Finite Group Databasebeta

← C31×#12194 #21468953

Subgroups — C31×#12194 #21468953

Order 7936.

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

normal subgroups only

This group has 574 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×#2555C211yesmaximal, characteristic, complemented
2C31×#2006C211yesmaximal, characteristic, complemented
3C31×#1604C211yesmaximal, characteristic, complemented
4C31×#1598C211yesmaximal, characteristic, complemented
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1318C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#537C2211yescharacteristic
10C2×C31×#124—21no—
11C2×C31×#121—21no—
12C31×#448—21no—
13C31×#447—21no—
14C2×C31×#104—21no—
15C31×#395C2211yescharacteristic
16C31×#394C2211yescharacteristic
17C31×#392—21no—
18C31×#390C2211yescharacteristic
19C31×#388—21no—
20C31×#386—21no—
21C4×C22⋊C4×C31—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31D411yescharacteristic
32C22×D4×C31D411yescharacteristic
33C23×C4×C31—21no—
34C31×AGL(1,ℤ/8)—42no—
35C31×AGL(1,ℤ/8)—42no—
36C2×SD16×C31—41no—
37C2×SD16×C31—41no—
38C2×D8×C31—41no—
39C2×D8×C31—41no—
40C2×M16×C31—21no—
41C31×#124—44no—
42C31×C22⋊Q8—41no—
43C31×#121—21no—
44C31×#121—21no—
45C31×#121—44no—
46C31×#121—41no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31C2311yescharacteristic
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×C31—21no—
57C2×C42×C31D411yescharacteristic
58C31×#104—42no—
59C31×#103—41no—
60C31×#102—41no—
61C31×C22⋊C8—21no—
62C31×C22⋊C8—21no—
63C31×#95—21no—
64C31×#95—21no—
65C31×#95D411yescharacteristic
66C31×#95D411yescharacteristic
67C24×C31—21no—
68C24×C31—21no—
69#41×C31—41no—
70#41×C31—41no—
71#41×C31—41no—
72#41×C31—41no—
73C2×Q8×C31—21no—
74C2×Q8×C31—42no—
75C2×Q8×C31—41no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—42no—
87C2×D4×C31—42no—
88C2×D4×C31—42no—
89C2×D4×C31—41no—
90C2×D4×C31—81no—
91C2×D4×C31—81no—
92C22×C4×C31—21no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—42no—
97C22×C4×C31—41no—
98C22×C4×C31—42no—
99C22×C4×C31—81no—
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103SD16×C31—44no—
104SD16×C31—82no—
105D8×C31—44no—
106D8×C31—82no—
107M16×C31—22no—
108M16×C31—41no—
109C2×C8×C31—41no—
110C2×C8×C31—41no—
111C2×C8×C31—41no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—42no—
116C22⋊C4×C31—24no—
117C22⋊C4×C31—42no—
118C22⋊C4×C31—42no—
119C22⋊C4×C31—42no—
120C22⋊C4×C31—42no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—42no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C42×C31—42no—
127C42×C31—41no—
128#12194C3111yesmaximal, characteristic, direct product factor, Sylow
129C23×C31—22no—
130C23×C31—41no—
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—41no—
146Q8×C31—41no—
147Q8×C31—42no—
148Q8×C31—81no—
149D4×C31—41no—
150D4×C31—42no—
151D4×C31—41no—
152D4×C31—42no—
153D4×C31—81no—
154D4×C31—81no—
155D4×C31—84no—
156D4×C31—84no—
157D4×C31—81no—
158D4×C31—81no—
159D4×C31—81no—
160D4×C31—81no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—22no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—21no—
167C2×C4×C31—21no—
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—41no—
180C2×C4×C31—81no—
181C2×C4×C31—81no—
182C2×C4×C31—84no—
183C2×C4×C31#13511yescharacteristic
184C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
185C8×C31—42no—
186C8×C31—81no—
187C8×C31—81no—
188#2555C2×C3111yescharacteristic, complemented
189#2006C2×C3111yescharacteristic, complemented
190#1604C2×C3111yescharacteristic, complemented
191#1598C2×C3111yescharacteristic, complemented
192#1354C2×C3111yescharacteristic, complemented
193#1345C2×C3111yescharacteristic, complemented
194#1318C2×C3111yescharacteristic, complemented
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—42no—
198C22×C31—42no—
199C22×C31—42no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31—81no—
