Finite Group Databasebeta

← C31×#12129 #21468888

Subgroups — C31×#12129 #21468888

Order 7936.

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

normal subgroups only

This group has 456 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×#2784C211yesmaximal, characteristic, complemented
2C31×#2006C211yesmaximal, characteristic
3C31×#1616C211yesmaximal, characteristic, complemented
4C31×#1604C211yesmaximal, characteristic, complemented
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1353C211yesmaximal, characteristic, complemented
7C31×#1318C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#537C2211yescharacteristic
10C31×#481—21no—
11C31×#448—21no—
12C2×C31×#104—21no—
13C31×#400C2211yescharacteristic
14C31×#396—21no—
15C31×#395C2211yescharacteristic
16C31×#393—21no—
17C31×#390C2211yescharacteristic
18C31×#387—21no—
19C31×#386—21no—
20C31×#385—21no—
21C31×#380—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C23×C4×C31—21no—
31C31×AGL(1,ℤ/8)—42no—
32C31×AGL(1,ℤ/8)—42no—
33C2×SD16×C31—41no—
34C2×SD16×C31—41no—
35C2×D8×C31—41no—
36C2×M16×C31—21no—
37C31×C22⋊Q8—41no—
38C31×#121—21no—
39C31×#121—21no—
40C31×#121—41no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31D411yescharacteristic
45C2×C4⋊C4×C31C2311yescharacteristic
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C42×C31D411yescharacteristic
52C31×#106—41no—
53C31×#104—42no—
54C31×#103—41no—
55C31×#102—41no—
56C31×C22⋊C8—21no—
57C31×C22⋊C8—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×Q8×C31—41no—
72C2×Q8×C31—81no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31SD1612yescomplemented
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—41no—
87C22×C4×C31—42no—
88C22×C4×C31—81no—
89C22×C4×C31—81no—
90C22×C4×C31C2×D411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93SD16×C31—44no—
94SD16×C31—82no—
95D8×C31—44no—
96M16×C31—22no—
97M16×C31—41no—
98C2×C8×C31—41no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C4⋊C4×C31—21no—
102C4⋊C4×C31—21no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—42no—
105C4⋊C4×C31—41no—
106C4⋊C4×C31—81no—
107C4⋊C4×C31—81no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C22⋊C4×C31—81no—
114C22⋊C4×C31—81no—
115C42×C31—41no—
116#12129C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—22no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31#12011yescharacteristic
126Q8×C31—41no—
127Q8×C31—41no—
128Q8×C31—42no—
129Q8×C31—81no—
130D4×C31—42no—
131D4×C31—41no—
132D4×C31—41no—
133D4×C31—81no—
134D4×C31—81no—
135D4×C31—81no—
136D4×C31—81no—
137D4×C31—81no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—22no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—81no—
161C2×C4×C31—81no—
162C2×C4×C31—84no—
163C2×C4×C31C2×SD1611yescharacteristic
164C2×C4×C31#13711yescharacteristic
165C8×C31—42no—
166C8×C31—81no—
167C8×C31—81no—
168#2784C2×C3111yescharacteristic, complemented
169#2006C2×C3111yescharacteristic
170#1616C2×C3111yescharacteristic, complemented
171#1604C2×C3111yescharacteristic, complemented
172#1354C2×C3111yescharacteristic, complemented
173#1353C2×C3111yescharacteristic, complemented
174#1318C2×C3111yescharacteristic, complemented
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—42no—
178C22×C31—42no—
179C22×C31—42no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—81no—
187C22×C31#44811yescharacteristic
188C22×C31#45311yescharacteristic
189C22×C31B(4,𝔽2)11yescharacteristic
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—42no—
193C4×C31—41no—
194C4×C31—41no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198C4×C31—81no—
199C4×C31—81no—
200C2×AGL(1,ℤ/8)—21no—
201#537C22×C3111yescharacteristic
202#481—21no—
203#448—21no—
204C2×#104—21no—
205#400C22×C3111yescharacteristic
206#396—21no—
207#395C22×C3111yescharacteristic
208#393—21no—
209#390C22×C3111yescharacteristic
210#387—21no—
211#386—21no—
212#385—21no—
213#380—21no—
214#343C22×C3111yescharacteristic
215#327—21no—
216#327—21no—
217#324C22×C3111yescharacteristic
218#324C22×C3111yescharacteristic
219C2×C31—21no—
220C2×C31—21no—
221C2×C31—82no—
222C2×C31—161no—
223C2×C31#159611yescharacteristic
224C2×C31#220011yescharacteristic
225C2×C31#220411yescharacteristic
226C2×#41—21no—
227C22×Q8D4×C3111yescharacteristic
228C22×D4D4×C3111yescharacteristic
229C23×C4—21no—
230AGL(1,ℤ/8)—42no—
231AGL(1,ℤ/8)—42no—
232C2×SD16—41no—
233C2×SD16—41no—
234C2×D8—41no—
235C2×M16—21no—
236C22⋊Q8—41no—
237#121—21no—
238#121—21no—
239#121—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4D4×C3111yescharacteristic
244C2×C4⋊C4C23×C3111yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42D4×C3111yescharacteristic
251#106—41no—
252#104—42no—
253#103—41no—
254#102—41no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95D4×C3111yescharacteristic
263#95D4×C3111yescharacteristic
264C31#1212911yescharacteristic, direct product factor, Sylow
265C24—21no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—81no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4SD16×C3112yescomplemented
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292C22×C4C2×D4×C3111yescharacteristic
293SD16—44no—
294SD16—82no—
295D8—44no—
296M16—22no—
297M16—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C42—41no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23C31×#12011yescharacteristic
325Q8—41no—
326Q8—41no—
327Q8—42no—
328Q8—81no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4C2×SD16×C3111yescharacteristic
363C2×C4C31×#13711yescharacteristic
364C8—42no—
365C8—81no—
366C8—81no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C31×#44811yescharacteristic
380C22C31×#45311yescharacteristic
381C22C31×B(4,𝔽2)11yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C2—21no—
393C2—21no—
394C2—82no—
395C2—161no—
396C2C31×#159611yescharacteristic
397C2C31×#220011yescharacteristic
398C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2006C2
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×#2784C2semidirect
3. C31×#1616C2semidirect
4. C31×#1604C2semidirect
5. C31×#1354C2semidirect
6. C31×#1353C2semidirect
7. C31×#1318C2semidirect
80. C2×D4×C31SD16semidirect
116. #12129C31direct (N × H)
168. #2784C2×C31semidirect
170. #1616C2×C31semidirect
171. #1604C2×C31semidirect
172. #1354C2×C31semidirect
173. #1353C2×C31semidirect
174. #1318C2×C31semidirect
264. C31#12129direct (N × H)
280. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27842
C31×#20062
C31×#16162
C31×#16042
C31×#13542
C31×#13532
C31×#13182
#1212931