Finite Group Databasebeta

← #26185498

Subgroups — #26185498

Order 12032.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252502C211yesmaximal, characteristic
2#20252501C211yesmaximal, characteristic
3#20252027C211yesmaximal, characteristic
4#20242178C211yesmaximal, characteristic, complemented
5#20242173C211yesmaximal, characteristic, complemented
6#20242170C211yesmaximal, characteristic, complemented
7#20242166C211yesmaximal, characteristic, complemented
8#20242165C211yesmaximal, characteristic, complemented
9#20242164C211yesmaximal, characteristic, complemented
10#20242161C211yesmaximal, characteristic, complemented
11#20242154C211yesmaximal, characteristic, complemented
12#20242152C211yesmaximal, characteristic, complemented
13#20242148C211yesmaximal, characteristic, complemented
14#20242145C211yesmaximal, characteristic, complemented
15C47×#1574C211yesmaximal, characteristic, complemented
16C4×#10625350C2211yescharacteristic
17#15689253—21no—
18#15689252—21no—
19#15689202—21no—
20#15689199—21no—
21#15689193—21no—
22#15689192—21no—
23#15689191—21no—
24#15689191—21no—
25#15689183C412yes—
26#15689181C412yes—
27#15689180C2211yescharacteristic
28#15689179C2211yescharacteristic
29#15689179C2211yescharacteristic
30#15689179C2211yescharacteristic
31#15689176C2211yescharacteristic
32#15689175C2211yescharacteristic
33#15689174C2211yescharacteristic
34#15689174C2211yescharacteristic
35#15689148C412yes—
36#15689145C2211yescharacteristic
37#15689145C2211yescharacteristic
38#15689145C2211yescharacteristic
39C2×C4⋊(C47⋊C8)C2211yescharacteristic
40C4⋊C8×D47C2211yescharacteristic
41#15689088C412yes—
42C4×#10625324C2211yescharacteristic
43C4×C8×D47C2211yescharacteristic
44#15688876C2211yescharacteristic
45#15688876C2211yescharacteristic
46#15688875C2211yescharacteristic
47#15688875C2211yescharacteristic
48#15688875C2211yescharacteristic
49#15688875C2211yescharacteristic
50#15688875C2211yescharacteristic
51#15688875C2211yescharacteristic
52#15688874C2211yescharacteristic
53#15688874C2211yescharacteristic
54#15688873C2211yescharacteristic
55#15688873C2211yescharacteristic
56C47×#431C2211yescharacteristic
57C47×#431C2211yescharacteristic
58C47×#423C2211yescharacteristic
59C47×#335C2211yescharacteristic
60C47×#335C2211yescharacteristic
61C47×#334C2211yescharacteristic
62C47×#334C2211yescharacteristic
63C2×#10625350C2×C411yescharacteristic
64C2×#10625324—21no—
65C2×#10625324—21no—
66C2×#10625324—21no—
67C2×#10625324C2×C412yes—
68C22×C47⋊C8—21no—
69C2×C8×D47—21no—
70C2×C8×D47C2×C412yes—
71C4×Dic94D411yescharacteristic
72C4×Dic94Q811yescharacteristic
73C4×D188D411yescharacteristic
74C4×D188Q811yescharacteristic
75C4×C22⋊D47D411yescharacteristic
76C4×C22⋊D47D411yescharacteristic
77C4×C22⋊D47Q811yescharacteristic
78C4×C22⋊D47Q811yescharacteristic
79#13093983C2×C411yescharacteristic
80#13093981C2×C411yescharacteristic
81#13093981C2×C411yescharacteristic
82C42×D47C2311yescharacteristic
83C42×D47C2311yescharacteristic
84C4⋊(C47⋊C8)—21no—
85C4⋊(C47⋊C8)C2×C412yes—
86C4⋊(C47⋊C8)C2311yescharacteristic
87C4⋊(C47⋊C8)C2311yescharacteristic
88C4⋊(C47⋊C8)C2311yescharacteristic
89C4⋊(C47⋊C8)C2311yescharacteristic
90Dic47⋊C8—21no—
91Dic47⋊C8—21no—
92Dic47⋊C8—21no—
