Finite Group Databasebeta

← C31×#8992 #21465751

Subgroups — C31×#8992 #21465751

Order 7936.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3103C211yesmaximal, characteristic
2Q8×M16×C31C211yesmaximal, characteristic, complemented
3C31×#1980C211yesmaximal, characteristic
4C4×C31×#103C211yesmaximal, characteristic, complemented
5C31×#1492C211yesmaximal, characteristic
6C31×#1303C211yesmaximal, characteristic, complemented
7C31×#1278C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C4×Q8×C31C2211yescharacteristic
10C31×#481—21no—
11C31×#475C411yescharacteristic, complemented
12C31×#475C411yescharacteristic, complemented
13C31×#460C411yescharacteristic, complemented
14C31×#460C411yescharacteristic, complemented
15C31×#448—21no—
16C31×#446—21no—
17C31×#446—22no—
18C8×Q8×C31—21no—
19C4×SD16×C31—21no—
20C4×SD16×C31—21no—
21C31×#431C2211yescharacteristic
22C31×#423—21no—
23C31×#423C2211yescharacteristic
24C2×C31×#104—21no—
25C2×C31×#104—21no—
26C2×C31×#103—21no—
27C4×M16×C31—21no—
28C4×M16×C31C2211yescharacteristic
29C2×C4×C8×C31C2211yescharacteristic
30C4×C4⋊C4×C31C2211yescharacteristic
31C31×#340—21no—
32C31×D4⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C2×SD16×C31—21no—
36C2×SD16×C31—21no—
37C2×M16×C31—21no—
38C2×M16×C31D411yescharacteristic
39C2×M16×C31D411yescharacteristic
40C22×C8×C31—21no—
41C31×C4⋊Q8C2×C411yescharacteristic
42C31×#127C2×C411yescharacteristic
43C31×#124—21no—
44C31×#121—21no—
45C31×#121—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31C2×C411yescharacteristic
49C4×Q8×C31C2×C411yescharacteristic
50C4×Q8×C31D411yescharacteristic
51C4×Q8×C31D411yescharacteristic
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C42×C31—21no—
58C2×C42×C31—21no—
59C2×C42×C31C2311yescharacteristic
60C31×#106—21no—
61C31×#106—21no—
62C31×C4⋊C8—22no—
63C31×C4⋊C8—22no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8C2×C411yescharacteristic
66C31×C4⋊C8C2×C411yescharacteristic
67C31×C4⋊C8D412yes—
68C31×#104—42no—
69C31×#104—42no—
70C31×#103—24no—
71C31×#103—21no—
72C31×#103—21no—
73C31×#102—21no—
74C31×#102—21no—
75C31×C22⋊C8—21no—
76C31×#97—22no—
77C31×#97—21no—
78C4×C8×C31—22no—
79C4×C8×C31—21no—
80C4×C8×C31—21no—
81C4×C8×C31—21no—
82C4×C8×C31—21no—
83C31×#95—21no—
84#41×C31—41no—
85#41×C31—41no—
86#41×C31—41no—
87#41×C31—41no—
88C2×Q8×C31—21no—
89C2×Q8×C31—22no—
90C2×Q8×C31—21no—
91C2×Q8×C31—41no—
92C2×Q8×C31C22⋊C411yescharacteristic
93C2×Q8×C31C22⋊C411yescharacteristic
94C2×D4×C31—21no—
95C2×D4×C31—21no—
96C2×D4×C31—21no—
97C2×D4×C31—41no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31#4111yescharacteristic
105SD16×C31—42no—
106SD16×C31—42no—
107M16×C31—21no—
108M16×C31—21no—
109M16×C31—21no—
110M16×C31—41no—
111M16×C31—41no—
112M16×C31SD1612yescomplemented
113C2×C8×C31—21no—
114C2×C8×C31—22no—
115C2×C8×C31—21no—
116C2×C8×C31—21no—
117C2×C8×C31—21no—
118C2×C8×C31—22no—
119C2×C8×C31—41no—
120C2×C8×C31—41no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—41no—
130C4⋊C4×C31C22⋊C411yescharacteristic
131C4⋊C4×C31C22⋊C411yescharacteristic
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C42×C31—21no—
135C42×C31—21no—
136C42×C31—21no—
137C42×C31—21no—
138C42×C31—41no—
139C42×C31—41no—
140C42×C31C22×C411yescharacteristic
141C42×C31C2×D411yescharacteristic
