Finite Group Databasebeta

← C31×#10001 #21466760

Subgroups — C31×#10001 #21466760

Order 7936.

568 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3101C211yesmaximal, characteristic
2C31×#3078C211yesmaximal, characteristic, complemented
3C31×#2536C211yesmaximal, characteristic
4C31×#1518C211yesmaximal, characteristic, complemented
5C31×#1486C211yesmaximal, characteristic
6C31×#1479C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×C22⋊Q8—21no—
10C2×C4×Q8×C31C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#483—21no—
13C31×#479—22no—
14C31×#477—21no—
15C31×#476—21no—
16C31×#475—21no—
17C31×#461—21no—
18C31×#450—21no—
19C4×SD16×C31—22no—
20C31×#431C2211yescharacteristic
21C31×#431C2211yescharacteristic
22C31×#428—21no—
23C31×#423C2211yescharacteristic
24C31×#419—21no—
25C31×#398—21no—
26C31×#394—21no—
27C31×#388—22no—
28C31×#382C2211yescharacteristic
29C31×#380—21no—
30C31×#328—22no—
31C31×Q8⋊C8—22no—
32C31×D4⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C2×SD16×C31—22no—
37C2×M16×C31—21no—
38C31×C4⋊Q8D411yescharacteristic
39C31×C4⋊Q8D411yescharacteristic
40C31×#125—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—42no—
44C31×C22⋊Q8—42no—
45C4×Q8×C31—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C31×#117—21no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—21no—
64C2×C42×C31—21no—
65C2×C42×C31C2311yescharacteristic
66C31×#107—22no—
67C31×#106—22no—
68C31×#106—22no—
69C31×C4⋊C8—21no—
70C31×C4⋊C8—21no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8D412yes—
73C31×#103—22no—
74C31×#103—22no—
75C31×#103—22no—
76C31×#102—22no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C4×C8×C31—21no—
80C4×C8×C31D412yes—
81C31×#95—22no—
82C31×#95—21no—
83C31×#95D412yes—
84C24×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—21no—
89C2×Q8×C31—21no—
90C2×Q8×C31—21no—
91C2×Q8×C31—21no—
92C2×Q8×C31SD1612yescomplemented
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—42no—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31—41no—
103C22×C4×C31—41no—
104C22×C4×C31—41no—
105C22×C4×C31—41no—
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109SD16×C31—44no—
110M16×C31—41no—
111M16×C31—41no—
112C2×C8×C31—22no—
113C2×C8×C31—22no—
114C2×C8×C31—41no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—44no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C4⋊C4×C31—41no—
127C4⋊C4×C31—41no—
128C4⋊C4×C31—41no—
129C4⋊C4×C31—41no—
130C22⋊C4×C31—44no—
131C22⋊C4×C31—44no—
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C22⋊C4×C31—41no—
135C42×C31—41no—
136C42×C31—41no—
137C42×C31—41no—
138C42×C31C2×D411yescharacteristic
139C42×C31C2×D411yescharacteristic
140C42×C31C2×D411yescharacteristic
141C42×C31#4111yescharacteristic
142#10001C3111yesmaximal, characteristic, direct product factor, Sylow
143C23×C31—22no—
144C23×C31—41no—
145C23×C31—41no—
146C23×C31—41no—
147C23×C31—41no—
148C23×C31—41no—
149C23×C31—41no—
150C23×C31#12011yescharacteristic
151Q8×C31—22no—
152Q8×C31—22no—
153Q8×C31—42no—
154Q8×C31—41no—
155Q8×C31—41no—
156Q8×C31—81no—
157D4×C31—22no—
158D4×C31—22no—
159D4×C31—41no—
160D4×C31—41no—
161C2×C4×C31—22no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—42no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—81no—
177C2×C4×C31—81no—
178C2×C4×C31—81no—
179C2×C4×C31—81no—
180C2×C4×C31#12011yescharacteristic
181C2×C4×C31#12011yescharacteristic
182C2×C4×C31#12111yescharacteristic
183C2×C4×C31#12111yescharacteristic
184C2×C4×C31#12111yescharacteristic
185C2×C4×C31#12711yescharacteristic
186C2×C4×C31C2×SD1611yescharacteristic
187C2×C4×C31#13511yescharacteristic
188C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C31#13711yescharacteristic
190C8×C31—42no—
191C8×C31—42no—
192C8×C31—81no—
193#3101C2×C3111yescharacteristic
194#3078C2×C3111yescharacteristic, complemented
195#2536C2×C3111yescharacteristic
196#1518C2×C3111yescharacteristic, complemented
197#1486C2×C3111yescharacteristic
198#1479C2×C3111yescharacteristic, complemented
199#1286C2×C3111yescharacteristic, complemented
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—42no—
203C22×C31—42no—
