Finite Group Databasebeta

← #244150

Subgroups — #244150

Order 512.

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

normal subgroups only

This group has 687 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
1#21709C211yesmaximal, characteristic, complemented
2#21638C211yesmaximal, characteristic, complemented
3#17929C212yesmaximal, complemented
4#17900C212yesmaximal, complemented
5#10665C212yesmaximal, complemented
6#10662C211yesmaximal, characteristic, complemented
7#10661C211yesmaximal, characteristic, complemented
8#10034C211yesmaximal, characteristic, complemented
9#10032C211yesmaximal, characteristic
10#10013C212yesmaximal, complemented
11#8137C211yesmaximal, characteristic, complemented
12#3150C2211yescharacteristic
13#3103C2212yes—
14#3100C2212yes—
15C2×#446C2211yescharacteristic
16C2×#444—22no—
17#2931C2212yes—
18C2×C8×D4C2211yescharacteristic
19#2398C2211yescharacteristic
20#1985C2211yescharacteristic
21#1974—22no—
22#1963C2212yes—
23#1955C2211yescharacteristic
24#1923—22no—
25#1849C2211yescharacteristic
26#1832C2211yescharacteristic
27#1752—22no—
28#1726C2211yescharacteristic
29#1590—24no—
30#1582—24no—
31#1581—24no—
32C8×D8—24no—
33#1571C2212yes—
34#1570C2212yes—
35#1487C2212yes—
36#1486C2212yes—
37#1480C2212yes—
38#1480C2212yes—
39#1450C2212yes—
40#1275C2211yescharacteristic
41#1275C2211yescharacteristic
42#1274C2212yes—
43C2×#135—22no—
44C2×#131—22no—
45C23×C8—21no—
46#532C2312yes—
47C2×C4×Q8C2311yescharacteristic
48C2×C4×D4C2311yescharacteristic
49#497—22no—
50#496—22no—
51#481—22no—
52#480—22no—
53#475—22no—
54#474—22no—
55#460—22no—
56#459—22no—
57#448—22no—
58#447—22no—
59#446—22no—
60#446—22no—
61#444—48no—
62C4×SD16—24no—
63C4×D8—24no—
64#436—22no—
65#436—22no—
66C8×D4—22no—
67C8×D4—22no—
68C8×D4—22no—
69C8×D4—22no—
70#431C2312yes—
71#431C2312yes—
72C2×#108—22no—
73#423C2312yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×#104—24no—
78#408—22no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—22no—
81C2×#97—21no—
82C2×#97D412yes—
83C2×#97C2311yescharacteristic
84C2×C4×C8D412yes—
85C2×C4×C8C2311yescharacteristic
86#400—21no—
87#397—21no—
88#394—21no—
89#393—21no—
90C4×C22⋊C4—22no—
91#376C2311yescharacteristic
92#336—22no—
93#336—22no—
94#335—24no—
95#334—24no—
96Q8⋊C8—24no—
97D4⋊C8—24no—
98D4⋊C8—24no—
99D4⋊C8—24no—
100#322D414yes—
101C82D414yes—
102C2×#41—22no—
103C22×Q8#4111yescharacteristic
104C22×D4#4111yescharacteristic
105C23×C4—21no—
106#135—44no—
107#135—44no—
108C2×Q16—22no—
109C2×SD16—24no—
110C2×SD16—24no—
111C2×D8—22no—
112C2×D8—24no—
113#131—48no—
114C2×M16—22no—
115C2×M16—42no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—44no—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D412yes—
123C22×C8C2×D411yescharacteristic
124C4⋊Q8#4112yes—
125#127#4112yes—
126#124—22no—
127#121—22no—
128#121—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4#4111yescharacteristic
137C2×C4⋊C4#4111yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C42—21no—
142C2×C42C2×D412yes—
143C2×C42C2411yescharacteristic
144#108—44no—
145#107—24no—
146#106—24no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155#104—48no—
156#103—24no—
157#102—24no—
158#102—24no—
159#102—24no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166C22⋊C8—44no—
167#97—24no—
168#97—22no—
169#97—42no—
170C4×C8—24no—
171C4×C8C2×D414yes—
172C4×C8C2×D414yes—
173C4×C8C2×D412yes—
174C4×C8C2×D412yes—
175#95—22no—
176#95—22no—
177#95—21no—
178#95—21no—
179C24—21no—
180#41—44no—
181#41—44no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—81no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—42no—
190C2×D4—42no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C22×C4C2×#4111yescharacteristic
203Q16—44no—
204SD16—48no—
205SD16—84no—
206D8—44no—
207D8—48no—
208M16—44no—
209M16—84no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—84no—
221C2×C8—84no—
222C2×C8#12711yescharacteristic
223C2×C8#12711yescharacteristic
224C2×C8#12712yes—
225C2×C8#13514yes—
226C2×C8AGL(1,ℤ/8)14yes—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C42—22no—
238C42—22no—
239C42—41no—
240C42—41no—
241C42C22×D412yes—
242C42C2×#4112yes—
243C23—22no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23—41no—
248C23—42no—
249C23#52511yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—44no—
253D4—24no—
254D4—42no—
255D4—44no—
256D4—44no—
257D4—44no—
258D4—84no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4#52512yes—
278C2×C4#52512yes—
279C2×C4#52511yescharacteristic
280C2×C4C2×#12711yescharacteristic
281C2×C4C2×#13512yes—
282C2×C4C2×AGL(1,ℤ/8)12yes—
283C8—24no—
284C8—21no—
285C8—21no—
286C8—44no—
287C8—44no—
288C8—81no—
289C8—81no—
290C8—82no—
291C8#49614yes—
292C22—21no—
293C22—21no—
294C22—44no—
295C22—42no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22#239711yescharacteristic
303C22#310612yes—
304C4—22no—
305C4—22no—
306C4—82no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—82no—
311C4#310512yes—
312C4#314711yescharacteristic
313C4C2×#49611yescharacteristic
314C2—22no—
315C2—82no—
316C2—162no—
317C2#2379411yescharacteristic
318C2#2379511yescharacteristic
319C2#2379611yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutator, Frattini, cosocleC2×C42C24
socleC22#2397
prime radical#10032C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21709C2semidirect
2. #21638C2semidirect
3. #17929C2semidirect
4. #17900C2semidirect
5. #10665C2semidirect
6. #10662C2semidirect
7. #10661C2semidirect
8. #10034C2semidirect
10. #10013C2semidirect
11. #8137C2semidirect

Maximal subgroups

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

subgroupindex
#217092
#216382
#179292
#179002
#106652
#106622
#106612
#100342
#100322
#100132
#81372