Finite Group Databasebeta

← #26272408

Subgroups — #26272408

Order 12032.

1056 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×#15689833C211yesmaximal, characteristic
2C2×#15689529C211yesmaximal, characteristic, complemented
3C2×#15689400C211yesmaximal, characteristic
4#20244726C212yesmaximal, complemented
5#20244716C212yesmaximal, complemented
6#20244345C211yesmaximal, characteristic, complemented
7#20244329C211yesmaximal, characteristic, complemented
8#20243648C212yesmaximal
9#20243634C212yesmaximal
10#20243216C211yesmaximal, characteristic
11C47×#2567C211yesmaximal, characteristic, complemented
12#15689833—24no—
13#15689833—24no—
14C2×#13094024C2211yescharacteristic
15C2×#13094016C2211yescharacteristic
16C2×#13094016C2211yescharacteristic
17C2×#13094002—21no—
18C2×#13094002—21no—
19C2×#13094000—21no—
20C2×C4×D188C2211yescharacteristic
21C2×C4×D188C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23C2×C4⋊C4×D47—21no—
24#15689529—24no—
25#15689400—24no—
26C2×#13093957C2212yes—
27C2×#13093957C2212yes—
28C2×#13093955C2212yes—
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×#13093944C2212yes—
31C2×#13093944C2212yes—
32C2×#13093935C2211yescharacteristic
33#15688885C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688878C2212yes—
37#15688878C2212yes—
38C2×C4⋊Q8×C47C2211yescharacteristic
39C2×C4×Q8×C47C2211yescharacteristic
40C47×#395—21no—
41C47×#395—21no—
42C47×#391—21no—
43C47×#389C2212yes—
44C47×#389C2212yes—
45C47×#384—21no—
46C4×C4⋊C4×C47C2211yescharacteristic
47C22×Dic94—21no—
48C22×D188C2311yescharacteristic
49C22×C4×D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—21no—
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2312yes—
54#13094024—22no—
55#13094024—22no—
56#13094016—22no—
57#13094016—22no—
58#13094016—22no—
59#13094016D414yes—
60#13094002—28no—
61#13094002—28no—
62#13094000—28no—
63C4×D188—22no—
64C4×D188—22no—
65C4×D188—22no—
66C4×D188—22no—
67C4×D188—22no—
68C4×D188D414yes—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C4⋊C4×D47—28no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81#13093957—24no—
82#13093957—24no—
83#13093955—24no—
84C4⋊(C47⋊C8)—22no—
85#13093944—24no—
86#13093944Q818yescomplemented
87#13093935—22no—
88C22×Q8×C47—21no—
89C22×Q8×C47D411yescharacteristic
90C4⋊Q8×C47—22no—
91C4⋊Q8×C47—22no—
92C4×Q8×C47—22no—
93C4×Q8×C47—22no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C4⋊C4×C47C2312yes—
102C2×C42×C47—21no—
103C2×C42×C47C2311yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C2×C42×C47C2311yescharacteristic
106#95×C47—21no—
107#95×C47—22no—
108#95×C47—22no—
109C23×D47—21no—
110C2×Q8×C47—22no—
111C2×Q8×C47—22no—
112C2×Q8×C47—41no—
113C2×Q8×C47—42no—
114C2×Q8×C47—82no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2411yescharacteristic
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C2×D188—21no—
128C2×D188—22no—
129C2×D188C2×D412yes—
130C2×D188C2×Q814yes—
131C2×C4×D47—24no—
132C2×C4×D47—24no—
133C2×C4×D47—24no—
134C2×C4×D47—42no—
135C2×C4×D47—42no—
136C2×C4×D47—82no—
137C2×C4×D47—82no—
138C2×Dic94—24no—
139C2×Dic94—82no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—42no—
147C4⋊C4×C47—82no—
148C4⋊C4×C47—82no—
149C4⋊C4×C47C2×Q814yes—
150C42×C47—21no—
151C42×C47—21no—
152C42×C47—21no—
153C42×C47—21no—
154C42×C47—42no—
155C42×C47C2×D412yes—
156C42×C47C2×D412yes—
157(C2×D47)⋊C4—24no—
158(C2×D47)⋊C4—24no—
159(C2×D47)⋊C4—24no—
160C4⋊Dic47—24no—
161C4⋊Dic47—21no—
