Finite Group Databasebeta

← C31×#18292 #21475051

Subgroups — C31×#18292 #21475051

Order 7936.

1282 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic, complemented
2C31×#3186C212yesmaximal, complemented
3C31×#2618C211yesmaximal, characteristic, complemented
4C31×#1610C214yesmaximal, complemented
5C31×#1598C214yesmaximal, complemented
6C31×#1519C212yesmaximal, complemented
7C2×C31×#324C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#537—24no—
10C31×#537—22no—
11C31×#537C2214yes—
12C2×C31×#123—21no—
13C2×C31×C22⋊Q8—22no—
14C2×C31×#121—22no—
15C2×C31×#121—21no—
16C2×C31×#121C2212yes—
17C2×C31×#121C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C31×#480—22no—
20C31×#480C2214yescomplemented
21C31×#466—24no—
22C31×#447—22no—
23C31×#447C2214yescomplemented
24C2×C31×C22⋊C8C2212yes—
25C31×#398C2212yescomplemented
26C31×#396—22no—
27C31×#394C2212yescomplemented
28C31×#386—21no—
29C31×#380C2211yescharacteristic
30C31×#327C2218yescomplemented
31C31×#324C2214yescomplemented
32C2×#41×C31—24no—
33C2×#41×C31—24no—
34C2×#41×C31—44no—
35C2×#41×C31—44no—
36C22×Q8×C31—21no—
37C22×D4×C31—21no—
38C22×D4×C31—22no—
39C22×D4×C31—21no—
40C22×D4×C31—21no—
41C22×D4×C31D412yes—
42C22×D4×C31C2311yescharacteristic
43C23×C4×C31—22no—
44C23×C4×C31D411yescharacteristic
45C23×C4×C31D411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C2×D8×C31—24no—
48C22×C8×C31—22no—
49C31×#123—44no—
50C31×C22⋊Q8—28no—
51C31×C22⋊Q8—28no—
52C31×#121—24no—
53C31×#121—28no—
54C31×#121—28no—
55C31×#121—24no—
56C31×#121—22no—
57C31×#121—22no—
58C31×#121—22no—
59C31×#121—22no—
60C31×#121D414yes—
61C31×#121C2314yes—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31D412yes—
68C2×C4⋊C4×C31D412yes—
69C2×C4⋊C4×C31C2311yescharacteristic
70C2×C4⋊C4×C31C2312yes—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—21no—
75C2×C22⋊C4×C31—22no—
76C2×C22⋊C4×C31—22no—
77C31×#107—24no—
78C31×#102—24no—
79C31×#102—24no—
80C31×C22⋊C8—22no—
81C31×C22⋊C8C2314yes—
82C31×#95—22no—
83C31×#95C2312yes—
84C24×C31—22no—
85C24×C31—21no—
86C24×C31C2×D411yescharacteristic
87#41×C31—48no—
88#41×C31—416no—
89#41×C31—48no—
90C2×Q8×C31—24no—
91C2×Q8×C31—44no—
92C2×D4×C31—24no—
93C2×D4×C31—24no—
94C2×D4×C31—24no—
95C2×D4×C31—24no—
96C2×D4×C31—22no—
97C2×D4×C31—22no—
98C2×D4×C31—44no—
99C2×D4×C31—44no—
100C2×D4×C31—44no—
101C2×D4×C31—41no—
102C2×D4×C31—41no—
103C2×D4×C31—42no—
104C2×D4×C31—42no—
105C2×D4×C31—81no—
106C2×D4×C31—82no—
107C2×D4×C31—82no—
108C2×D4×C31C2×D414yes—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—28no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—48no—
119C22×C4×C31—81no—
120C22×C4×C31—81no—
121C22×C4×C31D814yescomplemented
122C22×C4×C31C2×D412yes—
123C22×C4×C31C2×D412yes—
124C22×C4×C31C2×D411yescharacteristic
125C22×C4×C31C2×D412yes—
126C22×C4×C31C2×D412yes—
127C22×C4×C31C2411yescharacteristic
128D8×C31—84no—
129C2×C8×C31—24no—
130C2×C8×C31—82no—
131C4⋊C4×C31—24no—
132C4⋊C4×C31—24no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—24no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—44no—
137C4⋊C4×C31—81no—
138C4⋊C4×C31C2×D414yes—
139C22⋊C4×C31—28no—
140C22⋊C4×C31—24no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—42no—
143C22⋊C4×C31—42no—
