Finite Group Databasebeta

← #141571

Subgroups — #141571

Order 512.

371 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 72 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#28706C211yesmaximal, characteristic
2#11787C211yesmaximal, characteristic
3#11777C211yesmaximal, characteristic
4#11569C211yesmaximal, characteristic
5#11564C211yesmaximal, characteristic
6#8611C211yesmaximal, characteristic
7C8×#99C211yesmaximal, characteristic
8C2×#498C411yescharacteristic, complemented
9C2×C8×D4C2211yescharacteristic
10#2563C2211yescharacteristic
11#2547C2211yescharacteristic
12#2077—21no—
13#2077—21no—
14#1988C411yescharacteristic, complemented
15#1987C411yescharacteristic, complemented
16#1810—21no—
17#1807—21no—
18#1807—21no—
19C4×#106C411yescharacteristic, complemented
20C4×#99C2211yescharacteristic
21C8×C22⋊C4—21no—
22#1465C2211yescharacteristic
23#1465C2211yescharacteristic
24#1459C2211yescharacteristic
25#1351—22no—
26#1344—22no—
27#1324—22no—
28#1319—22no—
29#1293—21no—
30#1282—21no—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×#124—21no—
36C2×C4×D4C2311yescharacteristic
37#498—41no—
38#498—41no—
39C8×D4—41no—
40C8×D4—41no—
41C2×#108—41no—
42C2×#106—21no—
43C2×#106—21no—
44C2×#106—21no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×#99—21no—
47#407—21no—
48#407—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8D411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C2×C4×C8—41no—
53C2×C4×C8C2×C411yescharacteristic
54#397—21no—
55#397—21no—
56#393—21no—
57#393—21no—
58#391—21no—
59#390—21no—
60#382—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65C4×C4⋊C4C2×C411yescharacteristic
66C4×C4⋊C4C2×C411yescharacteristic
67C4×C22⋊C4—21no—
68#342D412yes—
69#341—41no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—41no—
76#324—21no—
77#324—21no—
78#324D412yes—
79#323—21no—
80#323—21no—
81#323—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×D4#4111yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—41no—
88C2×M16—41no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—41no—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104#124—82no—
105C4×D4—41no—
106C4×D4—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2×D411yescharacteristic
114C2×C22⋊C4—22no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4C2×D411yescharacteristic
120C2×C22⋊C4#4111yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—41no—
124C2×C42C22×C411yescharacteristic
125#108—82no—
126#106—42no—
127#106—42no—
128#106—41no—
129#106—41no—
130C4⋊C8—41no—
131#99—24no—
132C22⋊C8—22no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—82no—
143#95—21no—
144#95—21no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24#12111yescharacteristic
151C24#12411yescharacteristic
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8—161no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—41no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12711yescharacteristic
178M16—81no—
179M16—81no—
180M16—81no—
181M16—81no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8#9911yescharacteristic
199C2×C8#9911yescharacteristic
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C22⋊C4—24no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—161no—
218C22⋊C4—161no—
219C42—21no—
220C42—21no—
221C42—42no—
222C42—81no—
223C42—81no—
224C42—81no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23#39011yescharacteristic
235C23#47112yes—
236Q8—162no—
237Q8—162no—
238D4—44no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4C2×#9911yescharacteristic
271C2×C4#41011yescharacteristic
272C2×C4#44312yes—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—82no—
278C8—161no—
279C8—161no—
280C8#35412yes—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22#196411yescharacteristic
294C22#197411yescharacteristic
295C22#197711yescharacteristic
296C4—22no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—84no—
302C4—161no—
303C4—161no—
304C4—161no—
305C4—161no—
306C4C2×#35411yescharacteristic
307C4#212911yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#981311yescharacteristic
313C2#1322411yescharacteristic
314C2#1323211yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#498C4semidirect
14. #1988C4semidirect
15. #1987C4semidirect
19. C4×#106C4semidirect

Maximal subgroups

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

subgroupindex
#287062
#117872
#117772
#115692
#115642
#86112
C8×#992