Finite Group Databasebeta

← #26250390

Subgroups — #26250390

Order 12032.

1076 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689415C211yesmaximal, characteristic, complemented
2C2×#15689228C211yesmaximal, characteristic, complemented
3#20248297C211yesmaximal, characteristic
4#20244588C212yesmaximal, complemented
5#20244396C212yesmaximal, complemented
6#20244362C211yesmaximal, characteristic, complemented
7#20244187C211yesmaximal, characteristic
8#20243700C212yesmaximal, complemented
9#20243587C212yesmaximal, complemented
10#20243403C211yesmaximal, characteristic, complemented
11C2×C47×#556C211yesmaximal, characteristic
12C22×#10625337C2211yescharacteristic
13C2×#13094003C2212yes—
14C2×C4×Dic94C2211yescharacteristic
15#15689415—24no—
16#15689228—24no—
17C2×#13093957C2212yes—
18C2×#13093955C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2212yes—
22C2×#13093947C2211yescharacteristic
23C2×#13093947C2212yes—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×#13093935C2211yescharacteristic
27#15689054—21no—
28#15689037—21no—
29#15689011—22no—
30#15688989C2211yescharacteristic
31#15688983C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C47×#556—24no—
36C47×#556—24no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#126×C47—22no—
39C2×#124×C47—21no—
40C2×C22⋊Q8×C47C2212yes—
41C2×C4×Q8×C47—21no—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C2×#117×C47C2212yes—
45C22×Dic94C2311yescharacteristic
46C2×#10625337—22no—
47C2×#10625337—22no—
48C2×#10625337—22no—
49C22×C47⋊C8C2311yescharacteristic
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2311yescharacteristic
52#13094003—24no—
53#13094003—42no—
54C4×Dic94—22no—
55C4×Dic94—22no—
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4—22no—
59C2×Dic47⋊C4—21no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C4×Dic47—21no—
63#13093957—24no—
64#13093955—22no—
65C4⋊(C47⋊C8)—22no—
66#13093948—22no—
67#13093948—42no—
68#13093947—22no—
69#13093947—24no—
70C22⋊(C47⋊C8)—24no—
71C22⋊(C47⋊C8)—42no—
72#13093935—22no—
73C47⋊#95—22no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C22×Q8×C47—21no—
77C22×Q8×C47C2311yescharacteristic
78C22×D4×C47D411yescharacteristic
79C23×C4×C47C2312yes—
80C4⋊Q8×C47—22no—
81C4⋊Q8×C47D414yes—
82#126×C47—216no—
83#124×C47—28no—
84C22⋊Q8×C47—24no—
85C22⋊Q8×C47—24no—
86C4×Q8×C47—28no—
87C4×Q8×C47—22no—
88C4×Q8×C47D414yes—
89C4×D4×C47—22no—
90C4×D4×C47—22no—
91#117×C47—24no—
92#117×C47—24no—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47D411yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2312yes—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47D412yes—
102C2×C42×C47—22no—
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C24×C47C2×D412yes—
106C2×Q8×C47—24no—
107C2×Q8×C47—21no—
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—44no—
113C2×Q8×C47C2×D412yes—
114C2×D4×C47—22no—
115C2×D4×C47—42no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—21no—
130C22×Dic47—22no—
131C22×Dic47—21no—
132C2×Dic94—21no—
133C2×Dic94—21no—
134C2×Dic94—21no—
135C2×Dic94—21no—
136C2×Dic94—21no—
137C2×Dic94#4112yes—
138C4⋊C4×C47—28no—
139C4⋊C4×C47—24no—
140C4⋊C4×C47—24no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47C2×D412yes—
145C4⋊C4×C47C2×D414yes—
146C22⋊C4×C47—28no—
147C22⋊C4×C47—28no—
148C22⋊C4×C47—24no—
149C42×C47—28no—
150C42×C47—24no—
151C42×C47C2×D412yes—
152C42×C47C2×D412yes—
153C22⋊Dic47—44no—
154C22⋊Dic47—44no—
155#10625337—28no—
