Finite Group Databasebeta

← #26111139

Subgroups — #26111139

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#10625329C211yesmaximal, characteristic, complemented
2#20244308C211yesmaximal, characteristic, complemented
3C4×#13093945C211yesmaximal, characteristic, complemented
4#20241839C211yesmaximal, characteristic, complemented
5#20241290C211yesmaximal, characteristic, complemented
6#20241224C211yesmaximal, characteristic, complemented
7C47×#3100C211yesmaximal, characteristic
8C22×#10625329—21no—
9#15689201—21no—
10#15689195—22no—
11D4×C47⋊C8—21no—
12#15689147C2211yescharacteristic
13C2×#13093951—21no—
14C2×#13093951—21no—
15C2×#13093945—21no—
16#15689087—21no—
17C4×#10625329C2211yescharacteristic
18C2×C4×C47⋊C8C2211yescharacteristic
19C4×C4⋊Dic47C2211yescharacteristic
20#15688885—21no—
21D4⋊(C47⋊C8)—22no—
22C47×#532C2211yescharacteristic
23C2×C4×D4×C47C2211yescharacteristic
24C47×#480—21no—
25C47×#474C411yescharacteristic, complemented
26C47×#474C411yescharacteristic, complemented
27C47×#459C411yescharacteristic, complemented
28C47×#459C411yescharacteristic, complemented
29C47×#447—21no—
30C4×D8×C47—21no—
31C4×D8×C47—21no—
32C47×#431C2211yescharacteristic
33C2×#10625329—22no—
34C2×#10625329—21no—
35C2×#10625329—42no—
36C2×#10625329—42no—
37C2×#10625329D411yescharacteristic
38C2×#10625329D411yescharacteristic
39C22×C47⋊C8—22no—
40C22×C47⋊C8—21no—
41C2×C4⋊Dic47—21no—
42C2×C4×Dic47—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)D412yes—
45#13093951—42no—
46#13093951—42no—
47#13093945—24no—
48C22⋊(C47⋊C8)—22no—
49C22⋊(C47⋊C8)—22no—
50#13093935—21no—
51C4×C47⋊C8—21no—
52C4×C47⋊C8—21no—
53C4×C47⋊C8—21no—
54C47⋊#95—21no—
55C2×#41×C47—21no—
56C22×D4×C47D411yescharacteristic
57C23×C4×C47—21no—
58C2×D8×C47—21no—
59C2×D8×C47—21no—
60C4⋊Q8×C47C2×C411yescharacteristic
61#127×C47C2×C411yescharacteristic
62#124×C47—21no—
63#121×C47—21no—
64#121×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C4×D4×C47—21no—
68C4×D4×C47—21no—
69C4×D4×C47C2×C411yescharacteristic
70C4×D4×C47C2×C411yescharacteristic
71C4×D4×C47D411yescharacteristic
72C4×D4×C47D411yescharacteristic
73C2×C4⋊C4×C47D411yescharacteristic
74C2×C22⋊C4×C47—21no—
75C2×C42×C47C2311yescharacteristic
76#107×C47—21no—
77#107×C47—21no—
78C4⋊C8×C47C2×C411yescharacteristic
79C4⋊C8×C47C2×C411yescharacteristic
80#102×C47—21no—
81#102×C47—21no—
82#102×C47—21no—
83#102×C47—21no—
84C22⋊C8×C47—21no—
85C4×C8×C47—21no—
86C24×C47—21no—
87#41×C47—41no—
88#41×C47—41no—
89#41×C47—41no—
90#41×C47—41no—
91C2×Q8×C47—21no—
92C2×D4×C47—22no—
93C2×D4×C47—21no—
94C2×D4×C47—21no—
95C2×D4×C47—21no—
96C2×D4×C47—21no—
97C2×D4×C47—41no—
98C2×D4×C47—41no—
99C2×D4×C47C22⋊C411yescharacteristic
100C2×D4×C47C22⋊C411yescharacteristic
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—41no—
107C22×C4×C47—41no—
108C22×C4×C47—41no—
109C22×C4×C47—41no—
110C22×C4×C47—41no—
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×Dic47—41no—
115D8×C47—42no—
116D8×C47—42no—
117C2×C8×C47—21no—
118C2×C8×C47—21no—
119C4⋊C4×C47—21no—
120C4⋊C4×C47—21no—
121C4⋊C4×C47—21no—
122C4⋊C4×C47C22⋊C411yescharacteristic
123C4⋊C4×C47C22⋊C411yescharacteristic
124C22⋊C4×C47—21no—
125C22⋊C4×C47—21no—
126C22⋊C4×C47—21no—
127C22⋊C4×C47—21no—
128C22⋊C4×C47—41no—
129C22⋊C4×C47—41no—
130C42×C47C22×C411yescharacteristic
131C42×C47C2×D411yescharacteristic
132C42×C47C2×D411yescharacteristic
133C42×C47#4111yescharacteristic
134C4⋊Dic47—22no—
135C4⋊Dic47—41no—
136C4×Dic47—41no—
137C4×Dic47—41no—
138#10625329—21no—
139#10625329—21no—
140#10625329—21no—
141#10625329—44no—
142#10625329—41no—
143#10625329—41no—
144#10625329D812yescomplemented
145C2×C47⋊C8—21no—
