Finite Group Databasebeta

← C31×#10184 #21466943

Subgroups — C31×#10184 #21466943

Order 7936.

582 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3226C211yesmaximal, characteristic, complemented
2C31×#3036C211yesmaximal, characteristic
3C2×C31×#461C211yesmaximal, characteristic, complemented
4C31×#1546C211yesmaximal, characteristic, complemented
5C31×#1480C211yesmaximal, characteristic, complemented
6C31×#1460C211yesmaximal, characteristic, complemented
7C31×#1301C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C22×SD16×C31—21no—
10C31×#532C2211yescharacteristic
11C2×C31×C4⋊Q8C2211yescharacteristic
12C2×C4×D4×C31C2211yescharacteristic
13C31×#500—21no—
14C31×#486—21no—
15C31×#480—22no—
16C31×#474—22no—
17C31×#463—22no—
18C31×#461—22no—
19C31×#461—21no—
20C31×#461—22no—
21C31×#459—21no—
22C31×#431C2211yescharacteristic
23C31×#431C2211yescharacteristic
24C31×#423C2211yescharacteristic
25C2×C31×C4⋊C8C2211yescharacteristic
26C31×#417—21no—
27C2×C31×#103—21no—
28C31×#332—22no—
29C31×D4⋊C8—22no—
30C31×#323—22no—
31C2×#41×C31—21no—
32C22×Q8×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C31×AGL(1,ℤ/8)—42no—
36C31×AGL(1,ℤ/8)—42no—
37C2×SD16×C31—24no—
38C2×SD16×C31—22no—
39C2×SD16×C31—42no—
40C2×SD16×C31—42no—
41C2×D8×C31—22no—
42C2×M16×C31—21no—
43C22×C8×C31—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8D412yes—
48C31×C4⋊Q8D411yescharacteristic
49C31×#127D411yescharacteristic
50C31×#124—21no—
51C31×#121—21no—
52C31×#121—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C4×D4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C22⋊C4×C31—21no—
63C2×C42×C31C2311yescharacteristic
64C31×#107—24no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8—21no—
69C31×C4⋊C8D412yes—
70C31×C4⋊C8D412yes—
71C31×#103—24no—
72C31×#103—22no—
73C31×#102—22no—
74C31×#102—24no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8D412yes—
77C4×C8×C31—21no—
78C4×C8×C31—21no—
79C24×C31—21no—
80#41×C31—41no—
81#41×C31—41no—
82#41×C31—41no—
83#41×C31—41no—
84C2×Q8×C31—21no—
85C2×Q8×C31—22no—
86C2×Q8×C31—41no—
87C2×Q8×C31—41no—
88C2×Q8×C31—41no—
89C2×Q8×C31—41no—
90C2×Q8×C31—41no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—21no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—41no—
100C2×D4×C31D812yes—
101C22×C4×C31—21no—
102C22×C4×C31—41no—
103C22×C4×C31—41no—
104C22×C4×C31—41no—
105C22×C4×C31—41no—
106C22×C4×C31—82no—
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31#4111yescharacteristic
110SD16×C31—48no—
111SD16×C31—44no—
112D8×C31—44no—
113M16×C31—41no—
114M16×C31—41no—
115C2×C8×C31—22no—
116C2×C8×C31—22no—
117C2×C8×C31—22no—
118C2×C8×C31—41no—
119C2×C8×C31—41no—
120C2×C8×C31—41no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—41no—
128C4⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—41no—
131C22⋊C4×C31—41no—
132C22⋊C4×C31—41no—
133C42×C31C2×D411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31C2×D411yescharacteristic
136C42×C31C2×D411yescharacteristic
137#10184C3111yesmaximal, characteristic, direct product factor, Sylow
138C23×C31—22no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—81no—
143C23×C31—81no—
144C23×C31#12111yescharacteristic
145Q8×C31—42no—
146Q8×C31—41no—
147Q8×C31—42no—
148Q8×C31—41no—
149Q8×C31—81no—
150Q8×C31—81no—
151D4×C31—22no—
152D4×C31—22no—
153D4×C31—42no—
154D4×C31—41no—
155D4×C31—41no—
156D4×C31—41no—
157D4×C31—41no—
158D4×C31—42no—
159D4×C31—81no—
160D4×C31—81no—
161C2×C4×C31—22no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—42no—
166C2×C4×C31—82no—
167C2×C4×C31—81no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31—82no—
172C2×C4×C31—81no—
173C2×C4×C31#12011yescharacteristic
174C2×C4×C31#12011yescharacteristic
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12111yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C2×C4×C31#12711yescharacteristic
179C2×C4×C31C2×D811yescharacteristic
180C2×C4×C31#13511yescharacteristic
181C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
183C8×C31—42no—
184C8×C31—42no—
185C8×C31—42no—
186C8×C31—81no—
187#3226C2×C3111yescharacteristic, complemented
188#3036C2×C3111yescharacteristic
189C2×#461C2×C3111yescharacteristic, complemented
190#1546C2×C3111yescharacteristic, complemented
191#1480C2×C3111yescharacteristic, complemented
192#1460C2×C3111yescharacteristic, complemented
193#1301C2×C3111yescharacteristic, complemented
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—42no—
197C22×C31—82no—
