Finite Group Databasebeta

← C31×#8393 #21465152

Subgroups — C31×#8393 #21465152

Order 7936.

998 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C31×#1800C211yesmaximal, characteristic, complemented
3C31×#1548C212yesmaximal, complemented
4C31×#1486C212yesmaximal, complemented
5C31×#1460C211yesmaximal, characteristic, complemented
6C22×#41×C31C412yes—
7C31×#547—24no—
8C31×#547—24no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#123—22no—
11C2×C31×C22⋊Q8C412yes—
12C2×C31×#121—21no—
13C2×C4×D4×C31C2211yescharacteristic
14C2×C4×D4×C31C2211yescharacteristic
15C22×C4⋊C4×C31—21no—
16C31×#431C2212yes—
17C31×#423C2212yes—
18C2×C31×#100—21no—
19C31×#343—21no—
20C31×#333—24no—
21C31×D4⋊C8—24no—
22C31×#323—22no—
23C2×#41×C31—24no—
24C2×#41×C31—24no—
25C2×#41×C31—48no—
26C22×Q8×C31C2×C412yes—
27C22×D4×C31—22no—
28C22×D4×C31C2×C411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C23×C4×C31—22no—
31C23×C4×C31—21no—
32C23×C4×C31C2×C412yes—
33C2×M16×C31—22no—
34C31×C4⋊Q8—22no—
35C31×C4⋊Q8D414yes—
36C31×#123—216no—
37C31×C22⋊Q8—24no—
38C31×C22⋊Q8—24no—
39C31×#121—28no—
40C4×D4×C31—22no—
41C4×D4×C31—22no—
42C4×D4×C31—22no—
43C4×D4×C31D414yes—
44C2×C4⋊C4×C31—28no—
45C2×C4⋊C4×C31—22no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31C2×C411yescharacteristic
49C2×C4⋊C4×C31D411yescharacteristic
50C2×C22⋊C4×C31—22no—
51C2×C22⋊C4×C31—21no—
52C2×C22⋊C4×C31D412yes—
53C2×C42×C31C2311yescharacteristic
54C31×C4⋊C8—22no—
55C31×C4⋊C8—22no—
56C31×C4⋊C8—22no—
57C31×#100—42no—
58C31×C22⋊C8—22no—
59C4×C8×C31—22no—
60C24×C31—21no—
61C24×C31C22⋊C412yes—
62#41×C31—416no—
63#41×C31—48no—
64#41×C31—48no—
65C2×Q8×C31—22no—
66C2×Q8×C31—44no—
67C2×Q8×C31C22⋊C414yes—
68C2×D4×C31—24no—
69C2×D4×C31—28no—
70C2×D4×C31—21no—
71C2×D4×C31—22no—
72C2×D4×C31—22no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—48no—
76C2×D4×C31C22⋊C412yes—
77C22×C4×C31—28no—
78C22×C4×C31—21no—
79C22×C4×C31—22no—
80C22×C4×C31—22no—
81C22×C4×C31—22no—
82C22×C4×C31—24no—
83C22×C4×C31—44no—
84C22×C4×C31—48no—
85C22×C4×C31—42no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31C22⋊C412yes—
89C22×C4×C31C22×C411yescharacteristic
90C22×C4×C31C2×D411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92M16×C31—42no—
93M16×C31—42no—
94C2×C8×C31—42no—
95C2×C8×C31—42no—
96C4⋊C4×C31—28no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—22no—
99C4⋊C4×C31—24no—
100C4⋊C4×C31—24no—
101C4⋊C4×C31C22⋊C412yes—
102C22⋊C4×C31—28no—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—24no—
105C42×C31C2×D412yes—
106C42×C31C2×D412yes—
107#8393C3111yesmaximal, characteristic, direct product factor, Sylow
108C23×C31—24no—
109C23×C31—22no—
110C23×C31—22no—
111C23×C31—22no—
112C23×C31—22no—
113C23×C31—42no—
114C23×C31—42no—
115C23×C31—42no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31C2×C22⋊C411yescharacteristic
119Q8×C31—44no—
120Q8×C31—44no—
121D4×C31—24no—
122D4×C31—22no—
123D4×C31—416no—
124D4×C31—42no—
125D4×C31—42no—
126D4×C31#10014yescomplemented
127C2×C4×C31—22no—
128C2×C4×C31—22no—
129C2×C4×C31—22no—
130C2×C4×C31—24no—
131C2×C4×C31—41no—
132C2×C4×C31—44no—
133C2×C4×C31—44no—
134C2×C4×C31—416no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—44no—
138C2×C4×C31—44no—
139C2×C4×C31—44no—
140C2×C4×C31—42no—
141C2×C4×C31C2×C22⋊C411yescharacteristic
142C2×C4×C31C2×C22⋊C411yescharacteristic
143C2×C4×C31#12012yes—
144C2×C4×C31#12012yes—
145C2×C4×C31#13512yes—
146C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C31#13711yescharacteristic
148C8×C31—82no—
149C8×C31—82no—
150C2×#547C2×C3111yescharacteristic
151#1800C2×C3111yescharacteristic, complemented
152#1548C2×C3112yescomplemented
153#1486C2×C3112yescomplemented
154#1460C2×C3111yescharacteristic, complemented
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—24no—
158C22×C31—44no—
159C22×C31—44no—
