Finite Group Databasebeta

← #131113

Subgroups — #131113

Order 512.

549 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 100 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×#3383C211yesmaximal, characteristic
2#11969C212yesmaximal
3#11664C212yesmaximal
4#11494C211yesmaximal, characteristic
5#8615C211yesmaximal, characteristic
6C2×#557—21no—
7#3383—24no—
8#3383—24no—
9C2×#500C411yescharacteristic
10C2×#499C411yescharacteristic
11C2×#499C2211yescharacteristic
12C2×#479—21no—
13C2×#479—21no—
14C2×#475—21no—
15C2×C8×Q8C2211yescharacteristic
16C2×C4×Q16C411yescharacteristic
17C2×C4×Q16C411yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19#2163C2212yes—
20#1801—21no—
21#1566C2212yes—
22#1363—24no—
23#1334—24no—
24#1322—21no—
25#1322—21no—
26#1292—21no—
27C22×Q16C2×C411yescharacteristic
28#557—28no—
29#557—44no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—41no—
32C2×#125—21no—
33C2×#125—21no—
34C2×#125—41no—
35C2×#125—41no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—41no—
40C2×C4×Q8D411yescharacteristic
41#500—22no—
42#500—22no—
43#499—22no—
44#499—22no—
45#499—22no—
46#499D414yes—
47#479—28no—
48#479—28no—
49#475—28no—
50C8×Q8—22no—
51C8×Q8—22no—
52C4×Q16—22no—
53C4×Q16—22no—
54C4×Q16—22no—
55C4×Q16—22no—
56C4×Q16—22no—
57C4×Q16D414yes—
58C2×#107—21no—
59C2×#107C2×C411yescharacteristic
60C2×#107C2×C411yescharacteristic
61C2×#107C2×C411yescharacteristic
62C2×#106—21no—
63C2×#106—21no—
64#423—22no—
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68C2×#103—21no—
69C2×#103—21no—
70C2×#103—21no—
71C2×#101—41no—
72C2×C4×C8C2×C411yescharacteristic
73C2×C4×C8C2×C411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C4⋊C16—22no—
76C22⋊C16—22no—
77#346—22no—
78#343—41no—
79#334—22no—
80#334—22no—
81C22×Q8—21no—
82C22×Q8—41no—
83C22×Q8D811yescharacteristic
84C2×Q16—24no—
85C2×Q16—21no—
86C2×Q16C22⋊C412yes—
87C2×M16—42no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22×C411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C4⋊Q8—24no—
93C4⋊Q8—42no—
94C4⋊Q8—48no—
95#125—24no—
96#125—24no—
97#125—42no—
98#125—48no—
99#125—48no—
100#125—42no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×Q8—24no—
104C4×Q8—24no—
105C4×Q8—24no—
106C4×Q8—42no—
107C4×Q8—42no—
108C4×Q8—48no—
109C4×Q8—42no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4D811yescharacteristic
122C2×C4⋊C4SD1611yescharacteristic
123C2×C4⋊C4SD1611yescharacteristic
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22⋊C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C16—42no—
131#107—24no—
132#107—21no—
133#107—21no—
134#107—21no—
135#107C22⋊C412yes—
136#107C22⋊C412yes—
137#107C22⋊C412yes—
138#106—24no—
139#106—24no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145#103—24no—
146#103—24no—
147#103—24no—
148#101—82no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×D412yes—
154C4×C8C2×D412yes—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—24no—
158C2×Q8—22no—
159C2×Q8—41no—
160C2×Q8—42no—
161C2×Q8—44no—
162C2×Q8—41no—
163C2×Q8—82no—
164C2×Q8—82no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4#10211yescharacteristic
170C22×C4#10211yescharacteristic
171C22×C4#10211yescharacteristic
172C22×C4#10211yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C2×SD1611yescharacteristic
176Q16—24no—
177Q16—22no—
178Q16#10114yescomplemented
179M16—82no—
180M16—82no—
181C2×C8—22no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—82no—
188C2×C8C2×C22⋊C411yescharacteristic
189C2×C8C2×C22⋊C411yescharacteristic
190C2×C8#12012yes—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—44no—
213C42—44no—
214C42—44no—
215C42#12012yes—
216C42AGL(1,ℤ/8)12yes—
217C16—82no—
218C23C2×#10211yescharacteristic
219Q8—24no—
220Q8—44no—
221Q8—44no—
222Q8—42no—
223Q8—88no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4#36012yes—
249C2×C4#37911yescharacteristic
250C2×C4#41811yescharacteristic
251C2×C4#44712yes—
252C2×C4#45012yes—
253C8—22no—
254C8—21no—
255C8—41no—
256C8—41no—
257C8—162no—
258C8C2×#10112yes—
259C22—21no—
260C22—21no—
261C22#178611yescharacteristic
262C22C2×#36011yescharacteristic
263C22#214511yescharacteristic
264C4—22no—
265C4—22no—
266C4—21no—
267C4—22no—
268C4—41no—
269C4—44no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—82no—
275C4#178411yescharacteristic
276C4#179611yescharacteristic
277C2—21no—
278C2—21no—
279C2#838811yescharacteristic
280C2#1273511yescharacteristic
281C2#1274411yescharacteristic
subgroupG/H
center, socleC22#1786
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
178. Q16#101semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
#119692
#116642
#114942
#86152