Finite Group Databasebeta

← #511697

Subgroups — #511697

Order 512.

462 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 128 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#33261C211yesmaximal, characteristic, complemented
2#33247C211yesmaximal, characteristic
3#13530C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic
5#13483C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic
7#13475C212yesmaximal, complemented
8#13474C212yesmaximal
9#13451C212yesmaximal, complemented
10#13408C212yesmaximal
11#12880C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#495C2211yescharacteristic
14C2×#489—21no—
15#2647—21no—
16#2640—21no—
17C2×#366C2212yes—
18C2×#366C2212yes—
19C2×C4⋊C16C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24#1973—21no—
25#1973—21no—
26#1970C2211yescharacteristic
27#1970C2211yescharacteristic
28#1969C2211yescharacteristic
29#1969C2211yescharacteristic
30#1944—21no—
31#1943—21no—
32#1942C2212yes—
33#1942C2212yes—
34#1846C2211yescharacteristic
35#1845C2211yescharacteristic
36#1836—21no—
37#1726C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1374C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#124—21no—
48C22×C16C2312yes—
49C22×C16C2312yes—
50#500—42no—
51#495—42no—
52#489—44no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×#107C2311yescharacteristic
58C2×#107C2311yescharacteristic
59C2×#106—21no—
60C2×#103—21no—
61C2×#103—21no—
62C2×#102—21no—
63C2×#102—21no—
64C2×#97D411yescharacteristic
65C2×#97D411yescharacteristic
66C2×#97D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#400—42no—
71#398—42no—
72#397—42no—
73#393—42no—
74#391—21no—
75#391—21no—
76#390—21no—
77#390—21no—
78#389—22no—
79#389—22no—
80#384—21no—
81#382—21no—
82#376D411yescharacteristic
83#366—42no—
84#366—42no—
85C4⋊C16—42no—
86#358—42no—
87#358—42no—
88#357—42no—
89#357—42no—
90#340—22no—
91#340—22no—
92C22×Q8—21no—
93C22×Q8—41no—
94C22×D4—21no—
95C2×Q16—21no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—21no—
99C2×SD16—42no—
100C2×SD16—42no—
101C2×SD16—42no—
102C2×D8—21no—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—21no—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2411yescharacteristic
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112#124—42no—
113#124—42no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C16—22no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135#107—22no—
136#107—22no—
137#107—22no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—42no—
142#107—42no—
143#106—42no—
144#103—42no—
145#103—42no—
146#102—42no—
147#102—42no—
148#97—22no—
149#97—22no—
150#97—41no—
151C4×C8—21no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—21no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95D812yes—
161#95D812yes—
162C24—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—82no—
170C2×Q8—162no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4C2×D812yes—
183C22×C4C2×D812yes—
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D811yescharacteristic
186C22×C4C22×D411yescharacteristic
187Q16—42no—
188Q16—42no—
189SD16—84no—
190D8—42no—
191D8—42no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8AGL(1,ℤ/8)12yes—
200C2×C8#13712yes—
201C2×C8H(5,𝔽2)11yescharacteristic
202C2×C8H(5,𝔽2)12yes—
203C2×C82−1+412yes—
204C2×C82−1+411yescharacteristic
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—82no—
214C4⋊C4—162no—
215C4⋊C4—162no—
216C4⋊C4—162no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—162no—
220C22⋊C4—162no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C16—42no—
228C16—42no—
229C23—82no—
230C23—82no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23C22×D811yescharacteristic
235Q8—81no—
236Q8—81no—
237Q8—81no—
238Q8—81no—
239Q8—164no—
240D4—81no—
241D4—81no—
242D4—81no—
243D4—81no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4#53511yescharacteristic
268C2×C4#53611yescharacteristic
269C2×C4#53711yescharacteristic
270C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
271C2×C4C2×#13711yescharacteristic
272C2×C4#57511yescharacteristic
273C8—22no—
274C8—22no—
275C8—41no—
276C8#56212yes—
277C8#56312yes—
278C22—162no—
279C22—162no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22#320511yescharacteristic
284C22#322611yescharacteristic
285C22#324511yescharacteristic
286C22#341512yes—
287C22#341712yes—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—162no—
293C4—161no—
294C4—162no—
295C4—161no—
296C4—161no—
297C4#282011yescharacteristic
298C4#321511yescharacteristic
299C4#323411yescharacteristic
300C4#325411yescharacteristic
301C2—321no—
302C2#3281811yescharacteristic
303C2#3378611yescharacteristic
304C2#3378911yescharacteristic
305C2#3380912yes—
306C2#3382812yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33261C2semidirect
3. #13530C2semidirect
5. #13483C2semidirect
7. #13475C2semidirect
9. #13451C2semidirect
11. #12880C2semidirect

Maximal subgroups

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

subgroupindex
#332612
#332472
#135302
#135262
#134832
#134822
#134752
#134742
#134512
#134082
#128802