Finite Group Databasebeta

← C31×#8458 #21465217

Subgroups — C31×#8458 #21465217

Order 7936.

766 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#1955C211yesmaximal, characteristic, complemented
3C31×#1552C211yesmaximal, characteristic, complemented
4C31×#1551C211yesmaximal, characteristic, complemented
5C31×#1531C211yesmaximal, characteristic, complemented
6C31×#1530C211yesmaximal, characteristic, complemented
7C31×#1466C211yesmaximal, characteristic, complemented
8C31×#559—22no—
9C31×#556—21no—
10C31×#555—21no—
11C31×#553—22no—
12C31×#550—21no—
13C31×#547—21no—
14C31×#538—22no—
15C31×#535C412yes—
16C31×#533C2211yescharacteristic
17C31×#532C2211yescharacteristic
18C31×#525—22no—
19C31×#520C412yes—
20C2×C4×D4×C31C2211yescharacteristic
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×C22⋊C8—22no—
25C2×C31×#97C2211yescharacteristic
26C31×#333—21no—
27C31×#332—21no—
28C31×#332—21no—
29C31×#331—21no—
30C31×#330—22no—
31C31×#329—22no—
32C31×#323—22no—
33C2×#41×C31—22no—
34C2×#41×C31C2×C412yes—
35C22×D4×C31C2×C411yescharacteristic
36C23×C4×C31D412yes—
37C22×C8×C31—21no—
38C22×C8×C31—21no—
39C31×C4⋊Q8D411yescharacteristic
40C31×C4⋊Q8D411yescharacteristic
41C31×C4⋊Q8D411yescharacteristic
42C31×#127—22no—
43C31×#127D411yescharacteristic
44C31×#126—24no—
45C31×#125D412yes—
46C31×#124—24no—
47C31×#124—22no—
48C31×#124D412yes—
49C31×#123—24no—
50C31×#123—24no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8—21no—
54C31×#121—22no—
55C31×#121—24no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#121—22no—
59C31×#121—22no—
60C31×#120—24no—
61C4×Q8×C31—22no—
62C4×Q8×C31—21no—
63C4×Q8×C31—21no—
64C4×D4×C31—24no—
65C4×D4×C31—22no—
66C4×D4×C31—24no—
67C4×D4×C31—21no—
68C4×D4×C31—21no—
69C4×D4×C31—21no—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C4×D4×C31—21no—
73C31×#117—22no—
74C31×#117—22no—
75C31×#117C2×C412yes—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31C2×C411yescharacteristic
78C2×C22⋊C4×C31D412yes—
79C2×C42×C31C2311yescharacteristic
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8—21no—
84C31×C22⋊C8—21no—
85C31×C22⋊C8—21no—
86C31×C22⋊C8—44no—
87C31×#97—21no—
88C31×#97—21no—
89C31×#97—21no—
90C31×#97—21no—
91C24×C31C22⋊C412yes—
92#41×C31—44no—
93#41×C31—82no—
94#41×C31—82no—
95C2×Q8×C31—21no—
96C2×Q8×C31—21no—
97C2×Q8×C31C22⋊C412yes—
98C2×D4×C31—22no—
99C2×D4×C31—24no—
100C2×D4×C31—24no—
101C2×D4×C31—21no—
102C2×D4×C31—21no—
103C2×D4×C31—21no—
104C2×D4×C31—21no—
105C2×D4×C31—42no—
106C2×D4×C31C22⋊C412yes—
107C22×C4×C31—24no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—42no—
112C22×C4×C31—42no—
113C22×C4×C31C22⋊C412yes—
114C22×C4×C31C22×C411yescharacteristic
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117C2×C8×C31—22no—
118C2×C8×C31—41no—
119C2×C8×C31—41no—
120C2×C8×C31—42no—
121C2×C8×C31—41no—
122C2×C8×C31—41no—
123C2×C8×C31—41no—
124C4⋊C4×C31—24no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31C22⋊C412yes—
134C4⋊C4×C31C22⋊C412yes—
135C22⋊C4×C31—24no—
136C22⋊C4×C31—24no—
137C22⋊C4×C31—24no—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—22no—
142C22⋊C4×C31—22no—
143C42×C31—24no—
144C42×C31—22no—
145C42×C31C2×D411yescharacteristic
146C42×C31C2×D411yescharacteristic
147C42×C31C2×D411yescharacteristic
148C42×C31C2×D411yescharacteristic
149#8458C3111yesmaximal, characteristic, direct product factor, Sylow
150C23×C31—22no—
151C23×C31—22no—
152C23×C31—22no—
153C23×C31—22no—
154C23×C31—42no—
155C23×C31—42no—
156C23×C31C2×C22⋊C411yescharacteristic
157Q8×C31—42no—
158Q8×C31—41no—
159Q8×C31—41no—
160Q8×C31—81no—
161D4×C31—42no—
162D4×C31—42no—
163D4×C31—42no—
164D4×C31—44no—
165D4×C31—41no—
166D4×C31—41no—
167D4×C31—81no—
168D4×C31—84no—
169D4×C31—84no—
170C2×C4×C31—22no—
171C2×C4×C31—21no—
172C2×C4×C31—21no—
173C2×C4×C31—22no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—41no—
178C2×C4×C31—44no—
179C2×C4×C31—42no—
180C2×C4×C31—42no—
181C2×C4×C31—42no—
182C2×C4×C31—82no—
183C2×C4×C31—84no—
184C2×C4×C31C2×C22⋊C411yescharacteristic
185C2×C4×C31C2×C22⋊C411yescharacteristic
186C2×C4×C31#12011yescharacteristic
187C2×C4×C31#12011yescharacteristic
188C2×C4×C31#12011yescharacteristic
189C2×C4×C31#12011yescharacteristic
190C8×C31—42no—
191C8×C31—81no—
192C8×C31—81no—
193C8×C31—81no—
194#3506C2×C3111yescharacteristic
195#1955C2×C3111yescharacteristic, complemented
196#1552C2×C3111yescharacteristic, complemented
197#1551C2×C3111yescharacteristic, complemented
198#1531C2×C3111yescharacteristic, complemented
199#1530C2×C3111yescharacteristic, complemented
