Finite Group Databasebeta

← C31×#9237 #21465996

Subgroups — C31×#9237 #21465996

Order 7936.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3100C211yesmaximal, characteristic
2C31×#2293C211yesmaximal, characteristic, complemented
3C31×#1962C211yesmaximal, characteristic, complemented
4C31×#1502C211yesmaximal, characteristic, complemented
5C31×#1486C211yesmaximal, characteristic, complemented
6C31×#1303C211yesmaximal, characteristic, complemented
7C31×#1276C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C4×D4×C31C2211yescharacteristic
10C2×C31×#117—21no—
11C31×#480—21no—
12C31×#474C412yescomplemented
13C31×#459C412yescomplemented
14C31×#447—21no—
15C4×D8×C31—22no—
16C31×#431C2211yescharacteristic
17C31×#431C2211yescharacteristic
18C31×#423C2211yescharacteristic
19C2×C31×#104—21no—
20C2×C31×#104—21no—
21C31×#390—21no—
22C31×#389—21no—
23C31×#388—22no—
24C4×C4⋊C4×C31C2211yescharacteristic
25C4×C22⋊C4×C31—21no—
26C43×C31C2211yescharacteristic
27C31×#327—22no—
28C31×D4⋊C8—22no—
29C2×#41×C31—21no—
30C22×D4×C31D411yescharacteristic
31C23×C4×C31—21no—
32C2×D8×C31—22no—
33C2×M16×C31—21no—
34C31×C4⋊Q8C2×C411yescharacteristic
35C31×#127C2×C411yescharacteristic
36C31×#124—21no—
37C31×#121—21no—
38C31×#121—21no—
39C4×D4×C31—21no—
40C4×D4×C31—21no—
41C4×D4×C31—21no—
42C4×D4×C31—21no—
43C4×D4×C31C2×C412yes—
44C4×D4×C31D412yes—
45C31×#117—42no—
46C31×#117—42no—
47C2×C4⋊C4×C31—22no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31D411yescharacteristic
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—22no—
52C2×C22⋊C4×C31—22no—
53C2×C42×C31—21no—
54C2×C42×C31—21no—
55C2×C42×C31—21no—
56C2×C42×C31D411yescharacteristic
57C2×C42×C31D411yescharacteristic
58C2×C42×C31C2311yescharacteristic
59C31×#107—22no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8C2×C412yes—
64C31×#104—42no—
65C31×#104—42no—
66C31×#102—22no—
67C31×#102—22no—
68C31×C22⋊C8—21no—
69C31×C22⋊C8—21no—
70C4×C8×C31—21no—
71C4×C8×C31—21no—
72C31×#95—22no—
73C31×#95D412yes—
74C24×C31—21no—
75#41×C31—41no—
76#41×C31—41no—
77#41×C31—41no—
78#41×C31—41no—
79C2×Q8×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—42no—
85C2×D4×C31—41no—
86C2×D4×C31C22⋊C412yes—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—22no—
91C22×C4×C31—22no—
92C22×C4×C31—22no—
93C22×C4×C31—41no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—41no—
97C22×C4×C31—41no—
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101D8×C31—44no—
102M16×C31—41no—
103M16×C31—41no—
104C2×C8×C31—22no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—42no—
111C4⋊C4×C31—44no—
112C4⋊C4×C31C22⋊C412yes—
113C22⋊C4×C31—22no—
114C22⋊C4×C31—22no—
115C22⋊C4×C31—44no—
116C22⋊C4×C31—44no—
117C22⋊C4×C31—41no—
118C22⋊C4×C31—41no—
119C42×C31—21no—
120C42×C31—21no—
121C42×C31—21no—
122C42×C31—42no—
123C42×C31—42no—
124C42×C31—41no—
125C42×C31—41no—
126C42×C31D812yescomplemented
127C42×C31C22×C411yescharacteristic
128C42×C31C2×D411yescharacteristic
129C42×C31#4111yescharacteristic
130C42×C31#4111yescharacteristic
131#9237C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—22no—
133C23×C31—42no—
134C23×C31—42no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31#12011yescharacteristic
139Q8×C31—41no—
140Q8×C31—41no—
141D4×C31—22no—
142D4×C31—22no—
143D4×C31—42no—
144D4×C31—42no—
145D4×C31—41no—
146D4×C31—41no—
147D4×C31—81no—
148D4×C31—81no—
149D4×C31—81no—
150C2×C4×C31—22no—
151C2×C4×C31—22no—
152C2×C4×C31—22no—
153C2×C4×C31—22no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—42no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31C2×C22⋊C411yescharacteristic
172C2×C4×C31C4×D411yescharacteristic
173C2×C4×C31C4×D411yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31#12111yescharacteristic
176C2×C4×C31#12311yescharacteristic
177C2×C4×C31C2×D811yescharacteristic
178C2×C4×C31#13511yescharacteristic
179C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
181C8×C31—42no—
182C8×C31—81no—
183C8×C31—81no—
184#3100C2×C3111yescharacteristic
185#2293C2×C3111yescharacteristic, complemented
186#1962C2×C3111yescharacteristic, complemented
187#1502C2×C3111yescharacteristic, complemented
188#1486C2×C3111yescharacteristic, complemented
189#1303C2×C3111yescharacteristic, complemented
190#1276C2×C3111yescharacteristic, complemented
