Finite Group Databasebeta

← C31×#18294 #21475053

Subgroups — C31×#18294 #21475053

Order 7936.

1002 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#544C211yesmaximal, characteristic, complemented
2C31×#3187C212yesmaximal
3C31×#2618C211yesmaximal, characteristic
4C31×#1611C214yesmaximal, complemented
5C31×#1599C214yesmaximal, complemented
6C31×#1520C212yesmaximal, complemented
7C2×C31×#324C211yesmaximal, characteristic, complemented
8C31×#544—24no—
9C31×#544—22no—
10C31×#544C2214yes—
11C2×C31×C4⋊Q8—22no—
12C2×C31×#125—22no—
13C2×C31×#123—21no—
14C2×C31×C22⋊Q8—21no—
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×#121C2211yescharacteristic
17C2×C31×#117—21no—
18C22×C4⋊C4×C31C2211yescharacteristic
19C31×#483C2214yes—
20C31×#481—22no—
21C31×#465—24no—
22C31×#450C2214yes—
23C31×#448—22no—
24C2×C31×C22⋊C8C2212yes—
25C31×#398C2212yes—
26C31×#396—22no—
27C31×#394C2212yes—
28C31×#386—21no—
29C31×#380C2211yescharacteristic
30C31×#328C2218yes—
31C31×#324C2214yes—
32C22×Q8×C31—21no—
33C22×Q8×C31D412yes—
34C22×D4×C31—21no—
35C22×D4×C31C2311yescharacteristic
36C23×C4×C31D411yescharacteristic
37C23×C4×C31D411yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C2×SD16×C31—24no—
40C22×C8×C31—22no—
41C31×C4⋊Q8—28no—
42C31×C4⋊Q8—28no—
43C31×#125—28no—
44C31×#125—28no—
45C31×#123—44no—
46C31×C22⋊Q8—24no—
47C31×C22⋊Q8—24no—
48C31×C22⋊Q8—22no—
49C31×C22⋊Q8—22no—
50C31×C22⋊Q8D414yes—
51C31×C22⋊Q8C2314yes—
52C31×#121—22no—
53C31×#121—22no—
54C31×#117—24no—
55C31×#117—24no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31D412yes—
66C2×C4⋊C4×C31D412yes—
67C2×C4⋊C4×C31C2311yescharacteristic
68C2×C4⋊C4×C31C2312yes—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—22no—
74C2×C42×C31—22no—
75C31×#106—24no—
76C31×#103—24no—
77C31×#102—24no—
78C31×C22⋊C8—22no—
79C31×C22⋊C8C2314yes—
80C31×#95—22no—
81C31×#95C2312yes—
82C24×C31—21no—
83C24×C31C2×D411yescharacteristic
84C2×Q8×C31—24no—
85C2×Q8×C31—22no—
86C2×Q8×C31—82no—
87C2×Q8×C31—82no—
88C2×Q8×C31C2×D414yes—
89C2×D4×C31—22no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—42no—
93C2×D4×C31—42no—
94C2×D4×C31—81no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—22no—
104C22×C4×C31—81no—
105C22×C4×C31—81no—
106C22×C4×C31SD1614yescomplemented
107C22×C4×C31C2×D412yes—
108C22×C4×C31C2×D412yes—
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D412yes—
111C22×C4×C31C2×D412yes—
112C22×C4×C31C2411yescharacteristic
113SD16×C31—84no—
114C2×C8×C31—24no—
115C2×C8×C31—82no—
116C4⋊C4×C31—24no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—28no—
120C4⋊C4×C31—24no—
121C4⋊C4×C31—24no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—24no—
124C4⋊C4×C31—24no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—44no—
127C4⋊C4×C31—81no—
128C4⋊C4×C31C2×D414yes—
129C22⋊C4×C31—24no—
130C22⋊C4×C31—24no—
131C22⋊C4×C31—42no—
132C22⋊C4×C31—42no—
133C22⋊C4×C31—44no—
134C22⋊C4×C31—82no—
135C42×C31—28no—
136#18294C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—42no—
140C23×C31—42no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31#12011yescharacteristic
145C23×C31#12011yescharacteristic
146C23×C31C2×SD1612yes—
147C23×C31#13712yes—
148C23×C31C22×D411yescharacteristic
149Q8×C31—44no—
150Q8×C31—82no—
151Q8×C31—84no—
152D4×C31—44no—
153D4×C31—82no—
154D4×C31—82no—
155C2×C4×C31—22no—
156C2×C4×C31—22no—
157C2×C4×C31—24no—
158C2×C4×C31—24no—
159C2×C4×C31—24no—
160C2×C4×C31—24no—
161C2×C4×C31—22no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—41no—
166C2×C4×C31—44no—
167C2×C4×C31—42no—
168C2×C4×C31—44no—
169C2×C4×C31—42no—
170C2×C4×C31—42no—
171C2×C4×C31—42no—
172C2×C4×C31—41no—
173C2×C4×C31—42no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—81no—
177C2×C4×C31#12012yes—
178C2×C4×C31C2×SD1614yes—
179C2×C4×C31C22×D412yes—
180C8×C31—44no—
181C2×#544C2×C3111yescharacteristic, complemented
182#3187C2×C3112yes—
183#2618C2×C3111yescharacteristic
184#1611C2×C3114yescomplemented
185#1599C2×C3114yescomplemented
