Finite Group Databasebeta

← #26144687

Subgroups — #26144687

Order 12032.

1346 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689022C212yesmaximal
2C2×#15689009C212yesmaximal
3#20241645C212yesmaximal
4#20241641C212yesmaximal
5#20241641C212yesmaximal
6#20241622C212yesmaximal
7#20241618C211yesmaximal, characteristic
8#20241615C211yesmaximal, characteristic
9C2×C47×#389C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C4⋊Dic47C2212yes—
12C22×C4⋊Dic47C2212yes—
13C22×C4×Dic47C2212yes—
14#15689022—28no—
15#15689009—28no—
16C2×C47⋊#95C2212yes—
17C2×C47⋊#95C2212yes—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C22×C4⋊C4×C47C2212yes—
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C42×C47C2211yescharacteristic
32C47×#389C418yescomplemented
33C2×#95×C47C2212yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C2×C4⋊Dic47—24no—
39C2×C4⋊Dic47—24no—
40C2×C4⋊Dic47—24no—
41C2×C4⋊Dic47—24no—
42C2×C4⋊Dic47—24no—
43C2×C4⋊Dic47—24no—
44C2×C4⋊Dic47—42no—
45C2×C4⋊Dic47—42no—
46C2×C4⋊Dic47—42no—
47C2×C4×Dic47—24no—
48C2×C4×Dic47—24no—
49C2×C4×Dic47—42no—
50C47⋊#95—24no—
51C47⋊#95—24no—
52C47⋊#95—24no—
53C47⋊#95—24no—
54C47⋊#95—22no—
55C47⋊#95—24no—
56C47⋊#95—24no—
57C47⋊#95—41no—
58C47⋊#95—42no—
59C47⋊#95—42no—
60C47⋊#95—41no—
61C47⋊#95D414yes—
62C23×C4×C47C2311yescharacteristic
63C23×C4×C47C2312yes—
64C23×C4×C47C2311yescharacteristic
65C23×C4×C47C2312yes—
66C23×C4×C47C2311yescharacteristic
67C2×C4⋊C4×C47—24no—
68C2×C4⋊C4×C47—24no—
69C2×C4⋊C4×C47—42no—
70C2×C4⋊C4×C47C2×C418yes—
71C2×C4⋊C4×C47C2×C414yes—
72C2×C4⋊C4×C47C2×C414yes—
73C2×C4⋊C4×C47D414yes—
74C2×C4⋊C4×C47Q814yes—
75C2×C42×C47—42no—
76C2×C42×C47C2×C414yes—
77#95×C47C2×C418yes—
78C24×C47C2411yescharacteristic
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47—22no—
88C22×C4×C47—22no—
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47C22×C412yes—
98C22×C4×C47C22×C414yes—
99C22×C4×C47C22×C412yes—
100C22×C4×C47C22×C414yes—
101C22×C4×C47C22×C412yes—
102C22×C4×C47C2×D412yes—
103C22×C4×C47C2×D414yes—
104C22×C4×C47C2×D412yes—
105C22×C4×C47C2×Q812yes—
106C22×C4×C47C2×Q814yes—
107C22×C4×C47#4112yes—
108C22×Dic47—22no—
109C22×Dic47—22no—
110C22×Dic47—22no—
111C22×Dic47—22no—
112C22×Dic47—22no—
113C22×Dic47—22no—
114C22×Dic47—22no—
115C22×Dic47—22no—
116C22×Dic47—22no—
117C22×Dic47—22no—
118C22×Dic47—22no—
119C22×Dic47—22no—
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47C2×D414yes—
135C22×Dic47#4114yes—
136C4⋊C4×C47—216no—
137C4⋊C4×C47—44no—
138C4⋊C4×C47C4⋊C4116yescomplemented
139C42×C47—44no—
140C4⋊Dic47—48no—
141C4⋊Dic47—48no—
142C4⋊Dic47—48no—
143C4×Dic47—48no—
144C23×C47C23×C411yescharacteristic
145C23×C47C22×D411yescharacteristic
146C23×C47C22×D411yescharacteristic
147C23×C47C22×Q811yescharacteristic
148C23×C47C2×#4112yes—
149C23×C47C2×#4111yescharacteristic
150C23×C47H(5,𝔽2)12yes—
151C23×C47H(5,𝔽2)11yescharacteristic
152C23×C472−1+411yescharacteristic
153C23×C472−1+412yes—
154C23×C472−1+412yes—
155C2×C4×C47—24no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—24no—
161C2×C4×C47—24no—
162C2×C4×C47—24no—
163C2×C4×C47—24no—
164C2×C4×C47—24no—
165C2×C4×C47—24no—
166C2×C4×C47—24no—
167C2×C4×C47—24no—
168C2×C4×C47—24no—
169C2×C4×C47—24no—
170C2×C4×C47—24no—
171C2×C4×C47—22no—
172C2×C4×C47—24no—
173C2×C4×C47—22no—
174C2×C4×C47—22no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
179C2×C4×C47—42no—
180C2×C4×C47—42no—
