Finite Group Databasebeta

← C31×#9971 #21466730

Subgroups — C31×#9971 #21466730

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#483C211yesmaximal, characteristic, complemented
2C2×C31×#480C211yesmaximal, characteristic
3C31×#2639C211yesmaximal, characteristic
4C2×C31×#328C211yesmaximal, characteristic, complemented
5C2×C31×#327C211yesmaximal, characteristic
6C31×#1445C211yesmaximal, characteristic, complemented
7C31×#1280C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C22×C4⋊C4×C31C2211yescharacteristic
11C31×#483—22no—
12C31×#483—22no—
13C31×#480—22no—
14C31×#480—22no—
15C2×C31×#107—21no—
16C2×C31×#106—21no—
17C2×C31×#103—21no—
18C2×C31×#102—21no—
19C2×C31×C22⋊C8C2211yescharacteristic
20C2×C31×C22⋊C8C2211yescharacteristic
21C31×#400—21no—
22C31×#397—21no—
23C31×#394—21no—
24C31×#393—21no—
25C31×#386—22no—
26C2×C31×#95—22no—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C31×#327—22no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C23×C4×C31—21no—
36C23×C4×C31D412yes—
37C23×C4×C31C2311yescharacteristic
38C22×C8×C31—21no—
39C22×C8×C31—21no—
40C31×C22⋊Q8—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#121—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31D411yescharacteristic
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D412yes—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—22no—
61C31×#107—24no—
62C31×#106—24no—
63C31×#103—24no—
64C31×#102—24no—
65C31×C22⋊C8—21no—
66C31×C22⋊C8—21no—
67C31×C22⋊C8D412yes—
68C31×C22⋊C8D412yes—
69C31×#95—22no—
70C31×#95—22no—
71C31×#95—21no—
72C31×#95—21no—
73C31×#95—21no—
74C31×#95—21no—
75C31×#95—44no—
76C24×C31—21no—
77C24×C31C2×D411yescharacteristic
78C2×Q8×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—42no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—42no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—42no—
87C2×D4×C31—42no—
88C22×C4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—22no—
94C22×C4×C31—22no—
95C22×C4×C31—22no—
96C22×C4×C31—42no—
97C22×C4×C31—42no—
98C22×C4×C31—42no—
99C22×C4×C31—41no—
100C22×C4×C31—42no—
101C22×C4×C31—41no—
102C22×C4×C31—41no—
103C22×C4×C31—41no—
104C22×C4×C31—42no—
105C22×C4×C31—42no—
106C22×C4×C31—42no—
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111C22×C4×C31C2×D411yescharacteristic
112C22×C4×C31#4111yescharacteristic
113C2×C8×C31—22no—
114C2×C8×C31—22no—
115C2×C8×C31—42no—
116C2×C8×C31—42no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—22no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—41no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C4⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—41no—
131C22⋊C4×C31—82no—
132C22⋊C4×C31—82no—
133#9971C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—21no—
135C23×C31—21no—
136C23×C31—22no—
137C23×C31—41no—
138C23×C31—42no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—42no—
142C23×C31#12011yescharacteristic
143C23×C31#12111yescharacteristic
144C23×C31#12711yescharacteristic
145C23×C31#13512yes—
146C23×C31AGL(1,ℤ/8)12yes—
147Q8×C31—42no—
148Q8×C31—81no—
149D4×C31—42no—
150D4×C31—81no—
151D4×C31—82no—
152D4×C31—81no—
153D4×C31—81no—
154C2×C4×C31—22no—
155C2×C4×C31—22no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—44no—
161C2×C4×C31—42no—
162C2×C4×C31—44no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—41no—
170C2×C4×C31—42no—
171C2×C4×C31—41no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—44no—
175C2×C4×C31—82no—
176C2×C4×C31—81no—
177C2×C4×C31—82no—
178C2×C4×C31—81no—
179C2×C4×C31—81no—
180C2×C4×C31—81no—
181C2×C4×C31—82no—
182C2×C4×C31#12011yescharacteristic
183C2×C4×C31#12011yescharacteristic
184C2×C4×C31#12111yescharacteristic
185C2×C4×C31#12111yescharacteristic
186C8×C31—42no—
187C8×C31—42no—
188C2×#483C2×C3111yescharacteristic, complemented
189C2×#480C2×C3111yescharacteristic
190#2639C2×C3111yescharacteristic
191C2×#328C2×C3111yescharacteristic, complemented
