Finite Group Databasebeta

← C31×#9015 #21465774

Subgroups — C31×#9015 #21465774

Order 7936.

800 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#395C211yesmaximal, characteristic
2C31×#2316C211yesmaximal, characteristic
3C31×#1787C212yesmaximal
4C2×C31×#324C211yesmaximal, characteristic
5C31×#1294C212yesmaximal
6C23×Q8×C31C2211yescharacteristic
7C2×C31×#117—21no—
8C22×C4⋊C4×C31—21no—
9C22×C4⋊C4×C31C2211yescharacteristic
10C2×C31×#103—24no—
11C2×C31×C22⋊C8C2212yes—
12C31×#395—22no—
13C31×#395C414yescomplemented
14C31×#389—24no—
15C31×#380—21no—
16C31×#380C2212yes—
17C2×C31×#95—21no—
18C2×C31×#95C2211yescharacteristic
19C31×#324—22no—
20C22×Q8×C31—21no—
21C22×Q8×C31—22no—
22C22×Q8×C31—22no—
23C22×Q8×C31—44no—
24C22×Q8×C31C2×C412yes—
25C23×C4×C31—21no—
26C23×C4×C31—21no—
27C23×C4×C31D411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C23×C4×C31C2311yescharacteristic
30C22×C8×C31—22no—
31C31×#117—42no—
32C31×#117—42no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31—22no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—24no—
37C2×C4⋊C4×C31—22no—
38C2×C4⋊C4×C31—44no—
39C2×C4⋊C4×C31C2×C412yes—
40C2×C4⋊C4×C31D412yes—
41C2×C4⋊C4×C31D414yes—
42C2×C22⋊C4×C31—22no—
43C2×C42×C31—22no—
44C31×#103—48no—
45C31×C22⋊C8—22no—
46C31×C22⋊C8—22no—
47C31×#95—22no—
48C31×#95—24no—
49C31×#95—21no—
50C31×#95—22no—
51C31×#95C2×C412yes—
52C24×C31C2×D411yescharacteristic
53C2×Q8×C31—28no—
54C2×Q8×C31—24no—
55C2×Q8×C31—44no—
56C2×Q8×C31—44no—
57C2×Q8×C31—44no—
58C2×Q8×C31—48no—
59C2×Q8×C31—42no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—24no—
65C22×C4×C31—24no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—24no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31—41no—
75C22×C4×C31—41no—
76C22×C4×C31—41no—
77C22×C4×C31—41no—
78C22×C4×C31—42no—
79C22×C4×C31C22⋊C412yes—
80C22×C4×C31C22⋊C412yes—
81C22×C4×C31SD1612yes—
82C22×C4×C31Q1612yes—
83C22×C4×C31C22×C411yescharacteristic
84C22×C4×C31C2×D412yes—
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31#4112yes—
87C2×C8×C31—44no—
88C2×C8×C31—42no—
89C4⋊C4×C31—24no—
90C4⋊C4×C31—24no—
91C4⋊C4×C31—48no—
92C4⋊C4×C31—44no—
93C4⋊C4×C31—44no—
94C4⋊C4×C31—82no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31—42no—
97C42×C31—44no—
98#9015C3111yesmaximal, characteristic, direct product factor, Sylow
99C23×C31—22no—
100C23×C31—22no—
101C23×C31C2×C22⋊C411yescharacteristic
102C23×C31#12011yescharacteristic
103C23×C31#12311yescharacteristic
104C23×C31C2×SD1611yescharacteristic
105C23×C31C2×Q1611yescharacteristic
106C23×C31AGL(1,ℤ/8)11yescharacteristic
107C23×C31#13711yescharacteristic
108Q8×C31—28no—
109Q8×C31—48no—
110Q8×C31—44no—
111C2×C4×C31—22no—
112C2×C4×C31—22no—
113C2×C4×C31—22no—
114C2×C4×C31—22no—
115C2×C4×C31—24no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
118C2×C4×C31—24no—
119C2×C4×C31—24no—
120C2×C4×C31—22no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—44no—
126C2×C4×C31—44no—
127C2×C4×C31—42no—
128C2×C4×C31—44no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31—42no—
132C2×C4×C31—44no—
133C2×C4×C31—42no—
134C2×C4×C31—44no—
135C2×C4×C31—44no—
136C2×C4×C31—44no—
137C2×C4×C31—81no—
138C2×C4×C31#10314yescomplemented
139C2×C4×C31C4×D412yes—
140C2×C4×C31#12112yes—
141C8×C31—82no—
142C2×#395C2×C3111yescharacteristic
143#2316C2×C3111yescharacteristic
144#1787C2×C3112yes—
145C2×#324C2×C3111yescharacteristic
146#1294C2×C3112yes—
147C22×C31—22no—
148C22×C31—22no—
149C22×C31—22no—
150C22×C31—22no—
151C22×C31—21no—
