Finite Group Databasebeta

← C31×#9521 #21466280

Subgroups — C31×#9521 #21466280

Order 7936.

472 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
1C31×#3092C211yesmaximal, characteristic
2C31×#2976C211yesmaximal, characteristic
3C31×#1824C211yesmaximal, characteristic, complemented
4C31×#1821C211yesmaximal, characteristic, complemented
5C31×#1513C211yesmaximal, characteristic
6C31×#1305C211yesmaximal, characteristic, complemented
7C31×#1293C211yesmaximal, characteristic, complemented
8C31×#525C2211yescharacteristic
9C22×C4⋊C4×C31C2211yescharacteristic
10C31×#484—21no—
11C31×#483C412yescomplemented
12C31×#481—21no—
13C31×#480C412yescomplemented
14C31×#436—21no—
15C31×#435—21no—
16C31×#435—21no—
17C8×D4×C31—21no—
18C31×#433—21no—
19C31×#428—21no—
20C31×#417—21no—
21C31×#408—21no—
22C31×#407C2211yescharacteristic
23C31×#407C2211yescharacteristic
24C31×#407C2211yescharacteristic
25C2×C31×C22⋊C8C2211yescharacteristic
26C31×#380C2211yescharacteristic
27C31×#343—21no—
28C31×#343—21no—
29C31×#340—21no—
30C31×#340—21no—
31C31×#328—21no—
32C31×#327—21no—
33C2×#41×C31D411yescharacteristic
34C23×C4×C31—21no—
35C23×C4×C31C2311yescharacteristic
36C2×M16×C31—21no—
37C2×M16×C31—21no—
38C2×M16×C31—21no—
39C2×M16×C31D411yescharacteristic
40C2×M16×C31D411yescharacteristic
41C22×C8×C31—21no—
42C22×C8×C31D411yescharacteristic
43C22×C8×C31D411yescharacteristic
44C31×#123—21no—
45C31×C22⋊Q8C2×C411yescharacteristic
46C31×C22⋊Q8D411yescharacteristic
47C31×#121C2×C411yescharacteristic
48C31×#121D411yescharacteristic
49C31×#120—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C31×#117D411yescharacteristic
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31C2×C412yes—
60C2×C22⋊C4×C31—21no—
61C31×#107—22no—
62C31×#106—22no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×#103—22no—
66C31×#102—22no—
67C31×C22⋊C8—21no—
68C31×C22⋊C8—21no—
69C31×C22⋊C8—21no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8C2×C412yes—
77C31×#97—21no—
78C4×C8×C31—21no—
79C31×#95—21no—
80C24×C31#4111yescharacteristic
81#41×C31—41no—
82#41×C31—41no—
83C2×Q8×C31C22⋊C411yescharacteristic
84C2×D4×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31—21no—
87C2×D4×C31C22⋊C411yescharacteristic
88C22×C4×C31—21no—
89C22×C4×C31—22no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—41no—
93C22×C4×C31—41no—
94C22×C4×C31—41no—
95C22×C4×C31—82no—
96C22×C4×C31C22×C411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31#4111yescharacteristic
102M16×C31—41no—
103M16×C31—41no—
104M16×C31—41no—
105M16×C31—41no—
106M16×C31—41no—
107M16×C31—41no—
108M16×C31—81no—
109C2×C8×C31—22no—
110C2×C8×C31—21no—
111C2×C8×C31—21no—
112C2×C8×C31—21no—
113C2×C8×C31—21no—
114C2×C8×C31—41no—
115C2×C8×C31—41no—
116C2×C8×C31—41no—
117C2×C8×C31—81no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—22no—
122C4⋊C4×C31—41no—
123C4⋊C4×C31—41no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C4⋊C4×C31C22⋊C411yescharacteristic
127C4⋊C4×C31C22⋊C411yescharacteristic
128C22⋊C4×C31—21no—
129C22⋊C4×C31—21no—
130C22⋊C4×C31—21no—
131C22⋊C4×C31—21no—
132C22⋊C4×C31—21no—
133C22⋊C4×C31—81no—
134C42×C31—21no—
135#9521C3111yesmaximal, characteristic, direct product factor, Sylow
136C23×C31—21no—
137C23×C31—21no—
138C23×C31—21no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31C4×D411yescharacteristic
142C23×C31#12111yescharacteristic
143C23×C31#12311yescharacteristic
144Q8×C31—41no—
145D4×C31—41no—
146D4×C31—41no—
147D4×C31—41no—
148D4×C31—81no—
149D4×C31—81no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—42no—
165C2×C4×C31—82no—
166C2×C4×C31—82no—
167C2×C4×C31—82no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31C2×C22⋊C411yescharacteristic
172C2×C4×C31C4×D411yescharacteristic
173C2×C4×C31#12011yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C8×C31—42no—
176C8×C31—41no—
177C8×C31—41no—
178C8×C31—41no—
179C8×C31—41no—
180C8×C31—81no—
181#3092C2×C3111yescharacteristic
182#2976C2×C3111yescharacteristic
183#1824C2×C3111yescharacteristic, complemented
184#1821C2×C3111yescharacteristic, complemented
185#1513C2×C3111yescharacteristic
186#1305C2×C3111yescharacteristic, complemented
187#1293C2×C3111yescharacteristic, complemented
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—42no—
