Finite Group Databasebeta

← C31×#9062 #21465821

Subgroups — C31×#9062 #21465821

Order 7936.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#486C211yesmaximal, characteristic
2C31×#2365C211yesmaximal, characteristic, complemented
3C31×#1982C211yesmaximal, characteristic
4C31×#1982C211yesmaximal, characteristic
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C31×#1279C211yesmaximal, characteristic, complemented
7C31×#1279C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#127C2211yescharacteristic
10C2×C4×Q8×C31—21no—
11C31×#486—22no—
12C31×#486C414yescomplemented
13C2×C31×C4⋊C8C2211yescharacteristic
14C2×C31×C4⋊C8C2211yescharacteristic
15C2×C31×#102—22no—
16C2×C31×#102—21no—
17C2×C31×#102—21no—
18C2×C31×#102—21no—
19C2×C31×#102—21no—
20C2×C4×C8×C31C2211yescharacteristic
21C31×#391—21no—
22C31×#389—21no—
23C31×#387—22no—
24C4×C4⋊C4×C31—21no—
25C4×C4⋊C4×C31C2211yescharacteristic
26C4×C4⋊C4×C31C2211yescharacteristic
27C31×#332—22no—
28C22×Q8×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C31×C4⋊Q8—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8C2×C412yes—
38C31×C4⋊Q8D412yes—
39C31×#127—21no—
40C31×#127—21no—
41C31×#127—21no—
42C31×#127C2×C412yes—
43C4×Q8×C31—42no—
44C4×Q8×C31—42no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31—22no—
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C4⋊C4×C31D411yescharacteristic
49C2×C4⋊C4×C31D412yes—
50C2×C4⋊C4×C31D411yescharacteristic
51C2×C4⋊C4×C31D411yescharacteristic
52C2×C42×C31—22no—
53C2×C42×C31—21no—
54C2×C42×C31—21no—
55C2×C42×C31C2311yescharacteristic
56C31×C4⋊C8—21no—
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×#102—28no—
61C31×#102—42no—
62C31×#102—42no—
63C31×#102—42no—
64C31×#102—42no—
65C4×C8×C31—21no—
66C4×C8×C31C2×C412yes—
67C31×#95—22no—
68C31×#95—22no—
69C31×#95—21no—
70C31×#95—21no—
71C24×C31—41no—
72C2×Q8×C31—22no—
73C2×Q8×C31—22no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×D4×C31—22no—
79C2×D4×C31—22no—
80C2×D4×C31—22no—
81C2×D4×C31—42no—
82C2×D4×C31—42no—
83C2×D4×C31—41no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C22×C4×C31—21no—
95C22×C4×C31—22no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102C2×C8×C31—22no—
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×C31—41no—
110C2×C8×C31—41no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—22no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—44no—
121C4⋊C4×C31C22⋊C414yes—
122C4⋊C4×C31D812yes—
123C4⋊C4×C31SD1612yes—
124C42×C31—44no—
125C42×C31—41no—
126C42×C31—41no—
127C42×C31—41no—
128C42×C31—41no—
129C42×C31C22×C411yescharacteristic
130C42×C31C2×D411yescharacteristic
131C42×C31#4111yescharacteristic
132C42×C31#4111yescharacteristic
133#9062C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—42no—
135C23×C31—81no—
136C23×C31—81no—
137C23×C31—81no—
138C23×C31—81no—
139C23×C31—81no—
140C23×C31—81no—
141C23×C31#12011yescharacteristic
142Q8×C31—42no—
143Q8×C31—42no—
144Q8×C31—42no—
145Q8×C31—42no—
146D4×C31—44no—
147D4×C31—42no—
148D4×C31—42no—
149D4×C31—42no—
150D4×C31—42no—
151D4×C31—82no—
152D4×C31—81no—
153D4×C31—81no—
154D4×C31—81no—
155D4×C31—81no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—24no—
159C2×C4×C31—22no—
160C2×C4×C31—22no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—41no—
164C2×C4×C31—42no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31C2×C22⋊C411yescharacteristic
174C2×C4×C31C4×D411yescharacteristic
175C2×C4×C31C4×D411yescharacteristic
176C2×C4×C31#12111yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C2×C4×C31#12311yescharacteristic
179C2×C4×C31C2×D811yescharacteristic
180C2×C4×C31C2×SD1611yescharacteristic
181C2×C4×C31#13511yescharacteristic
182C2×C4×C31#13511yescharacteristic
183C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C31#13711yescharacteristic
187C8×C31—42no—
188C8×C31—81no—
189C8×C31—81no—
190C2×#486C2×C3111yescharacteristic
191#2365C2×C3111yescharacteristic, complemented
192#1982C2×C3111yescharacteristic
193#1982C2×C3111yescharacteristic
194C2×#332C2×C3111yescharacteristic, complemented
195#1279C2×C3111yescharacteristic, complemented
196#1279C2×C3111yescharacteristic, complemented
197C22×C31—82no—
198C22×C31—82no—
199C22×C31—82no—
200C22×C31—161no—
201C22×C31—161no—
202C22×C31—161no—
203C22×C31—161no—
204C22×C31#38611yescharacteristic
205C22×C31#44711yescharacteristic
206C22×C31#44811yescharacteristic
207C22×C31#44911yescharacteristic
208C22×C31#44911yescharacteristic
209C22×C31#45411yescharacteristic
210C22×C31#45411yescharacteristic
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—44no—
214C4×C31—42no—
