Finite Group Databasebeta

← #26107886

Subgroups — #26107886

Order 12032.

1074 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243332C212yesmaximal, complemented
2C4×#13093939C211yesmaximal, characteristic, complemented
3#20241908C212yesmaximal, complemented
4#20241726C211yesmaximal, characteristic, complemented
5C2×C47×#547C211yesmaximal, characteristic
6C2×#13093951—24no—
7C2×#13093939—21no—
8#15689087C2212yes—
9C4×C22⋊Dic47C2212yes—
10#15688890—21no—
11#15688860—24no—
12#15688849—21no—
13#15688847—21no—
14C22×#41×C47C411yescharacteristic, complemented
15C22×#41×C47C2211yescharacteristic
16C47×#547—24no—
17C47×#547—24no—
18C2×C4⋊Q8×C47C411yescharacteristic, complemented
19C2×#123×C47—21no—
20C2×#123×C47—21no—
21C2×C22⋊Q8×C47C411yescharacteristic, complemented
22C2×C22⋊Q8×C47C2211yescharacteristic
23C2×#121×C47—21no—
24C2×C4×D4×C47C411yescharacteristic, complemented
25C2×C4×D4×C47C2211yescharacteristic
26C22×C4⋊C4×C47—21no—
27C2×#10625329—22no—
28C2×C22⋊Dic47—22no—
29C2×C4×Dic47—22no—
30#13093951—48no—
31#13093939—24no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C2×#41×C47—22no—
36C2×#41×C47—22no—
37C2×#41×C47—22no—
38C2×#41×C47—44no—
39C2×#41×C47—44no—
40C2×#41×C47D414yes—
41C22×Q8×C47C2×C411yescharacteristic
42C22×Q8×C47C2×C411yescharacteristic
43C22×D4×C47—21no—
44C22×D4×C47—21no—
45C22×D4×C47C2×C411yescharacteristic
46C22×D4×C47D411yescharacteristic
47C23×C4×C47—21no—
48C23×C4×C47—21no—
49C23×C4×C47—21no—
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52C4⋊Q8×C47—22no—
53C4⋊Q8×C47—22no—
54#123×C47—28no—
55#123×C47—28no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47—22no—
58C22⋊Q8×C47—22no—
59C22⋊Q8×C47D414yes—
60#121×C47—28no—
61C4×D4×C47—22no—
62C4×D4×C47—22no—
63C4×D4×C47—22no—
64C4×D4×C47—22no—
65C2×C4⋊C4×C47—28no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47C2×C411yescharacteristic
71C2×C4⋊C4×C47D411yescharacteristic
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47—21no—
75C2×C22⋊C4×C47D411yescharacteristic
76C2×C22⋊C4×C47D411yescharacteristic
77C2×C42×C47C2×C411yescharacteristic
78C24×C47—21no—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C2×D411yescharacteristic
81#41×C47—28no—
82#41×C47—44no—
83#41×C47—48no—
84#41×C47—48no—
85#41×C47—44no—
86#41×C47—44no—
87C2×Q8×C47—21no—
88C2×Q8×C47—21no—
89C2×Q8×C47—42no—
90C2×Q8×C47—42no—
91C2×Q8×C47C22⋊C412yes—
92C2×Q8×C47C22⋊C412yes—
93C2×D4×C47—24no—
94C2×D4×C47—24no—
95C2×D4×C47—22no—
96C2×D4×C47—24no—
97C2×D4×C47—22no—
98C2×D4×C47—21no—
99C2×D4×C47—41no—
100C2×D4×C47—44no—
101C2×D4×C47—44no—
102C2×D4×C47C22⋊C412yes—
103C22×C4×C47—24no—
104C22×C4×C47—24no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—24no—
111C22×C4×C47—42no—
112C22×C4×C47—42no—
113C22×C4×C47—42no—
114C22×C4×C47—44no—
115C22×C4×C47—44no—
116C22×C4×C47—42no—
117C22×C4×C47C22⋊C411yescharacteristic
118C22×C4×C47C22⋊C411yescharacteristic
119C22×C4×C47C22⋊C411yescharacteristic
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D411yescharacteristic
124C22×Dic47—42no—
125C4⋊C4×C47—24no—
126C4⋊C4×C47—24no—
127C4⋊C4×C47—24no—
128C4⋊C4×C47—21no—
129C4⋊C4×C47—22no—
130C4⋊C4×C47—24no—
131C4⋊C4×C47C22⋊C412yes—
132C22⋊C4×C47—24no—
