Finite Group Databasebeta

← C31×#61828 #21518587

Subgroups — C31×#61828 #21518587

Order 7936.

1914 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3497C211yesmaximal, characteristic
2C2×C31×#557C211yesmaximal, characteristic, complemented
3C31×#3386C218yesmaximal, complemented
4C31×#3368C218yesmaximal, complemented
5C31×#3234C214yesmaximal, complemented
6C31×#3226C212yesmaximal, complemented
7C31×#3112C214yesmaximal, complemented
8C31×#3103C212yesmaximal, complemented
9C31×#2983C211yesmaximal, characteristic, complemented
10C2×C31×2−1+4—21no—
11C31×#557—24no—
12C31×#557C2218yes—
13C31×#550C2214yes—
14D4×Q8×C31C2214yes—
15C31×#548—24no—
16C31×#537—21no—
17C31×#537—22no—
18C31×#532C2212yes—
19C31×#532C2214yes—
20C2×C31×C4⋊Q8—21no—
21C2×C31×C4⋊Q8C2211yescharacteristic
22C2×C31×#125—22no—
23C2×C31×#125C2212yes—
24C2×C4×Q8×C31—21no—
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×Q8×C31C2212yes—
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#501C2218yes—
29C31×#500C2214yes—
30C31×#490C2218yes—
31C31×#486C2214yes—
32C31×#481C2214yes—
33C31×#480C2218yes—
34C31×#478C22116yes—
35C31×#477C2218yes—
36C31×#475C2214yes—
37C31×#474C2218yes—
38C31×#460C2214yes—
39C31×#460C2218yes—
40C31×#448C2214yes—
41C31×#446C2218yes—
42C31×#440C22116yes—
43C4×SD16×C31C2218yes—
44C31×#432C2214yes—
45C31×#431C2212yes—
46C31×2−1+4—48no—
47C2×#41×C31—24no—
48C2×#41×C31—24no—
49C2×#41×C31—22no—
50C22×Q8×C31—24no—
51C22×Q8×C31—21no—
52C22×Q8×C31D411yescharacteristic
53C22×Q8×C31C2311yescharacteristic
54C2×SD16×C31C2318yes—
55C31×C4⋊Q8—28no—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8C2314yes—
58C31×C4⋊Q8C2312yes—
59C31×C4⋊Q8C2314yes—
60C31×#127C2312yes—
61C31×#127C2314yes—
62C31×#125—216no—
63C31×#125—24no—
64C31×#125C2318yes—
65C31×#124—24no—
66C31×#124—22no—
67C31×C22⋊Q8—24no—
68C31×C22⋊Q8—28no—
69C31×#121—24no—
70C31×#121—22no—
71C31×#121—24no—
72C31×#121—28no—
73C31×#121C2314yes—
74C4×Q8×C31—28no—
75C4×Q8×C31—24no—
76C4×Q8×C31—22no—
77C4×Q8×C31D414yes—
78C4×Q8×C31C2314yes—
79C4×Q8×C31C2318yes—
80C4×Q8×C31C2314yes—
81C4×D4×C31—24no—
82C4×D4×C31—22no—
83C4×D4×C31C2314yes—
84C4×D4×C31C2318yes—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D412yes—
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C42×C31—22no—
95C2×C42×C31—21no—
96C2×C42×C31C2311yescharacteristic
97C2×C42×C31C2312yes—
98C31×#107C23116yes—
99C31×#106C2318yes—
100C31×C4⋊C8C2314yes—
101C31×C4⋊C8C2318yes—
102C31×#103C2318yes—
103C31×#102C2318yes—
104C31×#102C23116yes—
105C31×C22⋊C8C2314yes—
106C31×#97C2318yes—
107C4×C8×C31C2314yes—
108#41×C31—416no—
109#41×C31—416no—
110#41×C31—48no—
111C2×Q8×C31—24no—
112C2×Q8×C31—24no—
113C2×Q8×C31—21no—
114C2×Q8×C31—24no—
115C2×Q8×C31—24no—
116C2×Q8×C31—22no—
117C2×Q8×C31—416no—
118C2×Q8×C31—44no—
119C2×Q8×C31C2×D414yes—
120C2×Q8×C31C2412yes—
121C2×D4×C31—24no—
122C2×D4×C31—24no—
123C2×D4×C31—22no—
124C2×D4×C31—24no—
125C2×D4×C31—22no—
126C2×D4×C31C2414yes—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—28no—
131C22×C4×C31—24no—
132C22×C4×C31C2×D412yes—
133C22×C4×C31C2×D412yes—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2411yescharacteristic
137SD16×C31—216no—
138C2×C8×C31C2418yes—
139C4⋊C4×C31—28no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—28no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—28no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—24no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31C2×D414yes—
