Finite Group Databasebeta

← C31×#12696 #21469455

Subgroups — C31×#12696 #21469455

Order 7936.

1214 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C31×#535C211yesmaximal, characteristic
2C2×C31×#354C212yesmaximal, complemented
3C2×C31×#353C212yesmaximal, complemented
4C31×#1781C212yesmaximal, complemented
5C22×#41×C31C411yescharacteristic, complemented
6C23×D4×C31C2211yescharacteristic
7C31×#535—24no—
8C31×#535—24no—
9C2×C31×#127C411yescharacteristic, complemented
10C2×C31×#127C2211yescharacteristic
11C2×C31×#124C411yescharacteristic, complemented
12C2×C31×#124C2211yescharacteristic
13C2×C31×#121—22no—
14C2×C31×#120—22no—
15C2×C31×#117C411yescharacteristic, complemented
16C2×C31×#99—22no—
17C2×C31×#99C2214yes—
18C31×#379—22no—
19C31×#354—24no—
20C31×#353—24no—
21C25×C31D412yes—
22C2×#41×C31—22no—
23C2×#41×C31—22no—
24C2×#41×C31—42no—
25C2×#41×C31—42no—
26C22×Q8×C31C2×C411yescharacteristic
27C22×D4×C31—21no—
28C22×D4×C31—22no—
29C22×D4×C31—22no—
30C22×D4×C31—22no—
31C22×D4×C31—22no—
32C22×D4×C31—42no—
33C22×D4×C31—42no—
34C22×D4×C31C2×C411yescharacteristic
35C22×D4×C31C2×C411yescharacteristic
36C22×D4×C31D412yes—
37C22×D4×C31C2311yescharacteristic
38C23×C4×C31—21no—
39C23×C4×C31C2×C411yescharacteristic
40C31×#127—22no—
41C31×#127—22no—
42C31×#127—22no—
43C31×#127D414yes—
44C31×#124—22no—
45C31×#124—22no—
46C31×#124—22no—
47C31×#124D414yes—
48C31×#121—216no—
49C31×#120—216no—
50C31×#117—22no—
51C31×#117—22no—
52C2×C4⋊C4×C31—21no—
53C2×C22⋊C4×C31—22no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—22no—
56C2×C22⋊C4×C31—24no—
57C2×C22⋊C4×C31—44no—
58C2×C42×C31C2×C411yescharacteristic
59C2×C42×C31C2×C411yescharacteristic
60C31×#99—24no—
61C31×#99—24no—
62C31×#99—44no—
63C24×C31—21no—
64C24×C31—24no—
65C24×C31—24no—
66C24×C31—42no—
67C24×C31—84no—
68C24×C31C22⋊C412yes—
69C24×C31C22⋊C411yescharacteristic
70C24×C31C2×D412yes—
71#41×C31—48no—
72#41×C31—48no—
73C2×Q8×C31—21no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×Q8×C31C22⋊C412yescomplemented
77C2×D4×C31—24no—
78C2×D4×C31—24no—
79C2×D4×C31—28no—
80C2×D4×C31—28no—
81C2×D4×C31—24no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—42no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—48no—
91C2×D4×C31—48no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C2×D4×C31—42no—
95C2×D4×C31—42no—
96C2×D4×C31—84no—
97C2×D4×C31—84no—
98C2×D4×C31C22⋊C412yescomplemented
99C2×D4×C31C2×D412yes—
100C2×D4×C31C2×D412yes—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—24no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—42no—
110C22×C4×C31—42no—
111C22×C4×C31—44no—
112C22×C4×C31C22⋊C411yescharacteristic
113C22×C4×C31C22×C411yescharacteristic
114C4⋊C4×C31—24no—
115C22⋊C4×C31—24no—
116C22⋊C4×C31—28no—
117C22⋊C4×C31—28no—
118C22⋊C4×C31—44no—
119C22⋊C4×C31—44no—
120C22⋊C4×C31—88no—
121C42×C31—21no—
122C42×C31—21no—
123C42×C31C22⋊C412yescomplemented
124C42×C31C22⋊C412yescomplemented
125#12696C3111yesmaximal, characteristic, direct product factor, Sylow
126C23×C31—21no—
127C23×C31—21no—
128C23×C31—22no—
129C23×C31—22no—
130C23×C31—22no—
131C23×C31—22no—
132C23×C31—22no—
133C23×C31—22no—
134C23×C31—42no—
135C23×C31—48no—
136C23×C31—44no—
137C23×C31—44no—
138C23×C31—48no—
139C23×C31—44no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—84no—
143C23×C31—88no—
144C23×C31—88no—
145C23×C31#9912yes—
146C23×C31C2×C22⋊C411yescharacteristic
147C23×C31#12014yes—
148Q8×C31—42no—
149Q8×C31—42no—
150D4×C31—42no—
151D4×C31—42no—
152D4×C31—48no—
153D4×C31—42no—
154D4×C31—42no—
155D4×C31—42no—
156D4×C31—42no—
157D4×C31—48no—
158D4×C31—88no—
159D4×C31—88no—
160C2×C4×C31—24no—
161C2×C4×C31—24no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31—21no—
166C2×C4×C31—22no—
167C2×C4×C31—22no—
168C2×C4×C31—22no—
169C2×C4×C31—48no—
170C2×C4×C31—42no—
171C2×C4×C31—42no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—84no—
175C2×C4×C31—88no—
176C2×C4×C31#9912yes—
177C2×C4×C31C2×C22⋊C411yescharacteristic
178C2×C4×C31C2×C22⋊C411yescharacteristic
179C2×#535C2×C3111yescharacteristic
180C2×#354C2×C3112yescomplemented
181C2×#353C2×C3112yescomplemented
182#1781C2×C3112yescomplemented
183C22×C31—22no—
184C22×C31—24no—
185C22×C31—22no—
186C22×C31—24no—
187C22×C31—44no—
188C22×C31—42no—
189C22×C31—42no—
190C22×C31—44no—
191C22×C31—44no—
192C22×C31—44no—
