Finite Group Databasebeta

← C31×#9227 #21465986

Subgroups — C31×#9227 #21465986

Order 7936.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#474C211yesmaximal, characteristic
2C2×C31×#436C211yesmaximal, characteristic
3C31×#1956C211yesmaximal, characteristic, complemented
4C31×#1802C211yesmaximal, characteristic
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C31×#1296C211yesmaximal, characteristic, complemented
7C31×#1277C211yesmaximal, characteristic, complemented
8C22×M16×C31—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#474—22no—
12C31×#474C414yescomplemented
13C31×#436—22no—
14C31×#436—22no—
15C2×C31×#107—21no—
16C2×C31×C4⋊C8C2211yescharacteristic
17C2×C31×C4⋊C8C2211yescharacteristic
18C2×C31×C4⋊C8C2211yescharacteristic
19C2×C31×#103—21no—
20C2×C31×#103—21no—
21C2×C31×#102—21no—
22C2×C31×#102—21no—
23C2×C31×C22⋊C8—21no—
24C2×C4×C8×C31C2211yescharacteristic
25C31×#384C2211yescharacteristic
26C31×#340—21no—
27C31×#332—22no—
28C22×Q8×C31—21no—
29C22×D4×C31D411yescharacteristic
30C23×C4×C31—21no—
31C2×M16×C31—24no—
32C2×M16×C31—42no—
33C2×M16×C31—42no—
34C22×C8×C31—21no—
35C22×C8×C31—21no—
36C22×C8×C31—21no—
37C22×C8×C31D411yescharacteristic
38C22×C8×C31D411yescharacteristic
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8C2×C412yes—
43C4×D4×C31—21no—
44C4×D4×C31—21no—
45C4×D4×C31C2×C412yes—
46C4×D4×C31D412yes—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31D411yescharacteristic
53C2×C22⋊C4×C31—21no—
54C2×C42×C31C2311yescharacteristic
55C31×#107—24no—
56C31×C4⋊C8—21no—
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8C2×C412yes—
61C31×C4⋊C8D412yes—
62C31×#103—42no—
63C31×#103—42no—
64C31×#102—24no—
65C31×#102—42no—
66C31×C22⋊C8—24no—
67C4×C8×C31—21no—
68C4×C8×C31—21no—
69C24×C31—21no—
70C2×Q8×C31—22no—
71C2×Q8×C31—41no—
72C2×Q8×C31—41no—
73C2×Q8×C31—41no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31C22⋊C412yes—
82C22×C4×C31—21no—
83C22×C4×C31—22no—
84C22×C4×C31—22no—
85C22×C4×C31—41no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31—41no—
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31#4111yescharacteristic
94M16×C31—48no—
95C2×C8×C31—21no—
96C2×C8×C31—21no—
97C2×C8×C31—22no—
98C2×C8×C31—21no—
99C2×C8×C31—22no—
100C2×C8×C31—21no—
101C2×C8×C31—21no—
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C2×C8×C31—41no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—41no—
109C2×C8×C31D812yes—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—81no—
120C4⋊C4×C31C22⋊C412yes—
121C22⋊C4×C31—22no—
122C22⋊C4×C31—22no—
123C42×C31C22×C411yescharacteristic
124C42×C31C2×D411yescharacteristic
125C42×C31C2×D411yescharacteristic
126C42×C31#4111yescharacteristic
127#9227C3111yesmaximal, characteristic, direct product factor, Sylow
128C23×C31—22no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31#12111yescharacteristic
136Q8×C31—42no—
137Q8×C31—42no—
138Q8×C31—81no—
139Q8×C31—81no—
140D4×C31—22no—
141D4×C31—22no—
142D4×C31—41no—
143D4×C31—41no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—22no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—42no—
153C2×C4×C31—42no—
154C2×C4×C31—81no—
155C2×C4×C31—81no—
156C2×C4×C31—81no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31C2×C22⋊C411yescharacteristic
160C2×C4×C31C4×D411yescharacteristic
161C2×C4×C31C4×D411yescharacteristic
162C2×C4×C31#12011yescharacteristic
163C2×C4×C31#12111yescharacteristic
164C2×C4×C31#12311yescharacteristic
165C2×C4×C31C2×D811yescharacteristic
166C2×C4×C31#13511yescharacteristic
167C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C31#13711yescharacteristic
171C2×C4×C31#13711yescharacteristic
172C2×C4×C31#13711yescharacteristic
173C8×C31—22no—
174C8×C31—42no—
175C8×C31—42no—
176C8×C31—41no—
177C8×C31—81no—
178C2×#474C2×C3111yescharacteristic
179C2×#436C2×C3111yescharacteristic
180#1956C2×C3111yescharacteristic, complemented
181#1802C2×C3111yescharacteristic
182C2×#332C2×C3111yescharacteristic, complemented
183#1296C2×C3111yescharacteristic, complemented
184#1277C2×C3111yescharacteristic, complemented
185C22×C31—42no—
186C22×C31—42no—
187C22×C31—42no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31#38611yescharacteristic
193C22×C31#45911yescharacteristic
194C22×C31#46311yescharacteristic
195C22×C31#46811yescharacteristic
196C22×C31#47011yescharacteristic
197C22×C31#47111yescharacteristic
198C22×C31#47311yescharacteristic
199C4×C31—21no—
200C4×C31—21no—
201C4×C31—42no—
202C4×C31—82no—
203C4×C31—161no—
204C4×C31#41711yescharacteristic
