Finite Group Databasebeta

← C31×#12192 #21468951

Subgroups — C31×#12192 #21468951

Order 7936.

544 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2735C211yesmaximal, characteristic, complemented
2C31×#2021C211yesmaximal, characteristic
3C31×#1613C211yesmaximal, characteristic, complemented
4C31×#1607C211yesmaximal, characteristic, complemented
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1344C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C2×C31×#120—21no—
10C31×#481—21no—
11C31×#480—21no—
12C2×C31×#99—21no—
13C31×#399C2211yescharacteristic
14C31×#397—21no—
15C31×#397C2211yescharacteristic
16C31×#396—21no—
17C31×#388—21no—
18C31×#388—21no—
19C31×#388—21no—
20C31×#379—21no—
21C31×#379C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C25×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31—41no—
30C31×#123—41no—
31C31×#121—21no—
32C31×#121—21no—
33C31×#120—42no—
34C31×#120—41no—
35C31×#120—42no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31D411yescharacteristic, complemented
39C2×C4⋊C4×C31D411yescharacteristic, complemented
40C2×C22⋊C4×C31—21no—
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—41no—
50C2×C22⋊C4×C31D411yescharacteristic
51C2×C22⋊C4×C31C2311yescharacteristic
52C2×C42×C31—21no—
53C31×#107—41no—
54C31×#106—41no—
55C31×#102—41no—
56C31×#102—41no—
57C31×#99—42no—
58C31×C22⋊C8—21no—
59C31×C22⋊C8—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic, complemented
64C31×#95D411yescharacteristic, complemented
65C24×C31—21no—
66C24×C31—21no—
67C24×C31—21no—
68C24×C31—21no—
69C24×C31—41no—
70C24×C31—41no—
71C24×C31—42no—
72C24×C31—81no—
73C24×C31C2×D411yescharacteristic
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—44no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—82no—
84C2×D4×C31—82no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—41no—
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31C2×D411yescharacteristic
94C2×C8×C31—41no—
95C2×C8×C31—41no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—81no—
101C4⋊C4×C31—81no—
102C22⋊C4×C31—22no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—44no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—81no—
114C22⋊C4×C31—81no—
115C22⋊C4×C31—81no—
116C22⋊C4×C31—81no—
117C22⋊C4×C31—81no—
118C42×C31—81no—
119#12192C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—21no—
121C23×C31—21no—
122C23×C31—22no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—42no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—42no—
132C23×C31—42no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—81no—
138C23×C31—82no—
139C23×C31—81no—
140C23×C31—82no—
141C23×C31—84no—
142C23×C31—81no—
143C23×C31—81no—
144C23×C31—81no—
145C23×C31#12011yescharacteristic
146C23×C31#13511yescharacteristic
147C23×C31AGL(1,ℤ/8)11yescharacteristic
148D4×C31—81no—
149D4×C31—81no—
150D4×C31—81no—
151D4×C31—81no—
152D4×C31—88no—
153D4×C31—81no—
154D4×C31—81no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
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—81no—
179C2×C4×C31—81no—
180C8×C31—81no—
181C8×C31—81no—
182#2735C2×C3111yescharacteristic, complemented
183#2021C2×C3111yescharacteristic
184#1613C2×C3111yescharacteristic, complemented
185#1607C2×C3111yescharacteristic, complemented
186#1352C2×C3111yescharacteristic, complemented
187#1344C2×C3111yescharacteristic, complemented
188#1316C2×C3111yescharacteristic, complemented
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—41no—
194C22×C31—41no—
195C22×C31—42no—
196C22×C31—42no—
197C22×C31—41no—
198C22×C31—41no—
199C22×C31—44no—
200C22×C31—42no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—82no—
205C22×C31—82no—
206C22×C31—82no—
207C22×C31—82no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31—81no—
211C22×C31—81no—
212C22×C31—81no—
213C22×C31—161no—
