Finite Group Databasebeta

← C31×#19512 #21476271

Subgroups — C31×#19512 #21476271

Order 7936.

588 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3235C211yesmaximal, characteristic
2C31×#3229C211yesmaximal, characteristic, complemented
3C31×#3206C211yesmaximal, characteristic
4C31×#2981C211yesmaximal, characteristic
5C31×#1699C212yesmaximal, complemented
6C31×#1691C212yesmaximal, complemented
7C31×#1679C212yesmaximal, complemented
8C31×#1671C212yesmaximal, complemented
9C31×#1572C211yesmaximal, characteristic, complemented
10C31×#1556C211yesmaximal, characteristic, complemented
11C31×#1554C211yesmaximal, characteristic, complemented
12C31×#533C2211yescharacteristic
13C31×#532C2211yescharacteristic
14C2×C31×C4⋊Q8C2211yescharacteristic
15C31×#501—21no—
16C31×#498C2211yescharacteristic
17C31×#498C2211yescharacteristic
18C31×#492—21no—
19C31×#490—21no—
20C31×#487—21no—
21C31×#487C2211yescharacteristic
22C31×#487C2211yescharacteristic
23C31×#483—22no—
24C31×#482—22no—
25C31×#476C2212yes—
26C31×#475—22no—
27C31×#474C2212yes—
28C31×#461C2212yes—
29C31×#461C2212yes—
30C31×#448—22no—
31C31×#436C2212yes—
32C31×#435—22no—
33C8×D4×C31C2212yes—
34C31×#432C2211yescharacteristic
35C31×#432C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#408—22no—
39C31×#335C2212yes—
40C31×#334C2212yes—
41C31×#333C2212yes—
42C31×#332C2212yes—
43C31×#332C2212yes—
44C31×#332C2212yes—
45C2×#41×C31—21no—
46C22×Q8×C31—21no—
47C2×SD16×C31—22no—
48C2×SD16×C31—22no—
49C2×M16×C31—22no—
50C22×C8×C31—22no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8D411yescharacteristic
55C31×C4⋊Q8C2311yescharacteristic
56C31×C4⋊Q8C2311yescharacteristic
57C31×C4⋊Q8C2311yescharacteristic
58C31×C4⋊Q8C2311yescharacteristic
59C31×#127D411yescharacteristic
60C31×#125—21no—
61C31×#124—21no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×#121—21no—
65C31×#121D412yes—
66C4×Q8×C31—21no—
67C4×Q8×C31—21no—
68C4×D4×C31—21no—
69C4×D4×C31C2312yes—
70C31×#117—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C42×C31C2311yescharacteristic
74C31×#107—22no—
75C31×#106—22no—
76C31×#106—22no—
77C31×#106—22no—
78C31×C4⋊C8D412yes—
79C31×C4⋊C8C2312yes—
80C31×C4⋊C8C2312yes—
81C31×C4⋊C8C2312yes—
82C31×#103—22no—
83C31×#103—22no—
84C31×#103—22no—
85C31×#103—22no—
86C31×#102—22no—
87C31×#102—22no—
88C31×C22⋊C8—21no—
89C31×C22⋊C8—21no—
90C31×C22⋊C8—21no—
91C31×C22⋊C8—22no—
92C31×C22⋊C8—22no—
93C31×C22⋊C8D412yes—
94C31×#97—21no—
95C31×#97—21no—
96C4×C8×C31—21no—
97C4×C8×C31C2311yescharacteristic
98C4×C8×C31C2311yescharacteristic
99#41×C31—82no—
100C2×Q8×C31—21no—
101C2×Q8×C31—21no—
102C2×Q8×C31—21no—
103C2×Q8×C31—21no—
104C2×Q8×C31—21no—
105C2×Q8×C31—41no—
106C2×Q8×C31—82no—
107C2×D4×C31—22no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31C2×D412yes—
111C22×C4×C31—22no—
112C22×C4×C31—41no—
113C22×C4×C31C2×D411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31#4111yescharacteristic
116SD16×C31—82no—
117SD16×C31—82no—
118M16×C31—82no—
119C2×C8×C31—22no—
120C2×C8×C31—22no—
121C2×C8×C31—22no—
122C2×C8×C31—82no—
123C2×C8×C31C2×D412yes—
124C2×C8×C31C2×D412yes—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—41no—
135C4⋊C4×C31—41no—
136C4⋊C4×C31—41no—
137C4⋊C4×C31—41no—
138C4⋊C4×C31C2×D412yes—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—41no—
142C42×C31—41no—
143C42×C31C2×D411yescharacteristic
144C42×C31C2×D411yescharacteristic
145C42×C31#4111yescharacteristic
146C42×C31C2411yescharacteristic
147#19512C3111yesmaximal, characteristic, direct product factor, Sylow
148C23×C31—22no—
149C23×C31#12111yescharacteristic
150Q8×C31—42no—
151Q8×C31—42no—
152Q8×C31—42no—
153Q8×C31—81no—
154Q8×C31—81no—
155Q8×C31—82no—
156D4×C31—22no—
157D4×C31—22no—
158D4×C31—81no—
159D4×C31—81no—
160D4×C31—82no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—41no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—82no—
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31C22×D411yescharacteristic
176C2×C4×C31C22×D411yescharacteristic
177C2×C4×C31C2×#4111yescharacteristic
178C8×C31—22no—
179C8×C31—22no—
180C8×C31—42no—
181C8×C31—42no—
182C8×C31—42no—
183#3235C2×C3111yescharacteristic
184#3229C2×C3111yescharacteristic, complemented
185#3206C2×C3111yescharacteristic
186#2981C2×C3111yescharacteristic
187#1699C2×C3112yescomplemented
188#1691C2×C3112yescomplemented
189#1679C2×C3112yescomplemented
190#1671C2×C3112yescomplemented
191#1572C2×C3111yescharacteristic, complemented
192#1556C2×C3111yescharacteristic, complemented
193#1554C2×C3111yescharacteristic, complemented
