Finite Group Databasebeta

← C31×#10004 #21466763

Subgroups — C31×#10004 #21466763

Order 7936.

692 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3213C211yesmaximal, characteristic, complemented
2C31×#3109C211yesmaximal, characteristic, complemented
3C31×#2540C211yesmaximal, characteristic, complemented
4C31×#1980C211yesmaximal, characteristic, complemented
5C31×#1527C211yesmaximal, characteristic, complemented
6C31×#1490C211yesmaximal, characteristic, complemented
7C31×#1285C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C2×C31×C22⋊Q8—21no—
11C2×C31×#121—21no—
12C2×C4×D4×C31C2211yescharacteristic
13C31×#496—21no—
14C31×#489—21no—
15C31×#481—21no—
16C31×#476—21no—
17C31×#463—22no—
18C31×#459—21no—
19C31×#448—21no—
20C31×#447—21no—
21C31×#447—21no—
22C31×#442—22no—
23C31×#432C2211yescharacteristic
24C31×#432C2211yescharacteristic
25C2×C31×#104—22no—
26C4×M16×C31C2211yescharacteristic
27C31×#399—21no—
28C31×#392—21no—
29C31×#390—22no—
30C4×C4⋊C4×C31C2211yescharacteristic
31C31×#329—22no—
32C31×D4⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C22×D4×C31—22no—
36C22×D4×C31D411yescharacteristic
37C22×D4×C31D411yescharacteristic
38C23×C4×C31—21no—
39C2×SD16×C31—22no—
40C2×D8×C31—22no—
41C2×D8×C31—22no—
42C2×M16×C31—21no—
43C31×C4⋊Q8D411yescharacteristic, complemented
44C31×#127D411yescharacteristic, complemented
45C31×#124—21no—
46C31×#124—21no—
47C31×#124—21no—
48C31×#124—21no—
49C31×#124—21no—
50C31×C22⋊Q8—44no—
51C31×#121—21no—
52C31×#121—21no—
53C31×#121—44no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C4×D4×C31—21no—
59C4×D4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C42×C31D412yes—
69C2×C42×C31C2311yescharacteristic
70C31×#106—22no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8D412yescomplemented
73C31×#104—44no—
74C31×#103—22no—
75C31×#102—22no—
76C31×#102—22no—
77C31×#102—22no—
78C31×C22⋊C8—21no—
79C31×C22⋊C8—21no—
80C31×#97—21no—
81C31×#97—21no—
82C31×#97D412yescomplemented
83C4×C8×C31—21no—
84C31×#95—21no—
85C24×C31—21no—
86C24×C31—21no—
87#41×C31—42no—
88#41×C31—42no—
89C2×Q8×C31—21no—
90C2×Q8×C31—21no—
91C2×Q8×C31—21no—
92C2×Q8×C31—42no—
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—21no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—42no—
101C2×D4×C31—44no—
102C2×D4×C31—42no—
103C2×D4×C31—44no—
104C2×D4×C31—41no—
105C2×D4×C31—82no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—42no—
110C22×C4×C31—42no—
111C22×C4×C31—41no—
112C22×C4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117SD16×C31—82no—
118D8×C31—44no—
119D8×C31—82no—
120M16×C31—22no—
121M16×C31—41no—
122C2×C8×C31—22no—
123C2×C8×C31—22no—
124C2×C8×C31—41no—
125C4⋊C4×C31—24no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—44no—
131C22⋊C4×C31—44no—
132C22⋊C4×C31—44no—
133C22⋊C4×C31—41no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C22⋊C4×C31—41no—
139C22⋊C4×C31—41no—
140C22⋊C4×C31—41no—
141C42×C31—42no—
142C42×C31C2×D411yescharacteristic
143C42×C31C2×D411yescharacteristic
144C42×C31C2×D411yescharacteristic
145C42×C31#4111yescharacteristic
146#10004C3111yesmaximal, characteristic, direct product factor, Sylow
147C23×C31—22no—
148C23×C31—41no—
149C23×C31—41no—
150C23×C31—41no—
151C23×C31—42no—
152C23×C31—42no—
153C23×C31—41no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31—41no—
159C23×C31—41no—
160C23×C31#12011yescharacteristic
161Q8×C31—42no—
162Q8×C31—42no—
163Q8×C31—81no—
164D4×C31—22no—
165D4×C31—22no—
166D4×C31—41no—
167D4×C31—41no—
168D4×C31—42no—
169D4×C31—42no—
170D4×C31—42no—
171D4×C31—42no—
172D4×C31—81no—
173D4×C31—88no—
174D4×C31—81no—
175D4×C31—81no—
176C2×C4×C31—21no—
177C2×C4×C31—21no—
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—42no—
185C2×C4×C31—41no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—82no—
190C2×C4×C31—82no—
191C2×C4×C31—82no—
192C2×C4×C31#12011yescharacteristic
193C2×C4×C31#12011yescharacteristic
194C2×C4×C31#12111yescharacteristic
195C2×C4×C31#12111yescharacteristic
196C2×C4×C31#12111yescharacteristic
197C2×C4×C31#12711yescharacteristic
198C8×C31—42no—
199C8×C31—42no—
200C8×C31—42no—
201#3213C2×C3111yescharacteristic, complemented
202#3109C2×C3111yescharacteristic, complemented
203#2540C2×C3111yescharacteristic, complemented
204#1980C2×C3111yescharacteristic, complemented
205#1527C2×C3111yescharacteristic, complemented
206#1490C2×C3111yescharacteristic, complemented
207#1285C2×C3111yescharacteristic, complemented
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—42no—