149C4⋊C4×C31C2×D418yes—
150C4⋊C4×C31C2412yes—
151C4⋊C4×C31C2414yes—
152C4⋊C4×C31C2414yes—
153C22⋊C4×C31—24no—
154C22⋊C4×C31—28no—
155C22⋊C4×C31—24no—
156C42×C31—28no—
157C42×C31—24no—
158C42×C31C2×D414yes—
159C42×C31C2×D412yes—
160C42×C31C2412yes—
161C42×C31C2414yes—
162#61828C3111yesmaximal, characteristic, direct product factor, Sylow
163C23×C31—24no—
164C23×C31C22×D411yescharacteristic
165Q8×C31—24no—
166Q8×C31—28no—
167Q8×C31—22no—
168Q8×C31—48no—
169Q8×C31—48no—
170Q8×C31—44no—
171Q8×C312−1+414yescomplemented
172D4×C31—28no—
173D4×C31—48no—
174D4×C31—48no—
175D4×C31—44no—
176D4×C31—84no—
177D4×C31—82no—
178C2×C4×C31—24no—
179C2×C4×C31—24no—
180C2×C4×C31—22no—
181C2×C4×C31—22no—
182C2×C4×C31—22no—
183C2×C4×C31—24no—
184C2×C4×C31—24no—
185C2×C4×C31—42no—
186C2×C4×C31—41no—
187C2×C4×C31—416no—
188C2×C4×C31—42no—
189C2×C4×C31—44no—
190C2×C4×C31—48no—
191C2×C4×C31AGL(1,ℤ/8)14yes—
192C2×C4×C31C22×D414yes—
193C2×C4×C31C22×D411yescharacteristic
194C2×C4×C31C22×D412yes—
195C2×C4×C31C22×D414yes—
196C2×C4×C31C22×D412yes—
197C2×C4×C31C2511yescharacteristic
198C8×C31—28no—
199#3497C2×C3111yescharacteristic
200C2×#557C2×C3111yescharacteristic, complemented
201#3386C2×C3118yescomplemented
202#3368C2×C3118yescomplemented
203#3234C2×C3114yescomplemented
204#3226C2×C3112yescomplemented
205#3112C2×C3114yescomplemented
206#3103C2×C3112yescomplemented
207#2983C2×C3111yescharacteristic, complemented
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—44no—
211C22×C31—44no—
212C22×C31—44no—
213C22×C31C2×AGL(1,ℤ/8)12yes—
214C22×C31C23×D411yescharacteristic
215C4×C31—24no—
216C4×C31—22no—
217C4×C31—24no—
218C4×C31—21no—
219C4×C31—24no—
220C4×C31—42no—
221C4×C31—44no—
222C4×C31—44no—
223C4×C31C2×AGL(1,ℤ/8)14yes—
224C4×C31C2×2−1+412yes—
225C2×2−1+4—21no—
226#557—24no—
227#557C22×C3118yes—
228#550C22×C3114yes—
229D4×Q8C22×C3114yes—
230#548—24no—
231#537—21no—
232#537—22no—
233#532C22×C3112yes—
234#532C22×C3114yes—
235C2×C4⋊Q8—21no—
236C2×C4⋊Q8C22×C3111yescharacteristic
237C2×#125—22no—
238C2×#125C22×C3112yes—
239C2×C4×Q8—21no—
240C2×C4×Q8C22×C3111yescharacteristic
241C2×C4×Q8C22×C3112yes—
242C2×C4×Q8C22×C3111yescharacteristic
243#501C22×C3118yes—
244#500C22×C3114yes—
245#490C22×C3118yes—
246#486C22×C3114yes—
247#481C22×C3114yes—
248#480C22×C3118yes—
249#478C22×C31116yes—
250#477C22×C3118yes—
251#475C22×C3114yes—
252#474C22×C3118yes—
253#460C22×C3114yes—
254#460C22×C3118yes—
255#448C22×C3114yes—
256#446C22×C3118yes—
257#440C22×C31116yes—
258C4×SD16C22×C3118yes—
259#432C22×C3114yes—
260#431C22×C3112yes—
261C2×C31—22no—
262C2×C31—84no—
263C2×C31#348611yescharacteristic
264C2×C31C22×AGL(1,ℤ/8)11yescharacteristic
265C2×C31#358611yescharacteristic
2662−1+4—48no—
267C2×#41—24no—
268C2×#41—24no—
269C2×#41—22no—
270C22×Q8—24no—
271C22×Q8—21no—
272C22×Q8D4×C3111yescharacteristic
273C22×Q8C23×C3111yescharacteristic
274C2×SD16C23×C3118yes—
275C4⋊Q8—28no—
276C4⋊Q8—22no—
277C4⋊Q8C23×C3114yes—
278C4⋊Q8C23×C3112yes—
279C4⋊Q8C23×C3114yes—
280#127C23×C3112yes—
281#127C23×C3114yes—
282#125—216no—
283#125—24no—
284#125C23×C3118yes—
285#124—24no—
286#124—22no—
287C22⋊Q8—24no—
288C22⋊Q8—28no—
289#121—24no—
290#121—22no—
291#121—24no—
292#121—28no—
293#121C23×C3114yes—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—22no—