214C22×C31#44911yescharacteristic
215C22×C31#45311yescharacteristic
216C22×C31B(4,𝔽2)11yescharacteristic
217C4×C31—41no—
218C4×C31—41no—
219C4×C31—81no—
220C4×C31—81no—
221C4×C31—81no—
222C4×C31—82no—
223C4×C31—81no—
224C4×C31—81no—
225C4×C31—81no—
226#534C22×C3111yescharacteristic
227C2×#120—21no—
228#481—21no—
229#480—21no—
230C2×#99—21no—
231#399C22×C3111yescharacteristic
232#397—21no—
233#397C22×C3111yescharacteristic
234#396—21no—
235#388—21no—
236#388—21no—
237#388—21no—
238#379—21no—
239#379C22×C3111yescharacteristic
240#342C22×C3111yescharacteristic
241#327—21no—
242#327—21no—
243#323C22×C3111yescharacteristic
244#323C22×C3111yescharacteristic
245C2×C31—21no—
246C2×C31—21no—
247C2×C31—41no—
248C2×C31—41no—
249C2×C31—82no—
250C2×C31—161no—
251C2×C31#160011yescharacteristic
252C2×C31#219711yescharacteristic
253C2×C31#220011yescharacteristic
254C25D4×C3111yescharacteristic
255C22×D4—21no—
256C22×D4—41no—
257#123—41no—
258#121—21no—
259#121—21no—
260#120—42no—
261#120—41no—
262#120—42no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4D4×C3111yescharacteristic, complemented
266C2×C4⋊C4D4×C3111yescharacteristic, complemented
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4D4×C3111yescharacteristic
278C2×C22⋊C4C23×C3111yescharacteristic
279C2×C42—21no—
280#107—41no—
281#106—41no—
282#102—41no—
283#102—41no—
284#99—42no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95D4×C3111yescharacteristic, complemented
291#95D4×C3111yescharacteristic, complemented
292C31#1219211yescharacteristic, direct product factor, Sylow
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—81no—
301C24C2×D4×C3111yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—82no—
312C2×D4—82no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4C2×D4×C3111yescharacteristic
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—81no—
329C4⋊C4—81no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—81no—
342C22⋊C4—81no—
343C22⋊C4—81no—
344C22⋊C4—81no—
345C22⋊C4—81no—
346C42—81no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—81no—
365C23—82no—
366C23—81no—
367C23—82no—
368C23—84no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23C31×#12011yescharacteristic
373C23C31×#13511yescharacteristic
374C23C31×AGL(1,ℤ/8)11yescharacteristic
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—88no—
380D4—81no—
381D4—81no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
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—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C8—81no—
408C8—81no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—44no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—161no—
434C22C31×#44911yescharacteristic
435C22C31×#45311yescharacteristic
436C22C31×B(4,𝔽2)11yescharacteristic
437C4—41no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4—81no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—41no—
450C2—82no—
451C2—161no—
452C2C31×#160011yescharacteristic
453C2C31×#219711yescharacteristic
454C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2021C2
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
1. C31×#2735C2semidirect
3. C31×#1613C2semidirect
4. C31×#1607C2semidirect
5. C31×#1352C2semidirect
6. C31×#1344C2semidirect
7. C31×#1316C2semidirect
38. C2×C4⋊C4×C31D4semidirect
39. C2×C4⋊C4×C31D4semidirect
63. C31×#95D4semidirect
64. C31×#95D4semidirect
119. #12192C31direct (N × H)
182. #2735C2×C31semidirect
184. #1613C2×C31semidirect
185. #1607C2×C31semidirect
186. #1352C2×C31semidirect
187. #1344C2×C31semidirect
188. #1316C2×C31semidirect
265. C2×C4⋊C4D4×C31semidirect
266. C2×C4⋊C4D4×C31semidirect
290. #95D4×C31semidirect
291. #95D4×C31semidirect
292. C31#12192direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27352
C31×#20212
C31×#16132
C31×#16072
C31×#13522
C31×#13442
C31×#13162
#1219231