Finite Group Databasebeta

← C31×#9960 #21466719

Subgroups — C31×#9960 #21466719

Order 7936.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3187C211yesmaximal, characteristic
2C2×C31×#483C211yesmaximal, characteristic, complemented
3C31×#2501C211yesmaximal, characteristic
4C31×#1996C211yesmaximal, characteristic, complemented
5C31×#1823C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic
7C31×#1282C211yesmaximal, characteristic, complemented
8C2×C31×#123—22no—
9C2×C31×C22⋊Q8C2211yescharacteristic
10C2×C31×#121C2211yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C31×#483—22no—
13C31×#483—21no—
14C31×#483—22no—
15C31×#481—21no—
16C31×#465—22no—
17C31×#450—21no—
18C31×#448—21no—
19C2×C31×#106—21no—
20C2×C31×#103—21no—
21C2×C31×C22⋊C8C2211yescharacteristic
22C2×C31×C22⋊C8C2211yescharacteristic
23C31×#399—22no—
24C31×#390—21no—
25C31×#384—21no—
26C31×#380C2211yescharacteristic
27C4×C22⋊C4×C31C2211yescharacteristic
28C31×#342—22no—
29C31×#340—22no—
30C31×#327—22no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C23×C4×C31C2311yescharacteristic
36C2×SD16×C31—22no—
37C22×C8×C31—21no—
38C22×C8×C31D412yes—
39C31×#123—44no—
40C31×#123—44no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C31×C22⋊Q8—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#121—21no—
48C31×#121—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31D412yes—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31D412yes—
65C2×C42×C31—21no—
66C31×#106—24no—
67C31×#106—22no—
68C31×#103—24no—
69C31×#103—22no—
70C31×#102—22no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8D412yes—
75C31×#95—21no—
76C31×#95—21no—
77C24×C31—21no—
78C24×C31C2×D411yescharacteristic
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×Q8×C31—42no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—42no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—42no—
88C2×D4×C31—42no—
89C22×C4×C31—22no—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—21no—
94C22×C4×C31—42no—
95C22×C4×C31—41no—
96C22×C4×C31—41no—
97C22×C4×C31—42no—
98C22×C4×C31—42no—
99C22×C4×C31—41no—
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31#4111yescharacteristic
106SD16×C31—82no—
107C2×C8×C31—22no—
108C2×C8×C31—22no—
109C2×C8×C31—42no—
110C2×C8×C31—42no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—48no—
118C4⋊C4×C31—42no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—41no—
122C4⋊C4×C31—41no—
123C22⋊C4×C31—22no—
124C22⋊C4×C31—48no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—42no—
130C22⋊C4×C31SD1614yescomplemented
131C42×C31—42no—
132#9960C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—41no—
138C23×C31—42no—
139C23×C31#12011yescharacteristic
140C23×C31#12111yescharacteristic
141C23×C31#12711yescharacteristic
142C23×C31C2×SD1612yes—
143C23×C31AGL(1,ℤ/8)12yes—
144Q8×C31—42no—
145Q8×C31—42no—
146D4×C31—42no—
147D4×C31—81no—
148D4×C31—82no—
149D4×C31—81no—
150D4×C31—81no—
151C2×C4×C31—22no—
152C2×C4×C31—22no—
153C2×C4×C31—22no—
154C2×C4×C31—24no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—42no—
159C2×C4×C31—42no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—44no—
165C2×C4×C31—42no—
166C2×C4×C31—82no—
167C2×C4×C31—81no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C2×C4×C31#12011yescharacteristic
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12111yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C8×C31—42no—
179C8×C31—42no—
180#3187C2×C3111yescharacteristic
181C2×#483C2×C3111yescharacteristic, complemented
182#2501C2×C3111yescharacteristic
183#1996C2×C3111yescharacteristic, complemented
184#1823C2×C3111yescharacteristic, complemented