133C22⋊C4×C47—24no—
134C22⋊C4×C47—24no—
135C22⋊C4×C47—22no—
136C22⋊C4×C47—22no—
137C42×C47—21no—
138C42×C47—21no—
139C22⋊Dic47—24no—
140C22⋊Dic47—42no—
141C4×Dic47—44no—
142#10625329—44no—
143C2×C47⋊C8—42no—
144C23×C47—24no—
145C23×C47—22no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—21no—
151C23×C47—41no—
152C23×C47—41no—
153C23×C47—42no—
154C23×C47—41no—
155C23×C47—41no—
156C23×C47—41no—
157C23×C47—41no—
158C23×C47C2×C22⋊C411yescharacteristic
159C23×C47#12012yes—
160Q8×C47—24no—
161Q8×C47—42no—
162Q8×C47—42no—
163Q8×C47—42no—
164D4×C47—24no—
165D4×C47—22no—
166D4×C47—44no—
167D4×C47—48no—
168D4×C47—48no—
169D4×C47#9914yescomplemented
170C2×C4×C47—21no—
171C2×C4×C47—21no—
172C2×C4×C47—24no—
173C2×C4×C47—22no—
174C2×C4×C47—21no—
175C2×C4×C47—22no—
176C2×C4×C47—21no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—41no—
180C2×C4×C47—42no—
181C2×C4×C47—48no—
182C2×C4×C47—48no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—48no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—42no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47#10412yescomplemented
193C2×C4×C47C2×C22⋊C411yescharacteristic
194C2×C4×C47C2×C22⋊C411yescharacteristic
195C2×C4×C47#12012yes—
196C2×Dic47—44no—
197C2×Dic47—82no—
198C2×Dic47—82no—
199C47⋊C8—82no—
200#8364—471nomaximal, Sylow
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—22no—
204C22×C47—22no—
205C22×C47—24no—
206C22×C47—22no—
207C22×C47—44no—
208C22×C47—44no—
209C22×C47—42no—
210C22×C47—42no—
211C22×C47—41no—
212C22×C47—41no—
213C22×C47—41no—
214C22×C47—81no—
215C22×C47—81no—
216C22×C47—81no—
217C22×C47—81no—
218C22×C47#37911yescharacteristic
219C22×C47C2×#10411yescharacteristic
220C22×C47#42111yescharacteristic
221C4×C47—22no—
222C4×C47—21no—
223C4×C47—22no—
224C4×C47—41no—
225C4×C47—41no—
226C4×C47—42no—
227C4×C47—42no—
228C4×C47—42no—
229C4×C47C2×#9912yes—
230Dic47—84no—
231C2×#547D4711yescharacteristic
232#1789—472no—
233C4×#99—471no—
234#1508—472no—
235#1465—471no—
236C2×C47—21no—
237C2×C47—21no—
238C2×C47—42no—
239C2×C47—44no—
240C2×C47—81no—
241C2×C47#178111yescharacteristic
242C2×C47#178911yescharacteristic
243C2×C47#179611yescharacteristic
244C22×#41Dic4711yescharacteristic, complemented
245C22×#41C2×D4711yescharacteristic
246#547—24no—
247#547—24no—
248C2×C4⋊Q8Dic4711yescharacteristic, complemented
249C2×#123—21no—
250C2×#123—21no—
251C2×C22⋊Q8Dic4711yescharacteristic, complemented
252C2×C22⋊Q8C2×D4711yescharacteristic
253C2×#121—21no—
254C2×C4×D4Dic4711yescharacteristic, complemented
255C2×C4×D4C2×D4711yescharacteristic
256C22×C4⋊C4—21no—
257C2×#104—944no—
258C2×#99—941no—
259#407—472no—
260C4×C22⋊C4—472no—
261#342—941no—
262#328—944no—
263#324—941no—
264#323—941no—
265C47#836411yescharacteristic, complemented, Sylow
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—44no—
270C2×#41—44no—
271C2×#41C22⋊D4714yes—
272C22×Q8C2×Dic4711yescharacteristic
273C22×Q8C2×Dic4711yescharacteristic
274C22×D4—21no—
275C22×D4—21no—
276C22×D4C2×Dic4711yescharacteristic
277C22×D4C22⋊D4711yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4C2×Dic4711yescharacteristic
282C23×C4C22×D4711yescharacteristic
283C2×M16—942no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286#123—28no—
