Finite Group Databasebeta

← #499629

Subgroups — #499629

Order 512.

487 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 136 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#21736C211yesmaximal, characteristic, complemented
2#21286C211yesmaximal, characteristic
3#13541C211yesmaximal, characteristic, complemented
4#13535C211yesmaximal, characteristic
5#13521C212yesmaximal, complemented
6#13478C212yesmaximal
7#13454C211yesmaximal, characteristic, complemented
8#13367C211yesmaximal, characteristic
9#12677C212yesmaximal, complemented
10#12659C211yesmaximal, characteristic, complemented
11#12647C212yesmaximal
12#3162C2211yescharacteristic
13#2961—21no—
14C2×#442C2211yescharacteristic
15#2293—21no—
16#2159C2212yes—
17C2×#365C2212yes—
18C2×#364C2212yes—
19C2×#361C2212yes—
20#2140C2212yes—
21C2×#347C2212yes—
22#1994—21no—
23#1972C2211yescharacteristic
24#1970C2211yescharacteristic
25#1946—21no—
26#1945C2212yes—
27#1944C2211yescharacteristic
28#1941C2211yescharacteristic
29#1936—21no—
30#1934—21no—
31#1849C2211yescharacteristic
32#1776C2211yescharacteristic
33#1773C2212yes—
34#1772—21no—
35#1762—21no—
36C4×#97C2211yescharacteristic
37#1387C2212yes—
38#1386C2212yes—
39#1386C2212yes—
40#1383C2212yes—
41#1377C2211yescharacteristic
42#1377C2211yescharacteristic
43C22×D8C2311yescharacteristic
44#533—21no—
45C2×C4×D4—21no—
46C2×#117—41no—
47C2×M32C2312yes—
48C2×M32C2312yes—
49#501—22no—
50#501—22no—
51#446—42no—
52#446—42no—
53#442—22no—
54#442Q814yescomplemented
55C2×#108—21no—
56C2×#108C2312yes—
57#427—21no—
58#427C2312yes—
59C2×#107C2311yescharacteristic
60C2×#106C2311yescharacteristic
61C2×#106C2311yescharacteristic
62#424—22no—
63#423—21no—
64C2×#102—21no—
65C2×#102—21no—
66C2×#102—21no—
67#405—22no—
68C2×#97D411yescharacteristic
69C2×#97D411yescharacteristic
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#390—21no—
75#390—21no—
76#389—21no—
77#389—21no—
78#388—41no—
79#388—41no—
80C4×C4⋊C4—21no—
81C4×C22⋊C4—41no—
82C43D411yescharacteristic
83#367—22no—
84#366—22no—
85#365—24no—
86#364—24no—
87#361—24no—
88#357—22no—
89#357—22no—
90#347—24no—
91#340—21no—
92#340—21no—
93#340—21no—
94#339—22no—
95#339—22no—
96C22×D4—21no—
97C23×C4—41no—
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—21no—
101C2×D8C2×Q812yes—
102C2×M16—22no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2411yescharacteristic
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109#125—22no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×D4—24no—
115C4×D4—42no—
116#117—22no—
117#117—84no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—41no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134M32—22no—
135M32—22no—
136M32—22no—
137M32#4114yes—
138C2×C16—22no—
139C2×C16—22no—
140#108—22no—
141#108—22no—
142#108—42no—
143#107—21no—
144#107—21no—
145#107—22no—
146#107#4112yes—
147#107#4112yes—
148#106—22no—
149#106—21no—
150#106—22no—
151#106—22no—
152#106C2×Q812yes—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—44no—
156#102—24no—
157#102—42no—
158#102—42no—
159#97—22no—
160#97—22no—
161#97—41no—
162#97—42no—
163#97C2×Q812yes—
164#97#4112yes—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—42no—
168#95—41no—
169#95—41no—
170#95—41no—
171C24—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×D4—22no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—44no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4AGL(1,ℤ/8)11yescharacteristic
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4C22×D411yescharacteristic
191D8—24no—
192D8—42no—
193M16—42no—
194M16—42no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—42no—
202C2×C8C22⋊Q812yes—
203C2×C8AGL(1,ℤ/8)12yes—
204C2×C8#13712yes—
205C2×C8C22×Q811yescharacteristic
206C2×C8C2×#4111yescharacteristic
207C2×C8C2×#4112yes—
208C2×C8C2×#4112yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—42no—
235C42—42no—
236C42—81no—
237C42—82no—
238C42C22⋊Q812yes—
239C16—24no—
240C16—42no—
241C23—42no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23C2×AGL(1,ℤ/8)11yescharacteristic
249Q8—161no—
250Q8—161no—
251D4—42no—
252D4—42no—
253D4—81no—
254D4—81no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—88no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4C2×C22⋊Q811yescharacteristic
278C2×C4#52511yescharacteristic
279C2×C4#53211yescharacteristic
280C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
281C2×C4C2×#13711yescharacteristic
282C2×C4#57511yescharacteristic
283C8—22no—
284C8—22no—
285C8—22no—
286C8—82no—
287C8#52912yes—
288C8#53312yes—
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—161no—
295C22—161no—
296C22—161no—
297C22—161no—
298C22#307711yescharacteristic
299C22#310911yescharacteristic
300C22#318211yescharacteristic
301C4—42no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4—82no—
308C4—161no—
309C4—161no—
310C4#239611yescharacteristic
311C4#307911yescharacteristic
312C4#311711yescharacteristic
313C4#318011yescharacteristic
314C2—21no—
315C2—21no—
316C2—162no—
317C2#2378011yescharacteristic
318C2#3368811yescharacteristic
319C2#3369011yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21736C2semidirect
3. #13541C2semidirect
5. #13521C2semidirect
7. #13454C2semidirect
9. #12677C2semidirect
10. #12659C2semidirect
54. #442Q8semidirect

Maximal subgroups

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

subgroupindex
#217362
#212862
#135412
#135352
#135212
#134782
#134542
#133672
#126772
#126592
#126472