192C2×#327C2×C3111yescharacteristic
193#1445C2×C3111yescharacteristic, complemented
194#1280C2×C3111yescharacteristic, complemented
195C22×C31—22no—
196C22×C31—22no—
197C22×C31—22no—
198C22×C31—22no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—82no—
204C22×C31—82no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31#39211yescharacteristic
209C22×C31#44912yes—
210C22×C31#45511yescharacteristic
211C22×C31#45611yescharacteristic
212C22×C31#45812yes—
213C22×C31#46312yes—
214C22×C31#49511yescharacteristic
215C22×C31#49611yescharacteristic
216C4×C31—21no—
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—21no—
220C4×C31—82no—
221C4×C31—82no—
222C4×C31—81no—
223C4×C31—82no—
224C4×C31—81no—
225C4×C31—81no—
226C2×C22⋊Q8C22×C3111yescharacteristic
227C2×#121C22×C3111yescharacteristic
228C22×C4⋊C4C22×C3111yescharacteristic
229#483—22no—
230#483—22no—
231#480—22no—
232#480—22no—
233C2×#107—21no—
234C2×#106—21no—
235C2×#103—21no—
236C2×#102—21no—
237C2×C22⋊C8C22×C3111yescharacteristic
238C2×C22⋊C8C22×C3111yescharacteristic
239#400—21no—
240#397—21no—
241#394—21no—
242#393—21no—
243#386—22no—
244C2×#95—22no—
245C2×#95C22×C3111yescharacteristic
246C2×#95C22×C3111yescharacteristic
247#328—22no—
248#327—22no—
249C2×C31—21no—
250C2×C31—22no—
251C2×C31—21no—
252C2×C31—161no—
253C2×C31#160912yes—
254C2×C31#161212yes—
255C2×C31#200711yescharacteristic
256C2×C31#201211yescharacteristic
257C2×C31#202211yescharacteristic
258C22×Q8D4×C3111yescharacteristic
259C22×D4—21no—
260C22×D4D4×C3111yescharacteristic
261C23×C4—21no—
262C23×C4—21no—
263C23×C4D4×C3112yes—
264C23×C4C23×C3111yescharacteristic
265C22×C8—21no—
266C22×C8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4D4×C3112yes—
282C2×C4⋊C4D4×C3111yescharacteristic
283C2×C4⋊C4D4×C3111yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288#107—24no—
289#106—24no—
290#103—24no—
291#102—24no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8D4×C3112yes—
295C22⋊C8D4×C3112yes—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—44no—
303C31#997111yescharacteristic, direct product factor, Sylow
304C24—21no—
305C24C2×D4×C3111yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4C2×D4×C3111yescharacteristic
336C22×C4C2×D4×C3111yescharacteristic
337C22×C4C2×D4×C3111yescharacteristic
338C22×C4C2×D4×C3111yescharacteristic
339C22×C4C2×D4×C3111yescharacteristic
340C22×C4#41×C3111yescharacteristic
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23C31×#12011yescharacteristic
370C23C31×#12111yescharacteristic
371C23C31×#12711yescharacteristic
372C23C31×#13512yes—
373C23C31×AGL(1,ℤ/8)12yes—
374Q8—42no—
375Q8—81no—
376D4—42no—
377D4—81no—
378D4—82no—
379D4—81no—
380D4—81no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4C31×#12011yescharacteristic
410C2×C4C31×#12011yescharacteristic
411C2×C4C31×#12111yescharacteristic
412C2×C4C31×#12111yescharacteristic
413C8—42no—
414C8—42no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—82no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C31×#39211yescharacteristic
429C22C31×#44912yes—
430C22C31×#45511yescharacteristic
431C22C31×#45611yescharacteristic
432C22C31×#45812yes—
433C22C31×#46312yes—
434C22C31×#49511yescharacteristic
435C22C31×#49611yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—81no—
446C2—21no—
447C2—22no—
448C2—21no—
449C2—161no—
450C2C31×#160912yes—
451C2C31×#161212yes—
452C2C31×#200711yescharacteristic
453C2C31×#201211yescharacteristic
454C2C31×#202211yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#483C2semidirect
4. C2×C31×#328C2semidirect
6. C31×#1445C2semidirect
7. C31×#1280C2semidirect
133. #9971C31direct (N × H)
188. C2×#483C2×C31semidirect
191. C2×#328C2×C31semidirect
193. #1445C2×C31semidirect
194. #1280C2×C31semidirect
303. C31#9971direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4832
C2×C31×#4802
C31×#26392
C2×C31×#3282
C2×C31×#3272
C31×#14452
C31×#12802
#997131