Finite Group Databasebeta

← C31×#18399 #21475158

Subgroups — C31×#18399 #21475158

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic
2C31×#3230C211yesmaximal, characteristic, complemented
3C31×#3227C211yesmaximal, characteristic
4C31×#2568C211yesmaximal, characteristic, complemented
5C31×#1615C218yesmaximal, complemented
6C31×#1503C211yesmaximal, characteristic, complemented
7C31×#1502C211yesmaximal, characteristic
8C31×#1461C211yesmaximal, characteristic, complemented
9C31×#559—21no—
10C31×#557—21no—
11C31×#556—21no—
12C31×#556—22no—
13C31×#554—21no—
14C31×#552—22no—
15C31×#539C2214yes—
16C31×#536—21no—
17C31×#533C2211yescharacteristic
18C31×#532C2211yescharacteristic
19C2×C31×C4⋊Q8C2211yescharacteristic
20C31×#529—22no—
21C31×#520—21no—
22C2×C4×Q8×C31C2211yescharacteristic
23C31×#500—21no—
24C31×#498—21no—
25C31×#487—21no—
26C31×#486—21no—
27C31×#484C2214yes—
28C31×#481C2214yes—
29C31×#477—22no—
30C31×#476—22no—
31C31×#431C2211yescharacteristic
32C31×#431C2211yescharacteristic
33C31×#400C2214yes—
34C31×#391—21no—
35C31×#391—22no—
36C31×#382—21no—
37C4×C4⋊C4×C31C2211yescharacteristic
38C31×#328C2214yes—
39C31×#327C2214yes—
40C31×#324C2214yes—
41C2×#41×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31D411yescharacteristic
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8D411yescharacteristic
50C31×C4⋊Q8D411yescharacteristic
51C31×C4⋊Q8D411yescharacteristic
52C31×#127D411yescharacteristic
53C31×#126—24no—
54C31×#126—24no—
55C31×#125—22no—
56C31×#125—24no—
57C31×#125—21no—
58C31×#124—22no—
59C31×#124—22no—
60C31×#124—21no—
61C31×#123—24no—
62C31×#123—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—24no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8C2312yes—
68C31×#121—21no—
69C31×#121C2312yes—
70C4×Q8×C31—22no—
71C4×Q8×C31—22no—
72C4×Q8×C31—21no—
73C4×Q8×C31—21no—
74C4×Q8×C31—21no—
75C4×Q8×C31—21no—
76C4×Q8×C31—21no—
77C4×Q8×C31D412yes—
78C4×D4×C31—22no—
79C4×D4×C31—22no—
80C4×D4×C31—24no—
81C4×D4×C31—21no—
82C4×D4×C31D412yes—
83C31×#117—22no—
84C31×#117—24no—
85C31×#117—21no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2312yes—
92C2×C42×C31—21no—
93C2×C42×C31D412yes—
94C2×C42×C31C2311yescharacteristic
95C31×#107—24no—
96C31×#106—24no—
97C31×C4⋊C8—21no—
98C31×C4⋊C8—21no—
99C31×#103—24no—
100C31×#102—24no—
101C31×C22⋊C8C2312yes—
102C31×C22⋊C8C2312yes—
103C4×C8×C31—21no—
104C4×C8×C31—21no—
105C31×#95—24no—
106C31×#95—24no—
107C31×#95C2312yes—
108#41×C31—42no—
109#41×C31—42no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—44no—
117C2×Q8×C31—41no—
118C2×Q8×C31—41no—
119C2×Q8×C31C2×D412yes—
120C2×D4×C31—22no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31C2×D412yes—
124C22×C4×C31—24no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31C2×D412yes—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2411yescharacteristic
133C2×C8×C31—22no—
134C2×C8×C31—22no—
135C4⋊C4×C31—24no—
136C4⋊C4×C31—24no—
137C4⋊C4×C31—24no—
138C4⋊C4×C31—21no—
139C4⋊C4×C31—21no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—42no—
151C4⋊C4×C31—41no—
152C4⋊C4×C31—41no—
153C4⋊C4×C31C2×D412yes—
154C4⋊C4×C31C2×D412yes—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—24no—
157C22⋊C4×C31—24no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—22no—
160C22⋊C4×C31—22no—
161C42×C31—24no—
162C42×C31—22no—
163C42×C31—22no—
164C42×C31—22no—
165C42×C31—42no—
166C42×C31C2×D411yescharacteristic
167C42×C31C2×D411yescharacteristic
168C42×C31C2×D411yescharacteristic
169C42×C31C2×D411yescharacteristic
170#18399C3111yesmaximal, characteristic, direct product factor, Sylow
171C23×C31—22no—
172C23×C31C22×D411yescharacteristic
173Q8×C31—42no—
174Q8×C31—42no—
175Q8×C31—42no—
176Q8×C31—42no—
177Q8×C31—42no—
178Q8×C31—82no—
179Q8×C31—82no—
180D4×C31—42no—
181D4×C31—42no—
182D4×C31—42no—
183D4×C31—42no—
184D4×C31—42no—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—41no—
194C2×C4×C31—44no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—44no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—84no—
202C2×C4×C31#12011yescharacteristic
203C2×C4×C31#12011yescharacteristic
204C2×C4×C31#12011yescharacteristic
205C2×C4×C31#12011yescharacteristic
206C2×C4×C31#13512yes—
207C2×C4×C31#13712yes—
208C2×C4×C31C22×D411yescharacteristic
209C2×C4×C31C22×D411yescharacteristic
210C8×C31—42no—
211C8×C31—42no—
212#3507C2×C3111yescharacteristic
213#3230C2×C3111yescharacteristic, complemented
214#3227C2×C3111yescharacteristic
215#2568C2×C3111yescharacteristic, complemented
216#1615C2×C3118yescomplemented
217#1503C2×C3111yescharacteristic, complemented
218#1502C2×C3111yescharacteristic
219#1461C2×C3111yescharacteristic, complemented