93Dic47⋊C8—21no—
94Dic47⋊C8C2×C412yes—
95Dic47⋊C8C2×C412yes—
96C22⋊(C47⋊C8)—21no—
97C22⋊(C47⋊C8)C2×C412yes—
98(C2×D47)⋊C8—21no—
99(C2×D47)⋊C8—21no—
100(C2×D47)⋊C8—21no—
101(C2×D47)⋊C8—21no—
102(C2×D47)⋊C8C2×C412yes—
103(C2×D47)⋊C8C2×C412yes—
104#13093935—21no—
105#13093934—21no—
106#13093934—21no—
107#13093934—21no—
108#13093934C2×C412yes—
109C4×C47⋊C8C2311yescharacteristic
110C4×C47⋊C8C2311yescharacteristic
111C8×Dic47—21no—
112C8×Dic47C2×C412yes—
113C2×M16×C47—21no—
114C2×C42×C47C2311yescharacteristic
115C4⋊C8×C47—21no—
116C4⋊C8×C47C2×C412yes—
117C4⋊C8×C47C2311yescharacteristic
118C4⋊C8×C47C2311yescharacteristic
119C4⋊C8×C47C2311yescharacteristic
120C4⋊C8×C47C2311yescharacteristic
121C22⋊C8×C47—21no—
122C22⋊C8×C47C2×C412yes—
123C4×C8×C47—21no—
124C4×C8×C47C2311yescharacteristic
125C4×C8×C47C2311yescharacteristic
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×Q811yescharacteristic
129C2×C22⋊D47C4⋊C411yescharacteristic
130C2×C22⋊D47C4⋊C411yescharacteristic
131#10625350—42no—
132C2×D188C4⋊C411yescharacteristic
133C2×C4×D47C22×C411yescharacteristic
134C2×C4×D47C22×C411yescharacteristic
135C2×C4×D47C2×D411yescharacteristic
136C2×C4×D47C2×D411yescharacteristic
137C2×C4×D47C2×Q811yescharacteristic
138C2×C4×D47C2×Q811yescharacteristic
139C2×Dic94C4⋊C411yescharacteristic
140M16×C47—81no—
141C2×C8×C47—21no—
142C2×C8×C47—21no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C2×C8×C47C22×C412yes—
146C2×C8×C47C22×C412yes—
147C42×C47C22×C411yescharacteristic
148C42×C47C2×D411yescharacteristic
149C42×C47C2×Q811yescharacteristic
150C42×C47C2411yescharacteristic
151C22⋊Dic47C4⋊C411yescharacteristic
152C22⋊Dic47C4⋊C411yescharacteristic
153(C2×D47)⋊C4C4⋊C411yescharacteristic
154(C2×D47)⋊C4C4⋊C411yescharacteristic
155(C2×D47)⋊C4C4⋊C411yescharacteristic
156(C2×D47)⋊C4C4⋊C411yescharacteristic
157C4⋊Dic47C4⋊C411yescharacteristic
158C4⋊Dic47C4⋊C411yescharacteristic
159Dic47⋊C4C4⋊C411yescharacteristic
160Dic47⋊C4C4⋊C411yescharacteristic
161Dic47⋊C4C4⋊C411yescharacteristic
162Dic47⋊C4C4⋊C411yescharacteristic
163C4×Dic47C22×C411yescharacteristic
164C4×Dic47C22×C411yescharacteristic
165C4×Dic47C2×D411yescharacteristic
166C4×Dic47C2×D411yescharacteristic
167C4×Dic47C2×Q811yescharacteristic
168C4×Dic47C2×Q811yescharacteristic
169C2×C47⋊C8—21no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—81no—
174C2×C47⋊C8C22×C412yes—
175C2×C47⋊C8C22×C412yes—
176#10625324—42no—
177#10625324—81no—
178#10625324—81no—
179#10625324—81no—
180C8×D47—42no—
181C8×D47—81no—
182C23×C47C2×C4⋊C411yescharacteristic
183C22×D47C2×C4⋊C411yescharacteristic
184C22×D47C2×C4⋊C411yescharacteristic
185C2×C4×C47—22no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—41no—
189C2×C4×C47C2×C4⋊C411yescharacteristic
190C2×C4×C47C2×C4⋊C411yescharacteristic
191C2×C4×C47C2×C4⋊C411yescharacteristic
192C2×C4×C47C23×C411yescharacteristic
193C2×C4×C47C22×D411yescharacteristic
194C2×C4×C47C22×Q811yescharacteristic
195C22⋊D47—41no—
196C22⋊D47—41no—
197C2×Dic47C2×C4⋊C411yescharacteristic
198C2×Dic47C2×C4⋊C411yescharacteristic
199C2×Dic47C2×C4⋊C411yescharacteristic
200C2×Dic47C2×C4⋊C411yescharacteristic
201C2×Dic47C2×C4⋊C411yescharacteristic
202C2×Dic47C2×C4⋊C411yescharacteristic
