Finite Group Databasebeta

← C31×#10002 #21466761

Subgroups — C31×#10002 #21466761

Order 7936.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3104C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic
3C31×#2580C211yesmaximal, characteristic
4C31×#1518C211yesmaximal, characteristic, complemented
5C31×#1487C211yesmaximal, characteristic, complemented
6C31×#1478C211yesmaximal, characteristic
7C31×#1286C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C4×Q8×C31C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#484—21no—
12C31×#478—22no—
13C31×#477—21no—
14C31×#476—21no—
15C31×#474—21no—
16C31×#462—21no—
17C31×#451—21no—
18C4×Q16×C31—22no—
19C31×#431C2211yescharacteristic
20C31×#431C2211yescharacteristic
21C31×#428—21no—
22C31×#423C2211yescharacteristic
23C31×#418—21no—
24C31×#399—21no—
25C31×#393—21no—
26C31×#390—21no—
27C31×#389—21no—
28C31×#385—22no—
29C31×#382C2211yescharacteristic
30C31×#328—22no—
31C31×Q8⋊C8—22no—
32C31×D4⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C2×Q16×C31—22no—
37C2×M16×C31—21no—
38C31×C4⋊Q8D411yescharacteristic
39C31×C4⋊Q8D411yescharacteristic
40C31×#125—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C4×Q8×C31—21no—
44C4×Q8×C31—21no—
45C4×Q8×C31—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C31×#117—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—22no—
62C2×C42×C31—21no—
63C2×C42×C31C2311yescharacteristic
64C31×#107—22no—
65C31×#107—22no—
66C31×#106—22no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8—21no—
69C31×C4⋊C8—21no—
70C31×C4⋊C8D412yes—
71C31×#103—22no—
72C31×#103—22no—
73C31×#102—22no—
74C31×#102—22no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C4×C8×C31—21no—
78C4×C8×C31D412yes—
79C31×#95—22no—
80C31×#95—21no—
81C31×#95D412yes—
82C24×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—42no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31Q1612yescomplemented
94C22×C4×C31—21no—
95C22×C4×C31—22no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—41no—
99C22×C4×C31—41no—
100C22×C4×C31—82no—
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D411yescharacteristic
104Q16×C31—44no—
105M16×C31—41no—
106M16×C31—41no—
107C2×C8×C31—22no—
108C2×C8×C31—22no—
109C2×C8×C31—41no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—41no—
122C4⋊C4×C31—41no—
123C4⋊C4×C31—41no—
124C4⋊C4×C31—82no—
125C22⋊C4×C31—44no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—82no—
130C42×C31—41no—
131C42×C31—41no—
132C42×C31—41no—
133C42×C31C2×D411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31C2×D411yescharacteristic
136C42×C31#4111yescharacteristic
137#10002C3111yesmaximal, characteristic, direct product factor, Sylow
138C23×C31—22no—
139C23×C31—42no—
140C23×C31—42no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31#12011yescharacteristic
144Q8×C31—22no—
145Q8×C31—22no—
146Q8×C31—41no—
147Q8×C31—41no—
148Q8×C31—42no—
149Q8×C31—81no—
150D4×C31—22no—
151D4×C31—22no—
152D4×C31—41no—
153D4×C31—41no—
154C2×C4×C31—22no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—42no—
160C2×C4×C31—41no—
161C2×C4×C31—42no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—82no—
170C2×C4×C31—82no—
171C2×C4×C31#12011yescharacteristic
172C2×C4×C31#12011yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31#12111yescharacteristic
176C2×C4×C31#12711yescharacteristic
177C2×C4×C31C2×Q1611yescharacteristic
178C2×C4×C31#13511yescharacteristic
179C2×C4×C31#13711yescharacteristic
180C2×C4×C31#13711yescharacteristic
181C8×C31—42no—
182C8×C31—42no—
183C8×C31—81no—
184#3104C2×C3111yescharacteristic, complemented
185#3077C2×C3111yescharacteristic
186#2580C2×C3111yescharacteristic
187#1518C2×C3111yescharacteristic, complemented
188#1487C2×C3111yescharacteristic, complemented
189#1478C2×C3111yescharacteristic
190#1286C2×C3111yescharacteristic, complemented
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—44no—
194C22×C31—42no—
195C22×C31—82no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31#39211yescharacteristic
