Finite Group Databasebeta

← C31×#12186 #21468945

Subgroups — C31×#12186 #21468945

Order 7936.

682 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2668C211yesmaximal, characteristic, complemented
2C31×#2021C211yesmaximal, characteristic, complemented
3C31×#1613C211yesmaximal, characteristic, complemented
4C31×#1595C211yesmaximal, characteristic, complemented
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1343C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C2×C31×#121—21no—
10C2×C31×#120—21no—
11C2×C31×#120—21no—
12C31×#481—21no—
13C31×#447—21no—
14C2×C31×#99—21no—
15C31×#399C2211yescharacteristic
16C31×#397—21no—
17C31×#394—21no—
18C31×#392C2211yescharacteristic
19C31×#386—21no—
20C31×#386—21no—
21C31×#379C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C25×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×D4×C31—41no—
33C22×D4×C31D411yescharacteristic, complemented
34C23×C4×C31—21no—
35C2×D8×C31—41no—
36C31×#123—41no—
37C31×#121—21no—
38C31×#121—21no—
39C31×#121—44no—
40C31×#120—42no—
41C31×#120—41no—
42C31×#120—44no—
43C31×#120—42no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31D411yescharacteristic, complemented
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×C22⋊C4×C31—21no—
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—41no—
55C2×C22⋊C4×C31D411yescharacteristic
56C2×C22⋊C4×C31C2311yescharacteristic
57C31×#106—41no—
58C31×#102—41no—
59C31×#102—41no—
60C31×#99—42no—
61C31×C22⋊C8—21no—
62C31×C22⋊C8—21no—
63C31×#95—21no—
64C31×#95—21no—
65C31×#95D411yescharacteristic, complemented
66C31×#95D411yescharacteristic, complemented
67C24×C31—21no—
68C24×C31—21no—
69C24×C31—21no—
70C24×C31—21no—
71C24×C31—21no—
72C24×C31—41no—
73C24×C31—41no—
74C24×C31—42no—
75C24×C31—81no—
76C24×C31C2×D411yescharacteristic
77C2×D4×C31—24no—
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—41no—
81C2×D4×C31—42no—
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—41no—
87C2×D4×C31—42no—
88C2×D4×C31—44no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C2×D4×C31—81no—
95C2×D4×C31—81no—
96C2×D4×C31—81no—
97C2×D4×C31—81no—
98C2×D4×C31—82no—
99C2×D4×C31—82no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—42no—
106C22×C4×C31—41no—
107C22×C4×C31—81no—
108C22×C4×C31—81no—
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111D8×C31—82no—
112C2×C8×C31—41no—
113C2×C8×C31—41no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—42no—
117C4⋊C4×C31—41no—
118C22⋊C4×C31—22no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—42no—
121C22⋊C4×C31—42no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—42no—
125C22⋊C4×C31—42no—
126C22⋊C4×C31—42no—
127C22⋊C4×C31—44no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—81no—
131C22⋊C4×C31—81no—
132#12186C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—21no—
134C23×C31—21no—
135C23×C31—22no—
136C23×C31—22no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—42no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31—41no—
146C23×C31—41no—
147C23×C31—41no—
148C23×C31—42no—
149C23×C31—42no—
150C23×C31—41no—
151C23×C31—41no—
152C23×C31—41no—
153C23×C31—41no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—81no—
158C23×C31—82no—
159C23×C31—81no—
160C23×C31—82no—
161C23×C31—84no—
162C23×C31—81no—
163C23×C31—81no—
164C23×C31—81no—
165C23×C31#12011yescharacteristic
166C23×C31#13511yescharacteristic
167C23×C31AGL(1,ℤ/8)11yescharacteristic
168D4×C31—42no—
169D4×C31—84no—
170D4×C31—81no—
171D4×C31—81no—
172D4×C31—81no—
173D4×C31—81no—
174D4×C31—81no—
175D4×C31—82no—
176D4×C31—82no—
177D4×C31—84no—
178D4×C31—88no—
179D4×C31—81no—
180D4×C31—81no—
181C2×C4×C31—22no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—41no—
185C2×C4×C31—41no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—42no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—41no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—84no—
202C2×C4×C31—81no—
203C2×C4×C31—81no—
204C8×C31—81no—
205C8×C31—81no—
206#2668C2×C3111yescharacteristic, complemented
207#2021C2×C3111yescharacteristic, complemented
208#1613C2×C3111yescharacteristic, complemented
209#1595C2×C3111yescharacteristic, complemented
210#1352C2×C3111yescharacteristic, complemented
211#1343C2×C3111yescharacteristic, complemented
212#1316C2×C3111yescharacteristic, complemented
213C22×C31—21no—
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—21no—
217C22×C31—41no—
218C22×C31—41no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—42no—
