Finite Group Databasebeta

← C31×#18212 #21474971

Subgroups — C31×#18212 #21474971

Order 7936.

1058 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#3101C212yesmaximal, complemented
3C31×#2535C211yesmaximal, characteristic
4C31×#1608C214yesmaximal, complemented
5C31×#1596C214yesmaximal, complemented
6C31×#1518C212yesmaximal, complemented
7C31×#1460C211yesmaximal, characteristic, complemented
8C31×#554—22no—
9C31×#554—24no—
10D4×Q8×C31—22no—
11C31×#544—21no—
12C31×#543C2214yes—
13C31×#533C2212yes—
14C2×C31×C22⋊Q8—22no—
15C2×C31×#121—21no—
16C31×#518—21no—
17C2×C4×D4×C31C2211yescharacteristic
18C2×C4×D4×C31C2211yescharacteristic
19C31×#483C2214yes—
20C31×#476—22no—
21C31×#461—22no—
22C31×#450C2214yes—
23C4×SD16×C31—24no—
24C31×#431C2212yes—
25C31×#397C2212yes—
26C31×#392C2212yes—
27C31×#386—22no—
28C31×#382C2211yescharacteristic
29C31×#380—21no—
30C31×#328C2218yes—
31C31×#323C2214yes—
32C22×Q8×C31—22no—
33C22×D4×C31—22no—
34C22×D4×C31D411yescharacteristic
35C22×D4×C31C2311yescharacteristic
36C23×C4×C31—21no—
37C23×C4×C31D412yes—
38C2×SD16×C31—24no—
39C31×C4⋊Q8—24no—
40C31×C4⋊Q8D412yescomplemented
41C31×C4⋊Q8D412yescomplemented
42C31×#125—24no—
43C31×#125—22no—
44C31×C22⋊Q8—24no—
45C31×C22⋊Q8—24no—
46C31×C22⋊Q8—28no—
47C31×C22⋊Q8—22no—
48C31×C22⋊Q8—24no—
49C31×C22⋊Q8—28no—
50C31×C22⋊Q8C2314yes—
51C31×#121—44no—
52C4×Q8×C31—22no—
53C4×Q8×C31D414yescomplemented
54C4×D4×C31—24no—
55C4×D4×C31—24no—
56C4×D4×C31—22no—
57C4×D4×C31—22no—
58C4×D4×C31—22no—
59C4×D4×C31—22no—
60C31×#117—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—24no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31D411yescharacteristic
65C2×C4⋊C4×C31C2311yescharacteristic
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31C2312yes—
71C2×C42×C31C2311yescharacteristic
72C31×#106—24no—
73C31×C4⋊C8—22no—
74C31×#103—24no—
75C31×#102—24no—
76C31×C22⋊C8C2314yes—
77C4×C8×C31—22no—
78C31×#95—22no—
79C31×#95—21no—
80C31×#95C2312yes—
81C24×C31—21no—
82C24×C31C2×D412yes—
83C2×Q8×C31—24no—
84C2×Q8×C31—22no—
85C2×Q8×C31—84no—
86C2×Q8×C31C2×D414yes—
87C2×D4×C31—22no—
88C2×D4×C31—22no—
89C2×D4×C31—22no—
90C2×D4×C31—42no—
91C2×D4×C31—44no—
92C2×D4×C31—44no—
93C2×D4×C31—82no—
94C2×D4×C31SD1614yescomplemented
95C22×C4×C31—24no—
96C22×C4×C31—24no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—21no—
101C22×C4×C31—42no—
102C22×C4×C31—42no—
103C22×C4×C31—42no—
104C22×C4×C31—82no—
105C22×C4×C31C2×D412yes—
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2411yescharacteristic
109SD16×C31—48no—
110C2×C8×C31—24no—
111C4⋊C4×C31—24no—
112C4⋊C4×C31—28no—
113C4⋊C4×C31—22no—
114C4⋊C4×C31—24no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—42no—
121C4⋊C4×C31C2×D414yes—
122C22⋊C4×C31—28no—
123C22⋊C4×C31—24no—
124C22⋊C4×C31—24no—
125C22⋊C4×C31—44no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—82no—
129C42×C31—24no—
130C42×C31C2×D412yes—
131C42×C31C2×D412yes—
132#18212C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—22no—
134C23×C31—22no—
135C23×C31—22no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31C2×SD1614yes—
145C23×C31C22×D411yescharacteristic
146Q8×C31—44no—
147Q8×C31—44no—
148Q8×C31—84no—
149D4×C31—24no—
150D4×C31—28no—
151D4×C31—42no—
152D4×C31—88no—
153C2×C4×C31—24no—
154C2×C4×C31—24no—
155C2×C4×C31—22no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—44no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—42no—
165C2×C4×C31—41no—
166C2×C4×C31—82no—
167C2×C4×C31—84no—
168C2×C4×C31—84no—
169C2×C4×C31—84no—
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12011yescharacteristic
172C2×C4×C31#12012yes—
173C2×C4×C31C2×SD1612yes—
174C2×C4×C31#13712yes—
175C2×C4×C31C22×D412yes—
176C8×C31—44no—
177#3501C2×C3111yescharacteristic, complemented
178#3101C2×C3112yescomplemented
179#2535C2×C3111yescharacteristic
180#1608C2×C3114yescomplemented
181#1596C2×C3114yescomplemented
182#1518C2×C3112yescomplemented
183#1460C2×C3111yescharacteristic, complemented
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—24no—
187C22×C31—44no—
188C22×C31—42no—
189C22×C31—42no—
190C22×C31—42no—
191C22×C31—42no—
192C22×C31—42no—
193C22×C31—82no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31C2×#12011yescharacteristic
