Finite Group Databasebeta

← #345145

Subgroups — #345145

Order 512.

657 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 144 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#26296C212yesmaximal, complemented
2#24727C212yesmaximal
3#10588C212yesmaximal, complemented
4#10476C212yesmaximal, complemented
5#10066C212yesmaximal, complemented
6#9831C211yesmaximal, characteristic, complemented
7#9659C211yesmaximal, characteristic
8#9598C211yesmaximal, characteristic, complemented
9#8890C212yesmaximal, complemented
10C2×#500C2211yescharacteristic
11C2×#463—22no—
12C2×#461C2212yes—
13C2×C4×D8C2211yescharacteristic
14#2547C2212yes—
15D4×C4⋊C4C2212yes—
16#2075—22no—
17#2075—22no—
18#2068—22no—
19#2042—22no—
20#1990C2212yes—
21#1988C2212yes—
22#1956C2212yes—
23#1954C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1877—22no—
27#1852—22no—
28#1834C2211yescharacteristic
29#1823—22no—
30#1802C2212yes—
31C4×#107C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#332C2212yes—
35C2×D4⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×SD16—22no—
40C22×D8D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×#124—22no—
43C2×C4×D4—22no—
44C2×C4×D4C2312yes—
45C22×C4⋊C4—22no—
46#500—22no—
47#500—41no—
48#463—48no—
49#461—42no—
50#461—42no—
51C4×D8—41no—
52C4×D8—41no—
53C2×#107—22no—
54C2×#107—41no—
55C2×#107D411yescharacteristic
56C2×#107D412yes—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#102—22no—
64C2×#102D412yes—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#397—42no—
71#391—22no—
72#389—22no—
73#385—42no—
74#384C2312yes—
75#380—22no—
76#380—22no—
77#380—22no—
78C4×C4⋊C4C2312yes—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—41no—
86#340D411yescharacteristic
87#340D411yescharacteristic
88#336—22no—
89#335—41no—
90#334—41no—
91#332—42no—
92D4⋊C8—24no—
93#322—41no—
94C22×Q8—22no—
95C22×D4C2×D412yes—
96C23×C4—22no—
97C23×C4—42no—
98C2×SD16—44no—
99C2×SD16—44no—
100C2×SD16—44no—
101C2×D8—41no—
102C2×D8—41no—
103C2×D8D814yes—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115#124—44no—
116#124—82no—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—44no—
122C4×D4—82no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C42—22no—
139C2×C42C2411yescharacteristic
140#107—42no—
141#107—41no—
142#107—44no—
143#107—82no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—44no—
149#103—44no—
150#103—44no—
151#103—82no—
152#102—42no—
153#102—44no—
154C22⋊C8—44no—
155C22⋊C8—82no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8#4112yes—
162#95—22no—
163#95—42no—
164#95—42no—
165#95—42no—
166C24—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4C2×D814yes—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—84no—
191C22×C4—84no—
192C22×C4—162no—
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4H(5,𝔽2)11yescharacteristic
196SD16—88no—
197D8—44no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—21no—
206C2×C8—22no—
207C2×C8—21no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8C2×D812yes—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C22⋊C4—84no—
237C42—42no—
238C42—42no—
239C42C2×#4111yescharacteristic
240C42H(5,𝔽2)11yescharacteristic
241C42H(5,𝔽2)11yescharacteristic
242C422−1+411yescharacteristic
243C23—24no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23D4211yescharacteristic
251Q8—82no—
252Q8—82no—
253Q8—82no—
254Q8—82no—
255D4—24no—
256D4—24no—
257D4—42no—
258D4—42no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—168no—
280C2×C4#53411yescharacteristic
281C2×C4#53711yescharacteristic
282C2×C4#53811yescharacteristic
283C2×C4#54711yescharacteristic
284C2×C4#55011yescharacteristic
285C2×C4#56211yescharacteristic
286C2×C4C22×D811yescharacteristic
287C2×C4#57511yescharacteristic
288C2×C4#57512yes—
289C2×C4#57611yescharacteristic
290C2×C4#57712yes—
291C2×C4#57811yescharacteristic
292C8—22no—
293C8—42no—
294C8—41no—
295C8—41no—
296C8—41no—
297C8—82no—
298C8—82no—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22#268211yescharacteristic
307C22#326712yes—
308C22#327711yescharacteristic
309C22D4×D811yescharacteristic
310C22#328912yes—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—82no—
315C4—82no—
316C4—162no—
317C4—162no—
318C4#318611yescharacteristic
319C4#320311yescharacteristic
320C4#325611yescharacteristic
321C4#326311yescharacteristic
322C4#333411yescharacteristic
323C4#335511yescharacteristic
324C4#340112yes—
325C2—84no—
326C2#1975812yes—
327C2#2024812yes—
328C2#3118811yescharacteristic
329C2#3122311yescharacteristic
330C2#3123311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26296C2semidirect
3. #10588C2semidirect
4. #10476C2semidirect
5. #10066C2semidirect
6. #9831C2semidirect
8. #9598C2semidirect
9. #8890C2semidirect

Maximal subgroups

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

subgroupindex
#262962
#247272
#105882
#104762
#100662
#98312
#96592
#95982
#88902