Finite Group Databasebeta

← C31×#9948 #21466707

Subgroups — C31×#9948 #21466707

Order 7936.

812 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3186C211yesmaximal, characteristic
2C2×C31×#448C211yesmaximal, characteristic, complemented
3C31×#2436C211yesmaximal, characteristic, complemented
4C31×#1964C211yesmaximal, characteristic, complemented
5C31×#1787C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C23×Q8×C31C2211yescharacteristic
9C22×SD16×C31—21no—
10C2×C31×#124—21no—
11C2×C31×#121C2211yescharacteristic
12C2×C31×#121C2211yescharacteristic
13C31×#480—21no—
14C31×#480—21no—
15C31×#466—22no—
16C31×#448—22no—
17C31×#448—22no—
18C31×#447—21no—
19C31×#447—21no—
20C2×C31×#103—21no—
21C2×C31×#103—22no—
22C2×C31×#99—22no—
23C2×C31×C22⋊C8C2211yescharacteristic
24C2×C31×C22⋊C8C2211yescharacteristic
25C31×#393—22no—
26C31×#391—21no—
27C31×#386—22no—
28C31×#380C2211yescharacteristic
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#327—22no—
31C22×Q8×C31—21no—
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—21no—
36C22×Q8×C31—44no—
37C22×Q8×C31D412yes—
38C22×D4×C31—21no—
39C22×D4×C31—21no—
40C22×D4×C31D411yescharacteristic
41C22×D4×C31D411yescharacteristic
42C23×C4×C31—21no—
43C23×C4×C31C2311yescharacteristic
44C2×SD16×C31—24no—
45C2×SD16×C31—44no—
46C2×D8×C31—22no—
47C22×C8×C31—21no—
48C22×C8×C31D412yes—
49C31×#124—44no—
50C31×#121—21no—
51C31×#121—21no—
52C31×#121—21no—
53C31×#121—21no—
54C31×#121—21no—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31D412yes—
66C2×C42×C31—21no—
67C31×#107—22no—
68C31×#103—24no—
69C31×#103—44no—
70C31×#102—22no—
71C31×#102—22no—
72C31×#99—44no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8D412yes—
77C31×#95—24no—
78C31×#95—21no—
79C31×#95—21no—
80C24×C31—21no—
81C24×C31—21no—
82C24×C31C2×D411yescharacteristic
83C2×Q8×C31—24no—
84C2×Q8×C31—24no—
85C2×Q8×C31—24no—
86C2×Q8×C31—22no—
87C2×Q8×C31—22no—
88C2×Q8×C31—44no—
89C2×Q8×C31—42no—
90C2×Q8×C31—42no—
91C2×Q8×C31—42no—
92C2×Q8×C31—44no—
93C2×Q8×C31—44no—
94C2×Q8×C31—41no—
95C2×Q8×C31—41no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—42no—
101C2×D4×C31—42no—
102C2×D4×C31—41no—
103C2×D4×C31—41no—
104C2×D4×C31—41no—
105C2×D4×C31—41no—
106C2×D4×C31—42no—
107C2×D4×C31—42no—
108C2×D4×C31—42no—
109C2×D4×C31—42no—
110C22×C4×C31—22no—
111C22×C4×C31—21no—
112C22×C4×C31—22no—
113C22×C4×C31—21no—
114C22×C4×C31—42no—
115C22×C4×C31—44no—
116C22×C4×C31—41no—
117C22×C4×C31—41no—
118C22×C4×C31—41no—
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31C2×D411yescharacteristic
121C22×C4×C31C2×D411yescharacteristic
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31#4111yescharacteristic
125SD16×C31—48no—
126D8×C31—82no—
127C2×C8×C31—22no—
128C2×C8×C31—22no—
129C2×C8×C31—42no—
130C2×C8×C31—42no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—42no—
136C22⋊C4×C31—22no—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—48no—
139C22⋊C4×C31—41no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—42no—
144C42×C31—42no—
145#9948C3111yesmaximal, characteristic, direct product factor, Sylow
146C23×C31—41no—
147C23×C31—41no—
148C23×C31—42no—
149C23×C31—42no—
150C23×C31—41no—
151C23×C31—41no—
152C23×C31—42no—
153C23×C31—42no—
154C23×C31—81no—
155C23×C31—81no—
156C23×C31#12011yescharacteristic
157C23×C31#12111yescharacteristic
158C23×C31#12711yescharacteristic
159C23×C31#13512yes—
160C23×C31AGL(1,ℤ/8)12yes—
161Q8×C31—24no—
162Q8×C31—24no—
163Q8×C31—42no—
164Q8×C31—44no—
165Q8×C31—42no—
166Q8×C31—44no—
167D4×C31—42no—
168D4×C31—42no—
169D4×C31—42no—
170D4×C31—81no—
171D4×C31—82no—
172D4×C31—82no—
173D4×C31—81no—
174D4×C31—81no—
175D4×C31—81no—
176D4×C31—81no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—24no—
181C2×C4×C31—41no—
182C2×C4×C31—41no—
183C2×C4×C31—41no—
184C2×C4×C31—42no—
185C2×C4×C31—44no—
186C2×C4×C31—44no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—81no—
196C2×C4×C31—81no—
197C2×C4×C31—81no—
198C2×C4×C31#12011yescharacteristic
199C2×C4×C31#12011yescharacteristic
200C2×C4×C31#12111yescharacteristic
201C2×C4×C31#12111yescharacteristic
202C8×C31—42no—
203C8×C31—42no—
204#3186C2×C3111yescharacteristic
205C2×#448C2×C3111yescharacteristic, complemented
206#2436C2×C3111yescharacteristic, complemented
207#1964C2×C3111yescharacteristic, complemented
208#1787C2×C3111yescharacteristic, complemented
209#1519C2×C3111yescharacteristic, complemented
210#1282C2×C3111yescharacteristic, complemented
211C22×C31—21no—
212C22×C31—21no—
213C22×C31—21no—
