Finite Group Databasebeta

← C31×#12712 #21469471

Subgroups — C31×#12712 #21469471

Order 7936.

952 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3486C211yesmaximal, characteristic
2C31×#2135C211yesmaximal, characteristic, complemented
3C31×#2129C211yesmaximal, characteristic, complemented
4C31×#2120C212yesmaximal, complemented
5C31×#1796C212yesmaximal, complemented
6C2×C31×2−1+4C2211yescharacteristic
7C23×Q8×C31C411yescharacteristic, complemented
8D4×Q8×C31—22no—
9D4×Q8×C31—22no—
10C31×#548—22no—
11C31×#548—22no—
12C31×#536—22no—
13C31×#536C411yescharacteristic, complemented
14C31×#536C2211yescharacteristic
15C31×#525—22no—
16C31×#525C411yescharacteristic, complemented
17C31×#525C2211yescharacteristic
18C31×#519C411yescharacteristic, complemented
19C31×#421—22no—
20C31×#419—22no—
21C31×#413C2212yes—
22C31×#410C2212yes—
23C31×#356—22no—
24C31×#354—22no—
25C31×#352—22no—
26C31×AΣL(1,𝔽16)—22no—
27C31×2−1+4—44no—
28C2×#41×C31—22no—
29C2×#41×C31—21no—
30C2×#41×C31C2×C411yescharacteristic
31C2×#41×C31D412yes—
32C2×#41×C31C2311yescharacteristic
33C22×Q8×C31—22no—
34C22×Q8×C31—22no—
35C22×Q8×C31—21no—
36C22×Q8×C31—21no—
37C22×Q8×C31—44no—
38C22×Q8×C31C2×C411yescharacteristic
39C22×Q8×C31C2×C411yescharacteristic
40C22×Q8×C31D412yes—
41C23×C4×C31—21no—
42C23×C4×C31C2×C411yescharacteristic
43C2×M16×C31—22no—
44C31×C4⋊Q8—24no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8D412yes—
47C31×#124—24no—
48C31×#124—21no—
49C31×#124D412yes—
50C31×#123—22no—
51C31×#123—21no—
52C31×#123—22no—
53C31×#123D412yes—
54C31×C22⋊Q8—24no—
55C31×C22⋊Q8—24no—
56C31×C22⋊Q8—24no—
57C31×C22⋊Q8—21no—
58C31×C22⋊Q8—22no—
59C31×C22⋊Q8—21no—
60C31×C22⋊Q8—22no—
61C31×#121—24no—
62C31×#121—21no—
63C31×#121—21no—
64C31×#120—21no—
65C31×#120D412yes—
66C4×Q8×C31—22no—
67C4×Q8×C31—22no—
68C4×D4×C31—24no—
69C4×D4×C31—24no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117C2×C411yescharacteristic
75C31×#117C2×C411yescharacteristic
76C31×#104—44no—
77C31×#103—44no—
78C31×#101—22no—
79C31×#100—22no—
80C31×#99—22no—
81C31×#99—22no—
82C24×C31C22⋊C411yescharacteristic
83#41×C31—24no—
84#41×C31—42no—
85#41×C31—42no—
86#41×C31—42no—
87#41×C31—48no—
88#41×C31—44no—
89C2×Q8×C31—24no—
90C2×Q8×C31—24no—
91C2×Q8×C31—21no—
92C2×Q8×C31—21no—
93C2×Q8×C31—22no—
94C2×Q8×C31—24no—
95C2×Q8×C31—42no—
96C2×Q8×C31—44no—
97C2×Q8×C31—42no—
98C2×Q8×C31—48no—
99C2×Q8×C31—44no—
100C2×Q8×C31—44no—
101C2×Q8×C31—44no—
102C2×Q8×C31C22⋊C412yes—
103C2×Q8×C31C22⋊C411yescharacteristic
104C2×Q8×C31C22⋊C412yes—
105C2×Q8×C31C2×D411yescharacteristic
106C2×D4×C31—22no—
107C2×D4×C31—21no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31C22⋊C411yescharacteristic
111C2×D4×C31C2×D411yescharacteristic
112C2×D4×C31C2×D412yes—
113C22×C4×C31—21no—
114C22×C4×C31—24no—
115C22×C4×C31—21no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—44no—
122C22×C4×C31—42no—
123C22×C4×C31C22⋊C411yescharacteristic
124C22×C4×C31C22×C411yescharacteristic
125C22×C4×C31C2×D412yes—
