Finite Group Databasebeta

← C31×#12151 #21468910

Subgroups — C31×#12151 #21468910

Order 7936.

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

normal subgroups only

This group has 492 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×#2712C211yesmaximal, characteristic, complemented
2C31×#2044C211yesmaximal, characteristic, complemented
3C31×#1607C211yesmaximal, characteristic, complemented
4C31×#1595C211yesmaximal, characteristic
5C31×#1344C211yesmaximal, characteristic, complemented
6C31×#1343C211yesmaximal, characteristic, complemented
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C2×C31×#124—21no—
10C31×#480—21no—
11C31×#447—21no—
12C2×C31×#99—21no—
13C31×#397C2211yescharacteristic
14C31×#394—21no—
15C31×#393—21no—
16C31×#392C2211yescharacteristic
17C31×#391—21no—
18C31×#388—21no—
19C31×#386—21no—
20C31×#386—21no—
21C31×#385C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×Q8×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C23×C4×C31D411yescharacteristic
32C2×D8×C31—41no—
33C31×#124—42no—
34C31×#124—42no—
35C31×#123—41no—
36C31×#121—21no—
37C31×#121—21no—
38C31×#120—41no—
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—41no—
49C2×C22⋊C4×C31D411yescharacteristic
50C2×C22⋊C4×C31C2311yescharacteristic
51C2×C42×C31—21no—
52C31×#107—41no—
53C31×#102—41no—
54C31×#102—41no—
55C31×#99—42no—
56C31×C22⋊C8—21no—
57C31×C22⋊C8—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—41no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66C24×C31—21no—
67C24×C31C2×D411yescharacteristic
68C2×Q8×C31—42no—
69C2×Q8×C31—42no—
70C2×Q8×C31—81no—
71C2×D4×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—41no—
74C2×D4×C31—42no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—81no—
84C2×D4×C31—81no—
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×C31—41no—
93C22×C4×C31—81no—
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96D8×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—81no—
103C22⋊C4×C31—21no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—42no—
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—81no—
115C22⋊C4×C31—81no—
116C22⋊C4×C31—82no—
117C22⋊C4×C31D812yescomplemented
118C42×C31—42no—
119#12151C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—21no—
121C23×C31—21no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—81no—
135C23×C31—81no—
136C23×C31#12011yescharacteristic
137C23×C31C2×D811yescharacteristic
138C23×C31AGL(1,ℤ/8)11yescharacteristic
139Q8×C31—84no—
140D4×C31—42no—
141D4×C31—82no—
142D4×C31—82no—
143D4×C31—81no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148D4×C31—81no—
149D4×C31—81no—
150C2×C4×C31—22no—
151C2×C4×C31—22no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
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—42no—
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—81no—
174C2×C4×C31—81no—
175C2×C4×C31—81no—
176C2×C4×C31—81no—
177C8×C31—81no—
178C8×C31—81no—
179#2712C2×C3111yescharacteristic, complemented
180#2044C2×C3111yescharacteristic, complemented
181#1607C2×C3111yescharacteristic, complemented
182#1595C2×C3111yescharacteristic
183#1344C2×C3111yescharacteristic, complemented
184#1343C2×C3111yescharacteristic, complemented
185#1326C2×C3111yescharacteristic, complemented
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—41no—
191C22×C31—41no—
192C22×C31—41no—
193C22×C31—41no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—82no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—161no—
207C22×C31#44711yescharacteristic
208C22×C31#45411yescharacteristic
209C22×C31B(4,𝔽2)11yescharacteristic
210C4×C31—42no—
211C4×C31—42no—
212C4×C31—41no—
213C4×C31—41no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31—161no—
219#534C22×C3111yescharacteristic
220C2×#124—21no—
221#480—21no—
222#447—21no—
223C2×#99—21no—
224#397C22×C3111yescharacteristic
225#394—21no—
226#393—21no—
227#392C22×C3111yescharacteristic
228#391—21no—
229#388—21no—
230#386—21no—
231#386—21no—
232#385C22×C3111yescharacteristic
233#342C22×C3111yescharacteristic
234#327—21no—
235#327—21no—
236#323C22×C3111yescharacteristic
237#323C22×C3111yescharacteristic
238C2×C31—21no—
239C2×C31—21no—
240C2×C31—41no—
241C2×C31—41no—
242C2×C31—161no—
243C2×C31—161no—
244C2×C31#159811yescharacteristic
245C2×C31#219811yescharacteristic
246C2×C31#219911yescharacteristic
247C22×Q8—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4D4×C3111yescharacteristic
251C23×C4D4×C3111yescharacteristic
252C2×D8—41no—
253#124—42no—
254#124—42no—
255#123—41no—
256#121—21no—
257#121—21no—
258#120—41no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4D4×C3111yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4D4×C3111yescharacteristic
270C2×C22⋊C4C23×C3111yescharacteristic
271C2×C42—21no—
272#107—41no—
273#102—41no—
274#102—41no—
275#99—42no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—41no—
283#95D4×C3111yescharacteristic
284#95D4×C3111yescharacteristic
285C31#1215111yescharacteristic, direct product factor, Sylow
286C24—21no—
287C24—21no—
288C24C2×D4×C3111yescharacteristic
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—81no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—81no—
305C2×D4—81no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—81no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317D8—82no—
318C2×C8—41no—
319C2×C8—41no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—81no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—82no—
338C22⋊C4D8×C3112yescomplemented
339C42—42no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—81no—
355C23—81no—
356C23C31×#12011yescharacteristic
357C23C2×D8×C3111yescharacteristic
358C23C31×AGL(1,ℤ/8)11yescharacteristic
359Q8—84no—
360D4—42no—
361D4—82no—
362D4—82no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
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—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C8—81no—
398C8—81no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—82no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—161no—
420C22C31×#44711yescharacteristic
421C22C31×#45411yescharacteristic
422C22C31×B(4,𝔽2)11yescharacteristic
423C4—42no—
424C4—42no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—161no—
432C2—21no—
433C2—21no—
434C2—41no—
435C2—41no—
436C2—161no—
437C2—161no—
438C2C31×#159811yescharacteristic
439C2C31×#219811yescharacteristic
440C2C31×#219911yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#1595C2
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×#2712C2semidirect
2. C31×#2044C2semidirect
3. C31×#1607C2semidirect
5. C31×#1344C2semidirect
6. C31×#1343C2semidirect
7. C31×#1326C2semidirect
117. C22⋊C4×C31D8semidirect
119. #12151C31direct (N × H)
179. #2712C2×C31semidirect
180. #2044C2×C31semidirect
181. #1607C2×C31semidirect
183. #1344C2×C31semidirect
184. #1343C2×C31semidirect
185. #1326C2×C31semidirect
285. C31#12151direct (N × H)
338. C22⋊C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27122
C31×#20442
C31×#16072
C31×#15952
C31×#13442
C31×#13432
C31×#13262
#1215131