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—42no—
198C22×C31—42no—
199C22×C31—42no—
200C22×C31C2×#12111yescharacteristic
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—42no—
206C4×C31—81no—
207C4×C31—81no—
208C4×C31—81no—
209C4×C31—81no—
210C4×C31#57712yes—
211C4×C31#57712yes—
212#533C22×C3111yescharacteristic
213#532C22×C3111yescharacteristic
214C2×C4⋊Q8C22×C3111yescharacteristic
215#501—21no—
216#498C22×C3111yescharacteristic
217#498C22×C3111yescharacteristic
218#492—21no—
219#490—21no—
220#487—21no—
221#487C22×C3111yescharacteristic
222#487C22×C3111yescharacteristic
223#483—22no—
224#482—22no—
225#476C22×C3112yes—
226#475—22no—
227#474C22×C3112yes—
228#461C22×C3112yes—
229#461C22×C3112yes—
230#448—22no—
231#436C22×C3112yes—
232#435—22no—
233C8×D4C22×C3112yes—
234#432C22×C3111yescharacteristic
235#432C22×C3111yescharacteristic
236#431C22×C3111yescharacteristic
237#431C22×C3111yescharacteristic
238#408—22no—
239#335C22×C3112yes—
240#334C22×C3112yes—
241#333C22×C3112yes—
242#332C22×C3112yes—
243#332C22×C3112yes—
244#332C22×C3112yes—
245C2×C31—41no—
246C2×C31—82no—
247C2×C31#304211yescharacteristic
248C2×C31#305711yescharacteristic
249C2×C31#306911yescharacteristic
250C2×#41—21no—
251C22×Q8—21no—
252C2×SD16—22no—
253C2×SD16—22no—
254C2×M16—22no—
255C22×C8—22no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8D4×C3111yescharacteristic
260C4⋊Q8C23×C3111yescharacteristic
261C4⋊Q8C23×C3111yescharacteristic
262C4⋊Q8C23×C3111yescharacteristic
263C4⋊Q8C23×C3111yescharacteristic
264#127D4×C3111yescharacteristic
265#125—21no—
266#124—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269#121—21no—
270#121D4×C3112yes—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×D4—21no—
274C4×D4C23×C3112yes—
275#117—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C42C23×C3111yescharacteristic
279#107—22no—
280#106—22no—
281#106—22no—
282#106—22no—
283C4⋊C8D4×C3112yes—
284C4⋊C8C23×C3112yes—
285C4⋊C8C23×C3112yes—
286C4⋊C8C23×C3112yes—
287#103—22no—
288#103—22no—
289#103—22no—
290#103—22no—
291#102—22no—
292#102—22no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8D4×C3112yes—
299#97—21no—
300#97—21no—
301C4×C8—21no—
302C4×C8C23×C3111yescharacteristic
303C4×C8C23×C3111yescharacteristic
304C31#1951211yescharacteristic, direct product factor, Sylow
305#41—82no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—41no—
312C2×Q8—82no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4C2×D4×C3112yes—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4C2×D4×C3111yescharacteristic
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4#41×C3111yescharacteristic
322SD16—82no—
323SD16—82no—
324M16—82no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—82no—
329C2×C8C2×D4×C3112yes—
330C2×C8C2×D4×C3112yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4C2×D4×C3112yes—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—41no—
348C42—41no—
349C42C2×D4×C3111yescharacteristic
350C42C2×D4×C3111yescharacteristic
351C42#41×C3111yescharacteristic
352C42C24×C3111yescharacteristic
353C23—22no—
354C23C31×#12111yescharacteristic
355Q8—42no—
356Q8—42no—
357Q8—42no—
358Q8—81no—
359Q8—81no—
360Q8—82no—
361D4—22no—
362D4—22no—
363D4—81no—
364D4—81no—
365D4—82no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4C31×#12111yescharacteristic
378C2×C4C31×#12111yescharacteristic
379C2×C4C31×#12111yescharacteristic
380C2×C4C22×D4×C3111yescharacteristic
381C2×C4C22×D4×C3111yescharacteristic
382C2×C4C2×#41×C3111yescharacteristic
383C8—22no—
384C8—22no—
385C8—42no—
386C8—42no—
387C8—42no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22C2×C31×#12111yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4C31×#57712yes—
405C4C31×#57712yes—
406C2—41no—
407C2—82no—
408C2C31×#304211yescharacteristic
409C2C31×#305711yescharacteristic
410C2C31×#306911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3229C2semidirect
5. C31×#1699C2semidirect
6. C31×#1691C2semidirect
7. C31×#1679C2semidirect
8. C31×#1671C2semidirect
9. C31×#1572C2semidirect
10. C31×#1556C2semidirect
11. C31×#1554C2semidirect
147. #19512C31direct (N × H)
184. #3229C2×C31semidirect
187. #1699C2×C31semidirect
188. #1691C2×C31semidirect
189. #1679C2×C31semidirect
190. #1671C2×C31semidirect
191. #1572C2×C31semidirect
192. #1556C2×C31semidirect
193. #1554C2×C31semidirect
304. C31#19512direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32352
C31×#32292
C31×#32062
C31×#29812
C31×#16992
C31×#16912
C31×#16792
C31×#16712
C31×#15722
C31×#15562
C31×#15542
#1951231