Finite Group Databasebeta

← C31×#34976 #21491735

Subgroups — C31×#34976 #21491735

Order 7936.

2810 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41×C31C212yesmaximal, complemented
2C31×#2609C214yesmaximal, complemented
3C31×#2582C211yesmaximal, characteristic, complemented
4C31×#2538C212yesmaximal, complemented
5C31×#2537C211yesmaximal, characteristic, complemented
6C31×#2485C214yesmaximal, complemented
7C31×#2407C214yesmaximal, complemented
8C31×#2392C214yesmaximal, complemented
9C4×C31×C22⋊Q8C214yesmaximal, complemented
10C4×C31×#121C214yesmaximal, complemented
11C31×#2285C211yesmaximal, characteristic, complemented
12C22×#41×C31C2212yes—
13C2×C31×C22⋊Q8C2214yes—
14C2×C31×#121C2214yes—
15C4×#41×C31—216no—
16C2×C4×Q8×C31C2212yes—
17C2×C4×Q8×C31C2212yes—
18C2×C4×D4×C31C2214yes—
19C2×C4×D4×C31C2214yes—
20C2×C4×D4×C31C2212yes—
21C2×C4×D4×C31C2212yes—
22C2×C4×D4×C31C2214yes—
23C2×C31×#117C2214yes—
24C2×C31×#117C2212yes—
25C2×C31×#117C2212yes—
26C22×C42×C31C2214yes—
27C22×C42×C31C2211yescharacteristic
28C31×#398C2214yes—
29C31×#397C2214yes—
30C31×#396C2218yes—
31C31×#394C2214yes—
32C31×#394C2218yes—
33C31×#392C2214yes—
34C31×#391C2214yes—
35C31×#390C2214yes—
36C31×#389C2214yes—
37C31×#389C2214yes—
38C31×#388C2218yes—
39C31×#387C2218yes—
40C31×#386C2218yes—
41C31×#385C2218yes—
42C31×#384C2211yescharacteristic
43C31×#384C2211yescharacteristic
44C31×#382C2212yes—
45C31×#380C2214yes—
46C31×#380C2214yes—
47C31×#380C2214yes—
48C4×C4⋊C4×C31C2214yes—
49C4×C4⋊C4×C31C2214yes—
50C4×C22⋊C4×C31C2218yes—
51C2×#41×C31—28no—
52C2×#41×C31—28no—
53C2×#41×C31—28no—
54C22×Q8×C31C2312yes—
55C22×D4×C31C2314yes—
56C22×D4×C31C2314yes—
57C22×D4×C31C2312yes—
58C23×C4×C31C2314yes—
59C23×C4×C31C2314yes—
60C23×C4×C31C2314yes—
61C23×C4×C31C2312yes—
62C23×C4×C31C2311yescharacteristic
63C31×C22⋊Q8—216no—
64C31×#121—216no—
65C4×Q8×C31—28no—
66C4×Q8×C31—28no—
67C4×D4×C31—216no—
68C4×D4×C31—216no—
69C4×D4×C31—28no—
70C4×D4×C31—28no—
71C4×D4×C31—48no—
72C31×#117—28no—
73C31×#117—216no—
74C31×#117—44no—
75C2×C4⋊C4×C31C2314yes—
76C2×C4⋊C4×C31C2318yes—
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2312yes—
79C2×C4⋊C4×C31C2314yes—
80C2×C4⋊C4×C31C2314yes—
81C2×C4⋊C4×C31C2314yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2314yes—
84C2×C4⋊C4×C31C2312yes—
85C2×C22⋊C4×C31C2314yes—
86C2×C22⋊C4×C31C2318yes—
87C2×C22⋊C4×C31C2314yes—
88C2×C22⋊C4×C31C2314yes—
89C2×C22⋊C4×C31C2318yes—
90C2×C22⋊C4×C31C2318yes—
91C2×C22⋊C4×C31C2314yes—
92C2×C42×C31—216no—
93C2×C42×C31D418yescomplemented
94C2×C42×C31C2314yes—
95C2×C42×C31C2314yes—
96C2×C42×C31C2314yes—
97C2×C42×C31C2314yes—
98C2×C42×C31C2311yescharacteristic
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2312yes—
101C31×#95C2318yes—
102C31×#95C2314yes—
103C31×#95C2318yes—
104C31×#95C2314yes—
105C31×#95C2318yes—
106C24×C31C2411yescharacteristic
107C24×C31C2414yes—
108#41×C31—264no—
109C2×Q8×C31—24no—
110C2×Q8×C31—24no—
