Finite Group Databasebeta

← C31×#9998 #21466757

Subgroups — C31×#9998 #21466757

Order 7936.

602 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3103C211yesmaximal, characteristic, complemented
2C31×#3036C211yesmaximal, characteristic
3C31×#2580C211yesmaximal, characteristic, complemented
4C31×#1517C211yesmaximal, characteristic, complemented
5C31×#1487C211yesmaximal, characteristic, complemented
6C31×#1478C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#532C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#481—21no—
13C31×#475—21no—
14C31×#463—22no—
15C31×#461—21no—
16C31×#460—21no—
17C31×#459—21no—
18C31×#448—21no—
19C4×SD16×C31—22no—
20C31×#431C2211yescharacteristic
21C31×#431C2211yescharacteristic
22C31×#423C2211yescharacteristic
23C31×#417—21no—
24C31×#399—21no—
25C31×#393—21no—
26C31×#390—21no—
27C31×#389—21no—
28C31×#385—22no—
29C31×#382C2211yescharacteristic
30C31×#327—22no—
31C31×Q8⋊C8—22no—
32C31×D4⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C22×D4×C31D411yescharacteristic
36C23×C4×C31—21no—
37C31×AGL(1,ℤ/8)—42no—
38C31×AGL(1,ℤ/8)—42no—
39C2×SD16×C31—22no—
40C2×SD16×C31—22no—
41C2×D8×C31—22no—
42C2×M16×C31—21no—
43C31×C4⋊Q8D411yescharacteristic
44C31×#127D411yescharacteristic
45C31×#124—21no—
46C31×#121—21no—
47C31×#121—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31—22no—
66C2×C42×C31—21no—
67C2×C42×C31C2311yescharacteristic
68C31×#106—22no—
69C31×C4⋊C8—21no—
70C31×C4⋊C8—21no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8D412yes—
73C31×#103—22no—
74C31×#103—22no—
75C31×#102—22no—
76C31×#102—22no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C4×C8×C31—21no—
80C4×C8×C31D412yes—
81C31×#95—22no—
82C31×#95—21no—
83C31×#95D412yes—
84C24×C31—21no—
85#41×C31—41no—
86#41×C31—41no—
87#41×C31—41no—
88#41×C31—41no—
89C2×Q8×C31—21no—
90C2×Q8×C31—21no—
91C2×Q8×C31—21no—
92C2×Q8×C31—42no—
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—21no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—21no—
101C2×D4×C31—41no—
102C2×D4×C31SD1612yescomplemented
103C22×C4×C31—21no—
104C22×C4×C31—22no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—41no—
108C22×C4×C31—41no—
109C22×C4×C31—41no—
110C22×C4×C31—82no—
111C22×C4×C31C2×D411yescharacteristic
112C22×C4×C31C2×D411yescharacteristic
113C22×C4×C31C2×D411yescharacteristic
114SD16×C31—44no—
115SD16×C31—44no—
116D8×C31—44no—
117M16×C31—41no—
118M16×C31—41no—
119C2×C8×C31—22no—
120C2×C8×C31—22no—
121C2×C8×C31—41no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—41no—
129C4⋊C4×C31—41no—
130C4⋊C4×C31—41no—
131C4⋊C4×C31—41no—
132C4⋊C4×C31—82no—
133C22⋊C4×C31—44no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C22⋊C4×C31—82no—
139C42×C31—41no—
140C42×C31—41no—
141C42×C31C2×D411yescharacteristic
142C42×C31C2×D411yescharacteristic
143C42×C31C2×D411yescharacteristic
144C42×C31#4111yescharacteristic
145#9998C3111yesmaximal, characteristic, direct product factor, Sylow
146C23×C31—22no—
147C23×C31—42no—
148C23×C31—42no—
149C23×C31—41no—
150C23×C31—41no—
151C23×C31—41no—
152C23×C31#12011yescharacteristic
153Q8×C31—22no—
154Q8×C31—22no—
155Q8×C31—41no—
156Q8×C31—41no—
157Q8×C31—41no—
158Q8×C31—41no—
159D4×C31—22no—
160D4×C31—22no—
161D4×C31—42no—
162D4×C31—41no—
163D4×C31—41no—
164D4×C31—41no—
165D4×C31—41no—
166D4×C31—81no—
167D4×C31—81no—
168D4×C31—81no—
169C2×C4×C31—22no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—42no—
175C2×C4×C31—41no—
176C2×C4×C31—42no—
177C2×C4×C31—41no—
178C2×C4×C31—42no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—41no—
182C2×C4×C31—42no—
183C2×C4×C31—82no—
184C2×C4×C31—82no—
185C2×C4×C31—81no—
186C2×C4×C31—81no—
187C2×C4×C31#12011yescharacteristic
188C2×C4×C31#12011yescharacteristic
189C2×C4×C31#12111yescharacteristic
190C2×C4×C31#12111yescharacteristic
191C2×C4×C31#12111yescharacteristic
192C2×C4×C31#12711yescharacteristic
193C2×C4×C31C2×SD1611yescharacteristic
194C2×C4×C31#13511yescharacteristic
195C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C31#13711yescharacteristic
197C8×C31—42no—
198C8×C31—42no—
199C8×C31—81no—
200#3103C2×C3111yescharacteristic, complemented
201#3036C2×C3111yescharacteristic
202#2580C2×C3111yescharacteristic, complemented
203#1517C2×C3111yescharacteristic, complemented
204#1487C2×C3111yescharacteristic, complemented
205#1478C2×C3111yescharacteristic, complemented
206#1286C2×C3111yescharacteristic, complemented
207C22×C31—21no—
208C22×C31—21no—
209C22×C31—44no—
210C22×C31—42no—
211C22×C31—82no—
212C22×C31—81no—
213C22×C31—81no—
214C22×C31—81no—
215C22×C31—81no—