156C4⋊Dic47—21no—
157C4⋊Dic47—21no—
158C4⋊Dic47—22no—
159C4⋊Dic47—22no—
160Dic47⋊C4—44no—
161Dic47⋊C4—42no—
162C4×Dic47—42no—
163C2×C47⋊C8—21no—
164C2×C47⋊C8—21no—
165C2×C47⋊C8—21no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—21no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8#4112yes—
172C23×C47—22no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47C22×D411yescharacteristic
180Q8×C47—22no—
181Q8×C47—22no—
182Q8×C47—48no—
183Q8×C47—41no—
184Q8×C47—41no—
185D4×C47—44no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—24no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—44no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47#12012yes—
203C2×C4×C47#12012yes—
204C2×C4×C47C22×D411yescharacteristic
205C2×C4×C47C22×D411yescharacteristic
206C2×C4×C47C2×#4111yescharacteristic
207C2×C4×C47C2×#4111yescharacteristic
208C2×C4×C47H(5,𝔽2)11yescharacteristic
209C2×C4×C472−1+411yescharacteristic
210C2×Dic47—22no—
211C2×Dic47—41no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218Dic94—22no—
219Dic94—22no—
220Dic94—41no—
221Dic94—41no—
222C47⋊C8—22no—
223C47⋊C8—41no—
224C47⋊C8—42no—
225#24135—471nomaximal, Sylow
226C22×C47—42no—
227C22×C47—42no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47—42no—
231C22×C47—42no—
232C22×C47—42no—
233C22×C47C2×#12011yescharacteristic
234C22×C47#54611yescharacteristic
235C22×C47#54711yescharacteristic
236C22×C47#57512yes—
237C22×C47#57812yes—
238C4×C47—22no—
239C4×C47—22no—
240C4×C47—42no—
241C4×C47—42no—
242C4×C47—44no—
243C4×C47—42no—
244C4×C47#52511yescharacteristic
245C4×C47#53711yescharacteristic
246C4×C47#54811yescharacteristic
247C4×C47#55011yescharacteristic
248Dic47—42no—
249Dic47—81no—
250Dic47—82no—
251C2×#556D4711yescharacteristic
252C2×#477—471no—
253C2×#441—471no—
254#2536—471no—
255#2036—472no—
256#2001—472no—
257#1993—471no—
258#1955—471no—
259#1873—472no—
260#1854—472no—
261#1805—471no—
262C2×C47—82no—
263C2×C47#240711yescharacteristic
264C2×C47#300511yescharacteristic
265C2×C47#301111yescharacteristic
266C2×C47#331312yes—
267C2×C47#334912yes—
268C22×Q16—471no—
269#556—24no—
270#556—24no—
271C2×C4⋊Q8C2×D4711yescharacteristic
272C2×#126—22no—
273C2×#124—21no—
274C2×C22⋊Q8—472no—
275C2×C22⋊Q8C2×D4712yes—
276C2×C4×Q8—21no—
277C2×C4×Q8—471no—
278C2×C4×Q8C2×D4711yescharacteristic
279C2×C4×D4C2×D4711yescharacteristic
280C2×#117C2×D4712yes—
281#477—944no—
282#441—944no—
283C2×#107—472no—
284C2×#106—471no—
285C2×C4⋊C8—471no—
286C2×#103—471no—
287C2×#103—472no—
288C2×#103—471no—
289C2×#103—472no—
290C2×C22⋊C8—472no—
291C2×C22⋊C8—472no—
292C2×#97—471no—
293#398—941no—
294#394—941no—
295#388—942no—
296#382—471no—
297#380—472no—
298#340—472no—
299#340—472no—
300#340—472no—
301C47#2413511yescharacteristic, complemented, Sylow
302C22×Q8—21no—
303C22×Q8—471no—
304C22×Q8C22×D4711yescharacteristic
305C22×D4C22⋊D4711yescharacteristic
306C23×C4C22×D4712yes—
307C2×Q16—942no—
308C2×Q16—942no—
309C2×Q16—942no—
310C22×C8—471no—
311C22×C8—471no—
312C22×C8—472no—
313C4⋊Q8—22no—
314C4⋊Q8C22⋊D4714yes—
315#126—216no—
316#124—28no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—944no—
320C22⋊Q8—1882no—
321C4×Q8—28no—
322C4×Q8—22no—
323C4×Q8—942no—
324C4×Q8—942no—
325C4×Q8C22⋊D4714yes—
326C4×D4—22no—
327C4×D4—22no—
328#117—24no—
329#117—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—471no—
334C2×C4⋊C4—472no—
335C2×C4⋊C4—941no—
336C2×C4⋊C4—942no—
337C2×C4⋊C4C22⋊D4711yescharacteristic