220C22×C31—21no—
221C22×C31—21no—
222C22×C31—42no—
223C22×C31—42no—
224C22×C31—42no—
225C22×C31C2×#12011yescharacteristic
226C22×C31C2×#13511yescharacteristic
227C22×C31C2×#13711yescharacteristic
228C4×C31—21no—
229C4×C31—21no—
230C4×C31—21no—
231C4×C31—42no—
232C4×C31—42no—
233C4×C31—42no—
234C4×C31—42no—
235C4×C31—42no—
236C4×C31—82no—
237C4×C31—82no—
238C4×C31#45212yes—
239C4×C31#45212yes—
240C4×C31#45412yes—
241#559—21no—
242#557—21no—
243#556—21no—
244#556—22no—
245#554—21no—
246#552—22no—
247#539C22×C3114yes—
248#536—21no—
249#533C22×C3111yescharacteristic
250#532C22×C3111yescharacteristic
251C2×C4⋊Q8C22×C3111yescharacteristic
252#529—22no—
253#520—21no—
254C2×C4×Q8C22×C3111yescharacteristic
255#500—21no—
256#498—21no—
257#487—21no—
258#486—21no—
259#484C22×C3114yes—
260#481C22×C3114yes—
261#477—22no—
262#476—22no—
263#431C22×C3111yescharacteristic
264#431C22×C3111yescharacteristic
265#400C22×C3114yes—
266#391—21no—
267#391—22no—
268#382—21no—
269C4×C4⋊C4C22×C3111yescharacteristic
270#328C22×C3114yes—
271#327C22×C3114yes—
272#324C22×C3114yes—
273C2×C31—21no—
274C2×C31—21no—
275C2×C31—82no—
276C2×C31C2×#45211yescharacteristic
277C2×C31C2×#45411yescharacteristic
278C2×C31#302811yescharacteristic
279C2×#41—21no—
280C22×Q8—21no—
281C22×Q8D4×C3111yescharacteristic
282C4⋊Q8—22no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8D4×C3111yescharacteristic
288C4⋊Q8D4×C3111yescharacteristic
289C4⋊Q8D4×C3111yescharacteristic
290#127D4×C3111yescharacteristic
291#126—24no—
292#126—24no—
293#125—22no—
294#125—24no—
295#125—21no—
296#124—22no—
297#124—22no—
298#124—21no—
299#123—24no—
300#123—24no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—24no—
304C22⋊Q8—21no—
305C22⋊Q8C23×C3112yes—
306#121—21no—
307#121C23×C3112yes—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8D4×C3112yes—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—24no—
319C4×D4—21no—
320C4×D4D4×C3112yes—
321#117—22no—
322#117—24no—
323#117—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4D4×C3111yescharacteristic
328C2×C4⋊C4C23×C3112yes—
329C2×C4⋊C4C23×C3112yes—
330C2×C42—21no—
331C2×C42D4×C3112yes—
332C2×C42C23×C3111yescharacteristic
333#107—24no—
334#106—24no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337#103—24no—
338#102—24no—
339C22⋊C8C23×C3112yes—
340C22⋊C8C23×C3112yes—
341C4×C8—21no—
342C4×C8—21no—
343#95—24no—
344#95—24no—
345#95C23×C3112yes—
346C31#1839911yescharacteristic, direct product factor, Sylow
347#41—42no—
348#41—42no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—44no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8C2×D4×C3112yes—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4C2×D4×C3112yes—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4C2×D4×C3112yes—
368C22×C4C2×D4×C3111yescharacteristic
369C22×C4C2×D4×C3112yes—
370C22×C4C2×D4×C3111yescharacteristic
371C22×C4C24×C3111yescharacteristic
372C2×C8—22no—
373C2×C8—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4C2×D4×C3112yes—
393C4⋊C4C2×D4×C3112yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C42—24no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—42no—
405C42C2×D4×C3111yescharacteristic
406C42C2×D4×C3111yescharacteristic
407C42C2×D4×C3111yescharacteristic
408C42C2×D4×C3111yescharacteristic
409C23—22no—
410C23C22×D4×C3111yescharacteristic
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—82no—
417Q8—82no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—84no—
440C2×C4C31×#12011yescharacteristic
441C2×C4C31×#12011yescharacteristic
442C2×C4C31×#12011yescharacteristic
443C2×C4C31×#12011yescharacteristic
444C2×C4C31×#13512yes—
445C2×C4C31×#13712yes—
446C2×C4C22×D4×C3111yescharacteristic
447C2×C4C22×D4×C3111yescharacteristic
448C8—42no—
449C8—42no—
450C22—21no—
451C22—21no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22C2×C31×#12011yescharacteristic
456C22C2×C31×#13511yescharacteristic
457C22C2×C31×#13711yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—82no—
468C4C31×#45212yes—
469C4C31×#45212yes—
470C4C31×#45412yes—
471C2—21no—
472C2—21no—
473C2—82no—
474C2C2×C31×#45211yescharacteristic
475C2C2×C31×#45411yescharacteristic
476C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3230C2semidirect
4. C31×#2568C2semidirect
5. C31×#1615C2semidirect
6. C31×#1503C2semidirect
8. C31×#1461C2semidirect
170. #18399C31direct (N × H)
213. #3230C2×C31semidirect
215. #2568C2×C31semidirect
216. #1615C2×C31semidirect
217. #1503C2×C31semidirect
219. #1461C2×C31semidirect
346. C31#18399direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#32302
C31×#32272
C31×#25682
C31×#16152
C31×#15032
C31×#15022
C31×#14612
#1839931