Finite Group Databasebeta

← C31×#12205 #21468964

Subgroups — C31×#12205 #21468964

Order 7936.

544 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2812C211yesmaximal, characteristic
2C31×#2029C211yesmaximal, characteristic, complemented
3C31×#1608C211yesmaximal, characteristic, complemented
4C31×#1596C211yesmaximal, characteristic, complemented
5C31×#1344C211yesmaximal, characteristic, complemented
6C31×#1343C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic, complemented
8C31×#543C2211yescharacteristic
9C2×C31×#120—21no—
10C31×#483—21no—
11C31×#450—21no—
12C31×#397—21no—
13C31×#397C2211yescharacteristic
14C31×#394—21no—
15C31×#394—21no—
16C31×#394—21no—
17C31×#392—21no—
18C31×#392C2211yescharacteristic
19C31×#379—21no—
20C31×#379C2211yescharacteristic
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C25×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31D411yescharacteristic
32C2×SD16×C31—41no—
33C31×C22⋊Q8—21no—
34C31×C22⋊Q8—21no—
35C31×C22⋊Q8—41no—
36C31×C22⋊Q8—41no—
37C31×#120—42no—
38C31×#120—42no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—41no—
52C2×C22⋊C4×C31—41no—
53C2×C22⋊C4×C31D411yescharacteristic
54C2×C22⋊C4×C31C2311yescharacteristic
55C31×#106—41no—
56C31×#103—41no—
57C31×#102—41no—
58C31×C22⋊C8—21no—
59C31×C22⋊C8—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95D411yescharacteristic
63C31×#95D411yescharacteristic
64C24×C31—21no—
65C24×C31—21no—
66C24×C31—21no—
67C24×C31—41no—
68C24×C31—41no—
69C24×C31—81no—
70C24×C31—81no—
71C24×C31SD1612yescomplemented
72C24×C31C2×D411yescharacteristic
73C2×Q8×C31—21no—
74C2×Q8×C31—21no—
75C2×D4×C31—41no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—81no—
81C2×D4×C31—81no—
82C2×D4×C31—81no—
83C2×D4×C31—81no—
84C2×D4×C31—81no—
85C2×D4×C31—81no—
86C2×D4×C31—81no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—41no—
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96SD16×C31—82no—
97C2×C8×C31—41no—
98C2×C8×C31—41no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—81no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—81no—
116C22⋊C4×C31—81no—
117C22⋊C4×C31—81no—
118C22⋊C4×C31—81no—
119C22⋊C4×C31—81no—
120C22⋊C4×C31—82no—
121C22⋊C4×C31—82no—
122#12205C3111yesmaximal, characteristic, direct product factor, Sylow
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—22no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—42no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—42no—
138C23×C31—42no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—81no—
145C23×C31—82no—
146C23×C31—82no—
147C23×C31—81no—
148C23×C31—81no—
149C23×C31—84no—
150C23×C31#12011yescharacteristic
151C23×C31C2×SD1611yescharacteristic
152C23×C31AGL(1,ℤ/8)11yescharacteristic
153Q8×C31—81no—
154Q8×C31—81no—
155D4×C31—42no—
156D4×C31—81no—
157D4×C31—82no—
158D4×C31—82no—
159D4×C31—84no—
160D4×C31—84no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
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—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—81no—
180C2×C4×C31—81no—
181C2×C4×C31—81no—
182C2×C4×C31—81no—
183C2×C4×C31—81no—
184C8×C31—81no—
185C8×C31—81no—
186#2812C2×C3111yescharacteristic
187#2029C2×C3111yescharacteristic, complemented
188#1608C2×C3111yescharacteristic, complemented
189#1596C2×C3111yescharacteristic, complemented
190#1344C2×C3111yescharacteristic, complemented
191#1343C2×C3111yescharacteristic, complemented
192#1316C2×C3111yescharacteristic, complemented
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—41no—
198C22×C31—41no—
199C22×C31—42no—
200C22×C31—42no—
201C22×C31—41no—
202C22×C31—41no—
203C22×C31—44no—
204C22×C31—42no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31—81no—
211C22×C31—81no—
