Finite Group Databasebeta

← C31×#8496 #21465255

Subgroups — C31×#8496 #21465255

Order 7936.

676 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3550C211yesmaximal, characteristic
2C31×#1995C211yesmaximal, characteristic, complemented
3C31×#1559C211yesmaximal, characteristic, complemented
4C31×#1557C211yesmaximal, characteristic
5C31×#1537C211yesmaximal, characteristic, complemented
6C31×#1506C211yesmaximal, characteristic, complemented
7C31×#1505C211yesmaximal, characteristic
8C31×#563—22no—
9C31×#559—21no—
10Q82×C31—21no—
11C31×#556—21no—
12C31×#556C412yescomplemented
13D4×Q8×C31—21no—
14D4×Q8×C31C412yescomplemented
15C31×#542—22no—
16C31×#542—22no—
17C31×#536—22no—
18C31×#533C2211yescharacteristic
19C31×#533C2211yescharacteristic
20C31×#532C2211yescharacteristic
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C31×#376C2211yescharacteristic
25C31×#339—22no—
26C31×#333—21no—
27C31×#332—21no—
28C31×#332—21no—
29C31×#331—21no—
30C31×#330—22no—
31C31×#328—22no—
32C31×#327—22no—
33C2×#41×C31—21no—
34C22×Q8×C31—22no—
35C2×M16×C31—21no—
36C31×C4⋊Q8—24no—
37C31×C4⋊Q8—22no—
38C31×C4⋊Q8—22no—
39C31×C4⋊Q8C2×C411yescharacteristic
40C31×C4⋊Q8C2×C411yescharacteristic
41C31×C4⋊Q8D411yescharacteristic
42C31×C4⋊Q8D411yescharacteristic
43C31×C4⋊Q8D411yescharacteristic
44C31×#127D411yescharacteristic
45C31×#126—24no—
46C31×#126—24no—
47C31×#125—21no—
48C31×#125D412yes—
49C31×#124—24no—
50C31×#124—22no—
51C31×#124—22no—
52C31×#124—21no—
53C31×#123—24no—
54C31×#123—24no—
55C31×C22⋊Q8—24no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—21no—
58C31×C22⋊Q8—21no—
59C31×C22⋊Q8—21no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8D412yes—
63C31×#121—21no—
64C31×#121D412yes—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31—21no—
68C4×Q8×C31—21no—
69C4×Q8×C31—21no—
70C4×Q8×C31C2×C412yes—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—21no—
74C4×D4×C31C2×C412yes—
75C31×#117—21no—
76C31×#117—22no—
77C31×#117D412yes—
78C2×C42×C31—21no—
79C2×C42×C31C2311yescharacteristic
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8—21no—
84C31×C22⋊C8—21no—
85C31×C22⋊C8—21no—
86C31×#97—21no—
87C31×#97—21no—
88C31×#95—21no—
89C31×#95—21no—
90#41×C31—82no—
91C2×Q8×C31—24no—
92C2×Q8×C31—21no—
93C2×Q8×C31—21no—
94C2×Q8×C31—21no—
95C2×Q8×C31—21no—
96C2×Q8×C31—84no—
97C2×Q8×C31C22⋊C412yes—
98C2×D4×C31—22no—
99C2×D4×C31—21no—
100C2×D4×C31—21no—
101C2×D4×C31C22⋊C412yes—
102C22×C4×C31—24no—
103C22×C4×C31—22no—
104C22×C4×C31—41no—
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108M16×C31—42no—
109C2×C8×C31—41no—
110C2×C8×C31—41no—
111C2×C8×C31—41no—
112C4⋊C4×C31—24no—
113C4⋊C4×C31—24no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—22no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—22no—
124C4⋊C4×C31C22⋊C412yes—
125C4⋊C4×C31C22⋊C412yes—
126C4⋊C4×C31C22⋊C412yes—
127C4⋊C4×C31C22⋊C412yes—
128C22⋊C4×C31—24no—
129C22⋊C4×C31—24no—
130C22⋊C4×C31—24no—
131C22⋊C4×C31—22no—
132C22⋊C4×C31—22no—
133C22⋊C4×C31—22no—
134C22⋊C4×C31—22no—
135C42×C31—24no—
136C42×C31—22no—
137C42×C31—22no—
138C42×C31—22no—
139C42×C31—41no—
140C42×C31C22×C411yescharacteristic
141C42×C31C2×D411yescharacteristic
142C42×C31C2×D411yescharacteristic
143C42×C31C2×D411yescharacteristic
144#8496C3111yesmaximal, characteristic, direct product factor, Sylow
145C23×C31—22no—
146C23×C31#12011yescharacteristic
147Q8×C31—42no—
148Q8×C31—42no—
149Q8×C31—42no—
150Q8×C31—41no—
151Q8×C31—41no—
152Q8×C31—81no—
153Q8×C31—84no—
154D4×C31—42no—
155D4×C31—41no—
156D4×C31—41no—
157D4×C31—81no—
158D4×C31—82no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—44no—
169C2×C4×C31—42no—
170C2×C4×C31—84no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31C2×C22⋊C411yescharacteristic
174C2×C4×C31C2×C22⋊C411yescharacteristic
175C2×C4×C31C2×C22⋊C411yescharacteristic
176C2×C4×C31#12011yescharacteristic
177C2×C4×C31#12011yescharacteristic
178C2×C4×C31#12011yescharacteristic
179C8×C31—81no—
180C8×C31—81no—
181C8×C31—81no—
182#3550C2×C3111yescharacteristic
183#1995C2×C3111yescharacteristic, complemented
184#1559C2×C3111yescharacteristic, complemented
185#1557C2×C3111yescharacteristic
186#1537C2×C3111yescharacteristic, complemented
187#1506C2×C3111yescharacteristic, complemented