146C2×C47⋊C8—22no—
147C2×C47⋊C8—21no—
148C2×C47⋊C8—22no—
149C2×C47⋊C8—44no—
150C2×C47⋊C8—41no—
151C2×C47⋊C8—41no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—42no—
155C23×C47—41no—
156C23×C47—41no—
157C23×C47—41no—
158C23×C47—81no—
159C23×C47#12111yescharacteristic
160Q8×C47—41no—
161Q8×C47—41no—
162D4×C47—21no—
163D4×C47—21no—
164D4×C47—21no—
165D4×C47—42no—
166D4×C47—42no—
167D4×C47—41no—
168D4×C47—41no—
169D4×C47—41no—
170D4×C47—41no—
171D4×C47—81no—
172D4×C47—81no—
173D4×C47#10412yescomplemented
174C2×C4×C47—21no—
175C2×C4×C47—21no—
176C2×C4×C47—21no—
177C2×C4×C47—21no—
178C2×C4×C47—41no—
179C2×C4×C47—41no—
180C2×C4×C47—41no—
181C2×C4×C47—41no—
182C2×C4×C47—41no—
183C2×C4×C47—41no—
184C2×C4×C47—41no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—81no—
191C2×C4×C47—81no—
192C2×C4×C47C2×C22⋊C411yescharacteristic
193C2×C4×C47C4×D411yescharacteristic
194C2×C4×C47C4×D411yescharacteristic
195C2×C4×C47#12011yescharacteristic
196C2×C4×C47#12111yescharacteristic
197C2×C4×C47#12311yescharacteristic
198C2×C4×C47C2×D811yescharacteristic
199C2×C4×C47#13511yescharacteristic
200C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
202C2×Dic47—42no—
203C2×Dic47—81no—
204C2×Dic47—81no—
205C8×C47—41no—
206C8×C47—41no—
207C47⋊C8—22no—
208C47⋊C8—42no—
209C47⋊C8—42no—
210C47⋊C8—41no—
211#8991—471nomaximal, Sylow
212C22×C47—21no—
213C22×C47—21no—
214C22×C47—22no—
215C22×C47—41no—
216C22×C47—41no—
217C22×C47—41no—
218C22×C47—41no—
219C22×C47—41no—
220C22×C47—82no—
221C22×C47—82no—
222C22×C47—81no—
223C22×C47—81no—
224C22×C47—81no—
225C22×C47#38611yescharacteristic
226C22×C47#45911yescharacteristic
227C22×C47#46311yescharacteristic
228C4×C47—21no—
229C4×C47—21no—
230C4×C47—21no—
231C4×C47—41no—
232C4×C47—41no—
233C4×C47—81no—
234C4×C47C2×#10411yescharacteristic
235C4×C47#42111yescharacteristic
236C4×C47C4×D811yescharacteristic
237C4×C47#44211yescharacteristic
238C4×C47#44711yescharacteristic
239C4×C47#44911yescharacteristic
240Dic47—82no—
241#3100D4711yescharacteristic
242D4×M16—471no—
243#1980—471no—
244C4×#102—471no—
245#1492—471no—
246#1303—471no—
247#1278—471no—
248C2×C47—21no—
249C2×C47—21no—
250C2×C47—42no—
251C2×C47—81no—
252C2×C47—161no—
253C2×C47#187111yescharacteristic
254C2×C47#187911yescharacteristic
255C2×C47#188111yescharacteristic
256C22×M16—941no—
257#532C2×D4711yescharacteristic
258C2×C4×D4C2×D4711yescharacteristic
259#480—21no—
260#474Dic4711yescharacteristic, complemented
261#474Dic4711yescharacteristic, complemented
262#459Dic4711yescharacteristic, complemented
263#459Dic4711yescharacteristic, complemented
264#447—21no—
265C4×D8—21no—
266C4×D8—21no—
267#436—941no—
268#435—942no—
269C8×D4—941no—
270#431C2×D4711yescharacteristic
271#423—471no—
272C2×#104—941no—
273C2×#104—941no—
274C2×#102—941no—
275#407—941no—
276C4×M16—471no—
277C2×C4×C8—471no—
278C4×C4⋊C4—471no—
279#340—941no—
280D4⋊C8—942no—
281C47#899111yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×D4C22⋊D4711yescharacteristic
284C23×C4—21no—
285C2×D8—21no—
286C2×D8—21no—
287C2×M16—471no—
288C2×M16—471no—
289C2×M16—942no—
290C2×M16—941no—
291C2×M16—1882no—
292C2×M16—1882no—
293C22×C8—942no—
294C22×C8—941no—
295C4⋊Q8C2×Dic4711yescharacteristic
296#127C2×Dic4711yescharacteristic
297#124—21no—
298#121—21no—
299#121—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4C2×Dic4711yescharacteristic
305C4×D4C2×Dic4711yescharacteristic
306C4×D4C22⋊D4711yescharacteristic
307C4×D4C22⋊D4711yescharacteristic
308C2×C4⋊C4—941no—
309C2×C4⋊C4C22⋊D4711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C42—941no—
312C2×C42C22×D4711yescharacteristic
313#107—21no—
314#107—21no—
315C4⋊C8—472no—