186#1520C2×C3112yescomplemented
187C2×#324C2×C3111yescharacteristic, complemented
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—82no—
197C22×C31—82no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31#44812yes—
202C22×C31#44812yes—
203C22×C31C2×#12011yescharacteristic
204C22×C31C22×SD1611yescharacteristic
205C22×C31C2×#13711yescharacteristic
206C4×C31—24no—
207C4×C31—22no—
208C4×C31—22no—
209C4×C31—44no—
210C4×C31—42no—
211C4×C31—44no—
212C4×C31—82no—
213C4×C31—81no—
214#544—24no—
215#544—22no—
216#544C22×C3114yes—
217C2×C4⋊Q8—22no—
218C2×#125—22no—
219C2×#123—21no—
220C2×C22⋊Q8—21no—
221C2×C22⋊Q8C22×C3112yes—
222C2×#121C22×C3111yescharacteristic
223C2×#117—21no—
224C22×C4⋊C4C22×C3111yescharacteristic
225#483C22×C3114yes—
226#481—22no—
227#465—24no—
228#450C22×C3114yes—
229#448—22no—
230C2×C22⋊C8C22×C3112yes—
231#398C22×C3112yes—
232#396—22no—
233#394C22×C3112yes—
234#386—21no—
235#380C22×C3111yescharacteristic
236#328C22×C3118yes—
237#324C22×C3114yes—
238C2×C31—21no—
239C2×C31—21no—
240C2×C31—22no—
241C2×C31—22no—
242C2×C31—161no—
243C2×C31C2×#44811yescharacteristic
244C2×C31#302011yescharacteristic
245C2×C31#302611yescharacteristic
246C22×Q8—21no—
247C22×Q8D4×C3112yes—
248C22×D4—21no—
249C22×D4C23×C3111yescharacteristic
250C23×C4D4×C3111yescharacteristic
251C23×C4D4×C3111yescharacteristic
252C23×C4C23×C3111yescharacteristic
253C2×SD16—24no—
254C22×C8—22no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257#125—28no—
258#125—28no—
259#123—44no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8D4×C3114yes—
265C22⋊Q8C23×C3114yes—
266#121—22no—
267#121—22no—
268#117—24no—
269#117—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4D4×C3112yes—
280C2×C4⋊C4D4×C3112yes—
281C2×C4⋊C4C23×C3111yescharacteristic
282C2×C4⋊C4C23×C3112yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C42—22no—
289#106—24no—
290#103—24no—
291#102—24no—
292C22⋊C8—22no—
293C22⋊C8C23×C3114yes—
294#95—22no—
295#95C23×C3112yes—
296C31#1829411yescharacteristic, direct product factor, Sylow
297C24—21no—
298C24C2×D4×C3111yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×Q8C2×D4×C3114yes—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4SD16×C3114yescomplemented
322C22×C4C2×D4×C3112yes—
323C22×C4C2×D4×C3112yes—
324C22×C4C2×D4×C3111yescharacteristic
325C22×C4C2×D4×C3112yes—
326C22×C4C2×D4×C3112yes—
327C22×C4C24×C3111yescharacteristic
328SD16—84no—
329C2×C8—24no—
330C2×C8—82no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—81no—
343C4⋊C4C2×D4×C3114yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—82no—
350C42—28no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C31×#12011yescharacteristic
359C23C31×#12011yescharacteristic
360C23C2×SD16×C3112yes—
361C23C31×#13712yes—
362C23C22×D4×C3111yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366D4—44no—
367D4—82no—
368D4—82no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4C31×#12012yes—
392C2×C4C2×SD16×C3114yes—
393C2×C4C22×D4×C3112yes—
394C8—44no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C31×#44812yes—
409C22C31×#44812yes—
410C22C2×C31×#12011yescharacteristic
411C22C22×SD16×C3111yescharacteristic
412C22C2×C31×#13711yescharacteristic
413C4—24no—
414C4—22no—
415C4—22no—
416C4—44no—
417C4—42no—
418C4—44no—
419C4—82no—
420C4—81no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×C31×#44811yescharacteristic
427C2C31×#302011yescharacteristic
428C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#544C2semidirect
4. C31×#1611C2semidirect
5. C31×#1599C2semidirect
6. C31×#1520C2semidirect
7. C2×C31×#324C2semidirect
106. C22×C4×C31SD16semidirect
136. #18294C31direct (N × H)
181. C2×#544C2×C31semidirect
184. #1611C2×C31semidirect
185. #1599C2×C31semidirect
186. #1520C2×C31semidirect
187. C2×#324C2×C31semidirect
296. C31#18294direct (N × H)
321. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5442
C31×#31872
C31×#26182
C31×#16112
C31×#15992
C31×#15202
C2×C31×#3242
#1829431