Finite Group Databasebeta

← C31×#7612 #21464371

Subgroups — C31×#7612 #21464371

Order 7936.

656 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2297C211yesmaximal, characteristic
2C31×#1797C211yesmaximal, characteristic, complemented
3C31×#1749C211yesmaximal, characteristic, complemented
4C31×#1272C212yesmaximal, complemented
5C31×#1270C212yesmaximal, complemented
6C22×#41×C31C411yescharacteristic
7C2×C31×#117C411yescharacteristic
8C22×C42×C31C2211yescharacteristic
9C2×C31×C4⋊C8—21no—
10C2×C31×C22⋊C8—21no—
11C2×C31×C22⋊C8—21no—
12C2×C31×#97—21no—
13C31×#391C411yescharacteristic
14C31×#391C2211yescharacteristic
15C31×#390C411yescharacteristic
16C31×#390C2211yescharacteristic
17C31×#386—21no—
18C31×#386—21no—
19C4×C22⋊C4×C31—21no—
20C4×C22⋊C4×C31—21no—
21C31×#336C2212yes—
22C31×#336C2212yes—
23C2×#41×C31—22no—
24C2×#41×C31—22no—
25C2×#41×C31—42no—
26C2×#41×C31—42no—
27C22×Q8×C31C2×C411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31C2×C411yescharacteristic
30C23×C4×C31—21no—
31C23×C4×C31C2×C411yescharacteristic
32C23×C4×C31D411yescharacteristic
33C23×C4×C31D411yescharacteristic
34C22×C8×C31—22no—
35C22×C8×C31—22no—
36C31×#117—42no—
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C42×C31—21no—
45C2×C42×C31—42no—
46C2×C42×C31C2×C411yescharacteristic
47C2×C42×C31D411yescharacteristic
48C2×C42×C31D411yescharacteristic
49C2×C42×C31C2311yescharacteristic
50C31×C4⋊C8—42no—
51C31×C22⋊C8—24no—
52C31×C22⋊C8—42no—
53C31×#97—42no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C24×C31—21no—
59C24×C31C22⋊C411yescharacteristic
60#41×C31—48no—
61#41×C31—48no—
62C2×Q8×C31—21no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C2×Q8×C31M1612yes—
66C2×D4×C31—24no—
67C2×D4×C31—21no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31—42no—
71C2×D4×C31—42no—
72C2×D4×C31M1612yes—
73C22×C4×C31—22no—
74C22×C4×C31—22no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—24no—
78C22×C4×C31—21no—
79C22×C4×C31—22no—
80C22×C4×C31—22no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—41no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31—42no—
89C22×C4×C31—42no—
90C22×C4×C31C22⋊C411yescharacteristic
91C22×C4×C31C22⋊C411yescharacteristic
92C22×C4×C31C22⋊C411yescharacteristic
93C22×C4×C31C22×C411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96C2×C8×C31—24no—
97C2×C8×C31—42no—
98C2×C8×C31—42no—
99C2×C8×C31—42no—
100C2×C8×C31—42no—
101C2×C8×C31—42no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C42×C31—22no—
111C42×C31—22no—
112C42×C31—41no—
113C42×C31—41no—
114C42×C31—42no—
115#7612C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—21no—
117C23×C31—21no—
118C23×C31—21no—
119C23×C31—21no—
120C23×C31—21no—
121C23×C31—21no—
122C23×C31—22no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31#10012yes—
130C23×C31C2×C22⋊C411yescharacteristic
131Q8×C31—42no—
132Q8×C31—42no—
133D4×C31—42no—
134D4×C31—42no—
135D4×C31—48no—
136C2×C4×C31—24no—
137C2×C4×C31—24no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—22no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—44no—
153C2×C4×C31—42no—
154C2×C4×C31—42no—
155C2×C4×C31—48no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31#9912yes—
165C2×C4×C31#10412yes—
166C2×C4×C31C2×M1611yescharacteristic
167C2×C4×C31C2×M1611yescharacteristic
168C2×C4×C31#13512yes—
169C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C31#13711yescharacteristic
171C8×C31—44no—
172C8×C31—82no—
173#2297C2×C3111yescharacteristic
174#1797C2×C3111yescharacteristic, complemented
175#1749C2×C3111yescharacteristic, complemented
176#1272C2×C3112yescomplemented
177#1270C2×C3112yescomplemented
178C22×C31—22no—
179C22×C31—22no—
180C22×C31—22no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—42no—
186C22×C31—42no—
187C22×C31—42no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31#40711yescharacteristic
193C22×C31C2×#9911yescharacteristic
194C22×C31C2×#10011yescharacteristic
195C22×C31#41611yescharacteristic