199C22×C31#45111yescharacteristic
200C22×C31#45211yescharacteristic
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—42no—
206C4×C31—82no—
207C4×C31—81no—
208C4×C31—81no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31#45111yescharacteristic
212C4×C31#45211yescharacteristic
213C4×C31#46711yescharacteristic
214C4×C31#47011yescharacteristic
215#533C22×C3111yescharacteristic
216C2×C4×Q8C22×C3111yescharacteristic
217C2×C4×D4C22×C3111yescharacteristic
218#484—21no—
219#478—22no—
220#477—21no—
221#476—21no—
222#474—21no—
223#462—21no—
224#451—21no—
225C4×Q16—22no—
226#431C22×C3111yescharacteristic
227#431C22×C3111yescharacteristic
228#428—21no—
229#423C22×C3111yescharacteristic
230#418—21no—
231#399—21no—
232#393—21no—
233#390—21no—
234#389—21no—
235#385—22no—
236#382C22×C3111yescharacteristic
237#328—22no—
238Q8⋊C8—22no—
239D4⋊C8—22no—
240C2×C31—21no—
241C2×C31—21no—
242C2×C31—82no—
243C2×C31#200111yescharacteristic
244C2×C31#200811yescharacteristic
245C2×C31#201011yescharacteristic
246C22×Q8D4×C3111yescharacteristic
247C22×D4D4×C3111yescharacteristic
248C23×C4—21no—
249C2×Q16—22no—
250C2×M16—21no—
251C4⋊Q8D4×C3111yescharacteristic
252C4⋊Q8D4×C3111yescharacteristic
253#125—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266#117—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4D4×C3111yescharacteristic
271C2×C4⋊C4D4×C3111yescharacteristic
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C42—21no—
276C2×C42C23×C3111yescharacteristic
277#107—22no—
278#107—22no—
279#106—22no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8D4×C3112yes—
284#103—22no—
285#103—22no—
286#102—22no—
287#102—22no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C4×C8—21no—
291C4×C8D4×C3112yes—
292#95—22no—
293#95—21no—
294#95D4×C3112yes—
295C31#1000211yescharacteristic, direct product factor, Sylow
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—42no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4Q16×C3112yescomplemented
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318Q16—44no—
319M16—41no—
320M16—41no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—82no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—82no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42C2×D4×C3111yescharacteristic
348C42C2×D4×C3111yescharacteristic
349C42C2×D4×C3111yescharacteristic
350C42#41×C3111yescharacteristic
351C23—22no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23C31×#12011yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—41no—
360Q8—41no—
361Q8—42no—
362Q8—81no—
363D4—22no—
364D4—22no—
365D4—41no—
366D4—41no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4C31×#12011yescharacteristic
385C2×C4C31×#12011yescharacteristic
386C2×C4C31×#12111yescharacteristic
387C2×C4C31×#12111yescharacteristic
388C2×C4C31×#12111yescharacteristic
389C2×C4C31×#12711yescharacteristic
390C2×C4C2×Q16×C3111yescharacteristic
391C2×C4C31×#13511yescharacteristic
392C2×C4C31×#13711yescharacteristic
393C2×C4C31×#13711yescharacteristic
394C8—42no—
395C8—42no—
396C8—81no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22C31×#39211yescharacteristic
405C22C31×#45111yescharacteristic
406C22C31×#45211yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4C31×#45111yescharacteristic
418C4C31×#45211yescharacteristic
419C4C31×#46711yescharacteristic
420C4C31×#47011yescharacteristic
421C2—21no—
422C2—21no—
423C2—82no—
424C2C31×#200111yescharacteristic
425C2C31×#200811yescharacteristic
426C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
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×#3104C2semidirect
4. C31×#1518C2semidirect
5. C31×#1487C2semidirect
7. C31×#1286C2semidirect
93. C2×D4×C31Q16semidirect
137. #10002C31direct (N × H)
184. #3104C2×C31semidirect
187. #1518C2×C31semidirect
188. #1487C2×C31semidirect
190. #1286C2×C31semidirect
295. C31#10002direct (N × H)
307. C2×D4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31042
C31×#30772
C31×#25802
C31×#15182
C31×#14872
C31×#14782
C31×#12862
#1000231