162C4⋊Dic47—21no—
163C4⋊Dic47—21no—
164C4⋊Dic47C2×D412yes—
165C4⋊Dic47C2×D412yes—
166C4⋊Dic47C2×D412yes—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—24no—
175C2×C47⋊C8C2×Q814yes—
176C23×C47C22×D411yescharacteristic
177C22×D47—22no—
178C22×D47—24no—
179C22×D47—41no—
180C22×D47—41no—
181C22×D47—42no—
182Q8×C47—44no—
183Q8×C47—84no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—22no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—21no—
192C2×C4×C47—24no—
193C2×C4×C47—21no—
194C2×C4×C47—21no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
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×C47#13712yes—
205C2×C4×C47C22×D411yescharacteristic
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C22×Q812yes—
208C2×C4×C472−1+412yes—
209C2×Dic47—22no—
210C2×Dic47—22no—
211C2×Dic47—22no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216D188—22no—
217D188—24no—
218D188C4⋊Q814yes—
219C4×D47—48no—
220C4×D47—84no—
221C4×D47—84no—
222Dic94—84no—
223C47⋊C8—24no—
224C47⋊C8—42no—
225#25850—471nomaximal, Sylow
226C22×C47C2×#12011yescharacteristic
227C22×C47D4×Q812yes—
228C22×C47C2×#13711yescharacteristic
229C22×C47#57511yescharacteristic
230C22×C47#57612yes—
231C2×D47—24no—
232C2×D47—44no—
233C2×D47—44no—
234C2×D47—42no—
235C2×D47—42no—
236C4×C47—22no—
237C4×C47—22no—
238C4×C47—22no—
239C4×C47—44no—
240C4×C47—41no—
241C4×C47—41no—
242C4×C47—41no—
243C4×C47—82no—
244C4×C47C2×C4⋊Q811yescharacteristic
245C4×C47#53611yescharacteristic
246C4×C47D4×Q812yes—
247Dic47—42no—
248Dic47—42no—
249Dic47—42no—
250C2×#554—471no—
251C2×#501—471no—
252C2×#474—471no—
253#2567D4711yescharacteristic, complemented
254#2059—472no—
255#2057—472no—
256#1989—471no—
257#1985—471no—
258#1864—472no—
259#1862—472no—
260#1762—471no—
261C2×C47#243411yescharacteristic
262C2×C47#300711yescharacteristic
263C2×C47#300811yescharacteristic
264C2×C47#338512yes—
265C2×C47#338912yes—
266D47—44no—
267#554—944no—
268#554—944no—
269C2×C4⋊Q8—471no—
270C2×C4⋊Q8C2×D4711yescharacteristic
271C2×#125—471no—
272C2×#125—471no—
273C2×C22⋊Q8—941no—
274C2×C22⋊Q8—941no—
275C2×C22⋊Q8—941no—
276C2×C4×Q8C2×D4711yescharacteristic
277C2×C4×D4—471no—
278C2×C4×D4—471no—
279C2×C4×D4—471no—
280C22×C4⋊C4—941no—
281#501—944no—
282#474—944no—
283C2×#107—472no—
284C2×#107—472no—
285C2×#106—472no—
286C2×C4⋊C8—471no—
287C2×#102—472no—
288C2×#102—472no—
289C2×#97—471no—
290#395—21no—
291#395—21no—
292#391—21no—
293#389C2×D4712yes—
294#389C2×D4712yes—
295#384—21no—
296C4×C4⋊C4C2×D4711yescharacteristic
297#340—472no—
298#340—472no—
299#340—472no—
300#336—472no—
301#336—472no—
302C47#2585011yescharacteristic, complemented, Sylow
303C22×Q8—21no—
304C22×Q8—941no—
305C22×Q8C22⋊D4711yescharacteristic
306C22×D4—471no—
307C23×C4—941no—
308C23×C4—941no—
309C23×C4—941no—
310C22×C8—472no—
311C22×C8—472no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—942no—
315C4⋊Q8—942no—
316#125—474no—
317#125—942no—
318#125—942no—
319#125—942no—
320C22⋊Q8—948no—
321C22⋊Q8—948no—
322C22⋊Q8—948no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×D4—474no—
326C4×D4—942no—
327C4×D4—942no—
328C4×D4—942no—
329C4×D4—942no—
330C4×D4—942no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—471no—
335C2×C4⋊C4—471no—
336C2×C4⋊C4—471no—
337C2×C4⋊C4—948no—
338C2×C4⋊C4—941no—
339C2×C4⋊C4—941no—
340C2×C4⋊C4—941no—
341C2×C4⋊C4—941no—
342C2×C4⋊C4—941no—
343C2×C4⋊C4C22⋊D4711yescharacteristic
344C2×C4⋊C4C22⋊D4711yescharacteristic