152C22×C31—21no—
153C22×C31—21no—
154C22×C31—21no—
155C22×C31#38611yescharacteristic
156C22×C31C2×#10311yescharacteristic
157C22×C31#41711yescharacteristic
158C22×C31#45011yescharacteristic
159C22×C31#45111yescharacteristic
160C22×C31#45612yes—
161C22×C31B(4,𝔽2)12yes—
162C22×C31#48311yescharacteristic
163C22×C31#48411yescharacteristic
164C4×C31—24no—
165C4×C31—22no—
166C4×C31—22no—
167C4×C31—44no—
168C4×C31—44no—
169C4×C31—42no—
170C4×C31—84no—
171C23×Q8C22×C3111yescharacteristic
172C2×#117—21no—
173C22×C4⋊C4—21no—
174C22×C4⋊C4C22×C3111yescharacteristic
175C2×#103—24no—
176C2×C22⋊C8C22×C3112yes—
177#395—22no—
178#395C4×C3114yescomplemented
179#389—24no—
180#380—21no—
181#380C22×C3112yes—
182C2×#95—21no—
183C2×#95C22×C3111yescharacteristic
184#324—22no—
185C2×C31—21no—
186C2×C31—21no—
187C2×C31—21no—
188C2×C31—21no—
189C2×C31#160412yes—
190C2×C31#160512yes—
191C2×C31#187211yescharacteristic
192C2×C31#188611yescharacteristic
193C2×C31#188911yescharacteristic
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—44no—
198C22×Q8C2×C4×C3112yes—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4D4×C3111yescharacteristic
202C23×C4D4×C3111yescharacteristic
203C23×C4C23×C3111yescharacteristic
204C22×C8—22no—
205#117—42no—
206#117—42no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×C4×C3112yes—
214C2×C4⋊C4D4×C3112yes—
215C2×C4⋊C4D4×C3114yes—
216C2×C22⋊C4—22no—
217C2×C42—22no—
218#103—48no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221#95—22no—
222#95—24no—
223#95—21no—
224#95—22no—
225#95C2×C4×C3112yes—
226C31#901511yescharacteristic, direct product factor, Sylow
227C24C2×D4×C3111yescharacteristic
228C2×Q8—28no—
229C2×Q8—24no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—48no—
234C2×Q8—42no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4C22⋊C4×C3112yes—
255C22×C4C22⋊C4×C3112yes—
256C22×C4SD16×C3112yes—
257C22×C4Q16×C3112yes—
258C22×C4C22×C4×C3111yescharacteristic
259C22×C4C2×D4×C3112yes—
260C22×C4C2×D4×C3111yescharacteristic
261C22×C4#41×C3112yes—
262C2×C8—44no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—44no—
273C23—22no—
274C23—22no—
275C23C2×C22⋊C4×C3111yescharacteristic
276C23C31×#12011yescharacteristic
277C23C31×#12311yescharacteristic
278C23C2×SD16×C3111yescharacteristic
279C23C2×Q16×C3111yescharacteristic
280C23C31×AGL(1,ℤ/8)11yescharacteristic
281C23C31×#13711yescharacteristic
282Q8—28no—
283Q8—48no—
284Q8—44no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—81no—
312C2×C4C31×#10314yescomplemented
313C2×C4C4×D4×C3112yes—
314C2×C4C31×#12112yes—
315C8—82no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22C31×#38611yescharacteristic
325C22C2×C31×#10311yescharacteristic
326C22C31×#41711yescharacteristic
327C22C31×#45011yescharacteristic
328C22C31×#45111yescharacteristic
329C22C31×#45612yes—
330C22C31×B(4,𝔽2)12yes—
331C22C31×#48311yescharacteristic
332C22C31×#48411yescharacteristic
333C4—24no—
334C4—22no—
335C4—22no—
336C4—44no—
337C4—44no—
338C4—42no—
339C4—84no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2C31×#160412yes—
345C2C31×#160512yes—
346C2C31×#187211yescharacteristic
347C2C31×#188611yescharacteristic
348C2C31×#188911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
13. C31×#395C4semidirect
98. #9015C31direct (N × H)
138. C2×C4×C31#103semidirect
178. #395C4×C31semidirect
226. C31#9015direct (N × H)
312. C2×C4C31×#103semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#3952
C31×#23162
C31×#17872
C2×C31×#3242
C31×#12942
#901531