Finite Group Databasebeta

← C31×#9951 #21466710

Subgroups — C31×#9951 #21466710

Order 7936.

610 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3187C211yesmaximal, characteristic
2C2×C31×#481C211yesmaximal, characteristic
3C31×#2502C211yesmaximal, characteristic, complemented
4C31×#1964C211yesmaximal, characteristic
5C31×#1824C211yesmaximal, characteristic, complemented
6C31×#1520C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C22×C4⋊C4×C31C2211yescharacteristic
11C31×#483—21no—
12C31×#481—22no—
13C31×#481—21no—
14C31×#481—22no—
15C31×#465—22no—
16C31×#450—21no—
17C31×#448—21no—
18C2×C31×#106—21no—
19C2×C31×#102—21no—
20C2×C31×#99—22no—
21C2×C31×C22⋊C8C2211yescharacteristic
22C2×C31×C22⋊C8C2211yescharacteristic
23C31×#398—22no—
24C31×#391—21no—
25C31×#385—22no—
26C31×#384—21no—
27C31×#380C2211yescharacteristic
28C4×C22⋊C4×C31C2211yescharacteristic
29C31×#340—22no—
30C31×#328—22no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C23×C4×C31C2311yescharacteristic
36C2×SD16×C31—22no—
37C22×C8×C31—21no—
38C22×C8×C31D412yes—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×#121—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31D412yes—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31D412yes—
62C2×C42×C31—21no—
63C31×#106—24no—
64C31×#106—22no—
65C31×#103—22no—
66C31×#102—24no—
67C31×#102—22no—
68C31×#99—44no—
69C31×C22⋊C8—21no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8D412yes—
73C31×#95—24no—
74C31×#95—21no—
75C31×#95—21no—
76C24×C31—21no—
77C24×C31C2×D411yescharacteristic
78C2×Q8×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—42no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—42no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—42no—
87C2×D4×C31—42no—
88C22×C4×C31—22no—
89C22×C4×C31—21no—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—42no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—41no—
97C22×C4×C31—82no—
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31#4111yescharacteristic
104SD16×C31—82no—
105C2×C8×C31—22no—
106C2×C8×C31—22no—
107C2×C8×C31—42no—
108C2×C8×C31—42no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—22no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—42no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—84no—
121C22⋊C4×C31—22no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—42no—
127C22⋊C4×C31SD1614yescomplemented
128C42×C31—42no—
129#9951C3111yesmaximal, characteristic, direct product factor, Sylow
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—42no—
134C23×C31—81no—
135C23×C31—81no—
136C23×C31#12011yescharacteristic
137C23×C31#12111yescharacteristic
138C23×C31#12711yescharacteristic
139C23×C31C2×SD1612yes—
140C23×C31#13712yes—
141Q8×C31—42no—
142Q8×C31—81no—
143D4×C31—42no—
144D4×C31—42no—
145D4×C31—82no—
146D4×C31—81no—
147D4×C31—81no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—22no—
151C2×C4×C31—24no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—42no—
157C2×C4×C31—42no—
158C2×C4×C31—44no—
159C2×C4×C31—42no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—82no—
166C2×C4×C31—82no—
167C2×C4×C31—81no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12011yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C8×C31—42no—
175C8×C31—42no—
176#3187C2×C3111yescharacteristic
177C2×#481C2×C3111yescharacteristic
178#2502C2×C3111yescharacteristic, complemented
179#1964C2×C3111yescharacteristic
180#1824C2×C3111yescharacteristic, complemented
181#1520C2×C3111yescharacteristic, complemented
182#1282C2×C3111yescharacteristic, complemented
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—42no—