193C22×C31—44no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—82no—
197C22×C31—82no—
198C22×C31—82no—
199C22×C31—816no—
200C22×C31—82no—
201C22×C31#37911yescharacteristic
202C22×C31C2×#9911yescharacteristic
203C22×C31C2×#9911yescharacteristic
204C4×C31—22no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—42no—
208C4×C31—42no—
209C4×C31—42no—
210C4×C31—164no—
211C22×#41C4×C3111yescharacteristic, complemented
212C23×D4C22×C3111yescharacteristic
213#535—24no—
214#535—24no—
215C2×#127C4×C3111yescharacteristic, complemented
216C2×#127C22×C3111yescharacteristic
217C2×#124C4×C3111yescharacteristic, complemented
218C2×#124C22×C3111yescharacteristic
219C2×#121—22no—
220C2×#120—22no—
221C2×#117C4×C3111yescharacteristic, complemented
222C2×#99—22no—
223C2×#99C22×C3114yes—
224#379—22no—
225#354—24no—
226#353—24no—
227C2×C31—21no—
228C2×C31—21no—
229C2×C31—44no—
230C2×C31—42no—
231C2×C31—82no—
232C2×C31—82no—
233C2×C31#178111yescharacteristic
234C2×C31#212712yes—
235C25D4×C3112yes—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C22×Q8C2×C4×C3111yescharacteristic
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4C2×C4×C3111yescharacteristic
249C22×D4C2×C4×C3111yescharacteristic
250C22×D4D4×C3112yes—
251C22×D4C23×C3111yescharacteristic
252C23×C4—21no—
253C23×C4C2×C4×C3111yescharacteristic
254#127—22no—
255#127—22no—
256#127—22no—
257#127D4×C3114yes—
258#124—22no—
259#124—22no—
260#124—22no—
261#124D4×C3114yes—
262#121—216no—
263#120—216no—
264#117—22no—
265#117—22no—
266C2×C4⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C42C2×C4×C3111yescharacteristic
273C2×C42C2×C4×C3111yescharacteristic
274#99—24no—
275#99—24no—
276#99—44no—
277C31#1269611yescharacteristic, direct product factor, Sylow
278C24—21no—
279C24—24no—
280C24—24no—
281C24—42no—
282C24—84no—
283C24C22⋊C4×C3112yes—
284C24C22⋊C4×C3111yescharacteristic
285C24C2×D4×C3112yes—
286#41—48no—
287#41—48no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C22⋊C4×C3112yescomplemented
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4C22⋊C4×C3112yescomplemented
314C2×D4C2×D4×C3112yes—
315C2×D4C2×D4×C3112yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4C22⋊C4×C3111yescharacteristic
328C22×C4C22×C4×C3111yescharacteristic
329C4⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—88no—
336C42—21no—
337C42—21no—
338C42C22⋊C4×C3112yescomplemented
339C42C22⋊C4×C3112yescomplemented
340C23—21no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—48no—
350C23—44no—
351C23—44no—
352C23—48no—
353C23—44no—
354C23—41no—
355C23—41no—
356C23—84no—
357C23—88no—
358C23—88no—
359C23C31×#9912yes—
360C23C2×C22⋊C4×C3111yescharacteristic
361C23C31×#12014yes—
362Q8—42no—
363Q8—42no—
364D4—42no—
365D4—42no—
366D4—48no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—48no—
372D4—88no—
373D4—88no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—84no—
389C2×C4—88no—
390C2×C4C31×#9912yes—
391C2×C4C2×C22⋊C4×C3111yescharacteristic
392C2×C4C2×C22⋊C4×C3111yescharacteristic
393C22—22no—
394C22—24no—
395C22—22no—
396C22—24no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—816no—
410C22—82no—
411C22C31×#37911yescharacteristic
412C22C2×C31×#9911yescharacteristic
413C22C2×C31×#9911yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—164no—
421C2—21no—
422C2—21no—
423C2—44no—
424C2—42no—
425C2—82no—
426C2—82no—
427C2C31×#178111yescharacteristic
428C2C31×#212712yes—
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×#535C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C23×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#354C2semidirect
3. C2×C31×#353C2semidirect
4. C31×#1781C2semidirect
5. C22×#41×C31C4semidirect
9. C2×C31×#127C4semidirect
11. C2×C31×#124C4semidirect
15. C2×C31×#117C4semidirect
76. C2×Q8×C31C22⋊C4semidirect
98. C2×D4×C31C22⋊C4semidirect
123. C42×C31C22⋊C4semidirect
124. C42×C31C22⋊C4semidirect
125. #12696C31direct (N × H)
180. C2×#354C2×C31semidirect
181. C2×#353C2×C31semidirect
182. #1781C2×C31semidirect
211. C22×#41C4×C31semidirect
215. C2×#127C4×C31semidirect
217. C2×#124C4×C31semidirect
221. C2×#117C4×C31semidirect
277. C31#12696direct (N × H)
291. C2×Q8C22⋊C4×C31semidirect
313. C2×D4C22⋊C4×C31semidirect
338. C42C22⋊C4×C31semidirect
339. C42C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5352
C2×C31×#3542
C2×C31×#3532
C31×#17812
#1269631