197C22×C31—42no—
198C22×C31—41no—
199C22×C31—41no—
200C22×C31—41no—
201C22×C31#38611yescharacteristic
202C22×C31#47111yescharacteristic
203C22×C31#47311yescharacteristic
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—41no—
208C4×C31—41no—
209C4×C31—41no—
210C4×C31—82no—
211C4×C31—161no—
212C4×C31#45811yescharacteristic
213C4×C31#45811yescharacteristic
214#525C22×C3111yescharacteristic
215C22×C4⋊C4C22×C3111yescharacteristic
216#484—21no—
217#483C4×C3112yescomplemented
218#481—21no—
219#480C4×C3112yescomplemented
220#436—21no—
221#435—21no—
222#435—21no—
223C8×D4—21no—
224#433—21no—
225#428—21no—
226#417—21no—
227#408—21no—
228#407C22×C3111yescharacteristic
229#407C22×C3111yescharacteristic
230#407C22×C3111yescharacteristic
231C2×C22⋊C8C22×C3111yescharacteristic
232#380C22×C3111yescharacteristic
233#343—21no—
234#343—21no—
235#340—21no—
236#340—21no—
237#328—21no—
238#327—21no—
239C2×C31—21no—
240C2×C31—21no—
241C2×C31—41no—
242C2×C31—41no—
243C2×C31—81no—
244C2×C31#189011yescharacteristic
245C2×C31#190311yescharacteristic
246C2×C31#190811yescharacteristic
247C2×#41D4×C3111yescharacteristic
248C23×C4—21no—
249C23×C4C23×C3111yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16D4×C3111yescharacteristic
254C2×M16D4×C3111yescharacteristic
255C22×C8—21no—
256C22×C8D4×C3111yescharacteristic
257C22×C8D4×C3111yescharacteristic
258#123—21no—
259C22⋊Q8C2×C4×C3111yescharacteristic
260C22⋊Q8D4×C3111yescharacteristic
261#121C2×C4×C3111yescharacteristic
262#121D4×C3111yescharacteristic
263#120—21no—
264C4×D4—21no—
265C4×D4—21no—
266#117D4×C3111yescharacteristic
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×C4×C3112yes—
274C2×C22⋊C4—21no—
275#107—22no—
276#106—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279#103—22no—
280#102—22no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8C2×C4×C3112yes—
291#97—21no—
292C4×C8—21no—
293#95—21no—
294C31#952111yescharacteristic, direct product factor, Sylow
295C24#41×C3111yescharacteristic
296#41—41no—
297#41—41no—
298C2×Q8C22⋊C4×C3111yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4C22⋊C4×C3111yescharacteristic
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4C22×C4×C3111yescharacteristic
312C22×C4C2×D4×C3111yescharacteristic
313C22×C4C2×D4×C3111yescharacteristic
314C22×C4C2×D4×C3111yescharacteristic
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4#41×C3111yescharacteristic
317M16—41no—
318M16—41no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—81no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—81no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4C22⋊C4×C3111yescharacteristic
342C4⋊C4C22⋊C4×C3111yescharacteristic
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—81no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23C4×D4×C3111yescharacteristic
356C23C31×#12111yescharacteristic
357C23C31×#12311yescharacteristic
358Q8—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—81no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4C2×C22⋊C4×C3111yescharacteristic
386C2×C4C4×D4×C3111yescharacteristic
387C2×C4C31×#12011yescharacteristic
388C2×C4C31×#12111yescharacteristic
389C8—42no—
390C8—41no—
391C8—41no—
392C8—41no—
393C8—41no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22C31×#38611yescharacteristic
409C22C31×#47111yescharacteristic
410C22C31×#47311yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—161no—
419C4C31×#45811yescharacteristic
420C4C31×#45811yescharacteristic
421C2—21no—
422C2—21no—
423C2—41no—
424C2—41no—
425C2—81no—
426C2C31×#189011yescharacteristic
427C2C31×#190311yescharacteristic
428C2C31×#190811yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#525C22
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
3. C31×#1824C2semidirect
4. C31×#1821C2semidirect
6. C31×#1305C2semidirect
7. C31×#1293C2semidirect
11. C31×#483C4semidirect
13. C31×#480C4semidirect
135. #9521C31direct (N × H)
183. #1824C2×C31semidirect
184. #1821C2×C31semidirect
186. #1305C2×C31semidirect
187. #1293C2×C31semidirect
217. #483C4×C31semidirect
219. #480C4×C31semidirect
294. C31#9521direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30922
C31×#29762
C31×#18242
C31×#18212
C31×#15132
C31×#13052
C31×#12932
#952131