204C22×C31—42no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31#39211yescharacteristic
210C22×C31#45011yescharacteristic
211C22×C31#45211yescharacteristic
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—42no—
217C4×C31—82no—
218C4×C31—81no—
219C4×C31—81no—
220C4×C31—81no—
221C4×C31—81no—
222C4×C31#44811yescharacteristic
223C4×C31#44911yescharacteristic
224C4×C31#46511yescharacteristic
225C4×C31#46811yescharacteristic
226#533C22×C3111yescharacteristic
227C2×C22⋊Q8—21no—
228C2×C4×Q8C22×C3111yescharacteristic
229C2×C4×D4C22×C3111yescharacteristic
230#483—21no—
231#479—22no—
232#477—21no—
233#476—21no—
234#475—21no—
235#461—21no—
236#450—21no—
237C4×SD16—22no—
238#431C22×C3111yescharacteristic
239#431C22×C3111yescharacteristic
240#428—21no—
241#423C22×C3111yescharacteristic
242#419—21no—
243#398—21no—
244#394—21no—
245#388—22no—
246#382C22×C3111yescharacteristic
247#380—21no—
248#328—22no—
249Q8⋊C8—22no—
250D4⋊C8—22no—
251C2×C31—21no—
252C2×C31—21no—
253C2×C31—82no—
254C2×C31#200011yescharacteristic
255C2×C31#200911yescharacteristic
256C2×C31#201011yescharacteristic
257C22×Q8D4×C3111yescharacteristic
258C22×D4D4×C3111yescharacteristic
259C23×C4—21no—
260C2×SD16—22no—
261C2×M16—21no—
262C4⋊Q8D4×C3111yescharacteristic
263C4⋊Q8D4×C3111yescharacteristic
264#125—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279#117—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4D4×C3111yescharacteristic
284C2×C4⋊C4D4×C3111yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C42—21no—
289C2×C42C23×C3111yescharacteristic
290#107—22no—
291#106—22no—
292#106—22no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8D4×C3112yes—
297#103—22no—
298#103—22no—
299#103—22no—
300#102—22no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C4×C8—21no—
304C4×C8D4×C3112yes—
305#95—22no—
306#95—21no—
307#95D4×C3112yes—
308C31#1000111yescharacteristic, direct product factor, Sylow
309C24—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8SD16×C3112yescomplemented
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—42no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×D4×C3111yescharacteristic
332C22×C4C2×D4×C3111yescharacteristic
333C22×C4C2×D4×C3111yescharacteristic
334SD16—44no—
335M16—41no—
336M16—41no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—41no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—44no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42C2×D4×C3111yescharacteristic
364C42C2×D4×C3111yescharacteristic
365C42C2×D4×C3111yescharacteristic
366C42#41×C3111yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23C31×#12011yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—42no—
378Q8—41no—
379Q8—41no—
380Q8—81no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C31×#12011yescharacteristic
405C2×C4C31×#12011yescharacteristic
406C2×C4C31×#12111yescharacteristic
407C2×C4C31×#12111yescharacteristic
408C2×C4C31×#12111yescharacteristic
409C2×C4C31×#12711yescharacteristic
410C2×C4C2×SD16×C3111yescharacteristic
411C2×C4C31×#13511yescharacteristic
412C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
413C2×C4C31×#13711yescharacteristic
414C8—42no—
415C8—42no—
416C8—81no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C31×#39211yescharacteristic
427C22C31×#45011yescharacteristic
428C22C31×#45211yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4C31×#44811yescharacteristic
440C4C31×#44911yescharacteristic
441C4C31×#46511yescharacteristic
442C4C31×#46811yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2C31×#200011yescharacteristic
447C2C31×#200911yescharacteristic
448C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ 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×#3078C2semidirect
4. C31×#1518C2semidirect
6. C31×#1479C2semidirect
7. C31×#1286C2semidirect
92. C2×Q8×C31SD16semidirect
142. #10001C31direct (N × H)
194. #3078C2×C31semidirect
196. #1518C2×C31semidirect
198. #1479C2×C31semidirect
199. #1286C2×C31semidirect
308. C31#10001direct (N × H)
317. C2×Q8SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31012
C31×#30782
C31×#25362
C31×#15182
C31×#14862
C31×#14792
C31×#12862
#1000131