196C22×C31#41711yescharacteristic
197C22×C31C2×#10411yescharacteristic
198C22×C31#42111yescharacteristic
199C4×C31—22no—
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—22no—
203C4×C31—22no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—42no—
207C4×C31—41no—
208C4×C31—81no—
209C4×C31—81no—
210C22×#41C4×C3111yescharacteristic
211C2×#117C4×C3111yescharacteristic
212C22×C42C22×C3111yescharacteristic
213C2×C4⋊C8—21no—
214C2×C22⋊C8—21no—
215C2×C22⋊C8—21no—
216C2×#97—21no—
217#391C4×C3111yescharacteristic
218#391C22×C3111yescharacteristic
219#390C4×C3111yescharacteristic
220#390C22×C3111yescharacteristic
221#386—21no—
222#386—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225#336C22×C3112yes—
226#336C22×C3112yes—
227C2×C31—42no—
228C2×C31—82no—
229C2×C31#146211yescharacteristic
230C2×C31#148811yescharacteristic
231C2×C31#148911yescharacteristic
232C2×C31#150111yescharacteristic
233C2×C31#151311yescharacteristic
234C2×C31#153011yescharacteristic
235C2×C31#155111yescharacteristic
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C22×Q8C2×C4×C3111yescharacteristic
241C22×D4—21no—
242C22×D4C2×C4×C3111yescharacteristic
243C23×C4—21no—
244C23×C4C2×C4×C3111yescharacteristic
245C23×C4D4×C3111yescharacteristic
246C23×C4D4×C3111yescharacteristic
247C22×C8—22no—
248C22×C8—22no—
249#117—42no—
250C2×C4⋊C4C2×C4×C3111yescharacteristic
251C2×C4⋊C4C2×C4×C3111yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42C2×C4×C3111yescharacteristic
260C2×C42D4×C3111yescharacteristic
261C2×C42D4×C3111yescharacteristic
262C2×C42C23×C3111yescharacteristic
263C4⋊C8—42no—
264C22⋊C8—24no—
265C22⋊C8—42no—
266#97—42no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271C31#761211yescharacteristic, direct product factor, Sylow
272C24—21no—
273C24C22⋊C4×C3111yescharacteristic
274#41—48no—
275#41—48no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8M16×C3112yes—
280C2×D4—24no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4M16×C3112yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C22⋊C4×C3111yescharacteristic
305C22×C4C22⋊C4×C3111yescharacteristic
306C22×C4C22⋊C4×C3111yescharacteristic
307C22×C4C22×C4×C3111yescharacteristic
308C22×C4C2×D4×C3111yescharacteristic
309C22×C4C2×D4×C3111yescharacteristic
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—22no—
325C42—22no—
326C42—41no—
327C42—41no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23C31×#10012yes—
343C23C2×C22⋊C4×C3111yescharacteristic
344Q8—42no—
345Q8—42no—
346D4—42no—
347D4—42no—
348D4—48no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4C31×#9912yes—
378C2×C4C31×#10412yes—
379C2×C4C2×M16×C3111yescharacteristic
380C2×C4C2×M16×C3111yescharacteristic
381C2×C4C31×#13512yes—
382C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C31×#13711yescharacteristic
384C8—44no—
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C31×#40711yescharacteristic
401C22C2×C31×#9911yescharacteristic
402C22C2×C31×#10011yescharacteristic
403C22C31×#41611yescharacteristic
404C22C31×#41711yescharacteristic
405C22C2×C31×#10411yescharacteristic
406C22C31×#42111yescharacteristic
407C4—22no—
408C4—21no—
409C4—21no—
410C4—22no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—41no—
416C4—81no—
417C4—81no—
418C2—42no—
419C2—82no—
420C2C31×#146211yescharacteristic
421C2C31×#148811yescharacteristic
422C2C31×#148911yescharacteristic
423C2C31×#150111yescharacteristic
424C2C31×#151311yescharacteristic
425C2C31×#153011yescharacteristic
426C2C31×#155111yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×#41×C31C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1797C2semidirect
3. C31×#1749C2semidirect
4. C31×#1272C2semidirect
5. C31×#1270C2semidirect
115. #7612C31direct (N × H)
174. #1797C2×C31semidirect
175. #1749C2×C31semidirect
176. #1272C2×C31semidirect
177. #1270C2×C31semidirect
271. C31#7612direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#22972
C31×#17972
C31×#17492
C31×#12722
C31×#12702
#761231