211C22×C31—42no—
212C22×C31—42no—
213C22×C31—81no—
214C22×C31—81no—
215C22×C31—82no—
216C22×C31—82no—
217C22×C31—81no—
218C22×C31—81no—
219C22×C31—81no—
220C22×C31—81no—
221C22×C31—81no—
222C22×C31—81no—
223C22×C31—81no—
224C22×C31—81no—
225C22×C31#39211yescharacteristic
226C22×C31#45311yescharacteristic
227C22×C31#45411yescharacteristic
228C4×C31—21no—
229C4×C31—21no—
230C4×C31—21no—
231C4×C31—21no—
232C4×C31—21no—
233C4×C31—44no—
234C4×C31—81no—
235C4×C31—81no—
236C4×C31—81no—
237C4×C31#45312yes—
238#532C22×C3111yescharacteristic
239C2×#124C22×C3111yescharacteristic
240C2×C22⋊Q8—21no—
241C2×#121—21no—
242C2×C4×D4C22×C3111yescharacteristic
243#496—21no—
244#489—21no—
245#481—21no—
246#476—21no—
247#463—22no—
248#459—21no—
249#448—21no—
250#447—21no—
251#447—21no—
252#442—22no—
253#432C22×C3111yescharacteristic
254#432C22×C3111yescharacteristic
255C2×#104—22no—
256C4×M16C22×C3111yescharacteristic
257#399—21no—
258#392—21no—
259#390—22no—
260C4×C4⋊C4C22×C3111yescharacteristic
261#329—22no—
262D4⋊C8—22no—
263C2×C31—21no—
264C2×C31—21no—
265C2×C31—82no—
266C2×C31—161no—
267C2×C31—161no—
268C2×C31#200211yescharacteristic
269C2×C31#200611yescharacteristic
270C2×C31#201011yescharacteristic
271C2×#41—21no—
272C22×Q8D4×C3111yescharacteristic
273C22×D4—22no—
274C22×D4D4×C3111yescharacteristic
275C22×D4D4×C3111yescharacteristic
276C23×C4—21no—
277C2×SD16—22no—
278C2×D8—22no—
279C2×D8—22no—
280C2×M16—21no—
281C4⋊Q8D4×C3111yescharacteristic, complemented
282#127D4×C3111yescharacteristic, complemented
283#124—21no—
284#124—21no—
285#124—21no—
286#124—21no—
287#124—21no—
288C22⋊Q8—44no—
289#121—21no—
290#121—21no—
291#121—44no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4D4×C3111yescharacteristic
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C42D4×C3112yes—
307C2×C42C23×C3111yescharacteristic
308#106—22no—
309C4⋊C8—21no—
310C4⋊C8D4×C3112yescomplemented
311#104—44no—
312#103—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318#97—21no—
319#97—21no—
320#97D4×C3112yescomplemented
321C4×C8—21no—
322#95—21no—
323C31#1000411yescharacteristic, direct product factor, Sylow
324C24—21no—
325C24—21no—
326#41—42no—
327#41—42no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—42no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—82no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×D4×C3111yescharacteristic
354C22×C4C2×D4×C3111yescharacteristic
355C22×C4C2×D4×C3111yescharacteristic
356SD16—82no—
357D8—44no—
358D8—82no—
359M16—22no—
360M16—41no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—41no—
364C4⋊C4—24no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42—42no—
381C42C2×D4×C3111yescharacteristic
382C42C2×D4×C3111yescharacteristic
383C42C2×D4×C3111yescharacteristic
384C42#41×C3111yescharacteristic
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23C31×#12011yescharacteristic
399Q8—42no—
400Q8—42no—
401Q8—81no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—81no—
411D4—88no—
412D4—81no—
413D4—81no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4C31×#12011yescharacteristic
431C2×C4C31×#12011yescharacteristic
432C2×C4C31×#12111yescharacteristic
433C2×C4C31×#12111yescharacteristic
434C2×C4C31×#12111yescharacteristic
435C2×C4C31×#12711yescharacteristic
436C8—42no—
437C8—42no—
438C8—42no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—82no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C31×#39211yescharacteristic
457C22C31×#45311yescharacteristic
458C22C31×#45411yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4C31×#45312yes—
469C2—21no—
470C2—21no—
471C2—82no—
472C2—161no—
473C2—161no—
474C2C31×#200211yescharacteristic
475C2C31×#200611yescharacteristic
476C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3213C2semidirect
2. C31×#3109C2semidirect
3. C31×#2540C2semidirect
4. C31×#1980C2semidirect
5. C31×#1527C2semidirect
6. C31×#1490C2semidirect
7. C31×#1285C2semidirect
43. C31×C4⋊Q8D4semidirect
44. C31×#127D4semidirect
72. C31×C4⋊C8D4semidirect
82. C31×#97D4semidirect
146. #10004C31direct (N × H)
201. #3213C2×C31semidirect
202. #3109C2×C31semidirect
203. #2540C2×C31semidirect
204. #1980C2×C31semidirect
205. #1527C2×C31semidirect
206. #1490C2×C31semidirect
207. #1285C2×C31semidirect
281. C4⋊Q8D4×C31semidirect
282. #127D4×C31semidirect
310. C4⋊C8D4×C31semidirect
320. #97D4×C31semidirect
323. C31#10004direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32132
C31×#31092
C31×#25402
C31×#19802
C31×#15272
C31×#14902
C31×#12852
#1000431