181C2×C4×C47C2×C4⋊C414yes—
182C2×C4×C47C2×C4⋊C418yes—
183C2×C4×C47C4×D414yes—
184C2×Dic47—24no—
185C2×Dic47—24no—
186C2×Dic47—24no—
187C2×Dic47—24no—
188C2×Dic47—24no—
189C2×Dic47—24no—
190C2×Dic47—42no—
191C2×Dic47—42no—
192C2×Dic47—42no—
193C2×Dic47—42no—
194C2×Dic47—42no—
195C2×Dic47—42no—
196C2×Dic47—42no—
197C2×Dic47—42no—
198C2×Dic47—42no—
199C2×Dic47—42no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—42no—
203C2×Dic47—42no—
204C2×Dic47—42no—
205C2×Dic47—42no—
206C2×Dic47—42no—
207C2×Dic47—42no—
208C2×Dic47—42no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212#14439—471nomaximal, Sylow
213C22×C47C22×C4⋊C411yescharacteristic
214C22×C47C2×C4×D411yescharacteristic
215C22×C47C4×#4111yescharacteristic
216C22×C47#51811yescharacteristic
217C22×C47#51911yescharacteristic
218C22×C47#51911yescharacteristic
219C22×C47#52011yescharacteristic
220C22×C47#53612yes—
221C22×C47#53712yes—
222C22×C47#53912yes—
223C22×C47#54112yes—
224C22×C47#54112yes—
225C22×C47#54412yes—
226C22×C47D4211yescharacteristic
227C22×C47#54612yes—
228C22×C47#54711yescharacteristic
229C22×C47#54812yes—
230C22×C47D4×Q811yescharacteristic
231C22×C47#55312yes—
232C22×C47#55411yescharacteristic
233C22×C47#55712yes—
234C22×C47#56212yes—
235C22×C47#56312yes—
236C4×C47—24no—
237C4×C47—28no—
238C4×C47—24no—
239C4×C47—44no—
240C4×C47—42no—
241Dic47—44no—
242Dic47—44no—
243Dic47—82no—
244Dic47—82no—
245C2×#389—472no—
246C2×#389D4711yescharacteristic
247C2×#387—472no—
248#1442—472no—
249#1441—472no—
250#1441—472no—
251#1436—472no—
252C4⋊#95—471no—
253#1434—471no—
254C2×C47#231611yescharacteristic
255C2×C47#232711yescharacteristic
256C2×C47#233611yescharacteristic
257C2×C47D4×C4⋊C411yescharacteristic
258C2×C47#235111yescharacteristic
259C2×C47#236411yescharacteristic
260C2×C47#237811yescharacteristic
261C2×C47#268112yes—
262C2×C47#271312yes—
263C2×C47#272512yes—
264C2×C47#280212yes—
265C22×C4⋊C4—472no—
266C22×C4⋊C4—472no—
267C22×C4⋊C4—472no—
268C22×C4⋊C4C2×D4712yes—
269C22×C4⋊C4C2×D4711yescharacteristic
270C22×C4⋊C4C2×D4711yescharacteristic
271C22×C42—472no—
272C22×C42C2×D4711yescharacteristic
273#389—948no—
274#389Dic4718yescomplemented
275#387—948no—
276C2×#95—471no—
277C2×#95—471no—
278C2×#95—472no—
279C2×#95—472no—
280C2×#95—472no—
281C2×#95—472no—
282C2×#95—472no—
283C2×#95—472no—
284C2×#95—472no—
285C2×#95—471no—
286C2×#95—471no—
287C2×#95—472no—
288C2×#95C2×D4712yes—
289C47#1443911yescharacteristic, complemented, Sylow
290C23×C4—472no—
291C23×C4—472no—
292C23×C4—472no—
293C23×C4—472no—
294C23×C4C22×D4712yes—
295C23×C4C22×D4711yescharacteristic
296C23×C4C22×D4712yes—
297C23×C4C22×D4711yescharacteristic
298C23×C4C22×D4711yescharacteristic
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—944no—
303C2×C4⋊C4—944no—
304C2×C4⋊C4—944no—
305C2×C4⋊C4—944no—
306C2×C4⋊C4—944no—
307C2×C4⋊C4—944no—
308C2×C4⋊C4—1882no—
309C2×C4⋊C4—1882no—
310C2×C4⋊C4—1882no—
311C2×C4⋊C4Dic9414yes—
312C2×C4⋊C4D18814yes—
313C2×C4⋊C4C2×Dic4718yes—
314C2×C4⋊C4C2×Dic4714yes—
315C2×C4⋊C4C2×Dic4714yes—
316C2×C42—42no—
317C2×C42—944no—
318C2×C42—944no—
319C2×C42—1882no—
320C2×C42C2×Dic4714yes—
321#95—474no—
322#95—944no—
323#95—944no—
324#95—944no—
325#95—944no—
326#95—944no—
327#95—944no—
328#95—942no—
329#95—1882no—
330#95—1882no—
331#95—1881no—
332#95—1881no—
333#95C2×Dic4718yes—
334C24C23×D4711yescharacteristic
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—474no—