297C4×Q8D4×C3114yes—
298C4×Q8C23×C3114yes—
299C4×Q8C23×C3118yes—
300C4×Q8C23×C3114yes—
301C4×D4—24no—
302C4×D4—22no—
303C4×D4C23×C3114yes—
304C4×D4C23×C3118yes—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4D4×C3111yescharacteristic
311C2×C4⋊C4D4×C3112yes—
312C2×C4⋊C4C23×C3111yescharacteristic
313C2×C4⋊C4C23×C3112yes—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42C23×C3112yes—
317C2×C42C23×C3111yescharacteristic
318#107C23×C31116yes—
319#106C23×C3118yes—
320C4⋊C8C23×C3114yes—
321C4⋊C8C23×C3118yes—
322#103C23×C3118yes—
323#102C23×C3118yes—
324#102C23×C31116yes—
325C22⋊C8C23×C3114yes—
326#97C23×C3118yes—
327C4×C8C23×C3114yes—
328C31#6182811yescharacteristic, direct product factor, Sylow
329#41—416no—
330#41—416no—
331#41—48no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—21no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—416no—
339C2×Q8—44no—
340C2×Q8C2×D4×C3114yes—
341C2×Q8C24×C3112yes—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4C24×C3114yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4C2×D4×C3112yes—
354C22×C4C2×D4×C3112yes—
355C22×C4C2×D4×C3111yescharacteristic
356C22×C4C2×D4×C3111yescharacteristic
357C22×C4C24×C3111yescharacteristic
358SD16—216no—
359C2×C8C24×C3118yes—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4C2×D4×C3114yes—
370C4⋊C4C2×D4×C3118yes—
371C4⋊C4C24×C3112yes—
372C4⋊C4C24×C3114yes—
373C4⋊C4C24×C3114yes—
374C22⋊C4—24no—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C42—28no—
378C42—24no—
379C42C2×D4×C3114yes—
380C42C2×D4×C3112yes—
381C42C24×C3112yes—
382C42C24×C3114yes—
383C23—24no—
384C23C22×D4×C3111yescharacteristic
385Q8—24no—
386Q8—28no—
387Q8—22no—
388Q8—48no—
389Q8—48no—
390Q8—44no—
391Q8C31×2−1+414yescomplemented
392D4—28no—
393D4—48no—
394D4—48no—
395D4—44no—
396D4—84no—
397D4—82no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—416no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4C31×AGL(1,ℤ/8)14yes—
412C2×C4C22×D4×C3114yes—
413C2×C4C22×D4×C3111yescharacteristic
414C2×C4C22×D4×C3112yes—
415C2×C4C22×D4×C3114yes—
416C2×C4C22×D4×C3112yes—
417C2×C4C25×C3111yescharacteristic
418C8—28no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22C2×C31×AGL(1,ℤ/8)12yes—
425C22C23×D4×C3111yescharacteristic
426C4—24no—
427C4—21no—
428C4—24no—
429C4—24no—
430C4—22no—
431C4—44no—
432C4—44no—
433C4—42no—
434C4C2×C31×AGL(1,ℤ/8)14yes—
435C4C2×C31×2−1+412yes—
436C2—22no—
437C2—84no—
438C2C31×#348611yescharacteristic
439C2C22×C31×AGL(1,ℤ/8)11yescharacteristic
440C2C31×#358611yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC31×#3497C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#557C2semidirect
3. C31×#3386C2semidirect
4. C31×#3368C2semidirect
5. C31×#3234C2semidirect
6. C31×#3226C2semidirect
7. C31×#3112C2semidirect
8. C31×#3103C2semidirect
9. C31×#2983C2semidirect
162. #61828C31direct (N × H)
171. Q8×C312−1+4semidirect
200. C2×#557C2×C31semidirect
201. #3386C2×C31semidirect
202. #3368C2×C31semidirect
203. #3234C2×C31semidirect
204. #3226C2×C31semidirect
205. #3112C2×C31semidirect
206. #3103C2×C31semidirect
207. #2983C2×C31semidirect
328. C31#61828direct (N × H)
391. Q8C31×2−1+4semidirect

Maximal subgroups

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

subgroupindex
C31×#34972
C2×C31×#5572
C31×#33862
C31×#33682
C31×#32342
C31×#32262
C31×#31122
C31×#31032
C31×#29832
#6182831