Finite Group Databasebeta

← C31×#12180 #21468939

Subgroups — C31×#12180 #21468939

Order 7936.

472 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C31×#2551C211yesmaximal, characteristic, complemented
2C31×#2039C211yesmaximal, characteristic
3C31×#1604C212yesmaximal
4C31×#1354C212yesmaximal, complemented
5C31×#1325C211yesmaximal, characteristic, complemented
6C31×#537C2211yescharacteristic
7C2×C31×C4⋊Q8—22no—
8C31×#448—22no—
9C2×C31×#104—21no—
10C31×#417—21no—
11C31×#395—21no—
12C31×#395C2212yes—
13C31×#389—22no—
14C31×#384C2211yescharacteristic
15C4×C4⋊C4×C31—21no—
16C31×#343C2211yescharacteristic
17C31×#327—22no—
18C31×#324C2212yes—
19C2×#41×C31—21no—
20C22×Q8×C31—21no—
21C22×Q8×C31D412yes—
22C2×SD16×C31—42no—
23C2×M16×C31—21no—
24C31×C4⋊Q8—48no—
25C31×C22⋊Q8—41no—
26C31×#121—22no—
27C31×#121—41no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31—21no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31D411yescharacteristic
34C2×C4⋊C4×C31C2311yescharacteristic
35C2×C42×C31—22no—
36C2×C42×C31D411yescharacteristic
37C31×#104—42no—
38C31×#103—42no—
39C31×#103—41no—
40C31×#102—41no—
41C31×C22⋊C8—22no—
42C31×#95—21no—
43C31×#95—22no—
44C31×#95D412yes—
45#41×C31—41no—
46#41×C31—41no—
47#41×C31—41no—
48#41×C31—41no—
49C2×Q8×C31—21no—
50C2×Q8×C31—44no—
51C2×Q8×C31—44no—
52C2×Q8×C31—42no—
53C2×Q8×C31—81no—
54C2×D4×C31—21no—
55C2×D4×C31—22no—
56C2×D4×C31—41no—
57C22×C4×C31—21no—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—21no—
61C22×C4×C31—41no—
62C22×C4×C31C2×D411yescharacteristic
63C22×C4×C31C2×D412yes—
64SD16×C31—84no—
65M16×C31—41no—
66M16×C31—41no—
67C2×C8×C31—42no—
68C2×C8×C31—41no—
69C4⋊C4×C31—24no—
70C4⋊C4×C31—24no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—41no—
73C4⋊C4×C31—44no—
74C4⋊C4×C31—44no—
75C4⋊C4×C31—44no—
76C4⋊C4×C31—41no—
77C22⋊C4×C31—42no—
78C42×C31—21no—
79C42×C31—44no—
80C42×C31SD1612yescomplemented
81#12180C3111yesmaximal, characteristic, direct product factor, Sylow
82C23×C31—41no—
83C23×C31#12011yescharacteristic
84Q8×C31—41no—
85Q8×C31—41no—
86Q8×C31—44no—
87Q8×C31—82no—
88Q8×C31—84no—
89D4×C31—41no—
90D4×C31—41no—
91D4×C31—82no—
92D4×C31—81no—
93D4×C31—81no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31—21no—
98C2×C4×C31—21no—
99C2×C4×C31—24no—
100C2×C4×C31—24no—
101C2×C4×C31—22no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31—41no—
109C2×C4×C31—41no—
110C2×C4×C31—81no—
111C2×C4×C31—81no—
112C2×C4×C31C2×SD1611yescharacteristic
113C2×C4×C31#13711yescharacteristic
114C8×C31—81no—
115C8×C31—82no—
116#2551C2×C3111yescharacteristic, complemented
117#2039C2×C3111yescharacteristic
118#1604C2×C3112yes—
119#1354C2×C3112yescomplemented
120#1325C2×C3111yescharacteristic, complemented
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—81no—
124C22×C31—81no—
125C22×C31—81no—
126C22×C31#44811yescharacteristic
127C22×C31#45411yescharacteristic
128C22×C31B(4,𝔽2)11yescharacteristic
129C4×C31—24no—
130C4×C31—21no—
131C4×C31—21no—
132C4×C31—44no—
133C4×C31—42no—
134C4×C31—44no—
135C4×C31—81no—
136C4×C31—81no—
137#537C22×C3111yescharacteristic
138C2×C4⋊Q8—22no—
139#448—22no—
140C2×#104—21no—
141#417—21no—
142#395—21no—
143#395C22×C3112yes—
144#389—22no—
145#384C22×C3111yescharacteristic
146C4×C4⋊C4—21no—
147#343C22×C3111yescharacteristic
148#327—22no—
149#324C22×C3112yes—
150C2×C31—21no—
151C2×C31—21no—
152C2×C31—161no—
153C2×C31#159911yescharacteristic
154C2×C31#219611yescharacteristic
155C2×C31#220511yescharacteristic
156C2×#41—21no—
157C22×Q8—21no—
158C22×Q8D4×C3112yes—
159C2×SD16—42no—
160C2×M16—21no—
161C4⋊Q8—48no—
162C22⋊Q8—41no—
163#121—22no—
164#121—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4D4×C3111yescharacteristic
171C2×C4⋊C4C23×C3111yescharacteristic
172C2×C42—22no—
173C2×C42D4×C3111yescharacteristic
174#104—42no—
175#103—42no—
176#103—41no—
177#102—41no—
178C22⋊C8—22no—
179#95—21no—
180#95—22no—
181#95D4×C3112yes—
182C31#1218011yescharacteristic, direct product factor, Sylow
183#41—41no—
184#41—41no—
185#41—41no—
186#41—41no—
187C2×Q8—21no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—42no—
191C2×Q8—81no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—41no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4C2×D4×C3111yescharacteristic
201C22×C4C2×D4×C3112yes—
202SD16—84no—
203M16—41no—
204M16—41no—
205C2×C8—42no—
206C2×C8—41no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C22⋊C4—42no—
216C42—21no—
217C42—44no—
218C42SD16×C3112yescomplemented
219C23—41no—
220C23C31×#12011yescharacteristic
221Q8—41no—
222Q8—41no—
223Q8—44no—
224Q8—82no—
225Q8—84no—
226D4—41no—
227D4—41no—
228D4—82no—
229D4—81no—
230D4—81no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4C2×SD16×C3111yescharacteristic
250C2×C4C31×#13711yescharacteristic
251C8—81no—
252C8—82no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22C31×#44811yescharacteristic
259C22C31×#45411yescharacteristic
260C22C31×B(4,𝔽2)11yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2C31×#159911yescharacteristic
273C2C31×#219611yescharacteristic
274C2C31×#220511yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2551C2semidirect
4. C31×#1354C2semidirect
5. C31×#1325C2semidirect
80. C42×C31SD16semidirect
81. #12180C31direct (N × H)
116. #2551C2×C31semidirect
119. #1354C2×C31semidirect
120. #1325C2×C31semidirect
182. C31#12180direct (N × H)
218. C42SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#25512
C31×#20392
C31×#16042
C31×#13542
C31×#13252
#1218031