203D188—41no—
204C4×D47—22no—
205C4×D47—22no—
206C4×D47—21no—
207C4×D47—21no—
208C4×D47—21no—
209C4×D47—21no—
210C4×D47—41no—
211C4×D47—41no—
212Dic94—41no—
213C8×C47—22no—
214C8×C47—22no—
215C8×C47—41no—
216C8×C47—41no—
217C8×C47—41no—
218C8×C47—41no—
219C47⋊C8—22no—
220C47⋊C8—22no—
221C47⋊C8—41no—
222C47⋊C8—41no—
223C47⋊C8—41no—
224C47⋊C8—41no—
225#17825—471nomaximal, Sylow
226C22×C47—21no—
227C22×C47—21no—
228C22×C47—21no—
229C22×C47C22×C4⋊C411yescharacteristic
230C2×D47—21no—
231C2×D47—21no—
232C2×D47—21no—
233C2×D47—21no—
234C2×D47—21no—
235C2×D47—21no—
236C4×C47—21no—
237C4×C47—21no—
238C4×C47—21no—
239C4×C47—21no—
240C4×C47#56812yes—
241C4×C47#57611yescharacteristic
242C4×C47#57811yescharacteristic
243Dic47—21no—
244Dic47—21no—
245Dic47—21no—
246Dic47—21no—
247Dic47—21no—
248Dic47—21no—
249#2989—471no—
250#2989—471no—
251#2922—471no—
252#1574—471no—
253#1574D4711yescharacteristic, complemented
254#1573—471no—
255#1573—471no—
256#1572—471no—
257#1572—471no—
258#1571—471no—
259#1571—471no—
260#1569—471no—
261#1569—471no—
262#1568—471no—
263#1567—471no—
264C2×C47—41no—
265C2×C47#290511yescharacteristic
266C2×C47#291311yescharacteristic
267C2×C47#291611yescharacteristic
268D47—41no—
269D47—41no—
270C4×#41—471no—
271#446—941no—
272#446—941no—
273#436—941no—
274#436—941no—
275#435—941no—
276#435—941no—
277#435—941no—
278#435—941no—
279#433—472no—
280#433—472no—
281#432—471no—
282#432—471no—
283#432—471no—
284#432—471no—
285#431—471no—
286#431—471no—
287#431—471no—
288#431—471no—
289#431C2×D4711yescharacteristic
290#431C2×D4711yescharacteristic
291#424—472no—
292#423—471no—
293#423—471no—
294#423—471no—
295#423C2×D4711yescharacteristic
296C2×C4⋊C8—471no—
297C2×C4⋊C8—471no—
298#408—472no—
299C4×M16—471no—
300C2×C4×C8—471no—
301#335—471no—
302#335—471no—
303#335—471no—
304#335—471no—
305#335—471no—
306#335—471no—
307#335—471no—
308#335—471no—
309#335C2×D4711yescharacteristic
310#335C2×D4711yescharacteristic
311#334—471no—
312#334—471no—
313#334—471no—
314#334—471no—
315#334C2×D4711yescharacteristic
316#334C2×D4711yescharacteristic
317C47#1782511yescharacteristic, complemented, Sylow
318C2×#41—471no—
319C2×M16—21no—
320C2×M16—472no—
321C2×M16—941no—
322C2×M16—941no—
323C2×M16—941no—
324C22×C8—472no—
325C22×C8—941no—
326C22×C8—941no—
327C4×Q8—471no—
328C4×Q8—471no—
329C4×D4—471no—
330C4×D4—471no—
331C4×D4—471no—
332C4×D4—471no—
333C4×D4—471no—
334C4×D4—471no—
335#117—471no—
336#117—471no—
337#117—471no—
338C2×C42—471no—
339C2×C42—471no—
340C2×C42C22×D4711yescharacteristic
341C4⋊C8—21no—
342C4⋊C8—472no—
343C4⋊C8—472no—
344C4⋊C8—472no—
345C4⋊C8—471no—
346C4⋊C8—471no—
347C4⋊C8—471no—
348C4⋊C8—471no—
349C4⋊C8—941no—
350C4⋊C8—941no—
351C4⋊C8—941no—
352C4⋊C8—941no—
353C4⋊C8—941no—
354C4⋊C8C4×D4712yes—
355C4⋊C8C22×D4711yescharacteristic
356C4⋊C8C22×D4711yescharacteristic
357C4⋊C8C22×D4711yescharacteristic
358C4⋊C8C22×D4711yescharacteristic
359C22⋊C8—21no—
360C22⋊C8—472no—
361C22⋊C8—472no—
362C22⋊C8—472no—
363C22⋊C8—941no—
364C22⋊C8—941no—
365C22⋊C8—941no—
366C22⋊C8—941no—
367C22⋊C8—941no—
368C22⋊C8C4×D4712yes—
369#97—472no—
370#97—941no—
371#97—941no—
372#97—941no—
373#97—941no—
374C4×C8—21no—
375C4×C8—472no—
376C4×C8—471no—
377C4×C8—471no—