216C22×C31—81no—
217C22×C31#39211yescharacteristic
218C22×C31#44811yescharacteristic
219C22×C31#44911yescharacteristic
220C4×C31—21no—
221C4×C31—21no—
222C4×C31—21no—
223C4×C31—21no—
224C4×C31—42no—
225C4×C31—82no—
226C4×C31—81no—
227C4×C31—81no—
228C4×C31—81no—
229C4×C31#45011yescharacteristic
230C4×C31#45211yescharacteristic
231C4×C31#46511yescharacteristic
232C4×C31#46811yescharacteristic
233C2×AGL(1,ℤ/8)—21no—
234#532C22×C3111yescharacteristic
235C2×C4×Q8C22×C3111yescharacteristic
236C2×C4×D4C22×C3111yescharacteristic
237#481—21no—
238#475—21no—
239#463—22no—
240#461—21no—
241#460—21no—
242#459—21no—
243#448—21no—
244C4×SD16—22no—
245#431C22×C3111yescharacteristic
246#431C22×C3111yescharacteristic
247#423C22×C3111yescharacteristic
248#417—21no—
249#399—21no—
250#393—21no—
251#390—21no—
252#389—21no—
253#385—22no—
254#382C22×C3111yescharacteristic
255#327—22no—
256Q8⋊C8—22no—
257D4⋊C8—22no—
258C2×C31—21no—
259C2×C31—21no—
260C2×C31—82no—
261C2×C31—161no—
262C2×C31#200011yescharacteristic
263C2×C31#200811yescharacteristic
264C2×C31#201011yescharacteristic
265C2×#41—21no—
266C22×Q8D4×C3111yescharacteristic
267C22×D4D4×C3111yescharacteristic
268C23×C4—21no—
269AGL(1,ℤ/8)—42no—
270AGL(1,ℤ/8)—42no—
271C2×SD16—22no—
272C2×SD16—22no—
273C2×D8—22no—
274C2×M16—21no—
275C4⋊Q8D4×C3111yescharacteristic
276#127D4×C3111yescharacteristic
277#124—21no—
278#121—21no—
279#121—21no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4D4×C3111yescharacteristic
294C2×C4⋊C4D4×C3111yescharacteristic
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C42—21no—
299C2×C42C23×C3111yescharacteristic
300#106—22no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8D4×C3112yes—
305#103—22no—
306#103—22no—
307#102—22no—
308#102—22no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C4×C8—21no—
312C4×C8D4×C3112yes—
313#95—22no—
314#95—21no—
315#95D4×C3112yes—
316C31#999811yescharacteristic, direct product factor, Sylow
317C24—21no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—42no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4SD16×C3112yescomplemented
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—82no—
344C22×C4C2×D4×C3111yescharacteristic
345C22×C4C2×D4×C3111yescharacteristic
346C22×C4C2×D4×C3111yescharacteristic
347SD16—44no—
348SD16—44no—
349D8—44no—
350M16—41no—
351M16—41no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—41no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—82no—
366C22⋊C4—44no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—82no—
372C42—41no—
373C42—41no—
374C42C2×D4×C3111yescharacteristic
375C42C2×D4×C3111yescharacteristic
376C42C2×D4×C3111yescharacteristic
377C42#41×C3111yescharacteristic
378C23—22no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C31×#12011yescharacteristic
385Q8—22no—
386Q8—22no—
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—41no—
391D4—22no—
392D4—22no—
393D4—42no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—81no—
399D4—81no—
400D4—81no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4C31×#12011yescharacteristic
420C2×C4C31×#12011yescharacteristic
421C2×C4C31×#12111yescharacteristic
422C2×C4C31×#12111yescharacteristic
423C2×C4C31×#12111yescharacteristic
424C2×C4C31×#12711yescharacteristic
425C2×C4C2×SD16×C3111yescharacteristic
426C2×C4C31×#13511yescharacteristic
427C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
428C2×C4C31×#13711yescharacteristic
429C8—42no—
430C8—42no—
431C8—81no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22—42no—
436C22—82no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22C31×#39211yescharacteristic
443C22C31×#44811yescharacteristic
444C22C31×#44911yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4C31×#45011yescharacteristic
455C4C31×#45211yescharacteristic
456C4C31×#46511yescharacteristic
457C4C31×#46811yescharacteristic
458C2—21no—
459C2—21no—
460C2—82no—
461C2—161no—
462C2C31×#200011yescharacteristic
463C2C31×#200811yescharacteristic
464C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3103C2semidirect
3. C31×#2580C2semidirect
4. C31×#1517C2semidirect
5. C31×#1487C2semidirect
6. C31×#1478C2semidirect
7. C31×#1286C2semidirect
102. C2×D4×C31SD16semidirect
145. #9998C31direct (N × H)
200. #3103C2×C31semidirect
202. #2580C2×C31semidirect
203. #1517C2×C31semidirect
204. #1487C2×C31semidirect
205. #1478C2×C31semidirect
206. #1286C2×C31semidirect
316. C31#9998direct (N × H)
335. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31032
C31×#30362
C31×#25802
C31×#15172
C31×#14872
C31×#14782
C31×#12862
#999831