185#1519C2×C3111yescharacteristic
186#1282C2×C3111yescharacteristic, complemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31—84no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31#39211yescharacteristic
200C22×C31#45012yes—
201C22×C31#46012yes—
202C22×C31#49211yescharacteristic
203C22×C31#49611yescharacteristic
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—44no—
208C4×C31—81no—
209C4×C31—82no—
210C4×C31—81no—
211C4×C31—81no—
212C4×C31—161no—
213C4×C31#45812yes—
214C2×#123—22no—
215C2×C22⋊Q8C22×C3111yescharacteristic
216C2×#121C22×C3111yescharacteristic
217C22×C4⋊C4C22×C3111yescharacteristic
218#483—22no—
219#483—21no—
220#483—22no—
221#481—21no—
222#465—22no—
223#450—21no—
224#448—21no—
225C2×#106—21no—
226C2×#103—21no—
227C2×C22⋊C8C22×C3111yescharacteristic
228C2×C22⋊C8C22×C3111yescharacteristic
229#399—22no—
230#390—21no—
231#384—21no—
232#380C22×C3111yescharacteristic
233C4×C22⋊C4C22×C3111yescharacteristic
234#342—22no—
235#340—22no—
236#327—22no—
237C2×C31—22no—
238C2×C31—21no—
239C2×C31—22no—
240C2×C31—21no—
241C2×C31—161no—
242C2×C31#200911yescharacteristic
243C2×C31#201411yescharacteristic
244C2×C31#202211yescharacteristic
245C22×Q8D4×C3111yescharacteristic
246C22×D4—21no—
247C22×D4D4×C3111yescharacteristic
248C23×C4—21no—
249C23×C4C23×C3111yescharacteristic
250C2×SD16—22no—
251C22×C8—21no—
252C22×C8D4×C3112yes—
253#123—44no—
254#123—44no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4D4×C3112yes—
272C2×C4⋊C4D4×C3111yescharacteristic
273C2×C4⋊C4D4×C3111yescharacteristic
274C2×C22⋊C4—24no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4D4×C3112yes—
279C2×C42—21no—
280#106—24no—
281#106—22no—
282#103—24no—
283#103—22no—
284#102—22no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8D4×C3112yes—
289#95—21no—
290#95—21no—
291C31#996011yescharacteristic, direct product factor, Sylow
292C24—21no—
293C24C2×D4×C3111yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C2×D4×C3111yescharacteristic
319C22×C4C2×D4×C3111yescharacteristic
320C22×C4#41×C3111yescharacteristic
321SD16—82no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—48no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C22⋊C4—22no—
339C22⋊C4—48no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4SD16×C3114yescomplemented
346C42—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23C31×#12011yescharacteristic
354C23C31×#12111yescharacteristic
355C23C31×#12711yescharacteristic
356C23C2×SD16×C3112yes—
357C23C31×AGL(1,ℤ/8)12yes—
358Q8—42no—
359Q8—42no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4C31×#12011yescharacteristic
389C2×C4C31×#12011yescharacteristic
390C2×C4C31×#12111yescharacteristic
391C2×C4C31×#12111yescharacteristic
392C8—42no—
393C8—42no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C31×#39211yescharacteristic
407C22C31×#45012yes—
408C22C31×#46012yes—
409C22C31×#49211yescharacteristic
410C22C31×#49611yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—81no—
416C4—82no—
417C4—81no—
418C4—81no—
419C4—161no—
420C4C31×#45812yes—
421C2—22no—
422C2—21no—
423C2—22no—
424C2—21no—
425C2—161no—
426C2C31×#200911yescharacteristic
427C2C31×#201411yescharacteristic
428C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#483C2semidirect
4. C31×#1996C2semidirect
5. C31×#1823C2semidirect
7. C31×#1282C2semidirect
130. C22⋊C4×C31SD16semidirect
132. #9960C31direct (N × H)
181. C2×#483C2×C31semidirect
183. #1996C2×C31semidirect
184. #1823C2×C31semidirect
186. #1282C2×C31semidirect
291. C31#9960direct (N × H)
345. C22⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31872
C2×C31×#4832
C31×#25012
C31×#19962
C31×#18232
C31×#15192
C31×#12822
#996031