Finite Group Databasebeta

← #31885529

Subgroups — #31885529

Order 18176.

846 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 155 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#22588408C212yesmaximal, complemented
2#22586161C214yesmaximal, complemented
3C2×#390×C71C211yesmaximal, characteristic
4C2×C22⋊(C71⋊C8)—22no—
5#17985754—22no—
6#17985754C2214yescomplemented
7C23×D4×C71C411yescharacteristic
8C22×C4⋊C4×C71C411yescharacteristic
9C22×C22⋊C4×C71—22no—
10C22×C42×C71C2211yescharacteristic
11#390×C71—22no—
12#390×C71C412yes—
13#390×C71C2212yes—
14C22×C71⋊C8—24no—
15C22⋊(C71⋊C8)—44no—
16C25×C71—21no—
17C22×D4×C71—21no—
18C22×D4×C71—22no—
19C22×D4×C71—42no—
20C22×D4×C71—42no—
21C22×D4×C71C814yescomplemented
22C22×D4×C71C2×C412yes—
23C23×C4×C71—21no—
24C23×C4×C71C2×C411yescharacteristic
25C23×C4×C71D412yes—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71—42no—
28C2×C4⋊C4×C71C2×C412yes—
29C2×C22⋊C4×C71—24no—
30C2×C22⋊C4×C71—24no—
31C2×C22⋊C4×C71—48no—
32C2×C42×C71—42no—
33C2×C42×C71C2×C411yescharacteristic
34C2×C42×C71D412yes—
35C2×C42×C71C2311yescharacteristic
36C2×C71⋊C8—44no—
37C2×C71⋊C8—44no—
38C2×C71⋊C8—44no—
39C24×C71—24no—
40C24×C71—22no—
41C24×C71—44no—
42C24×C71—44no—
43C24×C71—42no—
44C24×C71—42no—
45C24×C71C22⋊C411yescharacteristic
46C2×D4×C71—24no—
47C2×D4×C71—22no—
48C2×D4×C71—48no—
49C2×D4×C71—42no—
50C2×D4×C71—42no—
51C2×D4×C71—48no—
52C2×D4×C71—42no—
53C2×D4×C71—42no—
54C2×D4×C71C2×C814yes—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71—24no—
58C22×C4×C71—24no—
59C22×C4×C71—22no—
60C22×C4×C71—42no—
61C22×C4×C71—44no—
62C22×C4×C71—42no—
63C22×C4×C71C22⋊C412yes—
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C2×C812yes—
66C22×C4×C71M1612yes—
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D412yes—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—42no—
71C22⋊C4×C71—48no—
72C22⋊C4×C71—48no—
73C42×C71—41no—
74C42×C71—42no—
75C42×C71—41no—
76C23×C71—22no—
77C23×C71—22no—
78C23×C71—24no—
79C23×C71—416no—
80C23×C71—48no—
81C23×C71—42no—
82C23×C71—42no—
83C23×C71—44no—
84C23×C71—44no—
85C23×C71—84no—
86C23×C71—88no—
87C23×C71—84no—
88C23×C71C22⋊C812yes—
89C23×C71#9912yes—
90C23×C71#10012yes—
91C23×C71C2×C22⋊C411yescharacteristic
92D4×C71—44no—
93D4×C71—48no—
94D4×C71—44no—
95C2×C4×C71—24no—
96C2×C4×C71—24no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—22no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—48no—
107C2×C4×C71—42no—
108C2×C4×C71—44no—
109C2×C4×C71—84no—
110C2×C4×C71C22⋊C812yes—
111C2×C4×C71C22×C811yescharacteristic
112C2×C4×C71C2×M1611yescharacteristic
113C2×C4×C71AGL(1,ℤ/8)14yes—
114C71⋊C8—84no—
115C22×C71—22no—
116C22×C71—24no—
117C22×C71—24no—
118C22×C71—22no—
119C22×C71—44no—
120C22×C71—48no—
121C22×C71—816no—
122C22×C71—82no—
123C22×C71—84no—
124C22×C71—82no—
125C22×C71#32314yes—
126C22×C71C2×C22⋊C811yescharacteristic
127C22×C71C2×#9911yescharacteristic
128C22×C71C2×#10011yescharacteristic
129C22×C71#41812yes—
130C22×C71#42112yes—
131C4×C71—22no—
132C4×C71—21no—
133C4×C71—21no—
134C4×C71—24no—
135C4×C71—42no—
136C4×C71—82no—
137#7592—711nomaximal, Sylow
138C2×C71—24no—
