Finite Group Databasebeta

← C31×#10180 #21466939

Subgroups — C31×#10180 #21466939

Order 7936.

562 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×#3206C211yesmaximal, characteristic, complemented
2C2×C31×#476C211yesmaximal, characteristic, complemented
3C31×#3036C211yesmaximal, characteristic
4C31×#1549C211yesmaximal, characteristic, complemented
5C31×#1480C211yesmaximal, characteristic, complemented
6C31×#1460C211yesmaximal, characteristic, complemented
7C31×#1301C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#532C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#492—21no—
13C31×#487—21no—
14C31×#476—22no—
15C31×#476—22no—
16C31×#463—22no—
17C31×#461—22no—
18C31×#461—21no—
19C31×#459—21no—
20C31×#448—22no—
21C31×#431C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C2×C31×#106—21no—
24C31×#423C2211yescharacteristic
25C2×C31×C4⋊C8C2211yescharacteristic
26C31×#417—21no—
27C2×C31×#102—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—22no—
38C2×SD16×C31—24no—
39C2×D8×C31—22no—
40C2×M16×C31—21no—
41C22×C8×C31—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8D412yes—
46C31×C4⋊Q8D411yescharacteristic
47C31×#127D411yescharacteristic
48C31×#124—21no—
49C31×#121—21no—
50C31×#121—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C22⋊C4×C31—21no—
61C2×C42×C31C2311yescharacteristic
62C31×#106—24no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8D412yes—
68C31×C4⋊C8D412yes—
69C31×#103—22no—
70C31×#103—24no—
71C31×#102—24no—
72C31×#102—22no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8D412yes—
75C4×C8×C31—21no—
76C4×C8×C31—21no—
77C24×C31—21no—
78#41×C31—41no—
79#41×C31—41no—
80#41×C31—41no—
81#41×C31—41no—
82C2×Q8×C31—21no—
83C2×Q8×C31—22no—
84C2×Q8×C31—41no—
85C2×Q8×C31—82no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—21no—
94C2×D4×C31—41no—
95C2×D4×C31SD1612yescomplemented
96C22×C4×C31—21no—
97C22×C4×C31—41no—
98C22×C4×C31—41no—
99C22×C4×C31—41no—
100C22×C4×C31—41no—
101C22×C4×C31—82no—
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31#4111yescharacteristic
105SD16×C31—44no—
106SD16×C31—84no—
107D8×C31—44no—
108M16×C31—41no—
109M16×C31—41no—
110C2×C8×C31—22no—
111C2×C8×C31—22no—
112C2×C8×C31—22no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C2×C8×C31—41no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—41no—
123C4⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C42×C31C2×D411yescharacteristic
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132#10180C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—22no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—81no—
138C23×C31—81no—
139C23×C31#12111yescharacteristic
140Q8×C31—44no—
141Q8×C31—41no—
142Q8×C31—41no—
143Q8×C31—82no—
144D4×C31—22no—
145D4×C31—22no—
146D4×C31—42no—
147D4×C31—41no—
148D4×C31—41no—
149D4×C31—41no—
150D4×C31—41no—
151D4×C31—42no—
152D4×C31—81no—
153D4×C31—81no—
154C2×C4×C31—22no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—42no—
159C2×C4×C31—82no—
160C2×C4×C31—82no—
161C2×C4×C31—81no—
162C2×C4×C31—81no—
163C2×C4×C31—82no—
164C2×C4×C31—81no—
165C2×C4×C31#12011yescharacteristic
166C2×C4×C31#12011yescharacteristic
167C2×C4×C31#12011yescharacteristic
168C2×C4×C31#12111yescharacteristic
169C2×C4×C31#12111yescharacteristic
170C2×C4×C31#12711yescharacteristic
171C2×C4×C31C2×SD1611yescharacteristic
172C2×C4×C31#13511yescharacteristic
173C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C31#13711yescharacteristic
175C8×C31—42no—
176C8×C31—42no—
177C8×C31—42no—
178C8×C31—81no—
179#3206C2×C3111yescharacteristic, complemented
180C2×#476C2×C3111yescharacteristic, complemented
181#3036C2×C3111yescharacteristic
182#1549C2×C3111yescharacteristic, complemented
183#1480C2×C3111yescharacteristic, complemented
184#1460C2×C3111yescharacteristic, complemented
185#1301C2×C3111yescharacteristic, complemented
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—42no—
189C22×C31—82no—
190C22×C31—82no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31#39211yescharacteristic
197C22×C31#46511yescharacteristic