142C42×C31C2×D411yescharacteristic
143C42×C31#4111yescharacteristic
144#8992C3111yesmaximal, characteristic, direct product factor, Sylow
145C23×C31—41no—
146C23×C31#12111yescharacteristic
147Q8×C31—21no—
148Q8×C31—21no—
149Q8×C31—21no—
150Q8×C31—42no—
151Q8×C31—41no—
152Q8×C31—41no—
153Q8×C31#10412yescomplemented
154D4×C31—42no—
155D4×C31—41no—
156D4×C31—41no—
157D4×C31—41no—
158D4×C31—41no—
159D4×C31—81no—
160D4×C31—81no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31—21no—
166C2×C4×C31—21no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C2×C4×C31—81no—
175C2×C4×C31—81no—
176C2×C4×C31C2×C22⋊C411yescharacteristic
177C2×C4×C31C4×D411yescharacteristic
178C2×C4×C31C4×D411yescharacteristic
179C2×C4×C31#12011yescharacteristic
180C2×C4×C31#12111yescharacteristic
181C2×C4×C31#12311yescharacteristic
182C2×C4×C31C2×SD1611yescharacteristic
183C2×C4×C31#13511yescharacteristic
184C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C31#13711yescharacteristic
186C8×C31—22no—
187C8×C31—42no—
188C8×C31—41no—
189C8×C31—41no—
190C8×C31—41no—
191C8×C31—42no—
192#3103C2×C3111yescharacteristic
193Q8×M16C2×C3111yescharacteristic, complemented
194#1980C2×C3111yescharacteristic
195C4×#103C2×C3111yescharacteristic, complemented
196#1492C2×C3111yescharacteristic
197#1303C2×C3111yescharacteristic, complemented
198#1278C2×C3111yescharacteristic, complemented
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31#38611yescharacteristic
205C22×C31#46011yescharacteristic
206C22×C31#46411yescharacteristic
207C4×C31—21no—
208C4×C31—22no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—41no—
212C4×C31—41no—
213C4×C31—41no—
214C4×C31—82no—
215C4×C31—81no—
216C4×C31C2×#10411yescharacteristic
217C4×C31#42111yescharacteristic
218C4×C31C4×SD1611yescharacteristic
219C4×C31#44011yescharacteristic
220C4×C31#44811yescharacteristic
221C4×C31#44911yescharacteristic
222#532C22×C3111yescharacteristic
223C2×C4×Q8C22×C3111yescharacteristic
224#481—21no—
225#475C4×C3111yescharacteristic, complemented
226#475C4×C3111yescharacteristic, complemented
227#460C4×C3111yescharacteristic, complemented
228#460C4×C3111yescharacteristic, complemented
229#448—21no—
230#446—21no—
231#446—22no—
232C8×Q8—21no—
233C4×SD16—21no—
234C4×SD16—21no—
235#431C22×C3111yescharacteristic
236#423—21no—
237#423C22×C3111yescharacteristic
238C2×#104—21no—
239C2×#104—21no—
240C2×#103—21no—
241C4×M16—21no—
242C4×M16C22×C3111yescharacteristic
243C2×C4×C8C22×C3111yescharacteristic
244C4×C4⋊C4C22×C3111yescharacteristic
245#340—21no—
246D4⋊C8—22no—
247C2×C31—21no—
248C2×C31—21no—
249C2×C31—161no—
250C2×C31#187111yescharacteristic
251C2×C31#188011yescharacteristic
252C2×C31#188111yescharacteristic
253C2×#41—21no—
254C22×Q8D4×C3111yescharacteristic
255C2×SD16—21no—
256C2×SD16—21no—
257C2×M16—21no—
258C2×M16D4×C3111yescharacteristic
259C2×M16D4×C3111yescharacteristic
260C22×C8—21no—
261C4⋊Q8C2×C4×C3111yescharacteristic
262#127C2×C4×C3111yescharacteristic
263#124—21no—
264#121—21no—
265#121—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8C2×C4×C3111yescharacteristic
269C4×Q8C2×C4×C3111yescharacteristic
270C4×Q8D4×C3111yescharacteristic
271C4×Q8D4×C3111yescharacteristic
272C4×D4—21no—