126M16×C31—42no—
127M16×C31—42no—
128C2×C8×C31—42no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—24no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—24no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—42no—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—24no—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—21no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—42no—
145C22⋊C4×C31C22⋊C412yes—
146C42×C31—24no—
147C42×C31—21no—
148C42×C31—42no—
149C42×C31C22⋊C412yes—
150#12712C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—21no—
152C23×C31—21no—
153C23×C31—41no—
154C23×C31—42no—
155C23×C31C2×C22⋊C411yescharacteristic
156C23×C31#12012yes—
157Q8×C31—24no—
158Q8×C31—22no—
159Q8×C31—22no—
160Q8×C31—41no—
161Q8×C31—44no—
162Q8×C31—42no—
163Q8×C31—48no—
164Q8×C31—42no—
165Q8×C31#9914yes—
166D4×C31—22no—
167D4×C31—42no—
168D4×C31—41no—
169D4×C31—44no—
170D4×C31—44no—
171D4×C31—44no—
172D4×C31—42no—
173C2×C4×C31—24no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—22no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—44no—
184C2×C4×C31—44no—
185C2×C4×C31—48no—
186C2×C4×C31—42no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—82no—
192C2×C4×C31C2×C22⋊C411yescharacteristic
193C2×C4×C31C2×C22⋊C411yescharacteristic
194C2×C4×C31#12012yes—
195C8×C31—82no—
196#3486C2×C3111yescharacteristic
197#2135C2×C3111yescharacteristic, complemented
198#2129C2×C3111yescharacteristic, complemented
199#2120C2×C3112yescomplemented
200#1796C2×C3112yescomplemented
201C22×C31—22no—
202C22×C31—21no—
203C22×C31—22no—
204C22×C31—42no—
205C22×C31—41no—
206C22×C31—41no—
207C22×C31—42no—
208C22×C31—42no—
209C22×C31—42no—
210C22×C31—41no—
211C22×C31—41no—
212C22×C31#37911yescharacteristic
213C4×C31—22no—
214C4×C31—24no—
215C4×C31—21no—
216C4×C31—42no—
217C4×C31—42no—
218C4×C31—41no—
219C4×C31—42no—
220C4×C31—41no—
221C4×C31—162no—
222C4×C31C2×#9912yes—
223C2×2−1+4C22×C3111yescharacteristic
224C23×Q8C4×C3111yescharacteristic, complemented
225D4×Q8—22no—
226D4×Q8—22no—
227#548—22no—
228#548—22no—
229#536—22no—
230#536C4×C3111yescharacteristic, complemented
231#536C22×C3111yescharacteristic
232#525—22no—
233#525C4×C3111yescharacteristic, complemented
234#525C22×C3111yescharacteristic
235#519C4×C3111yescharacteristic, complemented
236#421—22no—
237#419—22no—
238#413C22×C3112yes—
239#410C22×C3112yes—
240#356—22no—
241#354—22no—
242#352—22no—
243AΣL(1,𝔽16)—22no—
244C2×C31—21no—
245C2×C31—42no—
246C2×C31—42no—
247C2×C31—41no—
248C2×C31—81no—
249C2×C31#178111yescharacteristic
2502−1+4—44no—
251C2×#41—22no—
252C2×#41—21no—
253C2×#41C2×C4×C3111yescharacteristic
254C2×#41D4×C3112yes—
255C2×#41C23×C3111yescharacteristic
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—44no—
261C22×Q8C2×C4×C3111yescharacteristic
262C22×Q8C2×C4×C3111yescharacteristic
263C22×Q8D4×C3112yes—
264C23×C4—21no—
265C23×C4C2×C4×C3111yescharacteristic
266C2×M16—22no—
267C4⋊Q8—24no—
268C4⋊Q8—21no—
269C4⋊Q8D4×C3112yes—