316C4⋊C8—941no—
317C4⋊C8C2×Dic4711yescharacteristic
318C4⋊C8C2×Dic4711yescharacteristic
319#104—1882no—
320#104—1882no—
321#102—21no—
322#102—21no—
323#102—21no—
324#102—21no—
325#102—944no—
326C22⋊C8—21no—
327C22⋊C8—942no—
328C22⋊C8—942no—
329#97—941no—
330C4×C8—21no—
331C4×C8—941no—
332C4×C8—941no—
333C4×C8—941no—
334#95—941no—
335C24—21no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340C2×Q8—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C22⋊Dic4711yescharacteristic
349C2×D4C22⋊Dic4711yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—1881no—
361C22×C4D4×D4711yescharacteristic
362C22×C4#1062535211yescharacteristic
363C22×C4C2×C22⋊D4711yescharacteristic
364D8—42no—
365D8—42no—
366M16—472no—
367M16—941no—
368M16—941no—
369M16—941no—
370M16—1884no—
371M16—1881no—
372M16—1881no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—941no—
376C2×C8—942no—
377C2×C8—941no—
378C2×C8—942no—
379C2×C8—1884no—
380C2×C8—1881no—
381C2×C8—1881no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—942no—
386C4⋊C4—1881no—
387C4⋊C4C22⋊Dic4711yescharacteristic
388C4⋊C4C22⋊Dic4711yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—1881no—
396C42—1881no—
397C42D4×D4711yescharacteristic
398C42#1062535211yescharacteristic
399C42C22×Dic4711yescharacteristic
400C42C2×C22⋊D4711yescharacteristic
401C23—21no—
402C23—21no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—81no—
408C23#1309399611yescharacteristic
409Q8—41no—
410Q8—41no—
411D4—21no—
412D4—21no—
413D4—21no—
414D4—42no—
415D4—42no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—81no—
421D4—81no—
422D4#1309395112yescomplemented
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—3761no—
442C2×C4—3761no—
443C2×C4—1882no—
444C2×C4C2×C22⋊Dic4711yescharacteristic
445C2×C4D4×Dic4711yescharacteristic
446C2×C4D4×Dic4711yescharacteristic
447C2×C4#1309399211yescharacteristic
448C2×C4#1309399611yescharacteristic
449C2×C4#1309400511yescharacteristic
450C2×C4D8×D4711yescharacteristic
451C2×C4#1309404311yescharacteristic
452C2×C4#1309404811yescharacteristic
453C2×C4#1309404811yescharacteristic
454C8—41no—
455C8—41no—
456C8—942no—
457C8—1882no—
458C8—1882no—
459C8—1881no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—82no—
469C22—82no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22#1568900311yescharacteristic
474C22#1568931611yescharacteristic
475C22#1568933611yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—3762no—
483C4C2×#1309395111yescharacteristic
484C4#1568914011yescharacteristic
485C4D8×Dic4711yescharacteristic
486C4#1568923511yescharacteristic
487C4#1568925511yescharacteristic
488C4#1568926511yescharacteristic
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—161no—
494C2#2024369111yescharacteristic
495C2#2024373811yescharacteristic
496C2#2024375211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3100C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3100D47
hypocenterC47#8991
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. D4×#10625329C2semidirect
2. #20244308C2semidirect
3. C4×#13093945C2semidirect
4. #20241839C2semidirect
5. #20241290C2semidirect
6. #20241224C2semidirect
25. C47×#474C4semidirect
26. C47×#474C4semidirect
27. C47×#459C4semidirect
28. C47×#459C4semidirect
144. #10625329D8semidirect
173. D4×C47#104semidirect
260. #474Dic47semidirect
261. #474Dic47semidirect
262. #459Dic47semidirect
263. #459Dic47semidirect
281. C47#8991semidirect
422. D4#13093951semidirect

Maximal subgroups

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

subgroupindex
D4×#106253292
#202443082
C4×#130939452
#202418392
#202412902
#202412242
C47×#31002
#899147