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—82no—
197C22×C31—82no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31#38611yescharacteristic
202C22×C31#44711yescharacteristic
203C22×C31#44911yescharacteristic
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—42no—
208C4×C31—42no—
209C4×C31—42no—
210C4×C31—82no—
211C4×C31—81no—
212C4×C31#42112yes—
213C4×C31C4×D811yescharacteristic
214C4×C31#44211yescharacteristic
215C4×C31#45911yescharacteristic
216C4×C31#46311yescharacteristic
217#532C22×C3111yescharacteristic
218C2×C4×D4C22×C3111yescharacteristic
219C2×#117—21no—
220#480—21no—
221#474C4×C3112yescomplemented
222#459C4×C3112yescomplemented
223#447—21no—
224C4×D8—22no—
225#431C22×C3111yescharacteristic
226#431C22×C3111yescharacteristic
227#423C22×C3111yescharacteristic
228C2×#104—21no—
229C2×#104—21no—
230#390—21no—
231#389—21no—
232#388—22no—
233C4×C4⋊C4C22×C3111yescharacteristic
234C4×C22⋊C4—21no—
235C43C22×C3111yescharacteristic
236#327—22no—
237D4⋊C8—22no—
238C2×C31—21no—
239C2×C31—21no—
240C2×C31—82no—
241C2×C31—161no—
242C2×C31#187911yescharacteristic
243C2×C31#189711yescharacteristic
244C2×C31#190611yescharacteristic
245C2×#41—21no—
246C22×D4D4×C3111yescharacteristic
247C23×C4—21no—
248C2×D8—22no—
249C2×M16—21no—
250C4⋊Q8C2×C4×C3111yescharacteristic
251#127C2×C4×C3111yescharacteristic
252#124—21no—
253#121—21no—
254#121—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C2×C4×C3112yes—
260C4×D4D4×C3112yes—
261#117—42no—
262#117—42no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4D4×C3111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42D4×C3111yescharacteristic
273C2×C42D4×C3111yescharacteristic
274C2×C42C23×C3111yescharacteristic
275#107—22no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8C2×C4×C3112yes—
280#104—42no—
281#104—42no—
282#102—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288#95—22no—
289#95D4×C3112yes—
290C31#923711yescharacteristic, direct product factor, Sylow
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4C22⋊C4×C3112yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318D8—44no—
319M16—41no—
320M16—41no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4C22⋊C4×C3112yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—42no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42D8×C3112yescomplemented
344C42C22×C4×C3111yescharacteristic
345C42C2×D4×C3111yescharacteristic
346C42#41×C3111yescharacteristic
347C42#41×C3111yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C31×#12011yescharacteristic
355Q8—41no—
356Q8—41no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4C2×C22⋊C4×C3111yescharacteristic
388C2×C4C4×D4×C3111yescharacteristic
389C2×C4C4×D4×C3111yescharacteristic
390C2×C4C31×#12111yescharacteristic
391C2×C4C31×#12111yescharacteristic
392C2×C4C31×#12311yescharacteristic
393C2×C4C2×D8×C3111yescharacteristic
394C2×C4C31×#13511yescharacteristic
395C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
397C8—42no—
398C8—81no—
399C8—81no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C31×#38611yescharacteristic
411C22C31×#44711yescharacteristic
412C22C31×#44911yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—81no—
421C4C31×#42112yes—
422C4C4×D8×C3111yescharacteristic
423C4C31×#44211yescharacteristic
424C4C31×#45911yescharacteristic
425C4C31×#46311yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2C31×#187911yescharacteristic
431C2C31×#189711yescharacteristic
432C2C31×#190611yescharacteristic
subgroupG/H
centerC2×C4×C31#121
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#386
prime radicalC31×#3100C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2293C2semidirect
3. C31×#1962C2semidirect
4. C31×#1502C2semidirect
5. C31×#1486C2semidirect
6. C31×#1303C2semidirect
7. C31×#1276C2semidirect
12. C31×#474C4semidirect
13. C31×#459C4semidirect
126. C42×C31D8semidirect
131. #9237C31direct (N × H)
185. #2293C2×C31semidirect
186. #1962C2×C31semidirect
187. #1502C2×C31semidirect
188. #1486C2×C31semidirect
189. #1303C2×C31semidirect
190. #1276C2×C31semidirect
221. #474C4×C31semidirect
222. #459C4×C31semidirect
290. C31#9237direct (N × H)
343. C42D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31002
C31×#22932
C31×#19622
C31×#15022
C31×#14862
C31×#13032
C31×#12762
#923731