Finite Group Databasebeta

← #501429

Subgroups — #501429

Order 512.

560 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2234C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic, complemented
3#27932C211yesmaximal, characteristic
4#22613C211yesmaximal, characteristic, complemented
5#13507C212yesmaximal, complemented
6#13474C212yesmaximal, complemented
7#13178C212yesmaximal, complemented
8#13131C212yesmaximal, complemented
9#13061C211yesmaximal, characteristic
10#12913C211yesmaximal, characteristic, complemented
11#12866C211yesmaximal, characteristic, complemented
12C22×M32C2211yescharacteristic
13C2×#481—21no—
14C2×#466C2211yescharacteristic
15C2×#427C2211yescharacteristic
16C22×#107C2211yescharacteristic
17#2504—21no—
18#2346—21no—
19#2234—24no—
20#2159—24no—
21C2×#367C2212yes—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2212yes—
25C2×C4⋊C16C2212yes—
26C2×#357C2212yes—
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#2092—22no—
31#1970C2212yes—
32#1942C2212yes—
33#1920—22no—
34#1846C2212yes—
35#1825—21no—
36#1808C2211yescharacteristic
37C4×#107C2212yes—
38C8×C22⋊C4C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43C22×D8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C2×#121—21no—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C2×M32—22no—
49C2×M32—24no—
50C22×C16C2311yescharacteristic
51C22×C16C2311yescharacteristic
52C22×C16C2312yes—
53#481—44no—
54#466—22no—
55#466—22no—
56#427—22no—
57#427D414yes—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2311yescharacteristic
63C2×#107C2312yes—
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#106—21no—
67C2×#106—22no—
68C2×#102—22no—
69C2×#102—21no—
70C2×C22⋊C8D411yescharacteristic
71C2×C4×C8C2312yes—
72#397—41no—
73#397—41no—
74#390—22no—
75#389—22no—
76#388—42no—
77#386—42no—
78#382—22no—
79#382—41no—
80#380—21no—
81C4×C4⋊C4—22no—
82C4×C22⋊C4D411yescharacteristic
83#376—41no—
84#367—24no—
85#367—22no—
86#366—24no—
87#366—22no—
88C4⋊C16—24no—
89#357—24no—
90#357—22no—
91#357—22no—
92C22⋊C16—22no—
93#340—21no—
94#340—22no—
95#340—22no—
96#336D412yes—
97C22×D4—21no—
98C22×D4—41no—
99C23×C4—41no—
100C23×C4C2×D411yescharacteristic
101C2×D8—21no—
102C2×D8—22no—
103C2×D8—21no—
104C2×D8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C2×D412yes—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D412yes—
110C22×C8#4112yes—
111C22×C8C2411yescharacteristic
112#121—42no—
113#121—42no—
114#117—24no—
115#117—42no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4D812yes—
131C2×C42—41no—
132C2×C42—42no—
133C2×C42C2×D412yes—
134M32—28no—
135C2×C16—22no—
136C2×C16—22no—
137C2×C16—21no—
138C2×C16—21no—
139C2×C16—22no—
140C2×C16—22no—
141C2×C16#4114yes—
142#107—21no—
143#107—21no—
144#107—22no—
145#107—21no—
146#107—22no—
147#107—22no—
148#107—21no—
149#107—22no—
150#107—42no—
151#107C2×D412yes—
152#107C2×D412yes—
153#106—24no—
154#106—44no—
155#102—44no—
156#102—42no—
157C22⋊C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160#95—41no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—42no—
165#95D811yescharacteristic
166#95D811yescharacteristic
167C24—41no—
168C24C2×D811yescharacteristic
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—42no—
173C2×D4—84no—
174C2×D4—82no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—82no—
183C22×C4—161no—
184C22×C4—161no—
185C22×C4#12112yes—
186C22×C4C2×D812yes—
187C22×C4C2×D812yes—
188C22×C4C2×D811yescharacteristic
189C22×C4AGL(1,ℤ/8)12yes—
190C22×C4C22×D411yescharacteristic
191D8—41no—
192D8—41no—
193D8—42no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8#12112yes—
202C2×C8C22×D411yescharacteristic
203C2×C8C2×#4112yes—
204C2×C8C2×#4111yescharacteristic
205C2×C82−1+412yes—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—82no—
217C4⋊C4—162no—
218C22⋊C4—42no—
219C22⋊C4—44no—
220C22⋊C4—82no—
221C22⋊C4—162no—
222C22⋊C4—162no—
223C22⋊C4—161no—
224C42—22no—
225C42—22no—
226C42—44no—
227C42—84no—
228C16—24no—
229C16—41no—
230C16—41no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—82no—
236C23—82no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23#45912yes—
241C23C22×D811yescharacteristic
242D4—82no—
243D4—81no—
244D4—81no—
245D4—164no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—164no—
270C2×C4#45912yes—
271C2×C4C2×#12111yescharacteristic
272C2×C4#54712yes—
273C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C4#57812yes—
275C8—22no—
276C8—42no—
277C8#52511yescharacteristic
278C8#53611yescharacteristic
279C8#55212yes—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—42no—
284C22—42no—
285C22—161no—
286C22—161no—
287C22—161no—
288C22—162no—
289C22—162no—
290C22C2×#45911yescharacteristic
291C22#333412yes—
292C22#341412yes—
293C22#341511yescharacteristic
294C22#341611yescharacteristic
295C4—22no—
296C4—42no—
297C4—42no—
298C4—82no—
299C4—82no—
300C4—161no—
301C4—162no—
302C4—162no—
303C4#241311yescharacteristic
304C4#304411yescharacteristic
305C4#334312yes—
306C2—42no—
307C2—321no—
308C2#2457611yescharacteristic
309C2#3358411yescharacteristic
310C2#3358911yescharacteristic
311C2#3391412yes—
312C2#3391812yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2159C2semidirect
4. #22613C2semidirect
5. #13507C2semidirect
6. #13474C2semidirect
7. #13178C2semidirect
8. #13131C2semidirect
10. #12913C2semidirect
11. #12866C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C2×#21592
#279322
#226132
#135072
#134742
#131782
#131312
#130612
#129132
#128662