345C2×C4⋊C4C22⋊D4711yescharacteristic
346C2×C4⋊C4C22×D4712yes—
347C2×C4⋊C4C22×D4712yes—
348C2×C22⋊C4—941no—
349C2×C22⋊C4—941no—
350C2×C22⋊C4—941no—
351C2×C42—21no—
352C2×C42C22×D4711yescharacteristic
353C2×C42C22×D4711yescharacteristic
354C2×C42C22×D4711yescharacteristic
355#107—944no—
356#107—944no—
357#106—944no—
358C4⋊C8—942no—
359#102—478no—
360#102—944no—
361#97—942no—
362#95—21no—
363#95—22no—
364#95—22no—
365C24—941no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—41no—
369C2×Q8—42no—
370C2×Q8—82no—
371C2×Q8—944no—
372C2×Q8—3762no—
373C2×D4—472no—
374C2×D4—474no—
375C2×D4—941no—
376C2×D4—942no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—944no—
380C22×C4—944no—
381C22×C4—941no—
382C22×C4—941no—
383C22×C4—941no—
384C22×C4—944no—
385C22×C4—3762no—
386C22×C4—3762no—
387C22×C4—1882no—
388C22×C4—1882no—
389C22×C4C2×C22⋊D4711yescharacteristic
390C22×C4C2×C22⋊D4711yescharacteristic
391C22×C4C2×C22⋊D4711yescharacteristic
392C22×C4C2×C22⋊D4711yescharacteristic
393C22×C4C2×C22⋊D4711yescharacteristic
394C22×C4C2×C22⋊D4711yescharacteristic
395C22×C4C23×D4711yescharacteristic
396C2×C8—474no—
397C2×C8—942no—
398C2×C8—942no—
399C2×C8—942no—
400C2×C8—944no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—472no—
411C4⋊C4—472no—
412C4⋊C4—472no—
413C4⋊C4—944no—
414C4⋊C4—944no—
415C4⋊C4—944no—
416C4⋊C4—944no—
417C4⋊C4—944no—
418C4⋊C4—941no—
419C4⋊C4—941no—
420C4⋊C4—941no—
421C4⋊C4Q8×D4714yes—
422C22⋊C4—944no—
423C22⋊C4—944no—
424C22⋊C4—944no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—42no—
430C42D4×D4712yes—
431C42D4×D4712yes—
432C23—944no—
433C23—942no—
434C23—1881no—
435C23—1881no—
436C23—1882no—
437C23C22×C22⋊D4711yescharacteristic
438Q8—44no—
439Q8—84no—
440Q8—3764no—
441D4—474no—
442D4—944no—
443D4—942no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—942no—
463C2×C4—942no—
464C2×C4—942no—
465C2×C4—3764no—
466C2×C4—3764no—
467C2×C4—1888no—
468C2×C4—1881no—
469C2×C4—1881no—
470C2×C4—1882no—
471C2×C4—1882no—
472C2×C4#1309399212yes—
473C2×C4#1309399212yes—
474C2×C4#1309405512yes—
475C2×C4C2×D4×D4711yescharacteristic
476C2×C4C2×D4×D4711yescharacteristic
477C2×C4C2×Q8×D4712yes—
478C2×C4#1309407212yes—
479C8—944no—
480C8—1882no—
481C22—944no—
482C22—1884no—
483C22—1884no—
484C22—1882no—
485C22—1882no—
486C22C2×#1309399211yescharacteristic
487C22Q8×C22⋊D4712yes—
488C22C2×#1309405511yescharacteristic
489C22#1568992111yescharacteristic
490C22#1568992612yes—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—44no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—82no—
499C4—1882no—
500C4—1882no—
501C4—1882no—
502C4C4⋊Q8×D4711yescharacteristic
503C4#1568969811yescharacteristic
504C4#1568978812yes—
505C2—1884no—
506C2#2024702911yescharacteristic
507C2#2025264611yescharacteristic
508C2#2025265411yescharacteristic
509C2#2025629412yes—
510C2#2025634212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2567C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenter#2567D47
hypocenterC47#25850
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
2. C2×#15689529C2semidirect
4. #20244726C2semidirect
5. #20244716C2semidirect
6. #20244345C2semidirect
7. #20244329C2semidirect
11. C47×#2567C2semidirect
86. #13093944Q8semidirect
253. #2567D47semidirect
302. C47#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#156898332
C2×#156895292
C2×#156894002
#202447262
#202447162
#202443452
#202443292
#202436482
#202436342
#202432162
C47×#25672
#2585047