Finite Group Databasebeta

← C31×#10081 #21466840

Subgroups — C31×#10081 #21466840

Order 7936.

648 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×C31×#497C211yesmaximal, characteristic, complemented
2C2×C31×#489C211yesmaximal, characteristic, complemented
3C31×#2819C211yesmaximal, characteristic
4C31×#1993C211yesmaximal, characteristic, complemented
5C2×C31×#329C211yesmaximal, characteristic, complemented
6C2×C31×#329C211yesmaximal, characteristic, complemented
7C31×#1295C211yesmaximal, characteristic, complemented
8C22×Q16×C31—21no—
9C22×SD16×C31—21no—
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C31×#124C2211yescharacteristic
12C2×C31×#124C2211yescharacteristic
13C31×#497—22no—
14C31×#497—22no—
15C31×#489—22no—
16C31×#489—22no—
17C2×C31×#103—21no—
18C2×C31×#102—21no—
19C2×C31×#97C2211yescharacteristic
20C2×C31×#97C2211yescharacteristic
21C2×C31×#97C2211yescharacteristic
22C31×#394—22no—
23C31×#393—21no—
24C31×#393—21no—
25C31×#392—21no—
26C31×#392—21no—
27C31×#382C2211yescharacteristic
28C31×#340—22no—
29C31×#329—22no—
30C31×#329—22no—
31C22×Q8×C31—21no—
32C22×Q8×C31D411yescharacteristic
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31—22no—
35C22×D4×C31D411yescharacteristic
36C22×D4×C31D411yescharacteristic
37C2×Q16×C31—24no—
38C2×Q16×C31—44no—
39C2×SD16×C31—24no—
40C2×SD16×C31—44no—
41C22×C8×C31—21no—
42C22×C8×C31—21no—
43C22×C8×C31—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8D412yescomplemented
48C31×#124—21no—
49C31×#124—21no—
50C31×#124—21no—
51C31×#124—21no—
52C31×#124—21no—
53C31×#124—21no—
54C31×#124—21no—
55C31×#124—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31D412yes—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C42×C31C2311yescharacteristic
66C31×#103—24no—
67C31×#102—24no—
68C31×#97—21no—
69C31×#97—21no—
70C31×#97—21no—
71C31×#97—21no—
72C31×#97D412yescomplemented
73C31×#97D412yescomplemented
74C31×#95—22no—
75C31×#95—21no—
76C31×#95—21no—
77C24×C31—21no—
78C24×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×Q8×C31—22no—
82C2×Q8×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—42no—
85C2×Q8×C31—42no—
86C2×Q8×C31—42no—
87C2×Q8×C31—42no—
88C2×Q8×C31—41no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—42no—
94C2×D4×C31—42no—
95C2×D4×C31—82no—
96C2×D4×C31—82no—
97C2×D4×C31—82no—
98C22×C4×C31—22no—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—41no—
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105Q16×C31—48no—
106SD16×C31—48no—
107C2×C8×C31—24no—
108C2×C8×C31—22no—
109C2×C8×C31—22no—
110C2×C8×C31—22no—
111C2×C8×C31—41no—
112C2×C8×C31—42no—
113C2×C8×C31—41no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—42no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—82no—
128C22⋊C4×C31—82no—
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132C42×C31#4111yescharacteristic
133#10081C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—42no—
139C23×C31—42no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31#12011yescharacteristic
145Q8×C31—42no—
146Q8×C31—42no—
147Q8×C31—42no—
148Q8×C31—42no—
149Q8×C31—81no—
150Q8×C31—81no—
151Q8×C31—82no—
152D4×C31—42no—
153D4×C31—42no—
154D4×C31—81no—
155D4×C31—81no—
156D4×C31—88no—
157C2×C4×C31—22no—
158C2×C4×C31—21no—
159C2×C4×C31—22no—
160C2×C4×C31—21no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—82no—
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12011yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31#12711yescharacteristic
176C8×C31—24no—
177C8×C31—42no—
178C8×C31—42no—
179C2×#497C2×C3111yescharacteristic, complemented
180C2×#489C2×C3111yescharacteristic, complemented
181#2819C2×C3111yescharacteristic
182#1993C2×C3111yescharacteristic, complemented
183C2×#329C2×C3111yescharacteristic, complemented
184C2×#329C2×C3111yescharacteristic, complemented
185#1295C2×C3111yescharacteristic, complemented
186C22×C31—82no—
187C22×C31—82no—
