×C4C2×Q1612yes—
172C22×C4#13512yes—
173C22×C4AGL(1,ℤ/8)11yescharacteristic
174C22×C4AGL(1,ℤ/8)11yescharacteristic
175Q16—84no—
176Q16—84no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—24no—
182C2×C8—24no—
183C2×C8—24no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8#9912yes—
199C2×C8#10212yes—
200C2×C8#10212yes—
201C2×C8#10412yes—
202C2×C8#10412yes—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—41no—
218C42—41no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—42no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23#39011yescharacteristic
228C23#45911yescharacteristic
229C23#46011yescharacteristic
230C23#46712yes—
231C23#47312yes—
232Q8—82no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—28no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4C2×#9911yescharacteristic
272C2×C4#41011yescharacteristic
273C2×C4C2×#10211yescharacteristic
274C2×C4#41811yescharacteristic
275C2×C4C2×#10411yescharacteristic
276C2×C4C2×#10411yescharacteristic
277C2×C4C4×Q1612yes—
278C2×C4#44312yes—
279C2×C4#45911yescharacteristic
280C2×C4#46011yescharacteristic
281C2×C4#49411yescharacteristic
282C2×C4#49511yescharacteristic
283C8—22no—
284C8—22no—
285C8—44no—
286C8—41no—
287C8—162no—
288C8#32714yes—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—21no—
296C22—21no—
297C22#166912yes—
298C22#167012yes—
299C22#196211yescharacteristic
300C22#196411yescharacteristic
301C22#196911yescharacteristic
302C22#197411yescharacteristic
303C22#197611yescharacteristic
304C22#197911yescharacteristic
305C22#198211yescharacteristic
306C4—21no—
307C4—22no—
308C4—21no—
309C4—22no—
310C4—21no—
311C4—21no—
312C4—48no—
313C4—41no—
314C4—162no—
315C4—162no—
316C4C2×#32711yescharacteristic
317C4#150211yescharacteristic
318C4#150511yescharacteristic
319C4#150711yescharacteristic
320C2—22no—
321C2—22no—
322C2#971211yescharacteristic
323C2#973311yescharacteristic
324C2#975011yescharacteristic
325C2#976311yescharacteristic
326C2#978211yescharacteristic
327C2#981411yescharacteristic
328C2#984511yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#494C4semidirect
14. #1956C4semidirect
15. #1845C4semidirect
137. #95Q16semidirect

Maximal subgroups

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

subgroupindex
#331722
C8×#952
#99092
#85962
#76682