Finite Group Databasebeta

← #31906734

Subgroups — #31906734

Order 18176.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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#22586664C212yesmaximal
2#22586286C212yesmaximal
3#22586274C212yesmaximal
4C4×C4⋊Q8×C71C211yesmaximal, characteristic
5#17986052—22no—
6#17985725—22no—
7#17985725C2214yes—
8C2×C4⋊Q8×C71C411yescharacteristic
9C2×C4×Q8×C71C411yescharacteristic
10C2×C4×Q8×C71C2211yescharacteristic
11#391×C71C411yescharacteristic
12#391×C71C2211yescharacteristic
13#389×C71—22no—
14#384×C71C2211yescharacteristic
15C4×C4⋊C4×C71—22no—
16C43×C71C411yescharacteristic
17C4⋊(C71⋊C8)—42no—
18C22⋊(C71⋊C8)—24no—
19C4×C71⋊C8—42no—
20C22×Q8×C71C2×C411yescharacteristic
21C22×Q8×C71C2×C411yescharacteristic
22C4⋊Q8×C71—22no—
23C4⋊Q8×C71C814yescomplemented
24C4×Q8×C71—41no—
25C4×Q8×C71—41no—
26C4×Q8×C71—41no—
27C4×Q8×C71—41no—
28C2×C4⋊C4×C71—22no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71C2311yescharacteristic
34C2×C42×C71—21no—
35C2×C42×C71—21no—
36C2×C42×C71—21no—
37C2×C42×C71C2×C411yescharacteristic
38C2×C42×C71C2×C411yescharacteristic
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71D412yes—
41#95×C71—22no—
42#95×C71—22no—
43C2×C71⋊C8—44no—
44C2×Q8×C71—21no—
45C2×Q8×C71—41no—
46C2×Q8×C71—41no—
47C2×Q8×C71—41no—
48C2×Q8×C71—41no—
49C2×Q8×C71C2×C812yes—
50C2×Q8×C71C2×C812yes—
51C2×Q8×C71M1612yes—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71C22⋊C412yes—
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22×C411yescharacteristic
60C22×C4×C71C2×D412yes—
61C4⋊C4×C71—28no—
62C4⋊C4×C71—42no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—44no—
66C4⋊C4×C71—42no—
67C42×C71—24no—
68C42×C71—21no—
69C42×C71—21no—
70C42×C71—22no—
71C42×C71—22no—
72C42×C71—21no—
73C42×C71—41no—
74C42×C71—42no—
75C42×C71—41no—
76C42×C71—41no—
77C42×C71C2×C812yes—
78C23×C71C2×C22⋊C411yescharacteristic
79Q8×C71—42no—
80Q8×C71—42no—
81Q8×C71—42no—
82Q8×C71—42no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—41no—
103C2×C4×C71C22⋊C814yes—
104C2×C4×C71#9911yescharacteristic
105C2×C4×C71#9911yescharacteristic
106C2×C4×C71C22×C811yescharacteristic
107C2×C4×C71C2×M1611yescharacteristic
108C71⋊C8—84no—
109C22×C71—21no—
110C22×C71—21no—
111C22×C71C2×C22⋊C811yescharacteristic
112C22×C71C2×#9911yescharacteristic
113C22×C71#41311yescharacteristic
114C4×C71—24no—
115C4×C71—21no—
116C4×C71—24no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—42no—
120C4×C71—42no—
121C4×C71—41no—
122C4×C71—42no—
123C4×C71—81no—
124C4×C71#35612yes—
125#11478—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71#145911yescharacteristic
129C2×C71#212611yescharacteristic
130C2×C71C2×#35611yescharacteristic
131C4×C4⋊Q8D7111yescharacteristic
132#1461—712no—
133#1325—712no—
134#1319—712no—
135C71#1147811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8Dic7111yescharacteristic
137C2×C4×Q8Dic7111yescharacteristic
138C2×C4×Q8C2×D7111yescharacteristic
139#431—1422no—
140#391Dic7111yescharacteristic
141#391C2×D7111yescharacteristic
142#389—22no—
143#384C2×D7111yescharacteristic
144C4×C4⋊C4—22no—
145C43Dic7111yescharacteristic
146#324—714no—
147#324—1422no—
148C22×Q8C2×Dic7111yescharacteristic
149C22×Q8C2×Dic7111yescharacteristic
150C4⋊Q8—22no—
151C4⋊Q8C71⋊C814yescomplemented
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C2×Dic7111yescharacteristic
160C2×C4⋊C4C22⋊D7112yes—
161C2×C4⋊C4C22×D7111yescharacteristic
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42C2×Dic7111yescharacteristic
166C2×C42C2×Dic7111yescharacteristic
167C2×C42C2×Dic7111yescharacteristic
168C2×C42C22⋊D7112yes—
169C4⋊C8—2842no—
170C22⋊C8—1424no—
171C4×C8—2842no—
172#95—22no—
173#95—22no—
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8C2×C71⋊C812yes—
180C2×Q8C2×C71⋊C812yes—
181C2×Q8#1176584912yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4C22⋊Dic7112yes—
187C22×C4C22⋊Dic7111yescharacteristic
188C22×C4C22⋊Dic7111yescharacteristic
189C22×C4C22×Dic7111yescharacteristic
190C22×C4C2×C22⋊D7112yes—
191C2×C8—2844no—
192C4⋊C4—28no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C42—24no—
199C42—22no—
200C42—21no—
201C42—22no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—42no—
206C42—41no—
207C42—41no—
208C42C2×C71⋊C812yes—
209C23C2×C22⋊Dic7111yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4C22⋊(C71⋊C8)14yes—
235C2×C4#1455604311yescharacteristic
236C2×C4#1455604311yescharacteristic
237C2×C4C22×C71⋊C811yescharacteristic
238C2×C4C2×#1176584911yescharacteristic
239C8—5684no—
240C22—21no—
241C22—21no—
242C22C2×C22⋊(C71⋊C8)11yescharacteristic
243C22C2×#1455604311yescharacteristic
244C22#1798598311yescharacteristic
245C4—24no—
246C4—24no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—41no—
254C4—81no—
255C4#1798579812yes—
256C2—21no—
257C2—21no—
258C2#2258665811yescharacteristic
259C2#2259000511yescharacteristic
260C2C2×#1798579811yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC2×C4×C71C22×C8
FrattiniC2×C4⋊C4C22×D71
FittingC4×C4⋊Q8×C71C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenterC4×C4⋊Q8D71
hypocenterC71#11478
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
23. C4⋊Q8×C71C8semidirect
135. C71#11478semidirect
151. C4⋊Q8C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225866642
#225862862
#225862742
C4×C4⋊Q8×C712
#1147871