212C22×C31—82no—
213C22×C31—82no—
214C22×C31—82no—
215C22×C31—81no—
216C22×C31—81no—
217C22×C31—81no—
218C22×C31—81no—
219C22×C31—81no—
220C22×C31#45011yescharacteristic
221C22×C31#45511yescharacteristic
222C22×C31B(4,𝔽2)11yescharacteristic
223C4×C31—41no—
224C4×C31—41no—
225C4×C31—81no—
226C4×C31—81no—
227C4×C31—81no—
228C4×C31—81no—
229C4×C31—81no—
230C4×C31—81no—
231C4×C31—161no—
232#543C22×C3111yescharacteristic
233C2×#120—21no—
234#483—21no—
235#450—21no—
236#397—21no—
237#397C22×C3111yescharacteristic
238#394—21no—
239#394—21no—
240#394—21no—
241#392—21no—
242#392C22×C3111yescharacteristic
243#379—21no—
244#379C22×C3111yescharacteristic
245#342—21no—
246#342C22×C3111yescharacteristic
247#328—21no—
248#328—21no—
249#323C22×C3111yescharacteristic
250#323C22×C3111yescharacteristic
251C2×C31—21no—
252C2×C31—21no—
253C2×C31—41no—
254C2×C31—41no—
255C2×C31—82no—
256C2×C31—161no—
257C2×C31#160111yescharacteristic
258C2×C31#219911yescharacteristic
259C2×C31#220011yescharacteristic
260C25D4×C3111yescharacteristic
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4D4×C3111yescharacteristic
265C2×SD16—41no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—41no—
269C22⋊Q8—41no—
270#120—42no—
271#120—42no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4D4×C3111yescharacteristic
274C2×C22⋊C4—21no—
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×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4D4×C3111yescharacteristic
287C2×C22⋊C4C23×C3111yescharacteristic
288#106—41no—
289#103—41no—
290#102—41no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293#95—21no—
294#95—21no—
295#95D4×C3111yescharacteristic
296#95D4×C3111yescharacteristic
297C31#1220511yescharacteristic, direct product factor, Sylow
298C24—21no—
299C24—21no—
300C24—21no—
301C24—41no—
302C24—41no—
303C24—81no—
304C24—81no—
305C24SD16×C3112yescomplemented
306C24C2×D4×C3111yescharacteristic
307C2×Q8—21no—
308C2×Q8—21no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4—81no—
320C2×D4—81no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4C2×D4×C3111yescharacteristic
329C22×C4C2×D4×C3111yescharacteristic
330SD16—82no—
331C2×C8—41no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4—81no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—81no—
378C23—82no—
379C23—82no—
380C23—81no—
381C23—81no—
382C23—84no—
383C23C31×#12011yescharacteristic
384C23C2×SD16×C3111yescharacteristic
385C23C31×AGL(1,ℤ/8)11yescharacteristic
386Q8—81no—
387Q8—81no—
388D4—42no—
389D4—81no—
390D4—82no—
391D4—82no—
392D4—84no—
393D4—84no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C8—81no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—44no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22C31×#45011yescharacteristic
447C22C31×#45511yescharacteristic
448C22C31×B(4,𝔽2)11yescharacteristic
449C4—41no—
450C4—41no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—161no—
458C2—21no—
459C2—21no—
460C2—41no—
461C2—41no—
462C2—82no—
463C2—161no—
464C2C31×#160111yescharacteristic
465C2C31×#219911yescharacteristic
466C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2812C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2029C2semidirect
3. C31×#1608C2semidirect
4. C31×#1596C2semidirect
5. C31×#1344C2semidirect
6. C31×#1343C2semidirect
7. C31×#1316C2semidirect
71. C24×C31SD16semidirect
122. #12205C31direct (N × H)
187. #2029C2×C31semidirect
188. #1608C2×C31semidirect
189. #1596C2×C31semidirect
190. #1344C2×C31semidirect
191. #1343C2×C31semidirect
192. #1316C2×C31semidirect
297. C31#12205direct (N × H)
305. C24SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28122
C31×#20292
C31×#16082
C31×#15962
C31×#13442
C31×#13432
C31×#13162
#1220531