Finite Group Databasebeta

← C31×#10008 #21466767

Subgroups — C31×#10008 #21466767

Order 7936.

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

normal subgroups only

This group has 654 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×#3215C211yesmaximal, characteristic, complemented
2C31×#3110C211yesmaximal, characteristic, complemented
3C31×#2540C211yesmaximal, characteristic
4C31×#1986C211yesmaximal, characteristic, complemented
5C31×#1527C211yesmaximal, characteristic, complemented
6C31×#1490C211yesmaximal, characteristic, complemented
7C31×#1285C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C2×C31×C22⋊Q8—21no—
11C2×C31×#121—21no—
12C2×C4×D4×C31C2211yescharacteristic
13C31×#497—21no—
14C31×#489—21no—
15C31×#484—21no—
16C31×#474—21no—
17C31×#464—22no—
18C31×#461—21no—
19C31×#451—21no—
20C31×#450—21no—
21C31×#450—21no—
22C31×#440—22no—
23C31×#432C2211yescharacteristic
24C31×#432C2211yescharacteristic
25C4×M16×C31C2211yescharacteristic
26C31×#399—21no—
27C31×#392—21no—
28C31×#390—22no—
29C4×C4⋊C4×C31C2211yescharacteristic
30C31×#339—22no—
31C31×#329—22no—
32C31×D4⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31—22no—
35C22×D4×C31D411yescharacteristic
36C22×D4×C31D411yescharacteristic
37C23×C4×C31—21no—
38C2×Q16×C31—22no—
39C2×SD16×C31—22no—
40C2×SD16×C31—22no—
41C2×M16×C31—21no—
42C31×C4⋊Q8D411yescharacteristic, complemented
43C31×C4⋊Q8D411yescharacteristic, complemented
44C31×#125—21no—
45C31×#124—21no—
46C31×#124—21no—
47C31×#124—21no—
48C31×#124—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8—44no—
52C31×#121—44no—
53C4×Q8×C31—21no—
54C4×Q8×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C4×D4×C31—21no—
59C31×#117—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×#107—22no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8D412yescomplemented
73C31×#103—22no—
74C31×#103—22no—
75C31×#102—22no—
76C31×#102—22no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C31×#97—21no—
80C31×#97—21no—
81C31×#97D412yescomplemented
82C4×C8×C31—21no—
83C31×#95—21no—
84C24×C31—21no—
85C24×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—21no—
89C2×Q8×C31—21no—
90C2×Q8×C31—42no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—42no—
96C2×D4×C31—44no—
97C2×D4×C31—42no—
98C2×D4×C31—44no—
99C2×D4×C31—82no—
100C22×C4×C31—22no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—42no—
104C22×C4×C31—42no—
105C22×C4×C31—41no—
106C22×C4×C31—41no—
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110Q16×C31—82no—
111SD16×C31—44no—
112SD16×C31—82no—
113M16×C31—22no—
114M16×C31—41no—
115C2×C8×C31—22no—
116C2×C8×C31—22no—
117C2×C8×C31—41no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—44no—
126C4⋊C4×C31—41no—
127C4⋊C4×C31—41no—
128C22⋊C4×C31—44no—
129C22⋊C4×C31—44no—
130C22⋊C4×C31—41no—
131C22⋊C4×C31—41no—
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C42×C31—42no—
138C42×C31—41no—
139C42×C31C2×D411yescharacteristic
140C42×C31C2×D411yescharacteristic
141C42×C31C2×D411yescharacteristic
142C42×C31#4111yescharacteristic
143#10008C3111yesmaximal, characteristic, direct product factor, Sylow
144C23×C31—22no—
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#12011yescharacteristic
157Q8×C31—42no—
158Q8×C31—42no—
159Q8×C31—42no—
160Q8×C31—81no—
161D4×C31—22no—
162D4×C31—22no—
163D4×C31—41no—
164D4×C31—41no—
165D4×C31—42no—
166D4×C31—81no—
167D4×C31—88no—
168C2×C4×C31—21no—
169C2×C4×C31—21no—
170C2×C4×C31—22no—
171C2×C4×C31—24no—
172C2×C4×C31—41no—
173C2×C4×C31—42no—
174C2×C4×C31—42no—
175C2×C4×C31—41no—
176C2×C4×C31—42no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—41no—
182C2×C4×C31—82no—
183C2×C4×C31—82no—
184C2×C4×C31#12011yescharacteristic
185C2×C4×C31#12011yescharacteristic
186C2×C4×C31#12111yescharacteristic
187C2×C4×C31#12111yescharacteristic
188C2×C4×C31#12111yescharacteristic
189C2×C4×C31#12711yescharacteristic
190C8×C31—42no—
191C8×C31—42no—
192C8×C31—42no—
193#3215C2×C3111yescharacteristic, complemented
194#3110C2×C3111yescharacteristic, complemented
195#2540C2×C3111yescharacteristic
196#1986C2×C3111yescharacteristic, complemented
197#1527C2×C3111yescharacteristic, complemented
198#1490C2×C3111yescharacteristic, complemented
199#1285C2×C3111yescharacteristic, complemented
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—42no—
203C22×C31—42no—
204C22×C31—42no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—82no—
208C22×C31—82no—
209C22×C31—81no—
210C22×C31—81no—
211C22×C31—81no—
212C22×C31—81no—