338C2×C4⋊C4C22×D4711yescharacteristic
339C2×C4⋊C4C22×D4712yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—942no—
343C2×C22⋊C4—942no—
344C2×C22⋊C4C22⋊D4712yes—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—941no—
348C2×C42C22×D4711yescharacteristic
349#107—944no—
350#106—942no—
351C4⋊C8—942no—
352#103—942no—
353#103—942no—
354#103—944no—
355#103—1882no—
356C22⋊C8—944no—
357C22⋊C8—1882no—
358#97—942no—
359#95—942no—
360#95—941no—
361#95—941no—
362C24C2×C22⋊D4712yes—
363C2×Q8—24no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—44no—
370C2×Q8—472no—
371C2×Q8—941no—
372C2×Q8—941no—
373C2×Q8—941no—
374C2×Q8—941no—
375C2×Q8—941no—
376C2×Q8C2×C22⋊D4712yes—
377C2×D4—22no—
378C2×D4—42no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—941no—
389C22×C4—942no—
390C22×C4—941no—
391C22×C4C2×C22⋊D4712yes—
392C22×C4C2×C22⋊D4711yescharacteristic
393C22×C4C2×C22⋊D4711yescharacteristic
394C22×C4C23×D4711yescharacteristic
395Q16—948no—
396C2×C8—472no—
397C2×C8—941no—
398C2×C8—941no—
399C2×C8—941no—
400C2×C8—942no—
401C2×C8—942no—
402C2×C8—941no—
403C2×C8—942no—
404C2×C8—941no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—941no—
412C4⋊C4—941no—
413C4⋊C4—942no—
414C4⋊C4—942no—
415C4⋊C4—1884no—
416C4⋊C4—1882no—
417C4⋊C4C2×C22⋊D4712yes—
418C4⋊C4C2×C22⋊D4714yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—1884no—
423C22⋊C4—1884no—
424C42—28no—
425C42—24no—
426C42—1882no—
427C42C2×C22⋊D4712yes—
428C42C2×C22⋊D4712yes—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23C22×C22⋊D4711yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—48no—
440Q8—41no—
441Q8—41no—
442Q8—942no—
443Q8—942no—
444Q8—1881no—
445Q8—1881no—
446D4—44no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—942no—
464C2×C4—1882no—
465C2×C4—1881no—
466C2×C4—1882no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1881no—
470C2×C4—1881no—
471C2×C4#1309399312yes—
472C2×C4#1309399312yes—
473C2×C4C22×C22⋊D4711yescharacteristic
474C2×C4C22×C22⋊D4711yescharacteristic
475C2×C4#41×D4711yescharacteristic
476C2×C4#41×D4711yescharacteristic
477C2×C4#1309407011yescharacteristic
478C2×C4#1309407111yescharacteristic
479C8—942no—
480C8—1882no—
481C8—1881no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22C2×#1309399311yescharacteristic
490C22#1568976111yescharacteristic
491C22#1568977211yescharacteristic
492C22#1568992112yes—
493C22#1568993812yes—
494C4—22no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—44no—
500C4—3761no—
501C4—3762no—
502C4—1882no—
503C4#1568963011yescharacteristic
504C4#1568970111yescharacteristic
505C4#1568978011yescharacteristic
506C4#1568979211yescharacteristic
507C2—82no—
508C2#2024672711yescharacteristic
509C2#2025263011yescharacteristic
510C2#2025268311yescharacteristic
511C2#2025544812yes—
512C2#2025591212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#556C2
socleC23×C47C22×D4
prime radicalC22×D4×C47D4
hypercenterC2×#556D47
hypocenterC47#24135
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#15689415C2semidirect
2. C2×#15689228C2semidirect
4. #20244588C2semidirect
5. #20244396C2semidirect
6. #20244362C2semidirect
8. #20243700C2semidirect
9. #20243587C2semidirect
10. #20243403C2semidirect
301. C47#24135semidirect

Maximal subgroups

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

subgroupindex
C2×#156894152
C2×#156892282
#202482972
#202445882
#202443962
#202443622
#202441872
#202437002
#202435872
#202434032
C2×C47×#5562
#2413547