Finite Group Databasebeta

← C31×#12214 #21468973

Subgroups — C31×#12214 #21468973

Order 7936.

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

normal subgroups only

This group has 578 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×#2437C211yesmaximal, characteristic
2C31×#2029C211yesmaximal, characteristic
3C31×#1614C211yesmaximal, characteristic, complemented
4C31×#1602C211yesmaximal, characteristic, complemented
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1351C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic
8C31×#543C2211yescharacteristic
9C2×C31×C22⋊Q8—21no—
10C22×C22⋊C4×C31—21no—
11C31×#484—21no—
12C31×#451—21no—
13C31×#399C2211yescharacteristic
14C31×#396—21no—
15C31×#393C2211yescharacteristic
16C31×#385—21no—
17C31×#385—21no—
18C31×#380—21no—
19C31×#379—21no—
20C31×#379C2211yescharacteristic
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C25×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C23×C4×C31—21no—
30C23×C4×C31—21no—
31C2×Q16×C31—41no—
32C31×C22⋊Q8—21no—
33C31×C22⋊Q8—21no—
34C31×C22⋊Q8—44no—
35C31×C22⋊Q8—41no—
36C31×C22⋊Q8—41no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31D411yescharacteristic
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—44no—
51C2×C22⋊C4×C31—41no—
52C2×C22⋊C4×C31—41no—
53C2×C22⋊C4×C31D411yescharacteristic
54C2×C22⋊C4×C31C2311yescharacteristic
55C31×#107—41no—
56C31×#103—41no—
57C31×#103—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—42no—
67C24×C31—41no—
68C24×C31—42no—
69C24×C31—41no—
70C24×C31Q1612yescomplemented
71C24×C31C2×D411yescharacteristic
72C2×Q8×C31—21no—
73C2×Q8×C31—21no—
74C2×Q8×C31—42no—
75C2×Q8×C31—41no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—42no—
84C22×C4×C31—42no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31—41no—
88C22×C4×C31—81no—
89C22×C4×C31—81no—
90C22×C4×C31C2×D411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92Q16×C31—82no—
93C2×C8×C31—41no—
94C2×C8×C31—41no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—42no—
98C4⋊C4×C31—42no—
99C4⋊C4×C31—42no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—81no—
104C22⋊C4×C31—24no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—42no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—41no—
116C22⋊C4×C31—82no—
117C22⋊C4×C31—82no—
118#12214C3111yesmaximal, characteristic, direct product factor, Sylow
119C23×C31—22no—
120C23×C31—22no—
121C23×C31—22no—
122C23×C31—21no—
123C23×C31—21no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—44no—
128C23×C31—42no—
129C23×C31—41no—
130C23×C31—44no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—42no—
134C23×C31—42no—
135C23×C31—81no—
136C23×C31—84no—
137C23×C31—81no—
138C23×C31—81no—
139C23×C31#12011yescharacteristic
140C23×C31C2×Q1611yescharacteristic
141C23×C31#13711yescharacteristic
142Q8×C31—42no—
143Q8×C31—81no—
144Q8×C31—81no—
145Q8×C31—81no—
146C2×C4×C31—22no—
147C2×C4×C31—22no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
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—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—84no—
168C2×C4×C31—84no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C8×C31—81no—
175C8×C31—81no—
176#2437C2×C3111yescharacteristic
177#2029C2×C3111yescharacteristic
178#1614C2×C3111yescharacteristic, complemented
179#1602C2×C3111yescharacteristic, complemented
180#1352C2×C3111yescharacteristic, complemented
181#1351C2×C3111yescharacteristic, complemented
182#1316C2×C3111yescharacteristic
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—24no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—41no—
189C22×C31—41no—
190C22×C31—42no—
191C22×C31—42no—
192C22×C31—42no—
193C22×C31—44no—
194C22×C31—42no—
195C22×C31—41no—
196C22×C31—42no—
197C22×C31—41no—
198C22×C31—42no—
199C22×C31—81no—
200C22×C31—82no—
201C22×C31—82no—
202C22×C31—82no—
203C22×C31#45111yescharacteristic
204C22×C31#45511yescharacteristic
205C22×C31B(4,𝔽2)11yescharacteristic
206C4×C31—42no—
207C4×C31—41no—
208C4×C31—42no—
209C4×C31—41no—
210C4×C31—81no—
211C4×C31—81no—
212C4×C31—81no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31—161no—
216#543C22×C3111yescharacteristic
217C2×C22⋊Q8—21no—
218C22×C22⋊C4—21no—
219#484—21no—
220#451—21no—
221#399C22×C3111yescharacteristic
222#396—21no—
223#393C22×C3111yescharacteristic
224#385—21no—
225#385—21no—
226#380—21no—
227#379—21no—
228#379C22×C3111yescharacteristic
229#342—21no—
230#342C22×C3111yescharacteristic
231#328—21no—
232#328—21no—
233#323C22×C3111yescharacteristic
234#323C22×C3111yescharacteristic
235C2×C31—21no—
236C2×C31—21no—
237C2×C31—41no—
238C2×C31—44no—
239C2×C31—41no—
240C2×C31#160211yescharacteristic
241C2×C31#219611yescharacteristic
242C2×C31#219711yescharacteristic
243C25D4×C3111yescharacteristic
244C22×Q8D4×C3111yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C2×Q16—41no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—44no—
251C22⋊Q8—41no—
252C22⋊Q8—41no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C3111yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4D4×C3111yescharacteristic
270C2×C22⋊C4C23×C3111yescharacteristic
271#107—41no—
272#103—41no—
273#103—41no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276#95—21no—
277#95—21no—
278#95D4×C3111yescharacteristic
279#95D4×C3111yescharacteristic
280C31#1221411yescharacteristic, direct product factor, Sylow
281C24—21no—
282C24—21no—
283C24—42no—
284C24—41no—
285C24—42no—
286C24—41no—
287C24Q16×C3112yescomplemented
288C24C2×D4×C3111yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×Q8—41no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4C2×D4×C3111yescharacteristic
308C22×C4C2×D4×C3111yescharacteristic
309Q16—82no—
310C2×C8—41no—
311C2×C8—41no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C22⋊C4—24no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—44no—
344C23—42no—
345C23—41no—
346C23—44no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—84no—
353C23—81no—
354C23—81no—
355C23C31×#12011yescharacteristic
356C23C2×Q16×C3111yescharacteristic
357C23C31×#13711yescharacteristic
358Q8—42no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C8—81no—
391C8—81no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—41no—
405C22—42no—
406C22—41no—
407C22—42no—
408C22—81no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22C31×#45111yescharacteristic
413C22C31×#45511yescharacteristic
414C22C31×B(4,𝔽2)11yescharacteristic
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—161no—
425C2—21no—
426C2—21no—
427C2—41no—
428C2—44no—
429C2—41no—
430C2C31×#160211yescharacteristic
431C2C31×#219611yescharacteristic
432C2C31×#219711yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC25×C31D4
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
3. C31×#1614C2semidirect
4. C31×#1602C2semidirect
5. C31×#1352C2semidirect
6. C31×#1351C2semidirect
70. C24×C31Q16semidirect
118. #12214C31direct (N × H)
178. #1614C2×C31semidirect
179. #1602C2×C31semidirect
180. #1352C2×C31semidirect
181. #1351C2×C31semidirect
280. C31#12214direct (N × H)
287. C24Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#24372
C31×#20292
C31×#16142
C31×#16022
C31×#13522
C31×#13512
C31×#13162
#1221431