287#123—28no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8C22⋊D4714yes—
292#121—28no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C2×Dic4711yescharacteristic
303C2×C4⋊C4C22⋊D4711yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—942no—
308C2×C22⋊C4C22⋊D4711yescharacteristic
309C2×C22⋊C4C22⋊D4711yescharacteristic
310C2×C42—942no—
311C2×C42C2×Dic4711yescharacteristic
312#104—1888no—
313#99—944no—
314C22⋊C8—942no—
315C22⋊C8—942no—
316#95—942no—
317C24—21no—
318C24C22⋊Dic4711yescharacteristic
319C24C2×C22⋊D4711yescharacteristic
320#41—28no—
321#41—44no—
322#41—48no—
323#41—48no—
324#41—44no—
325#41—44no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8C22⋊Dic4712yes—
331C2×Q8C22⋊Dic4712yes—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4C22⋊Dic4712yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—1882no—
357C22×C4C22⋊Dic4711yescharacteristic
358C22×C4C22⋊Dic4711yescharacteristic
359C22×C4C22⋊Dic4711yescharacteristic
360C22×C4C22×Dic4711yescharacteristic
361C22×C4C2×C22⋊D4712yes—
362C22×C4C2×C22⋊D4712yes—
363C22×C4C2×C22⋊D4711yescharacteristic
364M16—1884no—
365C2×C8—1882no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4C22⋊Dic4712yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—944no—
379C22⋊C4—1882no—
380C42—21no—
381C42—21no—
382C42—1884no—
383C23—24no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23C2×C22⋊Dic4711yescharacteristic
398C23#1309399312yes—
399Q8—24no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403D4—24no—
404D4—22no—
405D4—44no—
406D4—48no—
407D4—48no—
408D4#1309393914yescomplemented
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—48no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—3762no—
432C2×C4—3762no—
433C2×C4—1884no—
434C2×C4#1309395112yescomplemented
435C2×C4C2×C22⋊Dic4711yescharacteristic
436C2×C4C2×C22⋊Dic4711yescharacteristic
437C2×C4#1309399312yes—
438C8—3762no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#1568897911yescharacteristic
457C22C2×#1309395111yescharacteristic
458C22#1568914011yescharacteristic
459C4—22no—
460C4—21no—
461C4—22no—
462C4—41no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—3764no—
468C4C2×#1309393912yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—44no—
473C2—81no—
474C2#2024330311yescharacteristic
475C2#2024333211yescharacteristic
476C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×C47×#547C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×#547D47
hypocenterC47#8364
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20243332C2semidirect
2. C4×#13093939C2semidirect
3. #20241908C2semidirect
4. #20241726C2semidirect
14. C22×#41×C47C4semidirect
18. C2×C4⋊Q8×C47C4semidirect
21. C2×C22⋊Q8×C47C4semidirect
24. C2×C4×D4×C47C4semidirect
169. D4×C47#99semidirect
192. C2×C4×C47#104semidirect
244. C22×#41Dic47semidirect
248. C2×C4⋊Q8Dic47semidirect
251. C2×C22⋊Q8Dic47semidirect
254. C2×C4×D4Dic47semidirect
265. C47#8364semidirect
408. D4#13093939semidirect
434. C2×C4#13093951semidirect

Maximal subgroups

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

subgroupindex
#202433322
C4×#130939392
#202419082
#202417262
C2×C47×#5472
#836447