111C2×Q8×C31#4118yescomplemented
112C2×D4×C31—28no—
113C2×D4×C31—28no—
114C2×D4×C31—216no—
115C2×D4×C31—24no—
116C2×D4×C31—28no—
117C2×D4×C31—24no—
118C2×D4×C31—44no—
119C2×D4×C31#4118yescomplemented
120C22×C4×C31—28no—
121C22×C4×C31—28no—
122C22×C4×C31—28no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—28no—
128C22×C4×C31—28no—
129C22×C4×C31—28no—
130C22×C4×C31—28no—
131C22×C4×C31—28no—
132C22×C4×C31—28no—
133C22×C4×C31C2×D418yes—
134C22×C4×C31C2×D414yes—
135C22×C4×C31C2414yes—
136C22×C4×C31C2414yes—
137C22×C4×C31C2414yes—
138C22×C4×C31C2414yes—
139C22×C4×C31C2412yes—
140C22×C4×C31C2411yescharacteristic
141C22×C4×C31C2414yes—
142C22×C4×C31C2412yes—
143C22×C4×C31C2411yescharacteristic
144C4⋊C4×C31—28no—
145C4⋊C4×C31—28no—
146C4⋊C4×C31—24no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—28no—
149C4⋊C4×C31—28no—
150C4⋊C4×C31—216no—
151C4⋊C4×C31—42no—
152C4⋊C4×C31—42no—
153C4⋊C4×C31—44no—
154C22⋊C4×C31—28no—
155C22⋊C4×C31—216no—
156C22⋊C4×C31—28no—
157C22⋊C4×C31—28no—
158C22⋊C4×C31—216no—
159C22⋊C4×C31—44no—
160C22⋊C4×C31—48no—
161C42×C31—28no—
162C42×C31—28no—
163C42×C31—28no—
164C42×C31—28no—
165C42×C31C2×D418yes—
166C42×C31C2×D414yes—
167C42×C31C2×D414yes—
168#34976C3111yesmaximal, characteristic, direct product factor, Sylow
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31—21no—
176C23×C31—24no—
177C23×C31—21no—
178C23×C31—24no—
179C23×C31C22×D412yes—
180C23×C31C2×#4118yes—
181C23×C31C2511yescharacteristic
182Q8×C31—216no—
183D4×C31—232no—
184D4×C31—216no—
185D4×C31—416no—
186C2×C4×C31—232no—
187C2×C4×C31—24no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—24no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—216no—
194C2×C4×C31—28no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—28no—
199C2×C4×C31—28no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—416no—
206C2×C4×C31—416no—
207C2×C4×C31C22×D414yes—
208C2×C4×C31C22×D412yes—
209C2×C4×C31C22×D414yes—
210C2×C4×C31C22×D412yes—
211C2×C4×C31C2×#4118yes—
212C2×C4×C31C2×#4114yes—
213C2×C4×C31C2×#4114yes—
214C2×C4×C31H(5,𝔽2)12yes—
215C2×C4×C312−1+412yes—
216C2×C4×#41C2×C3112yescomplemented
217#2609C2×C3114yescomplemented
218#2582C2×C3111yescharacteristic, complemented
219#2538C2×C3112yescomplemented
220#2537C2×C3111yescharacteristic, complemented
221#2485C2×C3114yescomplemented
222#2407C2×C3114yescomplemented
223#2392C2×C3114yescomplemented
224C4×C22⋊Q8C2×C3114yescomplemented
225C4×#121C2×C3114yescomplemented
226#2285C2×C3111yescharacteristic, complemented
227C22×C31—24no—
228C22×C31—28no—
229C22×C31—22no—
230C22×C31—28no—
231C22×C31—28no—
232C22×C31—44no—
233C22×C31—44no—
234C22×C31—41no—
235C22×C31—42no—
236C22×C31—44no—
237C22×C31—44no—
238C22×C31—41no—
239C22×C31C23×D411yescharacteristic
240C22×C31C22×#4112yes—
241C22×C31C22×#4112yes—
242C22×C31C2×H(5,𝔽2)11yescharacteristic
243C22×C31C2×2−1+411yescharacteristic
244C4×C31—24no—
245C4×C31—28no—
246C4×C31—22no—