144C22⋊C4×C31—44no—
145C22⋊C4×C31—82no—
146#18292C3111yesmaximal, characteristic, direct product factor, Sylow
147C23×C31—24no—
148C23×C31—41no—
149C23×C31—41no—
150C23×C31—44no—
151C23×C31—44no—
152C23×C31—42no—
153C23×C31—42no—
154C23×C31—44no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31#12011yescharacteristic
159C23×C31#12011yescharacteristic
160C23×C31C2×D812yes—
161C23×C31AGL(1,ℤ/8)12yes—
162C23×C31C22×D411yescharacteristic
163Q8×C31—48no—
164D4×C31—44no—
165D4×C31—48no—
166D4×C31—44no—
167D4×C31—44no—
168D4×C31—44no—
169D4×C31—48no—
170D4×C31—82no—
171D4×C31—84no—
172D4×C31—82no—
173D4×C31—82no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—24no—
177C2×C4×C31—24no—
178C2×C4×C31—22no—
179C2×C4×C31—24no—
180C2×C4×C31—41no—
181C2×C4×C31—42no—
182C2×C4×C31—42no—
183C2×C4×C31—41no—
184C2×C4×C31—416no—
185C2×C4×C31—42no—
186C2×C4×C31—44no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—42no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—81no—
195C2×C4×C31#12012yes—
196C2×C4×C31C2×D814yes—
197C2×C4×C31C22×D412yes—
198C8×C31—44no—
199C2×#537C2×C3111yescharacteristic, complemented
200#3186C2×C3112yescomplemented
201#2618C2×C3111yescharacteristic, complemented
202#1610C2×C3114yescomplemented
203#1598C2×C3114yescomplemented
204#1519C2×C3112yescomplemented
205C2×#324C2×C3111yescharacteristic, complemented
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—21no—
211C22×C31—21no—
212C22×C31—42no—
213C22×C31—42no—
214C22×C31—44no—
215C22×C31—44no—
216C22×C31—44no—
217C22×C31—88no—
218C22×C31—82no—
219C22×C31—82no—
220C22×C31—81no—
221C22×C31—81no—
222C22×C31—81no—
223C22×C31#44712yes—
224C22×C31#44712yes—
225C22×C31C2×#12011yescharacteristic
226C22×C31C22×D811yescharacteristic
227C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
228C4×C31—24no—
229C4×C31—22no—
230C4×C31—22no—
231C4×C31—44no—
232C4×C31—42no—
233C4×C31—82no—
234C4×C31—81no—
235C22×#41—21no—
236#537—24no—
237#537—22no—
238#537C22×C3114yes—
239C2×#123—21no—
240C2×C22⋊Q8—22no—
241C2×#121—22no—
242C2×#121—21no—
243C2×#121C22×C3112yes—
244C2×#121C22×C3111yescharacteristic
245C22×C4⋊C4C22×C3111yescharacteristic
246#480—22no—
247#480C22×C3114yescomplemented
248#466—24no—
249#447—22no—
250#447C22×C3114yescomplemented
251C2×C22⋊C8C22×C3112yes—
252#398C22×C3112yescomplemented
253#396—22no—
254#394C22×C3112yescomplemented
255#386—21no—
256#380C22×C3111yescharacteristic
257#327C22×C3118yescomplemented
258#324C22×C3114yescomplemented
259C2×C31—21no—
260C2×C31—22no—
261C2×C31—21no—
262C2×C31—22no—
263C2×C31—84no—
264C2×C31—161no—
265C2×C31C2×#44711yescharacteristic
266C2×C31#302011yescharacteristic
267C2×C31#302611yescharacteristic
268C2×#41—24no—
269C2×#41—24no—
270C2×#41—44no—
271C2×#41—44no—
272C22×Q8—21no—
273C22×D4—21no—
274C22×D4—22no—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4D4×C3112yes—
278C22×D4C23×C3111yescharacteristic
279C23×C4—22no—
280C23×C4D4×C3111yescharacteristic
281C23×C4D4×C3111yescharacteristic
282C23×C4C23×C3111yescharacteristic
283C2×D8—24no—
284C22×C8—22no—
285#123—44no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288#121—24no—
289#121—28no—
290#121—28no—
291#121—24no—
292#121—22no—
293#121—22no—
294#121—22no—
295#121—22no—