198C22×C31—82no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31#39211yescharacteristic
205C22×C31#46611yescharacteristic
206C22×C31#46911yescharacteristic
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—81no—
211C4×C31—82no—
212C4×C31—81no—
213C4×C31#44711yescharacteristic
214C4×C31#44711yescharacteristic
215C4×C31#44911yescharacteristic
216C4×C31#44911yescharacteristic
217C4×C31#47112yes—
218C2×AGL(1,ℤ/8)—21no—
219C22×SD16—21no—
220#532C22×C3111yescharacteristic
221C2×C4⋊Q8C22×C3111yescharacteristic
222C2×C4×D4C22×C3111yescharacteristic
223#500—21no—
224#486—21no—
225#480—22no—
226#474—22no—
227#463—22no—
228#461—22no—
229#461—21no—
230#461—22no—
231#459—21no—
232#431C22×C3111yescharacteristic
233#431C22×C3111yescharacteristic
234#423C22×C3111yescharacteristic
235C2×C4⋊C8C22×C3111yescharacteristic
236#417—21no—
237C2×#103—21no—
238#332—22no—
239D4⋊C8—22no—
240#323—22no—
241C2×C31—21no—
242C2×C31—21no—
243C2×C31—82no—
244C2×C31—161no—
245C2×C31#199911yescharacteristic
246C2×C31#200911yescharacteristic
247C2×C31#201911yescharacteristic
248C2×#41—21no—
249C22×Q8—21no—
250C22×D4D4×C3111yescharacteristic
251C23×C4—21no—
252AGL(1,ℤ/8)—42no—
253AGL(1,ℤ/8)—42no—
254C2×SD16—24no—
255C2×SD16—22no—
256C2×SD16—42no—
257C2×SD16—42no—
258C2×D8—22no—
259C2×M16—21no—
260C22×C8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8D4×C3112yes—
265C4⋊Q8D4×C3111yescharacteristic
266#127D4×C3111yescharacteristic
267#124—21no—
268#121—21no—
269#121—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4D4×C3111yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C23×C3111yescharacteristic
281#107—24no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8D4×C3112yes—
287C4⋊C8D4×C3112yes—
288#103—24no—
289#103—22no—
290#102—22no—
291#102—24no—
292C22⋊C8—21no—
293C22⋊C8D4×C3112yes—
294C4×C8—21no—
295C4×C8—21no—
296C31#1018411yescharacteristic, direct product factor, Sylow
297C24—21no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4D8×C3112yes—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—82no—
325C22×C4C2×D4×C3111yescharacteristic
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4#41×C3111yescharacteristic
328SD16—48no—
329SD16—44no—
330D8—44no—
331M16—41no—
332M16—41no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C42C2×D4×C3111yescharacteristic
352C42C2×D4×C3111yescharacteristic
353C42C2×D4×C3111yescharacteristic
354C42C2×D4×C3111yescharacteristic
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—81no—
360C23—81no—
361C23C31×#12111yescharacteristic
362Q8—42no—
363Q8—41no—
364Q8—42no—
365Q8—41no—
366Q8—81no—
367Q8—81no—
368D4—22no—
369D4—22no—
370D4—42no—
371D4—41no—
372D4—41no—
373D4—41no—
374D4—41no—
375D4—42no—
376D4—81no—
377D4—81no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4C31×#12011yescharacteristic
391C2×C4C31×#12011yescharacteristic
392C2×C4C31×#12011yescharacteristic
393C2×C4C31×#12111yescharacteristic
394C2×C4C31×#12111yescharacteristic
395C2×C4C31×#12711yescharacteristic
396C2×C4C2×D8×C3111yescharacteristic
397C2×C4C31×#13511yescharacteristic
398C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
399C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
400C8—42no—
401C8—42no—
402C8—42no—
403C8—81no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—82no—
408C22—82no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22C31×#39211yescharacteristic
415C22C31×#46611yescharacteristic
416C22C31×#46911yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—81no—
421C4—82no—
422C4—81no—
423C4C31×#44711yescharacteristic
424C4C31×#44711yescharacteristic
425C4C31×#44911yescharacteristic
426C4C31×#44911yescharacteristic
427C4C31×#47112yes—
428C2—21no—
429C2—21no—
430C2—82no—
431C2—161no—
432C2C31×#199911yescharacteristic
433C2C31×#200911yescharacteristic
434C2C31×#201911yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3036C2
LCSG ▷ C2×C42 ▷ 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×#3226C2semidirect
3. C2×C31×#461C2semidirect
4. C31×#1546C2semidirect
5. C31×#1480C2semidirect
6. C31×#1460C2semidirect
7. C31×#1301C2semidirect
137. #10184C31direct (N × H)
187. #3226C2×C31semidirect
189. C2×#461C2×C31semidirect
190. #1546C2×C31semidirect
191. #1480C2×C31semidirect
192. #1460C2×C31semidirect
193. #1301C2×C31semidirect
296. C31#10184direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32262
C31×#30362
C2×C31×#4612
C31×#15462
C31×#14802
C31×#14602
C31×#13012
#1018431