273C4×D4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4D4×C3111yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C23×C3111yescharacteristic
280#106—21no—
281#106—21no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—21no—
285C4⋊C8C2×C4×C3111yescharacteristic
286C4⋊C8C2×C4×C3111yescharacteristic
287C4⋊C8D4×C3112yes—
288#104—42no—
289#104—42no—
290#103—24no—
291#103—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295C22⋊C8—21no—
296#97—22no—
297#97—21no—
298C4×C8—22no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303#95—21no—
304C31#899211yescharacteristic, direct product factor, Sylow
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊C4×C3111yescharacteristic
314C2×Q8C22⋊C4×C3111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4C2×D4×C3111yescharacteristic
324C22×C4C2×D4×C3111yescharacteristic
325C22×C4#41×C3111yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16SD16×C3112yescomplemented
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—41no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—41no—
351C4⋊C4C22⋊C4×C3111yescharacteristic
352C4⋊C4C22⋊C4×C3111yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—41no—
360C42—41no—
361C42C22×C4×C3111yescharacteristic
362C42C2×D4×C3111yescharacteristic
363C42C2×D4×C3111yescharacteristic
364C42#41×C3111yescharacteristic
365C23—41no—
366C23C31×#12111yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8C31×#10412yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4C2×C22⋊C4×C3111yescharacteristic
397C2×C4C4×D4×C3111yescharacteristic
398C2×C4C4×D4×C3111yescharacteristic
399C2×C4C31×#12011yescharacteristic
400C2×C4C31×#12111yescharacteristic
401C2×C4C31×#12311yescharacteristic
402C2×C4C2×SD16×C3111yescharacteristic
403C2×C4C31×#13511yescharacteristic
404C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C31×#13711yescharacteristic
406C8—22no—
407C8—42no—
408C8—41no—
409C8—41no—
410C8—41no—
411C8—42no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22C31×#38611yescharacteristic
418C22C31×#46011yescharacteristic
419C22C31×#46411yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—81no—
429C4C2×C31×#10411yescharacteristic
430C4C31×#42111yescharacteristic
431C4C4×SD16×C3111yescharacteristic
432C4C31×#44011yescharacteristic
433C4C31×#44811yescharacteristic
434C4C31×#44911yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2C31×#187111yescharacteristic
439C2C31×#188011yescharacteristic
440C2C31×#188111yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#386
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×M16×C31C2semidirect
4. C4×C31×#103C2semidirect
6. C31×#1303C2semidirect
7. C31×#1278C2semidirect
11. C31×#475C4semidirect
12. C31×#475C4semidirect
13. C31×#460C4semidirect
14. C31×#460C4semidirect
112. M16×C31SD16semidirect
144. #8992C31direct (N × H)
153. Q8×C31#104semidirect
193. Q8×M16C2×C31semidirect
195. C4×#103C2×C31semidirect
197. #1303C2×C31semidirect
198. #1278C2×C31semidirect
225. #475C4×C31semidirect
226. #475C4×C31semidirect
227. #460C4×C31semidirect
228. #460C4×C31semidirect
304. C31#8992direct (N × H)
333. M16SD16×C31semidirect
373. Q8C31×#104semidirect

Maximal subgroups

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

subgroupindex
C31×#31032
Q8×M16×C312
C31×#19802
C4×C31×#1032
C31×#14922
C31×#13032
C31×#12782
#899231