Finite Group Databasebeta

← C31×#9497 #21466256

Subgroups — C31×#9497 #21466256

Order 7936.

1184 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2403C211yesmaximal, characteristic
2C31×#2309C211yesmaximal, characteristic, complemented
3C31×#1781C211yesmaximal, characteristic, complemented
4C31×#1781C212yesmaximal, complemented
5C31×#1304C212yesmaximal, complemented
6C23×D4×C31C2211yescharacteristic
7C23×D4×C31C2211yescharacteristic
8C2×C31×#120—21no—
9C2×C31×#120C412yescomplemented
10C2×C31×#117—21no—
11C2×C31×#99—22no—
12C2×C31×#99—24no—
13C31×#392—22no—
14C31×#390—24no—
15C31×#386—21no—
16C31×#386C412yescomplemented
17C31×#379—21no—
18C31×#379C2212yes—
19C31×#379C2211yescharacteristic
20C31×#379C2212yes—
21C25×C31C2×C411yescharacteristic
22C25×C31D411yescharacteristic
23C25×C31C2311yescharacteristic
24C22×D4×C31—22no—
25C22×D4×C31—21no—
26C22×D4×C31—22no—
27C22×D4×C31—22no—
28C22×D4×C31—21no—
29C22×D4×C31—22no—
30C22×D4×C31—22no—
31C22×D4×C31—44no—
32C22×D4×C31—44no—
33C22×D4×C31C2×C412yes—
34C22×D4×C31D412yes—
35C23×C4×C31C2×C411yescharacteristic
36C23×C4×C31D411yescharacteristic
37C31×#120—44no—
38C31×#120—44no—
39C31×#117—44no—
40C2×C4⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—22no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—22no—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31—22no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—22no—
52C2×C22⋊C4×C31C2×C412yes—
53C2×C22⋊C4×C31D414yescomplemented
54C2×C42×C31—22no—
55C31×#99—48no—
56C31×#99—44no—
57C31×#95—22no—
58C24×C31—21no—
59C24×C31—21no—
60C24×C31—22no—
61C24×C31—21no—
62C24×C31—42no—
63C24×C31—42no—
64C24×C31—42no—
65C24×C31—42no—
66C24×C31—42no—
67C24×C31—42no—
68C24×C31—81no—
69C24×C31—81no—
70C24×C31—81no—
71C24×C31C22⋊C412yes—
72C24×C31C22×C411yescharacteristic
73C24×C31C2×D411yescharacteristic
74C24×C31C2×D411yescharacteristic
75C24×C31C2×D412yes—
76C24×C31#4112yes—
77C2×D4×C31—28no—
78C2×D4×C31—24no—
79C2×D4×C31—24no—
80C2×D4×C31—24no—
81C2×D4×C31—24no—
82C2×D4×C31—24no—
83C2×D4×C31—42no—
84C2×D4×C31—42no—
85C2×D4×C31—44no—
86C2×D4×C31—44no—
87C2×D4×C31—48no—
88C2×D4×C31—42no—
89C2×D4×C31—44no—
90C2×D4×C31—42no—
91C2×D4×C31—44no—
92C2×D4×C31—42no—
93C2×D4×C31—44no—
94C2×D4×C31—41no—
95C2×D4×C31—42no—
96C2×D4×C31—41no—
97C2×D4×C31—44no—
98C2×D4×C31—44no—
99C2×D4×C31—82no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—24no—
103C22×C4×C31—21no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—24no—
107C22×C4×C31—42no—
108C22×C4×C31—42no—
109C22×C4×C31C22⋊C412yes—
110C4⋊C4×C31—44no—
111C22⋊C4×C31—44no—
112C22⋊C4×C31—44no—
113C22⋊C4×C31—42no—
114C22⋊C4×C31—44no—
115C22⋊C4×C31—44no—
116C22⋊C4×C31—42no—
117C22⋊C4×C31—44no—
118C22⋊C4×C31—44no—
119C22⋊C4×C31—82no—
120C22⋊C4×C31—82no—
121C22⋊C4×C31—81no—
122C22⋊C4×C31—82no—
123C22⋊C4×C31—82no—
124C42×C31—44no—
125#9497C3111yesmaximal, characteristic, direct product factor, Sylow
126C23×C31—22no—
127C23×C31—22no—
128C23×C31—22no—
129C23×C31—22no—
130C23×C31—24no—
131C23×C31—22no—
132C23×C31—24no—
133C23×C31—22no—
134C23×C31—22no—
135C23×C31—22no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—42no—
140C23×C31—41no—
141C23×C31—42no—
142C23×C31—41no—
143C23×C31—42no—
144C23×C31—42no—
145C23×C31—42no—
146C23×C31—42no—
147C23×C31—42no—
148C23×C31—42no—
149C23×C31—41no—
150C23×C31—42no—
151C23×C31—42no—
152C23×C31—42no—
153C23×C31—42no—
154C23×C31—41no—
155C23×C31—42no—
156C23×C31—42no—
157C23×C31—81no—
158C23×C31—81no—
159C23×C31—82no—
160C23×C31—82no—
161C23×C31—88no—
162C23×C31—88no—
163C23×C31—88no—
164C23×C31C2×C22⋊C411yescharacteristic
165C23×C31C4×D412yes—
166C23×C31#12011yescharacteristic
167C23×C31#12112yes—
168C23×C31#12311yescharacteristic
169D4×C31—48no—
170D4×C31—48no—
171D4×C31—48no—
172D4×C31—44no—
173D4×C31—44no—
174D4×C31—88no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—24no—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—22no—
181C2×C4×C31—24no—
182C2×C4×C31—41no—
183C2×C4×C31—42no—
184C2×C4×C31—41no—
185C2×C4×C31—44no—
186C2×C4×C31—42no—
187C2×C4×C31—44no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—44no—
191C2×C4×C31—82no—
192C2×C4×C31—82no—
193C2×C4×C31—82no—
194C2×C4×C31#9914yescomplemented
195#2403C2×C3111yescharacteristic