222C22×C31—42no—
223C22×C31—41no—
224C22×C31—41no—
225C22×C31—44no—
226C22×C31—42no—
227C22×C31—42no—
228C22×C31—81no—
229C22×C31—81no—
230C22×C31—81no—
231C22×C31—81no—
232C22×C31—81no—
233C22×C31—82no—
234C22×C31—82no—
235C22×C31—82no—
236C22×C31—82no—
237C22×C31—81no—
238C22×C31—81no—
239C22×C31—81no—
240C22×C31—81no—
241C22×C31—81no—
242C22×C31—81no—
243C22×C31—81no—
244C22×C31—161no—
245C22×C31#44911yescharacteristic
246C22×C31#45311yescharacteristic
247C22×C31B(4,𝔽2)11yescharacteristic
248C4×C31—42no—
249C4×C31—41no—
250C4×C31—41no—
251C4×C31—81no—
252C4×C31—81no—
253C4×C31—82no—
254C4×C31—81no—
255C4×C31—81no—
256#534C22×C3111yescharacteristic
257C2×#121—21no—
258C2×#120—21no—
259C2×#120—21no—
260#481—21no—
261#447—21no—
262C2×#99—21no—
263#399C22×C3111yescharacteristic
264#397—21no—
265#394—21no—
266#392C22×C3111yescharacteristic
267#386—21no—
268#386—21no—
269#379C22×C3111yescharacteristic
270#342C22×C3111yescharacteristic
271#327—21no—
272#327—21no—
273#323C22×C3111yescharacteristic
274#323C22×C3111yescharacteristic
275C2×C31—21no—
276C2×C31—21no—
277C2×C31—41no—
278C2×C31—41no—
279C2×C31—82no—
280C2×C31—82no—
281C2×C31—161no—
282C2×C31#160011yescharacteristic
283C2×C31#219611yescharacteristic
284C2×C31#220011yescharacteristic
285C25D4×C3111yescharacteristic
286C22×D4—21no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—41no—
291C22×D4D4×C3111yescharacteristic, complemented
292C23×C4—21no—
293C2×D8—41no—
294#123—41no—
295#121—21no—
296#121—21no—
297#121—44no—
298#120—42no—
299#120—41no—
300#120—44no—
301#120—42no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4D4×C3111yescharacteristic, complemented
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4D4×C3111yescharacteristic
314C2×C22⋊C4C23×C3111yescharacteristic
315#106—41no—
316#102—41no—
317#102—41no—
318#99—42no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321#95—21no—
322#95—21no—
323#95D4×C3111yescharacteristic, complemented
324#95D4×C3111yescharacteristic, complemented
325C31#1218611yescharacteristic, direct product factor, Sylow
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—41no—
332C24—41no—
333C24—42no—
334C24—81no—
335C24C2×D4×C3111yescharacteristic
336C2×D4—24no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—81no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4—82no—
358C2×D4—82no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4C2×D4×C3111yescharacteristic
369C22×C4C2×D4×C3111yescharacteristic
370D8—82no—
371C2×C8—41no—
372C2×C8—41no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C22⋊C4—22no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—81no—
390C22⋊C4—81no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—81no—
416C23—82no—
417C23—81no—
418C23—82no—
419C23—84no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23C31×#12011yescharacteristic
424C23C31×#13511yescharacteristic
425C23C31×AGL(1,ℤ/8)11yescharacteristic
426D4—42no—
427D4—84no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—82no—
434D4—82no—
435D4—84no—
436D4—88no—
437D4—81no—
438D4—81no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—84no—
460C2×C4—81no—
461C2×C4—81no—
462C8—81no—
463C8—81no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—41no—
469C22—41no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—161no—
496C22C31×#44911yescharacteristic
497C22C31×#45311yescharacteristic
498C22C31×B(4,𝔽2)11yescharacteristic
499C4—42no—
500C4—41no—
501C4—41no—
502C4—81no—
503C4—81no—
504C4—82no—
505C4—81no—
506C4—81no—
507C2—21no—
508C2—21no—
509C2—41no—
510C2—41no—
511C2—82no—
512C2—82no—
513C2—161no—
514C2C31×#160011yescharacteristic
515C2C31×#219611yescharacteristic
516C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2668C2semidirect
2. C31×#2021C2semidirect
3. C31×#1613C2semidirect
4. C31×#1595C2semidirect
5. C31×#1352C2semidirect
6. C31×#1343C2semidirect
7. C31×#1316C2semidirect
33. C22×D4×C31D4semidirect
45. C2×C4⋊C4×C31D4semidirect
65. C31×#95D4semidirect
66. C31×#95D4semidirect
132. #12186C31direct (N × H)
206. #2668C2×C31semidirect
207. #2021C2×C31semidirect
208. #1613C2×C31semidirect
209. #1595C2×C31semidirect
210. #1352C2×C31semidirect
211. #1343C2×C31semidirect
212. #1316C2×C31semidirect
291. C22×D4D4×C31semidirect
303. C2×C4⋊C4D4×C31semidirect
323. #95D4×C31semidirect
324. #95D4×C31semidirect
325. C31#12186direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26682
C31×#20212
C31×#16132
C31×#15952
C31×#13522
C31×#13432
C31×#13162
#1218631