Finite Group Databasebeta

← C31×#2009 #16841942

Subgroups — C31×#2009 #16841942

Order 3968.

348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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×#575C211yesmaximal, characteristic
2C2×C31×#137C211yesmaximal, characteristic, complemented
3C31×#536C211yesmaximal, characteristic, complemented
4C31×#421C211yesmaximal, characteristic, complemented
5C31×#419C211yesmaximal, characteristic, complemented
6C31×#410C211yesmaximal, characteristic, complemented
7C31×#344C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C2×#41×C31C2211yescharacteristic
10C22×Q8×C31C2211yescharacteristic
11C31×#137—22no—
12C31×#137—21no—
13C31×#137—21no—
14C31×#137—22no—
15C31×AGL(1,ℤ/8)—21no—
16C31×AGL(1,ℤ/8)—21no—
17C31×#135—22no—
18C2×Q16×C31—21no—
19C2×SD16×C31—21no—
20C2×M16×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic
22C31×C4⋊Q8—21no—
23C31×#124—21no—
24C31×#123—22no—
25C31×C22⋊Q8—22no—
26C31×#117C2211yescharacteristic
27C31×#117C2211yescharacteristic
28C31×#104—22no—
29C31×#103—22no—
30C31×#99—22no—
31#41×C31—21no—
32#41×C31—21no—
33#41×C31—21no—
34#41×C31—21no—
35#41×C31—21no—
36#41×C31—21no—
37#41×C31—21no—
38#41×C31—21no—
39C2×Q8×C31—21no—
40C2×Q8×C31—21no—
41C2×Q8×C31—21no—
42C2×Q8×C31—21no—
43C2×Q8×C31—21no—
44C2×Q8×C31D412yes—
45C2×Q8×C31D411yescharacteristic
46C2×Q8×C31D411yescharacteristic
47C2×D4×C31—21no—
48C2×D4×C31—21no—
49C2×D4×C31D411yescharacteristic
50C2×D4×C31D411yescharacteristic
51C22×C4×C31—21no—
52C22×C4×C31—21no—
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55Q16×C31—24no—
56Q16×C31—22no—
57SD16×C31—24no—
58SD16×C31—22no—
59SD16×C31—22no—
60D8×C31—22no—
61M16×C31—21no—
62M16×C31—21no—
63M16×C31—21no—
64M16×C31D412yes—
65C2×C8×C31—21no—
66C2×C8×C31D412yes—
67C4⋊C4×C31—24no—
68C4⋊C4×C31—21no—
69C4⋊C4×C31—21no—
70C22⋊C4×C31—24no—
71C22⋊C4×C31—21no—
72C22⋊C4×C31D412yes—
73C42×C31—21no—
74C42×C31—21no—
75C23×C31—21no—
76C23×C31—21no—
77C23×C31C2×D411yescharacteristic
78Q8×C31—22no—
79Q8×C31—21no—
80Q8×C31—21no—
81Q8×C31—21no—
82Q8×C31—22no—
83Q8×C31—21no—
84Q8×C31—41no—
85Q8×C31—41no—
86D4×C31—21no—
87D4×C31—21no—
88D4×C31—21no—
89D4×C31—21no—
90D4×C31—41no—
91D4×C31—41no—
92D4×C31—41no—
93D4×C31—41no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—21no—
97C2×C4×C31—21no—
98C2×C4×C31—41no—
99C2×C4×C31—41no—
100C2×C4×C31—41no—
101C2×C4×C31—41no—
102C2×C4×C31—41no—
103C2×C4×C31—41no—
104C2×C4×C31—41no—
105C2×C4×C31C2×D411yescharacteristic
106C2×C4×C31C2×D411yescharacteristic
107C2×C4×C31C2×D411yescharacteristic
108C2×C4×C31C2×D411yescharacteristic
109C2×C4×C31C2×D411yescharacteristic
110C2×C4×C31#4111yescharacteristic
111C8×C31—22no—
112C8×C31—22no—
113#2009C3111yesmaximal, characteristic, direct product factor, Sylow
114C22×C31—41no—
115C22×C31—42no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31—81no—
119C22×C31#12011yescharacteristic
120C22×C31#12111yescharacteristic
121C22×C31#12711yescharacteristic
122C4×C31—42no—
123C4×C31—41no—
124C4×C31—41no—
125C4×C31—42no—
126C4×C31—81no—
127C4×C31#12011yescharacteristic
128C4×C31#12011yescharacteristic
129C4×C31#12111yescharacteristic
