Finite Group Databasebeta

← #141159

Subgroups — #141159

Order 512.

366 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 88 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#33213C211yesmaximal, characteristic
2#11724C211yesmaximal, characteristic
3#11723C211yesmaximal, characteristic
4#11718C211yesmaximal, characteristic
5#11717C211yesmaximal, characteristic
6#10803C211yesmaximal, characteristic
7#8602C211yesmaximal, characteristic
8C2×#498C411yescharacteristic, complemented
9C2×#405C2211yescharacteristic
10#2628C2211yescharacteristic
11#2628C2211yescharacteristic
12#1993C411yescharacteristic, complemented
13#1993C411yescharacteristic, complemented
14#1943—21no—
15#1941—21no—
16#1846C411yescharacteristic, complemented
17#1809—21no—
18#1807—21no—
19C2×#344—21no—
20#1742C2211yescharacteristic
21#1740C2211yescharacteristic
22#1729—21no—
23#1564—21no—
24#1458C2211yescharacteristic
25#1458C2211yescharacteristic
26#1354—22no—
27#1353—22no—
28#1352—22no—
29#1351—22no—
30#1275—21no—
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×#117—21no—
37C2×#117C2311yescharacteristic
38#498—41no—
39#498—41no—
40#431—21no—
41#431—21no—
42C2×#107—21no—
43C2×#106—21no—
44C2×#106—21no—
45C2×#106—21no—
46#405—41no—
47#405—41no—
48C2×#97C2×C411yescharacteristic
49C2×#97D411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#400—21no—
53#400—21no—
54#399—21no—
55#399—21no—
56#395—21no—
57#395—21no—
58#393—21no—
59#393—21no—
60#389—21no—
61#389—21no—
62#382C2×C411yescharacteristic
63#382C2×C411yescharacteristic
64#380—21no—
65#380—21no—
66#344—42no—
67#343—21no—
68#343D412yes—
69#342—21no—
70#342D412yes—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—41no—
76#336—41no—
77#335—41no—
78#334—41no—
79#324—21no—
80#324—21no—
81#323—21no—
82#323—21no—
83C22×Q8—21no—
84C22×Q8—21no—
85C23×C4C2×D411yescharacteristic
86C2×M16—22no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16SD1612yes—
95C2×M16Q1612yes—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C4⋊Q8—41no—
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109C4⋊Q8—41no—
110#117—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C2×D411yescharacteristic
126C2×C4⋊C4#4111yescharacteristic
127C2×C22⋊C4—22no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4C2×D411yescharacteristic
131C2×C22⋊C4#4111yescharacteristic
132C2×C42—41no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135#107—42no—
136#106—42no—
137#106—42no—
138#106—41no—
139#106—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144#97—41no—
145#97—41no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—41no—
151C4×C8—41no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160C24#12111yescharacteristic
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8—161no—
164C2×Q8—161no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12411yescharacteristic
179C22×C4#12711yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4C2×Q1611yescharacteristic
182C22×C4#13711yescharacteristic
183C22×C4#13711yescharacteristic
184M16—42no—
185M16—42no—
186M16—42no—
187M16—42no—
188M16—81no—
189M16—81no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8#9911yescharacteristic
209C2×C8#9911yescharacteristic
210C2×C8#10312yes—
211C2×C8#10312yes—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—161no—
227C4⋊C4—161no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—161no—
233C22⋊C4—161no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—81no—
239C42—81no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23#39011yescharacteristic
245C23#46111yescharacteristic
246C23#46211yescharacteristic
247Q8—162no—
248Q8—162no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4C2×#9911yescharacteristic
283C2×C4#41011yescharacteristic
284C2×C4C2×#10311yescharacteristic
285C2×C4#41911yescharacteristic
286C2×C4#42111yescharacteristic
287C2×C4#42111yescharacteristic
288C2×C4#46111yescharacteristic
289C2×C4#46211yescharacteristic
290C8—21no—
291C8—21no—
292C8—22no—
293C8—22no—
294C8—41no—
295C8—82no—
296C8—161no—
297C8—161no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#196311yescharacteristic
308C22#196411yescharacteristic
309C22#198111yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—161no—
317C4—161no—
318C4—162no—
319C4—161no—
320C4—161no—
321C4#151111yescharacteristic
322C4#151511yescharacteristic
323C4#151811yescharacteristic
324C4#152011yescharacteristic
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2#971411yescharacteristic
330C2#1322911yescharacteristic
331C2#1323211yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#498C4semidirect
12. #1993C4semidirect
13. #1993C4semidirect
16. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332132
#117242
#117232
#117182
#117172
#108032
#86022