200#1466C2×C3111yescharacteristic, complemented
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—44no—
204C22×C31—42no—
205C22×C31—42no—
206C22×C31—42no—
207C22×C31—42no—
208C22×C31—42no—
209C22×C31—42no—
210C22×C31—82no—
211C22×C31#37911yescharacteristic
212C22×C31#41312yes—
213C4×C31—21no—
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—21no—
217C4×C31—21no—
218C4×C31—42no—
219C4×C31—42no—
220C4×C31—42no—
221C4×C31—42no—
222C4×C31#45311yescharacteristic
223C4×C31#45511yescharacteristic
224#559—22no—
225#556—21no—
226#555—21no—
227#553—22no—
228#550—21no—
229#547—21no—
230#538—22no—
231#535C4×C3112yes—
232#533C22×C3111yescharacteristic
233#532C22×C3111yescharacteristic
234#525—22no—
235#520C4×C3112yes—
236C2×C4×D4C22×C3111yescharacteristic
237#432C22×C3111yescharacteristic
238#432C22×C3111yescharacteristic
239C2×C4⋊C8C22×C3111yescharacteristic
240C2×C22⋊C8—22no—
241C2×#97C22×C3111yescharacteristic
242#333—21no—
243#332—21no—
244#332—21no—
245#331—21no—
246#330—22no—
247#329—22no—
248#323—22no—
249C2×C31—22no—
250C2×C31—82no—
251C2×C31—82no—
252C2×C31#178511yescharacteristic
253C2×C31#179211yescharacteristic
254C2×C31#179411yescharacteristic
255C2×#41—22no—
256C2×#41C2×C4×C3112yes—
257C22×D4C2×C4×C3111yescharacteristic
258C23×C4D4×C3112yes—
259C22×C8—21no—
260C22×C8—21no—
261C4⋊Q8D4×C3111yescharacteristic
262C4⋊Q8D4×C3111yescharacteristic
263C4⋊Q8D4×C3111yescharacteristic
264#127—22no—
265#127D4×C3111yescharacteristic
266#126—24no—
267#125D4×C3112yes—
268#124—24no—
269#124—22no—
270#124D4×C3112yes—
271#123—24no—
272#123—24no—
273C22⋊Q8—22no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276#121—22no—
277#121—24no—
278#121—21no—
279#121—21no—
280#121—22no—
281#121—22no—
282#120—24no—
283C4×Q8—22no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—21no—
295#117—22no—
296#117—22no—
297#117C2×C4×C3112yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C2×C4×C3111yescharacteristic
300C2×C22⋊C4D4×C3112yes—
301C2×C42C23×C3111yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—44no—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313C31#845811yescharacteristic, direct product factor, Sylow
314C24C22⋊C4×C3112yes—
315#41—44no—
316#41—82no—
317#41—82no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C22⋊C4×C3112yes—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4C22⋊C4×C3112yes—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C22⋊C4×C3112yes—
337C22×C4C22×C4×C3111yescharacteristic
338C22×C4C2×D4×C3111yescharacteristic
339C22×C4C2×D4×C3111yescharacteristic
340C2×C8—22no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C4⋊C4—24no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4C22⋊C4×C3112yes—
357C4⋊C4C22⋊C4×C3112yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C42—24no—
367C42—22no—
368C42C2×D4×C3111yescharacteristic
369C42C2×D4×C3111yescharacteristic
370C42C2×D4×C3111yescharacteristic
371C42C2×D4×C3111yescharacteristic
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—42no—
377C23—42no—
378C23C2×C22⋊C4×C3111yescharacteristic
379Q8—42no—
380Q8—41no—
381Q8—41no—
382Q8—81no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—44no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—84no—
391D4—84no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—84no—
406C2×C4C2×C22⋊C4×C3111yescharacteristic
407C2×C4C2×C22⋊C4×C3111yescharacteristic
408C2×C4C31×#12011yescharacteristic
409C2×C4C31×#12011yescharacteristic
410C2×C4C31×#12011yescharacteristic
411C2×C4C31×#12011yescharacteristic
412C8—42no—
413C8—81no—
414C8—81no—
415C8—81no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22C31×#37911yescharacteristic
427C22C31×#41312yes—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4C31×#45311yescharacteristic
438C4C31×#45511yescharacteristic
439C2—22no—
440C2—82no—
441C2—82no—
442C2C31×#178511yescharacteristic
443C2C31×#179211yescharacteristic
444C2C31×#179411yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3506C2
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×#1955C2semidirect
3. C31×#1552C2semidirect
4. C31×#1551C2semidirect
5. C31×#1531C2semidirect
6. C31×#1530C2semidirect
7. C31×#1466C2semidirect
149. #8458C31direct (N × H)
195. #1955C2×C31semidirect
196. #1552C2×C31semidirect
197. #1551C2×C31semidirect
198. #1531C2×C31semidirect
199. #1530C2×C31semidirect
200. #1466C2×C31semidirect
313. C31#8458direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#19552
C31×#15522
C31×#15512
C31×#15312
C31×#15302
C31×#14662
#845831