354C22×C4—474no—
355C22×C4—942no—
356C22×C4—942no—
357C22×C4—942no—
358C22×C4—942no—
359C22×C4—942no—
360C22×C4—942no—
361C22×C4—942no—
362C22×C4—942no—
363C22×C4—942no—
364C22×C4—942no—
365C22×C4—942no—
366C22×C4—942no—
367C22×C4—942no—
368C22×C4—942no—
369C22×C4—942no—
370C22×C4—942no—
371C22×C4—942no—
372C22×C4—942no—
373C22×C4—942no—
374C22×C4—942no—
375C22×C4—942no—
376C22×C4—942no—
377C22×C4—942no—
378C22×C4—942no—
379C22×C4—942no—
380C22×C4—942no—
381C22×C4C2×Dic9412yes—
382C22×C4C2×Dic9414yes—
383C22×C4C2×D18812yes—
384C22×C4C2×D18814yes—
385C22×C4D4×D4712yes—
386C22×C4#1062535212yes—
387C22×C4C22×Dic4712yes—
388C22×C4C22×Dic4714yes—
389C22×C4C22×Dic4714yes—
390C22×C4C22×Dic4712yes—
391C22×C4C22×Dic4712yes—
392C4⋊C4—216no—
393C4⋊C4—44no—
394C4⋊C4—1888no—
395C4⋊C4—1888no—
396C4⋊C4—1888no—
397C4⋊C4C4⋊Dic47116yescomplemented
398C42—44no—
399C42—1888no—
400C23C23×Dic4711yescharacteristic
401C23C2×D4×D4711yescharacteristic
402C23C22×D18811yescharacteristic
403C23C22×Dic9411yescharacteristic
404C23#41×D4712yes—
405C23C2×#1062535211yescharacteristic
406C23#1309406912yes—
407C23#1309407011yescharacteristic
408C23#1309407111yescharacteristic
409C23#1309407212yes—
410C23#1309407212yes—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—944no—
438C2×C4—944no—
439C2×C4—944no—
440C2×C4—944no—
441C2×C4—944no—
442C2×C4—944no—
443C2×C4—1882no—
444C2×C4—1882no—
445C2×C4—1882no—
446C2×C4—1882no—
447C2×C4—1882no—
448C2×C4—1882no—
449C2×C4—1882no—
450C2×C4—1882no—
451C2×C4—1882no—
452C2×C4—1882no—
453C2×C4—1882no—
454C2×C4—1882no—
455C2×C4—1882no—
456C2×C4—1882no—
457C2×C4—1882no—
458C2×C4—1882no—
459C2×C4—1882no—
460C2×C4—1882no—
461C2×C4—1882no—
462C2×C4—1882no—
463C2×C4—1882no—
464C2×C4—1882no—
465C2×C4C2×C4⋊Dic4718yes—
466C2×C4C2×C4⋊Dic4714yes—
467C2×C4D4×Dic4714yes—
468C22C22×C4⋊Dic4711yescharacteristic
469C22C2×D4×Dic4711yescharacteristic
470C22#41×Dic4711yescharacteristic
471C22#1568959811yescharacteristic
472C22#1568960111yescharacteristic
473C22#1568960111yescharacteristic
474C22#1568960611yescharacteristic
475C22#1568969812yes—
476C22#1568970412yes—
477C22#1568972112yes—
478C22#1568973212yes—
479C22#1568973412yes—
480C22#1568975412yes—
481C22D4×D18811yescharacteristic
482C22#1568976212yes—
483C22#1568977711yescharacteristic
484C22#1568978512yes—
485C22D4×Dic9411yescharacteristic
486C22#1568981612yes—
487C22#1568983011yescharacteristic
488C22#1568984912yes—
489C22#1568986912yes—
490C22#1568987412yes—
491C4—28no—
492C4—24no—
493C4—24no—
494C4—42no—
495C4—44no—
496C4—3762no—
497C4—3762no—
498C4—1884no—
499C4—1884no—
500C2#2024600611yescharacteristic
501C2#2024610811yescharacteristic
502C2#2024619411yescharacteristic
503C2D4×C4⋊Dic4711yescharacteristic
504C2#2024630911yescharacteristic
505C2#2024644011yescharacteristic
506C2#2024655411yescharacteristic
507C2#2024973512yes—
508C2#2025016612yes—
509C2#2025032612yes—
510C2#2025130712yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#389C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×#389D47
hypocenterC47#14439
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
32. C47×#389C4semidirect
138. C4⋊C4×C47C4⋊C4semidirect
274. #389Dic47semidirect
289. C47#14439semidirect
397. C4⋊C4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890222
C2×#156890092
#202416452
#202416412
#202416222
#202416182
#202416152
C2×C47×#3892
#1443947