247C4×C31—44no—
248C4×C31—44no—
249C4×C31—44no—
250C4×C31—44no—
251C4×C31#52518yes—
252C4×C31#53714yes—
253C22×#41C22×C3112yes—
254C2×C22⋊Q8C22×C3114yes—
255C2×#121C22×C3114yes—
256C4×#41—216no—
257C2×C4×Q8C22×C3112yes—
258C2×C4×Q8C22×C3112yes—
259C2×C4×D4C22×C3114yes—
260C2×C4×D4C22×C3114yes—
261C2×C4×D4C22×C3112yes—
262C2×C4×D4C22×C3112yes—
263C2×C4×D4C22×C3114yes—
264C2×#117C22×C3114yes—
265C2×#117C22×C3112yes—
266C2×#117C22×C3112yes—
267C22×C42C22×C3114yes—
268C22×C42C22×C3111yescharacteristic
269#398C22×C3114yes—
270#397C22×C3114yes—
271#396C22×C3118yes—
272#394C22×C3114yes—
273#394C22×C3118yes—
274#392C22×C3114yes—
275#391C22×C3114yes—
276#390C22×C3114yes—
277#389C22×C3114yes—
278#389C22×C3114yes—
279#388C22×C3118yes—
280#387C22×C3118yes—
281#386C22×C3118yes—
282#385C22×C3118yes—
283#384C22×C3111yescharacteristic
284#384C22×C3111yescharacteristic
285#382C22×C3112yes—
286#380C22×C3114yes—
287#380C22×C3114yes—
288#380C22×C3114yes—
289C4×C4⋊C4C22×C3114yes—
290C4×C4⋊C4C22×C3114yes—
291C4×C22⋊C4C22×C3118yes—
292C2×C31—42no—
293C2×C31—48no—
294C2×C31C2×#52512yes—
295C2×C31C2×#53711yescharacteristic
296C2×C31#348111yescharacteristic
297C2×C31#348212yes—
298C2×C31#348311yescharacteristic
299C2×#41—28no—
300C2×#41—28no—
301C2×#41—28no—
302C22×Q8C23×C3112yes—
303C22×D4C23×C3114yes—
304C22×D4C23×C3114yes—
305C22×D4C23×C3112yes—
306C23×C4C23×C3114yes—
307C23×C4C23×C3114yes—
308C23×C4C23×C3114yes—
309C23×C4C23×C3111yescharacteristic
310C23×C4C23×C3112yes—
311C22⋊Q8—216no—
312#121—216no—
313C4×Q8—28no—
314C4×Q8—28no—
315C4×D4—216no—
316C4×D4—216no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—48no—
320#117—28no—
321#117—216no—
322#117—44no—
323C2×C4⋊C4C23×C3112yes—
324C2×C4⋊C4C23×C3112yes—
325C2×C4⋊C4C23×C3114yes—
326C2×C4⋊C4C23×C3114yes—
327C2×C4⋊C4C23×C3114yes—
328C2×C4⋊C4C23×C3114yes—
329C2×C4⋊C4C23×C3118yes—
330C2×C4⋊C4C23×C3112yes—
331C2×C4⋊C4C23×C3112yes—
332C2×C4⋊C4C23×C3114yes—
333C2×C22⋊C4C23×C3114yes—
334C2×C22⋊C4C23×C3114yes—
335C2×C22⋊C4C23×C3114yes—
336C2×C22⋊C4C23×C3118yes—
337C2×C22⋊C4C23×C3118yes—
338C2×C22⋊C4C23×C3114yes—
339C2×C22⋊C4C23×C3118yes—
340C2×C42—216no—
341C2×C42D4×C3118yescomplemented
342C2×C42C23×C3114yes—
343C2×C42C23×C3114yes—
344C2×C42C23×C3114yes—
345C2×C42C23×C3114yes—
346C2×C42C23×C3111yescharacteristic
347C2×C42C23×C3111yescharacteristic
348C2×C42C23×C3112yes—
349#95C23×C3118yes—
350#95C23×C3118yes—
351#95C23×C3114yes—
352#95C23×C3118yes—
353#95C23×C3114yes—
354C31#3497611yescharacteristic, direct product factor, Sylow
355C24C24×C3111yescharacteristic
356C24C24×C3114yes—
357#41—264no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8#41×C3118yescomplemented
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—24no—
367C2×D4—44no—
368C2×D4#41×C3118yescomplemented
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4C2×D4×C3118yes—
383C22×C4C2×D4×C3114yes—
384C22×C4C24×C3114yes—
385C22×C4C24×C3114yes—
