Finite Group Databasebeta

← C31×#7762 #21464521

Subgroups — C31×#7762 #21464521

Order 7936.

542 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2682C211yesmaximal, characteristic
2C31×#2043C211yesmaximal, characteristic, complemented
3C31×#2023C211yesmaximal, characteristic, complemented
4C31×#2023C211yesmaximal, characteristic, complemented
5C31×#1271C211yesmaximal, characteristic, complemented
6C31×#1270C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#123—21no—
10C2×C31×#121—21no—
11C2×C31×#103—21no—
12C2×C31×#103—21no—
13C2×C31×#102—21no—
14C2×C31×#102—21no—
15C31×#396—21no—
16C31×#394—21no—
17C31×#392—21no—
18C31×#390C2211yescharacteristic
19C31×#390C2211yescharacteristic
20C31×#389C2211yescharacteristic
21C31×#385—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C22×C8×C31—21no—
36C31×C4⋊Q8—42no—
37C31×#123—42no—
38C31×#123—42no—
39C31×#121—42no—
40C31×#121—42no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31D411yescharacteristic
46C2×C4⋊C4×C31D411yescharacteristic
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C4⋊C4×C31D411yescharacteristic
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C42×C31C2311yescharacteristic
57C31×#103—42no—
58C31×#103—42no—
59C31×#102—42no—
60C31×#102—42no—
61C31×#95—21no—
62C31×#95—21no—
63C24×C31—21no—
64C24×C31—21no—
65C2×Q8×C31—22no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31—41no—
69C2×Q8×C31—41no—
70C2×Q8×C31—41no—
71C2×D4×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—21no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—42no—
79C2×D4×C31—42no—
80C2×D4×C31—42no—
81C2×D4×C31—81no—
82C2×D4×C31—81no—
83C2×D4×C31—81no—
84C2×D4×C31SD1612yes—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—42no—
90C22×C4×C31—42no—
91C22×C4×C31—41no—
92C22×C4×C31—81no—
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96C2×C8×C31—41no—
97C2×C8×C31—41no—
98C2×C8×C31—41no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—41no—
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C2×C8×C31—41no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—42no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31—41no—
111C4⋊C4×C31—41no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31D812yes—
117C22⋊C4×C31—42no—
118C22⋊C4×C31—42no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—81no—
128C42×C31—41no—
129#7762C3111yesmaximal, characteristic, direct product factor, Sylow
130C23×C31—22no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31#12011yescharacteristic
145Q8×C31—42no—
146Q8×C31—42no—
147Q8×C31—81no—
148Q8×C31—81no—
149D4×C31—42no—
150D4×C31—42no—
151D4×C31—81no—
152D4×C31—81no—
153D4×C31—84no—
154D4×C31—84no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—22no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—42no—
173C2×C4×C31—41no—
174C2×C4×C31—84no—
175C2×C4×C31—81no—
176C2×C4×C31—81no—
177C2×C4×C31C2×D811yescharacteristic
178C2×C4×C31C2×SD1611yescharacteristic
179C2×C4×C31#13511yescharacteristic
180C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C31#13711yescharacteristic
183C8×C31—81no—
184C8×C31—81no—
185C8×C31—81no—
186#2682C2×C3111yescharacteristic
187#2043C2×C3111yescharacteristic, complemented
188#2023C2×C3111yescharacteristic, complemented
189#2023C2×C3111yescharacteristic, complemented
190#1271C2×C3111yescharacteristic, complemented
191#1270C2×C3111yescharacteristic, complemented
192#1270C2×C3111yescharacteristic, complemented
193C22×C31—42no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31#44711yescharacteristic
208C22×C31#45011yescharacteristic
209C22×C31#45211yescharacteristic
210C22×C31#45411yescharacteristic
211C22×C31#45511yescharacteristic
212C22×C31#45511yescharacteristic
213C22×C31B(4,𝔽2)11yescharacteristic
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—21no—
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—21no—
220C4×C31—42no—
221C4×C31—82no—
222C4×C31—81no—
223C4×C31—81no—
224C4×C31—81no—
225C2×C4⋊Q8C22×C3111yescharacteristic
226C2×#123—21no—
227C2×#121—21no—
228C2×#103—21no—
229C2×#103—21no—
230C2×#102—21no—
231C2×#102—21no—
232#396—21no—
233#394—21no—
234#392—21no—
235#390C22×C3111yescharacteristic
236#390C22×C3111yescharacteristic
237#389C22×C3111yescharacteristic
238#385—21no—
239#340—21no—
240#340—21no—
241#336C22×C3111yescharacteristic
242#336C22×C3111yescharacteristic
243#336C22×C3111yescharacteristic
244C2×C31—82no—
245C2×C31—161no—
246C2×C31#159811yescharacteristic
247C2×C31#160111yescharacteristic
248C2×C31#160611yescharacteristic
249C2×C31#163511yescharacteristic
250C2×C31#163911yescharacteristic
251C2×C31#164111yescharacteristic
252C2×C31#166211yescharacteristic
253C22×Q8—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4D4×C3111yescharacteristic
257C22×D4D4×C3111yescharacteristic
258C23×C4—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C4⋊Q8—42no—
263#123—42no—
264#123—42no—
265#121—42no—
266#121—42no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4D4×C3111yescharacteristic
272C2×C4⋊C4D4×C3111yescharacteristic
273C2×C4⋊C4D4×C3111yescharacteristic
274C2×C4⋊C4D4×C3111yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C42C23×C3111yescharacteristic
283#103—42no—
284#103—42no—
285#102—42no—
286#102—42no—
287#95—21no—
288#95—21no—
289C31#776211yescharacteristic, direct product factor, Sylow
290C24—21no—
291C24—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4SD16×C3112yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—81no—
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4C2×D4×C3111yescharacteristic
322C22×C4C2×D4×C3111yescharacteristic
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4D8×C3112yes—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—81no—
355C42—41no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C31×#12011yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—84no—
380D4—84no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—84no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4C2×D8×C3111yescharacteristic
404C2×C4C2×SD16×C3111yescharacteristic
405C2×C4C31×#13511yescharacteristic
406C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C31×#13711yescharacteristic
409C8—81no—
410C8—81no—
411C8—81no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C31×#44711yescharacteristic
427C22C31×#45011yescharacteristic
428C22C31×#45211yescharacteristic
429C22C31×#45411yescharacteristic
430C22C31×#45511yescharacteristic
431C22C31×#45511yescharacteristic
432C22C31×B(4,𝔽2)11yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—82no—
441C4—81no—
442C4—81no—
443C4—81no—
444C2—82no—
445C2—161no—
446C2C31×#159811yescharacteristic
447C2C31×#160111yescharacteristic
448C2C31×#160611yescharacteristic
449C2C31×#163511yescharacteristic
450C2C31×#163911yescharacteristic
451C2C31×#164111yescharacteristic
452C2C31×#166211yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2682C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2043C2semidirect
3. C31×#2023C2semidirect
4. C31×#2023C2semidirect
5. C31×#1271C2semidirect
6. C31×#1270C2semidirect
7. C31×#1270C2semidirect
129. #7762C31direct (N × H)
187. #2043C2×C31semidirect
188. #2023C2×C31semidirect
189. #2023C2×C31semidirect
190. #1271C2×C31semidirect
191. #1270C2×C31semidirect
192. #1270C2×C31semidirect
289. C31#7762direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26822
C31×#20432
C31×#20232
C31×#12712
C31×#12702
#776231