198C22×C31#46811yescharacteristic
199C4×C31—21no—
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—81no—
203C4×C31—82no—
204C4×C31—81no—
205C4×C31#44811yescharacteristic
206C4×C31#44911yescharacteristic
207C4×C31#45011yescharacteristic
208C4×C31#45211yescharacteristic
209C4×C31#47112yes—
210C2×AGL(1,ℤ/8)—21no—
211#532C22×C3111yescharacteristic
212C2×C4⋊Q8C22×C3111yescharacteristic
213C2×C4×D4C22×C3111yescharacteristic
214#492—21no—
215#487—21no—
216#476—22no—
217#476—22no—
218#463—22no—
219#461—22no—
220#461—21no—
221#459—21no—
222#448—22no—
223#431C22×C3111yescharacteristic
224#431C22×C3111yescharacteristic
225C2×#106—21no—
226#423C22×C3111yescharacteristic
227C2×C4⋊C8C22×C3111yescharacteristic
228#417—21no—
229C2×#102—21no—
230#332—22no—
231D4⋊C8—22no—
232#323—22no—
233C2×C31—21no—
234C2×C31—21no—
235C2×C31—82no—
236C2×C31—161no—
237C2×C31#200011yescharacteristic
238C2×C31#200811yescharacteristic
239C2×C31#201911yescharacteristic
240C2×#41—21no—
241C22×Q8—21no—
242C22×D4D4×C3111yescharacteristic
243C23×C4—21no—
244AGL(1,ℤ/8)—42no—
245AGL(1,ℤ/8)—42no—
246C2×SD16—22no—
247C2×SD16—24no—
248C2×D8—22no—
249C2×M16—21no—
250C22×C8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8D4×C3112yes—
255C4⋊Q8D4×C3111yescharacteristic
256#127D4×C3111yescharacteristic
257#124—21no—
258#121—21no—
259#121—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4D4×C3111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C23×C3111yescharacteristic
271#106—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8D4×C3112yes—
277C4⋊C8D4×C3112yes—
278#103—22no—
279#103—24no—
280#102—24no—
281#102—22no—
282C22⋊C8—21no—
283C22⋊C8D4×C3112yes—
284C4×C8—21no—
285C4×C8—21no—
286C31#1018011yescharacteristic, direct product factor, Sylow
287C24—21no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4SD16×C3112yescomplemented
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4C2×D4×C3111yescharacteristic
313C22×C4C2×D4×C3111yescharacteristic
314C22×C4#41×C3111yescharacteristic
315SD16—44no—
316SD16—84no—
317D8—44no—
318M16—41no—
319M16—41no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42C2×D4×C3111yescharacteristic
339C42C2×D4×C3111yescharacteristic
340C42C2×D4×C3111yescharacteristic
341C42C2×D4×C3111yescharacteristic
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—81no—
347C23—81no—
348C23C31×#12111yescharacteristic
349Q8—44no—
350Q8—41no—
351Q8—41no—
352Q8—82no—
353D4—22no—
354D4—22no—
355D4—42no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—42no—
361D4—81no—
362D4—81no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4C31×#12011yescharacteristic
375C2×C4C31×#12011yescharacteristic
376C2×C4C31×#12011yescharacteristic
377C2×C4C31×#12111yescharacteristic
378C2×C4C31×#12111yescharacteristic
379C2×C4C31×#12711yescharacteristic
380C2×C4C2×SD16×C3111yescharacteristic
381C2×C4C31×#13511yescharacteristic
382C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C31×#13711yescharacteristic
384C8—42no—
385C8—42no—
386C8—42no—
387C8—81no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22C31×#39211yescharacteristic
399C22C31×#46511yescharacteristic
400C22C31×#46811yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4C31×#44811yescharacteristic
408C4C31×#44911yescharacteristic
409C4C31×#45011yescharacteristic
410C4C31×#45211yescharacteristic
411C4C31×#47112yes—
412C2—21no—
413C2—21no—
414C2—82no—
415C2—161no—
416C2C31×#200011yescharacteristic
417C2C31×#200811yescharacteristic
418C2C31×#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×#3206C2semidirect
2. C2×C31×#476C2semidirect
4. C31×#1549C2semidirect
5. C31×#1480C2semidirect
6. C31×#1460C2semidirect
7. C31×#1301C2semidirect
95. C2×D4×C31SD16semidirect
132. #10180C31direct (N × H)
179. #3206C2×C31semidirect
180. C2×#476C2×C31semidirect
182. #1549C2×C31semidirect
183. #1480C2×C31semidirect
184. #1460C2×C31semidirect
185. #1301C2×C31semidirect
286. C31#10180direct (N × H)
305. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32062
C2×C31×#4762
C31×#30362
C31×#15492
C31×#14802
C31×#14602
C31×#13012
#1018031