386C22×C4C24×C3114yes—
387C22×C4C24×C3114yes—
388C22×C4C24×C3112yes—
389C22×C4C24×C3111yescharacteristic
390C22×C4C24×C3114yes—
391C22×C4C24×C3112yes—
392C22×C4C24×C3111yescharacteristic
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—216no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C22⋊C4—28no—
404C22⋊C4—216no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—216no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C42—28no—
411C42—28no—
412C42—28no—
413C42—28no—
414C42C2×D4×C3118yes—
415C42C2×D4×C3114yes—
416C42C2×D4×C3114yes—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—21no—
424C23—24no—
425C23—21no—
426C23—24no—
427C23C22×D4×C3112yes—
428C23C2×#41×C3118yes—
429C23C25×C3111yescharacteristic
430Q8—216no—
431D4—232no—
432D4—216no—
433D4—416no—
434C2×C4—232no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—216no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—416no—
454C2×C4—416no—
455C2×C4C22×D4×C3114yes—
456C2×C4C22×D4×C3112yes—
457C2×C4C22×D4×C3114yes—
458C2×C4C22×D4×C3112yes—
459C2×C4C2×#41×C3118yes—
460C2×C4C2×#41×C3114yes—
461C2×C4C2×#41×C3114yes—
462C2×C4C31×H(5,𝔽2)12yes—
463C2×C4C31×2−1+412yes—
464C22—24no—
465C22—28no—
466C22—22no—
467C22—28no—
468C22—28no—
469C22—44no—
470C22—44no—
471C22—41no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—41no—
476C22C23×D4×C3111yescharacteristic
477C22C22×#41×C3112yes—
478C22C22×#41×C3112yes—
479C22C2×C31×H(5,𝔽2)11yescharacteristic
480C22C2×C31×2−1+411yescharacteristic
481C4—24no—
482C4—28no—
483C4—22no—
484C4—44no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4C31×#52518yes—
489C4C31×#53714yes—
490C2—42no—
491C2—48no—
492C2C2×C31×#52512yes—
493C2C2×C31×#53711yescharacteristic
494C2C31×#348111yescharacteristic
495C2C31×#348212yes—
496C2C31×#348311yescharacteristic
subgroupG/H
centerC22×C4×C31C24
commutator, Frattini, cosocleC23C25×C31
socleC23×C31C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×#41×C31C2semidirect
2. C31×#2609C2semidirect
3. C31×#2582C2semidirect
4. C31×#2538C2semidirect
5. C31×#2537C2semidirect
6. C31×#2485C2semidirect
7. C31×#2407C2semidirect
8. C31×#2392C2semidirect
9. C4×C31×C22⋊Q8C2semidirect
10. C4×C31×#121C2semidirect
11. C31×#2285C2semidirect
93. C2×C42×C31D4semidirect
111. C2×Q8×C31#41semidirect
119. C2×D4×C31#41semidirect
168. #34976C31direct (N × H)
216. C2×C4×#41C2×C31semidirect
217. #2609C2×C31semidirect
218. #2582C2×C31semidirect
219. #2538C2×C31semidirect
220. #2537C2×C31semidirect
221. #2485C2×C31semidirect
222. #2407C2×C31semidirect
223. #2392C2×C31semidirect
224. C4×C22⋊Q8C2×C31semidirect
225. C4×#121C2×C31semidirect
226. #2285C2×C31semidirect
341. C2×C42D4×C31semidirect
354. C31#34976direct (N × H)
360. C2×Q8#41×C31semidirect
368. C2×D4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#41×C312
C31×#26092
C31×#25822
C31×#25382
C31×#25372
C31×#24852
C31×#24072
C31×#23922
C4×C31×C22⋊Q82
C4×C31×#1212
C31×#22852
#3497631