190C22×C31—42no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—162no—
195C22×C31#39211yescharacteristic
196C22×C31#44812yes—
197C22×C31#46112yes—
198C22×C31#49211yescharacteristic
199C22×C31#49711yescharacteristic
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—44no—
204C4×C31—81no—
205C4×C31—82no—
206C4×C31—81no—
207C4×C31—81no—
208C4×C31—161no—
209C4×C31#45812yes—
210C2×C22⋊Q8C22×C3111yescharacteristic
211C2×#121C22×C3111yescharacteristic
212C22×C4⋊C4C22×C3111yescharacteristic
213#483—21no—
214#481—22no—
215#481—21no—
216#481—22no—
217#465—22no—
218#450—21no—
219#448—21no—
220C2×#106—21no—
221C2×#102—21no—
222C2×#99—22no—
223C2×C22⋊C8C22×C3111yescharacteristic
224C2×C22⋊C8C22×C3111yescharacteristic
225#398—22no—
226#391—21no—
227#385—22no—
228#384—21no—
229#380C22×C3111yescharacteristic
230C4×C22⋊C4C22×C3111yescharacteristic
231#340—22no—
232#328—22no—
233C2×C31—22no—
234C2×C31—21no—
235C2×C31—22no—
236C2×C31—21no—
237C2×C31—161no—
238C2×C31#200811yescharacteristic
239C2×C31#201311yescharacteristic
240C2×C31#202211yescharacteristic
241C22×Q8D4×C3111yescharacteristic
242C22×D4—21no—
243C22×D4D4×C3111yescharacteristic
244C23×C4—21no—
245C23×C4C23×C3111yescharacteristic
246C2×SD16—22no—
247C22×C8—21no—
248C22×C8D4×C3112yes—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4D4×C3112yes—
266C2×C4⋊C4D4×C3111yescharacteristic
267C2×C4⋊C4D4×C3111yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4D4×C3112yes—
272C2×C42—21no—
273#106—24no—
274#106—22no—
275#103—22no—
276#102—24no—
277#102—22no—
278#99—44no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8D4×C3112yes—
283#95—24no—
284#95—21no—
285#95—21no—
286C31#995111yescharacteristic, direct product factor, Sylow
287C24—21no—
288C24C2×D4×C3111yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4C2×D4×C3111yescharacteristic
310C22×C4C2×D4×C3111yescharacteristic
311C22×C4C2×D4×C3111yescharacteristic
312C22×C4C2×D4×C3111yescharacteristic
313C22×C4C2×D4×C3111yescharacteristic
314C22×C4#41×C3111yescharacteristic
315SD16—82no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—42no—
319C2×C8—42no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—84no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—42no—
338C22⋊C4SD16×C3114yescomplemented
339C42—42no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23C31×#12011yescharacteristic
347C23C31×#12111yescharacteristic
348C23C31×#12711yescharacteristic
349C23C2×SD16×C3112yes—
350C23C31×#13712yes—
351Q8—42no—
352Q8—81no—
353D4—42no—
354D4—42no—
355D4—82no—
356D4—81no—
357D4—81no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4C31×#12011yescharacteristic
381C2×C4C31×#12011yescharacteristic
382C2×C4C31×#12111yescharacteristic
383C2×C4C31×#12111yescharacteristic
384C8—42no—
385C8—42no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—162no—
398C22C31×#39211yescharacteristic
399C22C31×#44812yes—
400C22C31×#46112yes—
401C22C31×#49211yescharacteristic
402C22C31×#49711yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—161no—
412C4C31×#45812yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—161no—
418C2C31×#200811yescharacteristic
419C2C31×#201311yescharacteristic
420C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2502C2semidirect
5. C31×#1824C2semidirect
6. C31×#1520C2semidirect
7. C31×#1282C2semidirect
127. C22⋊C4×C31SD16semidirect
129. #9951C31direct (N × H)
178. #2502C2×C31semidirect
180. #1824C2×C31semidirect
181. #1520C2×C31semidirect
182. #1282C2×C31semidirect
286. C31#9951direct (N × H)
338. C22⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31872
C2×C31×#4812
C31×#25022
C31×#19642
C31×#18242
C31×#15202
C31×#12822
#995131