215C4×C31—42no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31C4×D811yescharacteristic
219C4×C31C4×SD1611yescharacteristic
220C4×C31#44011yescharacteristic
221C4×C31#44211yescharacteristic
222C4×C31#45911yescharacteristic
223C4×C31#46011yescharacteristic
224C4×C31#46311yescharacteristic
225C4×C31#46411yescharacteristic
226C2×C4⋊Q8C22×C3111yescharacteristic
227C2×#127C22×C3111yescharacteristic
228C2×C4×Q8—21no—
229#486—22no—
230#486C4×C3114yescomplemented
231C2×C4⋊C8C22×C3111yescharacteristic
232C2×C4⋊C8C22×C3111yescharacteristic
233C2×#102—22no—
234C2×#102—21no—
235C2×#102—21no—
236C2×#102—21no—
237C2×#102—21no—
238C2×C4×C8C22×C3111yescharacteristic
239#391—21no—
240#389—21no—
241#387—22no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C22×C3111yescharacteristic
244C4×C4⋊C4C22×C3111yescharacteristic
245#332—22no—
246C2×C31—162no—
247C2×C31#162511yescharacteristic
248C2×C31#162611yescharacteristic
249C2×C31#162711yescharacteristic
250C2×C31#162811yescharacteristic
251C2×C31#187911yescharacteristic
252C2×C31#188011yescharacteristic
253C2×C31#188811yescharacteristic
254C22×Q8—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8C2×C4×C3112yes—
264C4⋊Q8D4×C3112yes—
265#127—21no—
266#127—21no—
267#127—21no—
268#127C2×C4×C3112yes—
269C4×Q8—42no—
270C4×Q8—42no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4D4×C3111yescharacteristic
274C2×C4⋊C4D4×C3111yescharacteristic
275C2×C4⋊C4D4×C3112yes—
276C2×C4⋊C4D4×C3111yescharacteristic
277C2×C4⋊C4D4×C3111yescharacteristic
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C23×C3111yescharacteristic
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286#102—28no—
287#102—42no—
288#102—42no—
289#102—42no—
290#102—42no—
291C4×C8—21no—
292C4×C8C2×C4×C3112yes—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297C31#906211yescharacteristic, direct product factor, Sylow
298C24—41no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×D4×C3111yescharacteristic
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—44no—
348C4⋊C4C22⋊C4×C3114yes—
349C4⋊C4D8×C3112yes—
350C4⋊C4SD16×C3112yes—
351C42—44no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C22×C4×C3111yescharacteristic
357C42C2×D4×C3111yescharacteristic
358C42#41×C3111yescharacteristic
359C42#41×C3111yescharacteristic
360C23—42no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23C31×#12011yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372D4—44no—
373D4—42no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—82no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4C2×C22⋊C4×C3111yescharacteristic
400C2×C4C4×D4×C3111yescharacteristic
401C2×C4C4×D4×C3111yescharacteristic
402C2×C4C31×#12111yescharacteristic
403C2×C4C31×#12111yescharacteristic
404C2×C4C31×#12311yescharacteristic
405C2×C4C2×D8×C3111yescharacteristic
406C2×C4C2×SD16×C3111yescharacteristic
407C2×C4C31×#13511yescharacteristic
408C2×C4C31×#13511yescharacteristic
409C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
410C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
411C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C31×#13711yescharacteristic
413C8—42no—
414C8—81no—
415C8—81no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22C31×#38611yescharacteristic
424C22C31×#44711yescharacteristic
425C22C31×#44811yescharacteristic
426C22C31×#44911yescharacteristic
427C22C31×#44911yescharacteristic
428C22C31×#45411yescharacteristic
429C22C31×#45411yescharacteristic
430C4—21no—
431C4—21no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—81no—
436C4—81no—
437C4C4×D8×C3111yescharacteristic
438C4C4×SD16×C3111yescharacteristic
439C4C31×#44011yescharacteristic
440C4C31×#44211yescharacteristic
441C4C31×#45911yescharacteristic
442C4C31×#46011yescharacteristic
443C4C31×#46311yescharacteristic
444C4C31×#46411yescharacteristic
445C2—162no—
446C2C31×#162511yescharacteristic
447C2C31×#162611yescharacteristic
448C2C31×#162711yescharacteristic
449C2C31×#162811yescharacteristic
450C2C31×#187911yescharacteristic
451C2C31×#188011yescharacteristic
452C2C31×#188811yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
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
2. C31×#2365C2semidirect
5. C2×C31×#332C2semidirect
6. C31×#1279C2semidirect
7. C31×#1279C2semidirect
12. C31×#486C4semidirect
133. #9062C31direct (N × H)
191. #2365C2×C31semidirect
194. C2×#332C2×C31semidirect
195. #1279C2×C31semidirect
196. #1279C2×C31semidirect
230. #486C4×C31semidirect
297. C31#9062direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4862
C31×#23652
C31×#19822
C2×C31×#3322
C31×#12792
#906231