Finite Group Databasebeta

← #137144

Subgroups — #137144

Order 512.

810 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C2×#2001C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic
3#12732C212yesmaximal, complemented
4C2×#1314C211yesmaximal, characteristic
5#7258C212yesmaximal, complemented
6C23×Q16C2211yescharacteristic
7C22×C22⋊Q8—21no—
8C22×#103—21no—
9C22×C22⋊C8C2211yescharacteristic
10C2×#380—21no—
11C2×#358—24no—
12C2×C22⋊C16C2212yes—
13#2001—22no—
14#2001C414yescomplemented
15#1852—24no—
16#1808—21no—
17#1808C2212yes—
18C2×#340C2211yescharacteristic
19#1314—22no—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C22×Q16—22no—
23C22×Q16—21no—
24C22×Q16—22no—
25C22×Q16—44no—
26C22×Q16C2×C412yescomplemented
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×C22⋊Q8—24no—
30C2×C22⋊Q8—44no—
31C2×C22⋊Q8—44no—
32C2×C22⋊Q8—42no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—41no—
36C22×C22⋊C4—41no—
37C22×C22⋊C4—41no—
38C22×C16—22no—
39C2×#107—22no—
40C2×#107D414yes—
41C2×#103—24no—
42C2×#103—44no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8C2×C412yes—
46C2×C22⋊C8D412yes—
47#380—22no—
48#380—22no—
49#380—41no—
50#380—41no—
51C2×#95—41no—
52#358—48no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55#340—22no—
56#340—22no—
57#340—21no—
58#340C2×C412yescomplemented
59C25D811yescharacteristic
60C22×Q8—24no—
61C22×Q8—22no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—41no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4—42no—
79C23×C4C22⋊C412yes—
80C23×C4D811yescharacteristic
81C23×C4SD1612yes—
82C23×C4C2×D411yescharacteristic
83C2×Q16—28no—
84C2×Q16—24no—
85C2×Q16—44no—
86C2×Q16—44no—
87C2×Q16—48no—
88C2×Q16—44no—
89C2×Q16—42no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8C22⋊C412yes—
95C22×C8C22×C411yescharacteristic
96C22×C8C2×D412yes—
97C22×C8C2×D411yescharacteristic
98C22×C8#4112yes—
99C22⋊Q8—48no—
100C22⋊Q8—48no—
101C22⋊Q8—416no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4—82no—
112C2×C22⋊C4—48no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—82no—
117C2×C22⋊C4—82no—
118C2×C16—44no—
119C2×C16—42no—
120#107—44no—
121#107—82no—
122#103—48no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125#95—42no—
126#95—42no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—42no—
132C24—42no—
133C24#10212yes—
134C24SD3212yes—
135C24Q3212yes—
136C24C2×D811yescharacteristic
137C2×Q8—24no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—44no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—82no—
152C2×Q8—81no—
153C2×Q8—82no—
154C2×Q8—81no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—82no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—84no—
185C22×C4—84no—
186C22×C4—84no—
187C22×C4#10212yes—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4C2×SD1611yescharacteristic
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4#13711yescharacteristic
194Q16—28no—
195Q16—48no—
196Q16—44no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—81no—
205C2×C8C4×D412yes—
206C2×C8#12112yes—
207C4⋊C4—24no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—416no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—84no—
214C22⋊C4—416no—
215C22⋊C4—84no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C16—82no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—44no—
230C23—44no—
231C23—42no—
232C23—42no—
233C23—81no—
234C23—82no—
235C23—82no—
236C23—81no—
237C23#35814yescomplemented
238C23C2×#10211yescharacteristic
239C23#44711yescharacteristic
240C23#48011yescharacteristic
241C23C2×SD3211yescharacteristic
242C23C2×Q3211yescharacteristic
243C23#50911yescharacteristic
244C23#51011yescharacteristic
245Q8—44no—
246Q8—44no—
247Q8—82no—
248Q8—84no—
249Q8—84no—
250Q8—82no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—168no—
281C2×C4#38611yescharacteristic
282C2×C4#41711yescharacteristic
283C2×C4#44811yescharacteristic
284C2×C4#45312yes—
285C2×C4B(4,𝔽2)12yes—
286C2×C4#48111yescharacteristic
287C8—22no—
288C8—22no—
289C8—42no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—21no—
297C22—21no—
298C22—24no—
299C22—44no—
300C22—42no—
301C22—44no—
302C22—44no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22#159512yes—
308C22#187411yescharacteristic
309C22C2×#35811yescharacteristic
310C22#213911yescharacteristic
311C22#218711yescharacteristic
312C22#218811yescharacteristic
313C22#219112yes—
314C22#223511yescharacteristic
315C22#223611yescharacteristic
316C4—21no—
317C4—21no—
318C4—42no—
319C4—84no—
320C4—84no—
321C4—164no—
322C4#159612yes—
323C4#188711yescharacteristic
324C4#188911yescharacteristic
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—44no—
330C2#901711yescharacteristic
331C2#1210512yes—
332C2#1210612yes—
333C2#1299311yescharacteristic
334C2#1300011yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12732C2semidirect
5. #7258C2semidirect
14. #2001C4semidirect
26. C22×Q16C2×C4semidirect
58. #340C2×C4semidirect
237. C23#358semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
#219402
#127322
C2×#13142
#72582