188#1505C2×C3111yescharacteristic
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—42no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31#37911yescharacteristic
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—21no—
201C4×C31—42no—
202C4×C31—42no—
203C4×C31—42no—
204C4×C31—42no—
205C4×C31—42no—
206C4×C31—82no—
207C4×C31#45411yescharacteristic
208C4×C31#45611yescharacteristic
209#563—22no—
210#559—21no—
211Q82—21no—
212#556—21no—
213#556C4×C3112yescomplemented
214D4×Q8—21no—
215D4×Q8C4×C3112yescomplemented
216#542—22no—
217#542—22no—
218#536—22no—
219#533C22×C3111yescharacteristic
220#533C22×C3111yescharacteristic
221#532C22×C3111yescharacteristic
222#432C22×C3111yescharacteristic
223#432C22×C3111yescharacteristic
224#423C22×C3111yescharacteristic
225#376C22×C3111yescharacteristic
226#339—22no—
227#333—21no—
228#332—21no—
229#332—21no—
230#331—21no—
231#330—22no—
232#328—22no—
233#327—22no—
234C2×C31—41no—
235C2×C31—82no—
236C2×C31#179311yescharacteristic
237C2×C31#179511yescharacteristic
238C2×C31#179611yescharacteristic
239C2×#41—21no—
240C22×Q8—22no—
241C2×M16—21no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8C2×C4×C3111yescharacteristic
246C4⋊Q8C2×C4×C3111yescharacteristic
247C4⋊Q8D4×C3111yescharacteristic
248C4⋊Q8D4×C3111yescharacteristic
249C4⋊Q8D4×C3111yescharacteristic
250#127D4×C3111yescharacteristic
251#126—24no—
252#126—24no—
253#125—21no—
254#125D4×C3112yes—
255#124—24no—
256#124—22no—
257#124—22no—
258#124—21no—
259#123—24no—
260#123—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—22no—
267C22⋊Q8—24no—
268C22⋊Q8D4×C3112yes—
269#121—21no—
270#121D4×C3112yes—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8C2×C4×C3112yes—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—21no—
280C4×D4C2×C4×C3112yes—
281#117—21no—
282#117—22no—
283#117D4×C3112yes—
284C2×C42—21no—
285C2×C42C23×C3111yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292#97—21no—
293#97—21no—
294#95—21no—
295#95—21no—
296C31#849611yescharacteristic, direct product factor, Sylow
297#41—82no—
298C2×Q8—24no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—84no—
304C2×Q8C22⋊C4×C3112yes—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4C22⋊C4×C3112yes—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4C2×D4×C3111yescharacteristic
313C22×C4C2×D4×C3111yescharacteristic
314C22×C4C2×D4×C3111yescharacteristic
315M16—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4C22⋊C4×C3112yes—
332C4⋊C4C22⋊C4×C3112yes—
333C4⋊C4C22⋊C4×C3112yes—
334C4⋊C4C22⋊C4×C3112yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C42—24no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—41no—
347C42C22×C4×C3111yescharacteristic
348C42C2×D4×C3111yescharacteristic
349C42C2×D4×C3111yescharacteristic
350C42C2×D4×C3111yescharacteristic
351C23—22no—
352C23C31×#12011yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—42no—
356Q8—41no—
357Q8—41no—
358Q8—81no—
359Q8—84no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—82no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—84no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×C22⋊C4×C3111yescharacteristic
380C2×C4C2×C22⋊C4×C3111yescharacteristic
381C2×C4C2×C22⋊C4×C3111yescharacteristic
382C2×C4C31×#12011yescharacteristic
383C2×C4C31×#12011yescharacteristic
384C2×C4C31×#12011yescharacteristic
385C8—81no—
386C8—81no—
387C8—81no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22C31×#37911yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4C31×#45411yescharacteristic
407C4C31×#45611yescharacteristic
408C2—41no—
409C2—82no—
410C2C31×#179311yescharacteristic
411C2C31×#179511yescharacteristic
412C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×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×#1995C2semidirect
3. C31×#1559C2semidirect
5. C31×#1537C2semidirect
6. C31×#1506C2semidirect
12. C31×#556C4semidirect
14. D4×Q8×C31C4semidirect
144. #8496C31direct (N × H)
183. #1995C2×C31semidirect
184. #1559C2×C31semidirect
186. #1537C2×C31semidirect
187. #1506C2×C31semidirect
213. #556C4×C31semidirect
215. D4×Q8C4×C31semidirect
296. C31#8496direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35502
C31×#19952
C31×#15592
C31×#15572
C31×#15372
C31×#15062
C31×#15052
#849631