378C4×C8—941no—
379C4×C8C22×D4711yescharacteristic
380C4×C8C22×D4711yescharacteristic
381#41—1882no—
382C2×Q8—471no—
383C2×D4—471no—
384C2×D4—471no—
385C2×D4—471no—
386C22×C4—471no—
387C22×C4—471no—
388C22×C4—471no—
389C22×C4—471no—
390C22×C4—471no—
391C22×C4—471no—
392C22×C4C2×C4×D4711yescharacteristic
393C22×C4D4×D4711yescharacteristic
394C22×C4Q8×D4711yescharacteristic
395M16—81no—
396M16—3761no—
397M16—3761no—
398M16—3761no—
399M16—1882no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—472no—
405C2×C8—472no—
406C2×C8—941no—
407C2×C8—941no—
408C2×C8—941no—
409C2×C8—941no—
410C2×C8—3761no—
411C2×C8—3761no—
412C2×C8—1882no—
413C2×C8C2×C4×D4712yes—
414C2×C8C2×C4×D4712yes—
415C4⋊C4—471no—
416C4⋊C4—471no—
417C4⋊C4—471no—
418C4⋊C4—471no—
419C4⋊C4—471no—
420C4⋊C4—471no—
421C22⋊C4—471no—
422C22⋊C4—471no—
423C22⋊C4—471no—
424C22⋊C4—471no—
425C22⋊C4—471no—
426C22⋊C4—471no—
427C42—471no—
428C42—471no—
429C42—471no—
430C42—471no—
431C42—471no—
432C42—471no—
433C42C2×C4×D4711yescharacteristic
434C42D4×D4711yescharacteristic
435C42Q8×D4711yescharacteristic
436C42C23×D4711yescharacteristic
437C23—471no—
438C23—471no—
439C23C4⋊C4×D4711yescharacteristic
440Q8—1881no—
441D4—1881no—
442D4—1881no—
443D4—1881no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—471no—
449C2×C4—471no—
450C2×C4—471no—
451C2×C4—471no—
452C2×C4—471no—
453C2×C4—471no—
454C2×C4—942no—
455C2×C4—942no—
456C2×C4—941no—
457C2×C4—941no—
458C2×C4—941no—
459C2×C4—941no—
460C2×C4—1881no—
461C2×C4—1881no—
462C2×C4C4⋊C4×D4711yescharacteristic
463C2×C4C4⋊C4×D4711yescharacteristic
464C2×C4C4⋊C4×D4711yescharacteristic
465C2×C4C22×C4×D4711yescharacteristic
466C2×C4C2×D4×D4711yescharacteristic
467C2×C4C2×Q8×D4711yescharacteristic
468C8—22no—
469C8—22no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—942no—
475C8—942no—
476C8—1881no—
477C8—1881no—
478C8—1881no—
479C8—1881no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—941no—
484C22—941no—
485C22—941no—
486C22—941no—
487C22—941no—
488C22—941no—
489C22C2×C4⋊C4×D4711yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—941no—
495C4—941no—
496C4—941no—
497C4—941no—
498C4—941no—
499C4—941no—
500C4#1568988812yes—
501C4#1568992711yescharacteristic
502C4#1568993911yescharacteristic
503C2—41no—
504C2—1881no—
505C2—1881no—
506C2#2025194911yescharacteristic
507C2#2025198711yescharacteristic
508C2#2025199611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1574C2
socleC22×C47C22×C4⋊C4
prime radicalC2×#10625350C2×C4
cosocleC42×C47C24
hypercenter#1574D47
hypocenterC47#17825
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
4. #20242178C2semidirect
5. #20242173C2semidirect
6. #20242170C2semidirect
7. #20242166C2semidirect
8. #20242165C2semidirect
9. #20242164C2semidirect
10. #20242161C2semidirect
11. #20242154C2semidirect
12. #20242152C2semidirect
13. #20242148C2semidirect
14. #20242145C2semidirect
15. C47×#1574C2semidirect
253. #1574D47semidirect
317. C47#17825semidirect

Maximal subgroups

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

subgroupindex
#202525022
#202525012
#202520272
#202421782
#202421732
#202421702
#202421662
#202421652
#202421642
#202421612
#202421542
#202421522
#202421482
#202421452
C47×#15742
#1782547