211C22×C31—81no—
212C22×C31—81no—
213C22×C31—81no—
214C22×C31#44911yescharacteristic
215C22×C31#45311yescharacteristic
216C22×C31B(4,𝔽2)11yescharacteristic
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—42no—
220C4×C31—42no—
221C4×C31—41no—
222C4×C31—41no—
223C4×C31—42no—
224C4×C31—81no—
225C4×C31—81no—
226C2×AGL(1,ℤ/8)—21no—
227#537C22×C3111yescharacteristic
228C2×#124—21no—
229C2×#121—21no—
230#448—21no—
231#447—21no—
232C2×#104—21no—
233#395C22×C3111yescharacteristic
234#394C22×C3111yescharacteristic
235#392—21no—
236#390C22×C3111yescharacteristic
237#388—21no—
238#386—21no—
239C4×C22⋊C4—21no—
240#343C22×C3111yescharacteristic
241#327—21no—
242#327—21no—
243#324C22×C3111yescharacteristic
244#324C22×C3111yescharacteristic
245C2×C31—21no—
246C2×C31—21no—
247C2×C31—82no—
248C2×C31—161no—
249C2×C31—161no—
250C2×C31#160011yescharacteristic
251C2×C31#219611yescharacteristic
252C2×C31#220411yescharacteristic
253C2×#41—21no—
254C22×Q8D4×C3111yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4D4×C3111yescharacteristic
258C22×D4D4×C3111yescharacteristic
259C23×C4—21no—
260AGL(1,ℤ/8)—42no—
261AGL(1,ℤ/8)—42no—
262C2×SD16—41no—
263C2×SD16—41no—
264C2×D8—41no—
265C2×D8—41no—
266C2×M16—21no—
267#124—44no—
268C22⋊Q8—41no—
269#121—21no—
270#121—21no—
271#121—44no—
272#121—41no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C23×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×C42—21no—
283C2×C42D4×C3111yescharacteristic
284#104—42no—
285#103—41no—
286#102—41no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#95—21no—
290#95—21no—
291#95D4×C3111yescharacteristic
292#95D4×C3111yescharacteristic
293C31#1219411yescharacteristic, direct product factor, Sylow
294C24—21no—
295C24—21no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—42no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—81no—
318C2×D4—81no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—81no—
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×D4×C3111yescharacteristic
329C22×C4C2×D4×C3111yescharacteristic
330SD16—44no—
331SD16—82no—
332D8—44no—
333D8—82no—
334M16—22no—
335M16—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—42no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42—41no—
354C42—42no—
355C23—22no—
356C23—41no—
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—41no—
372Q8—41no—
373Q8—42no—
374Q8—81no—
375D4—41no—
376D4—42no—
377D4—41no—
378D4—42no—
379D4—81no—
380D4—81no—
381D4—84no—
382D4—84no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
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—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—84no—
409C2×C4C31×#13511yescharacteristic
410C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
411C8—42no—
412C8—81no—
413C8—81no—
414C22—21no—
415C22—21no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22C31×#44911yescharacteristic
434C22C31×#45311yescharacteristic
435C22C31×B(4,𝔽2)11yescharacteristic
436C4—21no—
437C4—21no—
438C4—42no—
439C4—42no—
440C4—41no—
441C4—41no—
442C4—42no—
443C4—81no—
444C4—81no—
445C2—21no—
446C2—21no—
447C2—82no—
448C2—161no—
449C2—161no—
450C2C31×#160011yescharacteristic
451C2C31×#219611yescharacteristic
452C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2555C2semidirect
2. C31×#2006C2semidirect
3. C31×#1604C2semidirect
4. C31×#1598C2semidirect
5. C31×#1354C2semidirect
6. C31×#1345C2semidirect
7. C31×#1318C2semidirect
128. #12194C31direct (N × H)
188. #2555C2×C31semidirect
189. #2006C2×C31semidirect
190. #1604C2×C31semidirect
191. #1598C2×C31semidirect
192. #1354C2×C31semidirect
193. #1345C2×C31semidirect
194. #1318C2×C31semidirect
293. C31#12194direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25552
C31×#20062
C31×#16042
C31×#15982
C31×#13542
C31×#13452
C31×#13182
#1219431