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—21no—
217C22×C31—42no—
218C22×C31—42no—
219C22×C31—84no—
220C22×C31—81no—
221C22×C31—81no—
222C22×C31—81no—
223C22×C31—81no—
224C22×C31—81no—
225C22×C31—81no—
226C22×C31—162no—
227C22×C31#39211yescharacteristic
228C22×C31#44912yes—
229C22×C31#46312yes—
230C22×C31#49511yescharacteristic
231C22×C31#49611yescharacteristic
232C4×C31—21no—
233C4×C31—21no—
234C4×C31—21no—
235C4×C31—44no—
236C4×C31—82no—
237C4×C31—81no—
238C4×C31—81no—
239C4×C31—161no—
240C4×C31B(4,𝔽2)12yes—
241C23×Q8C22×C3111yescharacteristic
242C22×SD16—21no—
243C2×#124—21no—
244C2×#121C22×C3111yescharacteristic
245C2×#121C22×C3111yescharacteristic
246#480—21no—
247#480—21no—
248#466—22no—
249#448—22no—
250#448—22no—
251#447—21no—
252#447—21no—
253C2×#103—21no—
254C2×#103—22no—
255C2×#99—22no—
256C2×C22⋊C8C22×C3111yescharacteristic
257C2×C22⋊C8C22×C3111yescharacteristic
258#393—22no—
259#391—21no—
260#386—22no—
261#380C22×C3111yescharacteristic
262C4×C22⋊C4C22×C3111yescharacteristic
263#327—22no—
264C2×C31—22no—
265C2×C31—21no—
266C2×C31—22no—
267C2×C31—21no—
268C2×C31—161no—
269C2×C31—161no—
270C2×C31#200911yescharacteristic
271C2×C31#201211yescharacteristic
272C2×C31#202111yescharacteristic
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—44no—
279C22×Q8D4×C3112yes—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4D4×C3111yescharacteristic
283C22×D4D4×C3111yescharacteristic
284C23×C4—21no—
285C23×C4C23×C3111yescharacteristic
286C2×SD16—24no—
287C2×SD16—44no—
288C2×D8—22no—
289C22×C8—21no—
290C22×C8D4×C3112yes—
291#124—44no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296#121—21no—
297#121—21no—
298#121—21no—
299#121—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4D4×C3111yescharacteristic
302C2×C4⋊C4D4×C3111yescharacteristic
303C2×C22⋊C4—24no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4D4×C3112yes—
308C2×C42—21no—
309#107—22no—
310#103—24no—
311#103—44no—
312#102—22no—
313#102—22no—
314#99—44no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8D4×C3112yes—
319#95—24no—
320#95—21no—
321#95—21no—
322C31#994811yescharacteristic, direct product factor, Sylow
323C24—21no—
324C24—21no—
325C24C2×D4×C3111yescharacteristic
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—44no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4C2×D4×C3111yescharacteristic
363C22×C4C2×D4×C3111yescharacteristic
364C22×C4C2×D4×C3111yescharacteristic
365C22×C4C2×D4×C3111yescharacteristic
366C22×C4C2×D4×C3111yescharacteristic
367C22×C4#41×C3111yescharacteristic
368SD16—48no—
369D8—82no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—42no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—48no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—42no—
387C42—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—81no—
397C23—81no—
398C23C31×#12011yescharacteristic
399C23C31×#12111yescharacteristic
400C23C31×#12711yescharacteristic
401C23C31×#13512yes—
402C23C31×AGL(1,ℤ/8)12yes—
403Q8—24no—
404Q8—24no—
405Q8—42no—
406Q8—44no—
407Q8—42no—
408Q8—44no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—81no—
413D4—82no—
414D4—82no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C31×#12011yescharacteristic
441C2×C4C31×#12011yescharacteristic
442C2×C4C31×#12111yescharacteristic
443C2×C4C31×#12111yescharacteristic
444C8—42no—
445C8—42no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—42no—
453C22—42no—
454C22—84no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—162no—
462C22C31×#39211yescharacteristic
463C22C31×#44912yes—
464C22C31×#46312yes—
465C22C31×#49511yescharacteristic
466C22C31×#49611yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—44no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—161no—
475C4C31×B(4,𝔽2)12yes—
476C2—22no—
477C2—21no—
478C2—22no—
479C2—21no—
480C2—161no—
481C2—161no—
482C2C31×#200911yescharacteristic
483C2C31×#201211yescharacteristic
484C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3186C2
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×#448C2semidirect
3. C31×#2436C2semidirect
4. C31×#1964C2semidirect
5. C31×#1787C2semidirect
6. C31×#1519C2semidirect
7. C31×#1282C2semidirect
145. #9948C31direct (N × H)
205. C2×#448C2×C31semidirect
206. #2436C2×C31semidirect
207. #1964C2×C31semidirect
208. #1787C2×C31semidirect
209. #1519C2×C31semidirect
210. #1282C2×C31semidirect
322. C31#9948direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31862
C2×C31×#4482
C31×#24362
C31×#19642
C31×#17872
C31×#15192
C31×#12822
#994831