130C4×C31#12111yescharacteristic
131#575C2×C3111yescharacteristic
132C2×#137C2×C3111yescharacteristic, complemented
133#536C2×C3111yescharacteristic, complemented
134#421C2×C3111yescharacteristic, complemented
135#419C2×C3111yescharacteristic, complemented
136#410C2×C3111yescharacteristic, complemented
137#344C2×C3111yescharacteristic, complemented
138C2×C31—21no—
139C2×C31—21no—
140C2×C31—21no—
141C2×C31—81no—
142C2×C31—81no—
143C2×C31#39211yescharacteristic
144C2×#41C22×C3111yescharacteristic
145C2×#41C22×C3111yescharacteristic
146C22×Q8C22×C3111yescharacteristic
147#137—22no—
148#137—21no—
149#137—21no—
150#137—22no—
151AGL(1,ℤ/8)—21no—
152AGL(1,ℤ/8)—21no—
153#135—22no—
154C2×Q16—21no—
155C2×SD16—21no—
156C2×M16C22×C3111yescharacteristic
157C2×M16C22×C3111yescharacteristic
158C4⋊Q8—21no—
159#124—21no—
160#123—22no—
161C22⋊Q8—22no—
162#117C22×C3111yescharacteristic
163#117C22×C3111yescharacteristic
164#104—22no—
165#103—22no—
166#99—22no—
167C31#200911yescharacteristic, direct product factor, Sylow
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8D4×C3112yes—
182C2×Q8D4×C3111yescharacteristic
183C2×Q8D4×C3111yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4D4×C3111yescharacteristic
187C2×D4D4×C3111yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4C23×C3111yescharacteristic
192Q16—24no—
193Q16—22no—
194SD16—24no—
195SD16—22no—
196SD16—22no—
197D8—22no—
198M16—21no—
199M16—21no—
200M16—21no—
201M16D4×C3112yes—
202C2×C8—21no—
203C2×C8D4×C3112yes—
204C4⋊C4—24no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C22⋊C4—24no—
208C22⋊C4—21no—
209C22⋊C4D4×C3112yes—
210C42—21no—
211C42—21no—
212C23—21no—
213C23—21no—
214C23C2×D4×C3111yescharacteristic
215Q8—22no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—22no—
220Q8—21no—
221Q8—41no—
222Q8—41no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4C2×D4×C3111yescharacteristic
243C2×C4C2×D4×C3111yescharacteristic
244C2×C4C2×D4×C3111yescharacteristic
245C2×C4C2×D4×C3111yescharacteristic
246C2×C4C2×D4×C3111yescharacteristic
247C2×C4#41×C3111yescharacteristic
248C8—22no—
249C8—22no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—81no—
255C22C31×#12011yescharacteristic
256C22C31×#12111yescharacteristic
257C22C31×#12711yescharacteristic
258C4—42no—
259C4—41no—
260C4—41no—
261C4—42no—
262C4—81no—
263C4C31×#12011yescharacteristic
264C4C31×#12011yescharacteristic
265C4C31×#12111yescharacteristic
266C4C31×#12111yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—81no—
272C2C31×#39211yescharacteristic
subgroupG/H
center, socleC2×C31#392
commutator, Frattini, cosocleC22×C4C23×C31
prime radicalC31×#575C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#137C2semidirect
3. C31×#536C2semidirect
4. C31×#421C2semidirect
5. C31×#419C2semidirect
6. C31×#410C2semidirect
7. C31×#344C2semidirect
113. #2009C31direct (N × H)
132. C2×#137C2×C31semidirect
133. #536C2×C31semidirect
134. #421C2×C31semidirect
135. #419C2×C31semidirect
136. #410C2×C31semidirect
137. #344C2×C31semidirect
167. C31#2009direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5752
C2×C31×#1372
C31×#5362
C31×#4212
C31×#4192
C31×#4102
C31×#3442
#200931