270#124—24no—
271#124—21no—
272#124D4×C3112yes—
273#123—22no—
274#123—21no—
275#123—22no—
276#123D4×C3112yes—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—21no—
281C22⋊Q8—22no—
282C22⋊Q8—21no—
283C22⋊Q8—22no—
284#121—24no—
285#121—21no—
286#121—21no—
287#120—21no—
288#120D4×C3112yes—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295#117—22no—
296#117—22no—
297#117C2×C4×C3111yescharacteristic
298#117C2×C4×C3111yescharacteristic
299#104—44no—
300#103—44no—
301#101—22no—
302#100—22no—
303#99—22no—
304#99—22no—
305C31#1271211yescharacteristic, direct product factor, Sylow
306C24C22⋊C4×C3111yescharacteristic
307#41—24no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—48no—
312#41—44no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8C22⋊C4×C3111yescharacteristic
327C2×Q8C22⋊C4×C3112yes—
328C2×Q8C22⋊C4×C3112yes—
329C2×Q8C2×D4×C3111yescharacteristic
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4C22⋊C4×C3111yescharacteristic
335C2×D4C2×D4×C3111yescharacteristic
336C2×D4C2×D4×C3112yes—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4C22⋊C4×C3111yescharacteristic
348C22×C4C22×C4×C3111yescharacteristic
349C22×C4C2×D4×C3112yes—
350M16—42no—
351M16—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C22⋊C4—22no—
362C22⋊C4—24no—
363C22⋊C4—22no—
364C22⋊C4—21no—
365C22⋊C4—24no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—42no—
369C22⋊C4C22⋊C4×C3112yes—
370C42—24no—
371C42—21no—
372C42—42no—
373C42C22⋊C4×C3112yes—
374C23—21no—
375C23—21no—
376C23—41no—
377C23—42no—
378C23C2×C22⋊C4×C3111yescharacteristic
379C23C31×#12012yes—
380Q8—24no—
381Q8—22no—
382Q8—22no—
383Q8—41no—
384Q8—44no—
385Q8—42no—
386Q8—48no—
387Q8—42no—
388Q8C31×#9914yes—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—42no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4C2×C22⋊C4×C3111yescharacteristic
416C2×C4C2×C22⋊C4×C3111yescharacteristic
417C2×C4C31×#12012yes—
418C8—82no—
419C22—22no—
420C22—21no—
421C22—22no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22C31×#37911yescharacteristic
431C4—22no—
432C4—24no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—162no—
440C4C2×C31×#9912yes—
441C2—21no—
442C2—42no—
443C2—42no—
444C2—41no—
445C2—81no—
446C2C31×#178111yescharacteristic
subgroupG/H
center, socleC2×C31#1781
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×C31×2−1+4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2135C2semidirect
3. C31×#2129C2semidirect
4. C31×#2120C2semidirect
5. C31×#1796C2semidirect
7. C23×Q8×C31C4semidirect
13. C31×#536C4semidirect
16. C31×#525C4semidirect
18. C31×#519C4semidirect
150. #12712C31direct (N × H)
197. #2135C2×C31semidirect
198. #2129C2×C31semidirect
199. #2120C2×C31semidirect
200. #1796C2×C31semidirect
224. C23×Q8C4×C31semidirect
230. #536C4×C31semidirect
233. #525C4×C31semidirect
235. #519C4×C31semidirect
305. C31#12712direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34862
C31×#21352
C31×#21292
C31×#21202
C31×#17962
#1271231