196#2309C2×C3111yescharacteristic, complemented
197#1781C2×C3112yescomplemented
198#1781C2×C3111yescharacteristic, complemented
199#1304C2×C3112yescomplemented
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—22no—
205C22×C31—22no—
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—22no—
209C22×C31—22no—
210C22×C31—22no—
211C22×C31—22no—
212C22×C31—21no—
213C22×C31—44no—
214C22×C31—44no—
215C22×C31—44no—
216C22×C31—44no—
217C22×C31—42no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—84no—
222C22×C31—82no—
223C22×C31—82no—
224C22×C31—84no—
225C22×C31—82no—
226C22×C31—84no—
227C22×C31—84no—
228C22×C31—84no—
229C22×C31—84no—
230C22×C31—81no—
231C22×C31—81no—
232C22×C31#38611yescharacteristic
233C22×C31C2×#9912yes—
234C22×C31B(4,𝔽2)12yes—
235C4×C31—24no—
236C4×C31—42no—
237C4×C31—84no—
238C4×C31—82no—
239C4×C31—162no—
240C23×D4C22×C3111yescharacteristic
241C23×D4C22×C3111yescharacteristic
242C2×#120—21no—
243C2×#120C4×C3112yescomplemented
244C2×#117—21no—
245C2×#99—22no—
246C2×#99—24no—
247#392—22no—
248#390—24no—
249#386—21no—
250#386C4×C3112yescomplemented
251#379—21no—
252#379C22×C3112yes—
253#379C22×C3111yescharacteristic
254#379C22×C3112yes—
255C2×C31—22no—
256C2×C31—22no—
257C2×C31—42no—
258C2×C31—42no—
259C2×C31—41no—
260C2×C31—82no—
261C2×C31—82no—
262C2×C31#188911yescharacteristic
263C2×C31#189811yescharacteristic
264C2×C31#191311yescharacteristic
265C25C2×C4×C3111yescharacteristic
266C25D4×C3111yescharacteristic
267C25C23×C3111yescharacteristic
268C22×D4—21no—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—21no—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—44no—
276C22×D4—44no—
277C22×D4C2×C4×C3112yes—
278C22×D4D4×C3112yes—
279C23×C4C2×C4×C3111yescharacteristic
280C23×C4D4×C3111yescharacteristic
281#120—44no—
282#120—44no—
283#117—44no—
284C2×C4⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4C2×C4×C3112yes—
297C2×C22⋊C4D4×C3114yescomplemented
298C2×C42—22no—
299#99—48no—
300#99—44no—
301#95—22no—
302C31#949711yescharacteristic, direct product factor, Sylow
303C24—21no—
304C24—21no—
305C24—22no—
306C24—21no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—81no—
314C24—81no—
315C24—81no—
316C24C22⋊C4×C3112yes—
317C24C22×C4×C3111yescharacteristic
318C24C2×D4×C3111yescharacteristic
319C24C2×D4×C3111yescharacteristic
320C24C2×D4×C3112yes—
321C24#41×C3112yes—
322C2×D4—28no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—82no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C22⋊C4×C3112yes—
355C4⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—81no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—44no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—88no—
406C23—88no—
407C23—88no—
408C23C2×C22⋊C4×C3111yescharacteristic
409C23C4×D4×C3112yes—
410C23C31×#12011yescharacteristic
411C23C31×#12112yes—
412C23C31×#12311yescharacteristic
413D4—48no—
414D4—48no—
415D4—48no—
416D4—44no—
417D4—44no—
418D4—88no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4C31×#9914yescomplemented
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—84no—
464C22—82no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—81no—
470C22—81no—
471C22C31×#38611yescharacteristic
472C22C2×C31×#9912yes—
473C22C31×B(4,𝔽2)12yes—
474C4—24no—
475C4—42no—
476C4—82no—
477C4—84no—
478C4—162no—
479C2—22no—
480C2—22no—
481C2—42no—
482C2—42no—
483C2—41no—
484C2—82no—
485C2—82no—
486C2C31×#188911yescharacteristic
487C2C31×#189811yescharacteristic
488C2C31×#191311yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC24C22×C4×C31
Frattini, cosocleC25C23×C31
prime radicalC31×#2403C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2309C2semidirect
3. C31×#1781C2semidirect
4. C31×#1781C2semidirect
5. C31×#1304C2semidirect
9. C2×C31×#120C4semidirect
16. C31×#386C4semidirect
53. C2×C22⋊C4×C31D4semidirect
125. #9497C31direct (N × H)
194. C2×C4×C31#99semidirect
196. #2309C2×C31semidirect
197. #1781C2×C31semidirect
198. #1781C2×C31semidirect
199. #1304C2×C31semidirect
243. C2×#120C4×C31semidirect
250. #386C4×C31semidirect
297. C2×C22⋊C4D4×C31semidirect
302. C31#9497direct (N × H)
438. C2×C4C31×#99semidirect

Maximal subgroups

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

subgroupindex
C31×#24032
C31×#23092
C31×#17812
C31×#13042
#949731