213C22×C31—81no—
214C22×C31#39211yescharacteristic
215C22×C31#45511yescharacteristic
216C22×C31#45611yescharacteristic
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—21no—
220C4×C31—21no—
221C4×C31—21no—
222C4×C31—44no—
223C4×C31—81no—
224C4×C31—81no—
225C4×C31—81no—
226C4×C31—81no—
227C4×C31#45312yes—
228#533C22×C3111yescharacteristic
229C2×#124C22×C3111yescharacteristic
230C2×C22⋊Q8—21no—
231C2×#121—21no—
232C2×C4×D4C22×C3111yescharacteristic
233#497—21no—
234#489—21no—
235#484—21no—
236#474—21no—
237#464—22no—
238#461—21no—
239#451—21no—
240#450—21no—
241#450—21no—
242#440—22no—
243#432C22×C3111yescharacteristic
244#432C22×C3111yescharacteristic
245C4×M16C22×C3111yescharacteristic
246#399—21no—
247#392—21no—
248#390—22no—
249C4×C4⋊C4C22×C3111yescharacteristic
250#339—22no—
251#329—22no—
252D4⋊C8—22no—
253C2×C31—21no—
254C2×C31—21no—
255C2×C31—82no—
256C2×C31—161no—
257C2×C31#200211yescharacteristic
258C2×C31#200711yescharacteristic
259C2×C31#201011yescharacteristic
260C22×Q8D4×C3111yescharacteristic
261C22×D4—22no—
262C22×D4D4×C3111yescharacteristic
263C22×D4D4×C3111yescharacteristic
264C23×C4—21no—
265C2×Q16—22no—
266C2×SD16—22no—
267C2×SD16—22no—
268C2×M16—21no—
269C4⋊Q8D4×C3111yescharacteristic, complemented
270C4⋊Q8D4×C3111yescharacteristic, complemented
271#125—21no—
272#124—21no—
273#124—21no—
274#124—21no—
275#124—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—44no—
279#121—44no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286#117—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4D4×C3111yescharacteristic
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C42D4×C3112yes—
296C2×C42C23×C3111yescharacteristic
297#107—22no—
298C4⋊C8—21no—
299C4⋊C8D4×C3112yescomplemented
300#103—22no—
301#103—22no—
302#102—22no—
303#102—22no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308#97D4×C3112yescomplemented
309C4×C8—21no—
310#95—21no—
311C31#1000811yescharacteristic, direct product factor, Sylow
312C24—21no—
313C24—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—42no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—82no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C2×D4×C3111yescharacteristic
336C22×C4C2×D4×C3111yescharacteristic
337C22×C4C2×D4×C3111yescharacteristic
338Q16—82no—
339SD16—44no—
340SD16—82no—
341M16—22no—
342M16—41no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—41no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—44no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—42no—
366C42—41no—
367C42C2×D4×C3111yescharacteristic
368C42C2×D4×C3111yescharacteristic
369C42C2×D4×C3111yescharacteristic
370C42#41×C3111yescharacteristic
371C23—22no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23C31×#12011yescharacteristic
384Q8—42no—
385Q8—42no—
386Q8—42no—
387Q8—81no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392D4—42no—
393D4—81no—
394D4—88no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4C31×#12011yescharacteristic
412C2×C4C31×#12011yescharacteristic
413C2×C4C31×#12111yescharacteristic
414C2×C4C31×#12111yescharacteristic
415C2×C4C31×#12111yescharacteristic
416C2×C4C31×#12711yescharacteristic
417C8—42no—
418C8—42no—
419C8—42no—
420C22—21no—
421C22—21no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—82no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22C31×#39211yescharacteristic
435C22C31×#45511yescharacteristic
436C22C31×#45611yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—44no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4C31×#45312yes—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—161no—
452C2C31×#200211yescharacteristic
453C2C31×#200711yescharacteristic
454C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2540C2
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×#3215C2semidirect
2. C31×#3110C2semidirect
4. C31×#1986C2semidirect
5. C31×#1527C2semidirect
6. C31×#1490C2semidirect
7. C31×#1285C2semidirect
42. C31×C4⋊Q8D4semidirect
43. C31×C4⋊Q8D4semidirect
72. C31×C4⋊C8D4semidirect
81. C31×#97D4semidirect
143. #10008C31direct (N × H)
193. #3215C2×C31semidirect
194. #3110C2×C31semidirect
196. #1986C2×C31semidirect
197. #1527C2×C31semidirect
198. #1490C2×C31semidirect
199. #1285C2×C31semidirect
269. C4⋊Q8D4×C31semidirect
270. C4⋊Q8D4×C31semidirect
299. C4⋊C8D4×C31semidirect
308. #97D4×C31semidirect
311. C31#10008direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32152
C31×#31102
C31×#25402
C31×#19862
C31×#15272
C31×#14902
C31×#12852
#1000831