205C4×C31#41911yescharacteristic
206C4×C31C4×D811yescharacteristic
207C4×C31#44211yescharacteristic
208C4×C31#44711yescharacteristic
209C4×C31#44911yescharacteristic
210C4×C31#46811yescharacteristic
211C4×C31#47011yescharacteristic
212C22×M16—21no—
213C2×C4⋊Q8C22×C3111yescharacteristic
214C2×C4×D4C22×C3111yescharacteristic
215#474—22no—
216#474C4×C3114yescomplemented
217#436—22no—
218#436—22no—
219C2×#107—21no—
220C2×C4⋊C8C22×C3111yescharacteristic
221C2×C4⋊C8C22×C3111yescharacteristic
222C2×C4⋊C8C22×C3111yescharacteristic
223C2×#103—21no—
224C2×#103—21no—
225C2×#102—21no—
226C2×#102—21no—
227C2×C22⋊C8—21no—
228C2×C4×C8C22×C3111yescharacteristic
229#384C22×C3111yescharacteristic
230#340—21no—
231#332—22no—
232C2×C31—82no—
233C2×C31#168511yescharacteristic
234C2×C31#168911yescharacteristic
235C2×C31#169911yescharacteristic
236C2×C31#170311yescharacteristic
237C2×C31#187911yescharacteristic
238C2×C31#189411yescharacteristic
239C2×C31#190811yescharacteristic
240C22×Q8—21no—
241C22×D4D4×C3111yescharacteristic
242C23×C4—21no—
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8D4×C3111yescharacteristic
250C22×C8D4×C3111yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8C2×C4×C3112yes—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4C2×C4×C3112yes—
258C4×D4D4×C3112yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4D4×C3111yescharacteristic
265C2×C22⋊C4—21no—
266C2×C42C23×C3111yescharacteristic
267#107—24no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8C2×C4×C3112yes—
273C4⋊C8D4×C3112yes—
274#103—42no—
275#103—42no—
276#102—24no—
277#102—42no—
278C22⋊C8—24no—
279C4×C8—21no—
280C4×C8—21no—
281C31#922711yescharacteristic, direct product factor, Sylow
282C24—21no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C22⋊C4×C3112yes—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×D4×C3111yescharacteristic
305C22×C4C2×D4×C3111yescharacteristic
306C22×C4#41×C3111yescharacteristic
307M16—48no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8D8×C3112yes—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—81no—
333C4⋊C4C22⋊C4×C3112yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C42C22×C4×C3111yescharacteristic
337C42C2×D4×C3111yescharacteristic
338C42C2×D4×C3111yescharacteristic
339C42#41×C3111yescharacteristic
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C31×#12111yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352D4—22no—
353D4—22no—
354D4—41no—
355D4—41no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4C2×C22⋊C4×C3111yescharacteristic
372C2×C4C4×D4×C3111yescharacteristic
373C2×C4C4×D4×C3111yescharacteristic
374C2×C4C31×#12011yescharacteristic
375C2×C4C31×#12111yescharacteristic
376C2×C4C31×#12311yescharacteristic
377C2×C4C2×D8×C3111yescharacteristic
378C2×C4C31×#13511yescharacteristic
379C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
381C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
382C2×C4C31×#13711yescharacteristic
383C2×C4C31×#13711yescharacteristic
384C2×C4C31×#13711yescharacteristic
385C8—22no—
386C8—42no—
387C8—42no—
388C8—41no—
389C8—81no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C31×#38611yescharacteristic
398C22C31×#45911yescharacteristic
399C22C31×#46311yescharacteristic
400C22C31×#46811yescharacteristic
401C22C31×#47011yescharacteristic
402C22C31×#47111yescharacteristic
403C22C31×#47311yescharacteristic
404C4—21no—
405C4—21no—
406C4—42no—
407C4—82no—
408C4—161no—
409C4C31×#41711yescharacteristic
410C4C31×#41911yescharacteristic
411C4C4×D8×C3111yescharacteristic
412C4C31×#44211yescharacteristic
413C4C31×#44711yescharacteristic
414C4C31×#44911yescharacteristic
415C4C31×#46811yescharacteristic
416C4C31×#47011yescharacteristic
417C2—82no—
418C2C31×#168511yescharacteristic
419C2C31×#168911yescharacteristic
420C2C31×#169911yescharacteristic
421C2C31×#170311yescharacteristic
422C2C31×#187911yescharacteristic
423C2C31×#189411yescharacteristic
424C2C31×#190811yescharacteristic
subgroupG/H
center, socleC23×C31#121
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1956C2semidirect
5. C2×C31×#332C2semidirect
6. C31×#1296C2semidirect
7. C31×#1277C2semidirect
12. C31×#474C4semidirect
127. #9227C31direct (N × H)
180. #1956C2×C31semidirect
182. C2×#332C2×C31semidirect
183. #1296C2×C31semidirect
184. #1277C2×C31semidirect
216. #474C4×C31semidirect
281. C31#9227direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4742
C2×C31×#4362
C31×#19562
C31×#18022
C2×C31×#3322
C31×#12962
C31×#12772
#922731