160C22×C31—42no—
161C22×C31—42no—
162C22×C31—42no—
163C22×C31—42no—
164C22×C31—42no—
165C22×C31—82no—
166C22×C31—82no—
167C22×C31#37911yescharacteristic
168C22×C31#41611yescharacteristic
169C22×C31#41711yescharacteristic
170C4×C31—22no—
171C4×C31—21no—
172C4×C31—42no—
173C4×C31—44no—
174C4×C31—42no—
175C4×C31C2×#10012yes—
176C4×C31#44912yes—
177C4×C31#45212yes—
178C22×#41C4×C3112yes—
179#547—24no—
180#547—24no—
181C2×C4⋊Q8C22×C3111yescharacteristic
182C2×#123—22no—
183C2×C22⋊Q8C4×C3112yes—
184C2×#121—21no—
185C2×C4×D4C22×C3111yescharacteristic
186C2×C4×D4C22×C3111yescharacteristic
187C22×C4⋊C4—21no—
188#431C22×C3112yes—
189#423C22×C3112yes—
190C2×#100—21no—
191#343—21no—
192#333—24no—
193D4⋊C8—24no—
194#323—22no—
195C2×C31—21no—
196C2×C31—21no—
197C2×C31—44no—
198C2×C31—82no—
199C2×C31#178311yescharacteristic
200C2×C31#178811yescharacteristic
201C2×C31#179611yescharacteristic
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—48no—
205C22×Q8C2×C4×C3112yes—
206C22×D4—22no—
207C22×D4C2×C4×C3111yescharacteristic
208C22×D4D4×C3111yescharacteristic
209C23×C4—22no—
210C23×C4—21no—
211C23×C4C2×C4×C3112yes—
212C2×M16—22no—
213C4⋊Q8—22no—
214C4⋊Q8D4×C3114yes—
215#123—216no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218#121—28no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4D4×C3114yes—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C2×C4×C3111yescharacteristic
228C2×C4⋊C4D4×C3111yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4D4×C3112yes—
232C2×C42C23×C3111yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236#100—42no—
237C22⋊C8—22no—
238C4×C8—22no—
239C31#839311yescharacteristic, direct product factor, Sylow
240C24—21no—
241C24C22⋊C4×C3112yes—
242#41—416no—
243#41—48no—
244#41—48no—
245C2×Q8—22no—
246C2×Q8—44no—
247C2×Q8C22⋊C4×C3114yes—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—48no—
256C2×D4C22⋊C4×C3112yes—
257C22×C4—28no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C22⋊C4×C3112yes—
269C22×C4C22×C4×C3111yescharacteristic
270C22×C4C2×D4×C3111yescharacteristic
271C22×C4C2×D4×C3111yescharacteristic
272M16—42no—
273M16—42no—
274C2×C8—42no—
275C2×C8—42no—
276C4⋊C4—28no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4C22⋊C4×C3112yes—
282C22⋊C4—28no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C42C2×D4×C3112yes—
286C42C2×D4×C3112yes—
287C23—24no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊C4×C3111yescharacteristic
298Q8—44no—
299Q8—44no—
300D4—24no—
301D4—22no—
302D4—416no—
303D4—42no—
304D4—42no—
305D4C31×#10014yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4C2×C22⋊C4×C3111yescharacteristic
321C2×C4C2×C22⋊C4×C3111yescharacteristic
322C2×C4C31×#12012yes—
323C2×C4C31×#12012yes—
324C2×C4C31×#13512yes—
325C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
326C2×C4C31×#13711yescharacteristic
327C8—82no—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—24no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22C31×#37911yescharacteristic
342C22C31×#41611yescharacteristic
343C22C31×#41711yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—44no—
348C4—42no—
349C4C2×C31×#10012yes—
350C4C31×#44912yes—
351C4C31×#45212yes—
352C2—21no—
353C2—21no—
354C2—44no—
355C2—82no—
356C2C31×#178311yescharacteristic
357C2C31×#178811yescharacteristic
358C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#547C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1800C2semidirect
3. C31×#1548C2semidirect
4. C31×#1486C2semidirect
5. C31×#1460C2semidirect
107. #8393C31direct (N × H)
126. D4×C31#100semidirect
151. #1800C2×C31semidirect
152. #1548C2×C31semidirect
153. #1486C2×C31semidirect
154. #1460C2×C31semidirect
239. C31#8393direct (N × H)
305. D4C31×#100semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C31×#18002
C31×#15482
C31×#14862
C31×#14602
#839331