188C22×C31—82no—
189C22×C31—82no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31#39211yescharacteristic
197C22×C31#45312yes—
198C22×C31#45511yescharacteristic
199C22×C31#45512yes—
200C22×C31#45611yescharacteristic
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—82no—
208C4×C31—81no—
209C4×C31—82no—
210C4×C31—81no—
211C22×Q16—21no—
212C22×SD16—21no—
213C2×C4⋊Q8C22×C3111yescharacteristic
214C2×#124C22×C3111yescharacteristic
215C2×#124C22×C3111yescharacteristic
216#497—22no—
217#497—22no—
218#489—22no—
219#489—22no—
220C2×#103—21no—
221C2×#102—21no—
222C2×#97C22×C3111yescharacteristic
223C2×#97C22×C3111yescharacteristic
224C2×#97C22×C3111yescharacteristic
225#394—22no—
226#393—21no—
227#393—21no—
228#392—21no—
229#392—21no—
230#382C22×C3111yescharacteristic
231#340—22no—
232#329—22no—
233#329—22no—
234C2×C31—161no—
235C2×C31—161no—
236C2×C31#165912yes—
237C2×C31#166112yes—
238C2×C31#200211yescharacteristic
239C2×C31#200411yescharacteristic
240C2×C31#200711yescharacteristic
241C22×Q8—21no—
242C22×Q8D4×C3111yescharacteristic
243C22×Q8D4×C3111yescharacteristic
244C22×D4—22no—
245C22×D4D4×C3111yescharacteristic
246C22×D4D4×C3111yescharacteristic
247C2×Q16—24no—
248C2×Q16—44no—
249C2×SD16—24no—
250C2×SD16—44no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8D4×C3112yescomplemented
258#124—21no—
259#124—21no—
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124—21no—
265#124—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4D4×C3112yes—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C42C23×C3111yescharacteristic
276#103—24no—
277#102—24no—
278#97—21no—
279#97—21no—
280#97—21no—
281#97—21no—
282#97D4×C3112yescomplemented
283#97D4×C3112yescomplemented
284#95—22no—
285#95—21no—
286#95—21no—
287C31#1008111yescharacteristic, direct product factor, Sylow
288C24—21no—
289C24—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4C2×D4×C3111yescharacteristic
314C22×C4C2×D4×C3111yescharacteristic
315C22×C4C2×D4×C3111yescharacteristic
316Q16—48no—
317SD16—48no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42C2×D4×C3111yescharacteristic
341C42C2×D4×C3111yescharacteristic
342C42C2×D4×C3111yescharacteristic
343C42#41×C3111yescharacteristic
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C31×#12011yescharacteristic
355Q8—42no—
356Q8—42no—
357Q8—42no—
358Q8—42no—
359Q8—81no—
360Q8—81no—
361Q8—82no—
362D4—42no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—88no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4C31×#12011yescharacteristic
381C2×C4C31×#12011yescharacteristic
382C2×C4C31×#12111yescharacteristic
383C2×C4C31×#12111yescharacteristic
384C2×C4C31×#12111yescharacteristic
385C2×C4C31×#12711yescharacteristic
386C8—24no—
387C8—42no—
388C8—42no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22C31×#39211yescharacteristic
400C22C31×#45312yes—
401C22C31×#45511yescharacteristic
402C22C31×#45512yes—
403C22C31×#45611yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4—81no—
414C2—161no—
415C2—161no—
416C2C31×#165912yes—
417C2C31×#166112yes—
418C2C31×#200211yescharacteristic
419C2C31×#200411yescharacteristic
420C2C31×#200711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2819C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#497C2semidirect
2. C2×C31×#489C2semidirect
4. C31×#1993C2semidirect
5. C2×C31×#329C2semidirect
6. C2×C31×#329C2semidirect
7. C31×#1295C2semidirect
47. C31×C4⋊Q8D4semidirect
72. C31×#97D4semidirect
73. C31×#97D4semidirect
133. #10081C31direct (N × H)
179. C2×#497C2×C31semidirect
180. C2×#489C2×C31semidirect
182. #1993C2×C31semidirect
183. C2×#329C2×C31semidirect
184. C2×#329C2×C31semidirect
185. #1295C2×C31semidirect
257. C4⋊Q8D4×C31semidirect
282. #97D4×C31semidirect
283. #97D4×C31semidirect
287. C31#10081direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4972
C2×C31×#4892
C31×#28192
C31×#19932
C2×C31×#3292
C31×#12952
#1008131