199C22×C31C22×SD1611yescharacteristic
200C22×C31C2×#13711yescharacteristic
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—22no—
204C4×C31—42no—
205C4×C31—44no—
206C4×C31—44no—
207C4×C31—82no—
208C4×C31—81no—
209C4×C31#44814yes—
210#554—22no—
211#554—24no—
212D4×Q8—22no—
213#544—21no—
214#543C22×C3114yes—
215#533C22×C3112yes—
216C2×C22⋊Q8—22no—
217C2×#121—21no—
218#518—21no—
219C2×C4×D4C22×C3111yescharacteristic
220C2×C4×D4C22×C3111yescharacteristic
221#483C22×C3114yes—
222#476—22no—
223#461—22no—
224#450C22×C3114yes—
225C4×SD16—24no—
226#431C22×C3112yes—
227#397C22×C3112yes—
228#392C22×C3112yes—
229#386—22no—
230#382C22×C3111yescharacteristic
231#380—21no—
232#328C22×C3118yes—
233#323C22×C3114yes—
234C2×C31—21no—
235C2×C31—21no—
236C2×C31—44no—
237C2×C31—82no—
238C2×C31C2×#44811yescharacteristic
239C2×C31#301911yescharacteristic
240C2×C31#302511yescharacteristic
241C22×Q8—22no—
242C22×D4—22no—
243C22×D4D4×C3111yescharacteristic
244C22×D4C23×C3111yescharacteristic
245C23×C4—21no—
246C23×C4D4×C3112yes—
247C2×SD16—24no—
248C4⋊Q8—24no—
249C4⋊Q8D4×C3112yescomplemented
250C4⋊Q8D4×C3112yescomplemented
251#125—24no—
252#125—22no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—28no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—28no—
259C22⋊Q8C23×C3114yes—
260#121—44no—
261C4×Q8—22no—
262C4×Q8D4×C3114yescomplemented
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269#117—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4D4×C3111yescharacteristic
274C2×C4⋊C4C23×C3111yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4C23×C3112yes—
280C2×C42C23×C3111yescharacteristic
281#106—24no—
282C4⋊C8—22no—
283#103—24no—
284#102—24no—
285C22⋊C8C23×C3114yes—
286C4×C8—22no—
287#95—22no—
288#95—21no—
289#95C23×C3112yes—
290C31#1821211yescharacteristic, direct product factor, Sylow
291C24—21no—
292C24C2×D4×C3112yes—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—84no—
296C2×Q8C2×D4×C3114yes—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—82no—
304C2×D4SD16×C3114yescomplemented
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4C2×D4×C3112yes—
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C24×C3111yescharacteristic
319SD16—48no—
320C2×C8—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4C2×D4×C3114yes—
332C22⋊C4—28no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—44no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C42—24no—
340C42C2×D4×C3112yes—
341C42C2×D4×C3112yes—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C2×SD16×C3114yes—
354C23C22×D4×C3111yescharacteristic
355Q8—44no—
356Q8—44no—
357Q8—84no—
358D4—24no—
359D4—28no—
360D4—42no—
361D4—88no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4C31×#12011yescharacteristic
380C2×C4C31×#12011yescharacteristic
381C2×C4C31×#12012yes—
382C2×C4C2×SD16×C3112yes—
383C2×C4C31×#13712yes—
384C2×C4C22×D4×C3112yes—
385C8—44no—
386C22—21no—
387C22—21no—
388C22—24no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—82no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×C31×#12011yescharacteristic
401C22C22×SD16×C3111yescharacteristic
402C22C2×C31×#13711yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—42no—
407C4—44no—
408C4—44no—
409C4—82no—
410C4—81no—
411C4C31×#44814yes—
412C2—21no—
413C2—21no—
414C2—44no—
415C2—82no—
416C2C2×C31×#44811yescharacteristic
417C2C31×#301911yescharacteristic
418C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2535C2
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
1. C31×#3501C2semidirect
2. C31×#3101C2semidirect
4. C31×#1608C2semidirect
5. C31×#1596C2semidirect
6. C31×#1518C2semidirect
7. C31×#1460C2semidirect
40. C31×C4⋊Q8D4semidirect
41. C31×C4⋊Q8D4semidirect
53. C4×Q8×C31D4semidirect
94. C2×D4×C31SD16semidirect
132. #18212C31direct (N × H)
177. #3501C2×C31semidirect
178. #3101C2×C31semidirect
180. #1608C2×C31semidirect
181. #1596C2×C31semidirect
182. #1518C2×C31semidirect
183. #1460C2×C31semidirect
249. C4⋊Q8D4×C31semidirect
250. C4⋊Q8D4×C31semidirect
262. C4×Q8D4×C31semidirect
290. C31#18212direct (N × H)
304. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#31012
C31×#25352
C31×#16082
C31×#15962
C31×#15182
C31×#14602
#1821231