296#121D4×C3114yes—
297#121C23×C3114yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4D4×C3112yes—
304C2×C4⋊C4D4×C3112yes—
305C2×C4⋊C4C23×C3111yescharacteristic
306C2×C4⋊C4C23×C3112yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313#107—24no—
314#102—24no—
315#102—24no—
316C22⋊C8—22no—
317C22⋊C8C23×C3114yes—
318#95—22no—
319#95C23×C3112yes—
320C31#1829211yescharacteristic, direct product factor, Sylow
321C24—22no—
322C24—21no—
323C24C2×D4×C3111yescharacteristic
324#41—416no—
325#41—48no—
326#41—48no—
327C2×Q8—24no—
328C2×Q8—44no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—81no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4C2×D4×C3114yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—48no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4D8×C3114yescomplemented
359C22×C4C2×D4×C3112yes—
360C22×C4C2×D4×C3112yes—
361C22×C4C2×D4×C3111yescharacteristic
362C22×C4C2×D4×C3112yes—
363C22×C4C2×D4×C3112yes—
364C22×C4C24×C3111yescharacteristic
365D8—84no—
366C2×C8—24no—
367C2×C8—82no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—44no—
374C4⋊C4—81no—
375C4⋊C4C2×D4×C3114yes—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C23—24no—
384C23—41no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C31×#12011yescharacteristic
395C23C31×#12011yescharacteristic
396C23C2×D8×C3112yes—
397C23C31×AGL(1,ℤ/8)12yes—
398C23C22×D4×C3111yescharacteristic
399Q8—48no—
400D4—44no—
401D4—48no—
402D4—44no—
403D4—48no—
404D4—44no—
405D4—44no—
406D4—82no—
407D4—84no—
408D4—82no—
409D4—82no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—416no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4C31×#12012yes—
432C2×C4C2×D8×C3114yes—
433C2×C4C22×D4×C3112yes—
434C8—44no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—88no—
447C22—82no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22C31×#44712yes—
453C22C31×#44712yes—
454C22C2×C31×#12011yescharacteristic
455C22C22×D8×C3111yescharacteristic
456C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
457C4—24no—
458C4—22no—
459C4—22no—
460C4—44no—
461C4—42no—
462C4—82no—
463C4—81no—
464C2—21no—
465C2—22no—
466C2—21no—
467C2—22no—
468C2—84no—
469C2—161no—
470C2C2×C31×#44711yescharacteristic
471C2C31×#302011yescharacteristic
472C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#537C2semidirect
2. C31×#3186C2semidirect
3. C31×#2618C2semidirect
4. C31×#1610C2semidirect
5. C31×#1598C2semidirect
6. C31×#1519C2semidirect
7. C2×C31×#324C2semidirect
20. C31×#480C22semidirect
23. C31×#447C22semidirect
25. C31×#398C22semidirect
27. C31×#394C22semidirect
30. C31×#327C22semidirect
31. C31×#324C22semidirect
121. C22×C4×C31D8semidirect
146. #18292C31direct (N × H)
199. C2×#537C2×C31semidirect
200. #3186C2×C31semidirect
201. #2618C2×C31semidirect
202. #1610C2×C31semidirect
203. #1598C2×C31semidirect
204. #1519C2×C31semidirect
205. C2×#324C2×C31semidirect
247. #480C22×C31semidirect
250. #447C22×C31semidirect
252. #398C22×C31semidirect
254. #394C22×C31semidirect
257. #327C22×C31semidirect
258. #324C22×C31semidirect
320. C31#18292direct (N × H)
358. C22×C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C31×#31862
C31×#26182
C31×#16102
C31×#15982
C31×#15192
C2×C31×#3242
#1829231