139C2×C71—84no—
140C2×C71C2×#32311yescharacteristic
141C2×C71#147812yes—
142C2×C71#151012yes—
143C2×C71#153311yescharacteristic
144C2×C71#155511yescharacteristic
145C2×#390D7111yescharacteristic
146#1817—712no—
147#1270—714no—
148C71#759211yescharacteristic, complemented, Sylow
149C23×D4Dic7111yescharacteristic
150C22×C4⋊C4Dic7111yescharacteristic
151C22×C22⋊C4—22no—
152C22×C42C2×D7111yescharacteristic
153C2×C22⋊C8—1422no—
154#390—22no—
155#390Dic7112yes—
156#390C2×D7112yes—
157#336—714no—
158#336—1422no—
159C25—21no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4C71⋊C814yescomplemented
165C22×D4C2×Dic7112yes—
166C23×C4—21no—
167C23×C4C2×Dic7111yescharacteristic
168C23×C4C22⋊D7112yes—
169C22×C8—1424no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×Dic7112yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C42—42no—
177C2×C42C2×Dic7111yescharacteristic
178C2×C42C22⋊D7112yes—
179C2×C42C22×D7111yescharacteristic
180C22⋊C8—2844no—
181C24—24no—
182C24—22no—
183C24—44no—
184C24—44no—
185C24—42no—
186C24—42no—
187C24C22⋊Dic7111yescharacteristic
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—48no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—48no—
196C2×D4C2×C71⋊C814yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4C22⋊Dic7112yes—
206C22×C4C22⋊Dic7111yescharacteristic
207C22×C4C2×C71⋊C812yes—
208C22×C4#1176584912yes—
209C22×C4C22×Dic7111yescharacteristic
210C22×C4C2×C22⋊D7112yes—
211C2×C8—2844no—
212C2×C8—2844no—
213C2×C8—2844no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C42—41no—
219C42—42no—
220C42—41no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—416no—
225C23—42no—
226C23—42no—
227C23—48no—
228C23—44no—
229C23—44no—
230C23—84no—
231C23—84no—
232C23—88no—
233C23C22⋊(C71⋊C8)12yes—
234C23#1455604312yes—
235C23#1455604512yes—
236C23C2×C22⋊Dic7111yescharacteristic
237D4—44no—
238D4—48no—
239D4—44no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—84no—
255C2×C4C22⋊(C71⋊C8)12yes—
256C2×C4C22×C71⋊C811yescharacteristic
257C2×C4C2×#1176584911yescharacteristic
258C2×C4#1455615414yes—
259C8—5684no—
260C22—22no—
261C22—24no—
262C22—22no—
263C22—24no—
264C22—44no—
265C22—48no—
266C22—816no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22#1798572314yes—
271C22C2×C22⋊(C71⋊C8)11yescharacteristic
272C22C2×#1455604311yescharacteristic
273C22C2×#1455604511yescharacteristic
274C22#1798600412yes—
275C22#1798601612yes—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—24no—
280C4—42no—
281C4—82no—
282C2—24no—
283C2—84no—
284C2C2×#1798572311yescharacteristic
285C2#2258673112yes—
286C2#2258686912yes—
287C2#2258696411yescharacteristic
288C2#2258705411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC2×#390×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D4×C71C4
cosocleC2×C42×C71C23
hypercenterC2×#390D71
hypocenterC71#7592
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #22588408C2semidirect
2. #22586161C2semidirect
6. #17985754C22semidirect
21. C22×D4×C71C8semidirect
148. C71#7592semidirect
164. C22×D4C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225884082
#225861612
C2×#390×C712
#759271