Finite Group Databasebeta

← #26107611

Subgroups — #26107611

Order 12032.

1158 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2#20243454C212yesmaximal, complemented
3C2×#15688849C211yesmaximal, characteristic, complemented
4C2×#15688846C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic, complemented
6C23×C4×D47C411yescharacteristic
7C2×#13093982—24no—
8C2×#13093982—24no—
9C2×#13093982—22no—
10C2×#13093982—22no—
11C22×C4⋊Dic47C411yescharacteristic
12C22×(C2×D47)⋊C4C411yescharacteristic
13C22×(C2×D47)⋊C4C2211yescharacteristic
14C22×C4×Dic47C411yescharacteristic
15C22×C4×Dic47C2211yescharacteristic
16C2×C22⋊(C47⋊C8)C2212yes—
17#15688877—24no—
18#15688849—22no—
19#15688846—24no—
20C22×C4⋊C4×C47C2211yescharacteristic
21C2×C22⋊C8×C47C2212yes—
22C47×#324—22no—
23C24×D47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C22×C4×D47—22no—
28C22×C4×D47—22no—
29C22×C4×D47—22no—
30C22×C4×D47—22no—
31C22×C4×D47—22no—
32C22×C4×D47—22no—
33C22×C4×D47C814yescomplemented
34C22×C47⋊C8—22no—
35#13093982—216no—
36#13093982—216no—
37C2×C4⋊Dic47—22no—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47—22no—
40C2×(C2×D47)⋊C4—22no—
41C2×(C2×D47)⋊C4—22no—
42C2×(C2×D47)⋊C4—22no—
43C2×(C2×D47)⋊C4—21no—
44C2×(C2×D47)⋊C4—21no—
45C2×(C2×D47)⋊C4—22no—
46C2×(C2×D47)⋊C4D414yes—
47C2×C4×Dic47—22no—
48C2×C4×Dic47—21no—
49C2×C4×Dic47—22no—
50C2×C4×Dic47—21no—
51C2×C4×Dic47—22no—
52C2×C4×Dic47—22no—
53C2×C4×Dic47D414yes—
54C22⋊(C47⋊C8)—22no—
55C22⋊(C47⋊C8)—22no—
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47C2×C411yescharacteristic
58C23×C4×C47C2311yescharacteristic
59C22×C8×C47—22no—
60C2×C4⋊C4×C47—22no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47D412yes—
63C2×C4⋊C4×C47D412yes—
64C22⋊C8×C47—22no—
65C22⋊C8×C47—22no—
66C24×C47C22×C411yescharacteristic
67C23×D47—21no—
68C23×D47—21no—
69C23×D47—21no—
70C23×D47—21no—
71C23×D47—42no—
72C23×D47—42no—
73C23×D47C22⋊C412yes—
74C23×D47C2×C812yes—
75C23×D47M1612yes—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—42no—
83C22×C4×C47—42no—
84C22×C4×C47C22⋊C412yes—
85C22×C4×C47C22⋊C412yes—
86C22×C4×C47C2×C812yes—
87C22×C4×C47M1612yes—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D411yescharacteristic
92C22×Dic47—21no—
93C22×Dic47—21no—
94C22×Dic47—21no—
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—21no—
98C22×Dic47—21no—
99C22×Dic47—21no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22×Dic47C22⋊C412yes—
103C22×Dic47C22⋊C412yes—
104C22×Dic47C22⋊C412yes—
105C22×Dic47C2×C812yes—
106C22×Dic47M1612yes—
107C2×C4×D47—28no—
108C2×C4×D47—28no—
109C2×C4×D47—28no—
110C2×C4×D47—28no—
111C2×C4×D47—28no—
112C2×C4×D47—28no—
113C2×C4×D47—28no—
114C2×C8×C47—44no—
115C2×C8×C47—42no—
116C4⋊C4×C47—24no—
117C4⋊C4×C47—24no—
118(C2×D47)⋊C4—28no—
119(C2×D47)⋊C4—24no—
120(C2×D47)⋊C4—24no—
121C4⋊Dic47—28no—
122C4×Dic47—28no—
123C4×Dic47—24no—
124C4×Dic47—24no—
125C2×C47⋊C8—44no—
126C2×C47⋊C8—42no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47C2×C22⋊C411yescharacteristic
130C23×C47C2×C22⋊C411yescharacteristic
131C23×C47C2×C22⋊C411yescharacteristic
132C23×C47C22×C811yescharacteristic
133C23×C47C2×M1611yescharacteristic
134C23×C47C2×M1611yescharacteristic
135C23×C47C2×M1611yescharacteristic
136C22×D47—22no—
137C22×D47—22no—
138C22×D47—22no—
139C22×D47—22no—
140C22×D47—22no—
141C22×D47—22no—
142C22×D47—44no—
143C22×D47—44no—
144C22×D47—42no—
145C22×D47—42no—
146C22×D47—42no—
147C22×D47—42no—
148C22×D47—42no—
149C22×D47—42no—
150C22×D47—44no—
151C22×D47—44no—
152C22×D47C22⋊C814yes—
153C2×C4×C47—22no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—24no—
160C2×C4×C47—22no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47C22⋊C814yes—
168C2×C4×C47#12012yes—
169C2×C4×C47#12012yes—
170C2×Dic47—22no—
171C2×Dic47—22no—
172C2×Dic47—22no—
173C2×Dic47—22no—
174C2×Dic47—22no—
175C2×Dic47—22no—
176C2×Dic47—22no—
177C2×Dic47—22no—
178C2×Dic47—24no—
179C2×Dic47—42no—
180C2×Dic47—42no—
181C2×Dic47—42no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47C22⋊C814yes—
185C4×D47—232no—
186C8×C47—82no—
187C47⋊C8—82no—
188#8300—471nomaximal, Sylow
189C22×C47—22no—
190C22×C47—22no—
191C22×C47—22no—
192C22×C47—22no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47#37911yescharacteristic
198C22×C47C2×C22⋊C811yescharacteristic
199C22×C47C2×C22⋊C811yescharacteristic
200C22×C47C2×C22⋊C811yescharacteristic
201C22×C47#40711yescharacteristic
202C22×C47#40711yescharacteristic
203C22×C47#40711yescharacteristic
204C22×C47#41012yes—
205C22×C47#41312yes—
206C2×D47—24no—
207C2×D47—44no—
208C2×D47—44no—
209C2×D47—44no—
210C2×D47—44no—
211C2×D47—44no—
212C2×D47—42no—
213C2×D47—42no—
214C2×D47—44no—
215C4×C47—24no—
216C4×C47—22no—
217C4×C47—22no—
218C4×C47—42no—
219Dic47—24no—
220Dic47—42no—
221Dic47—42no—
222C22×#117—471no—
223#1817—472no—
224C2×#324—471no—
225C2×#324D4711yescharacteristic, complemented
226C2×#323—472no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47—21no—
230C2×C47—21no—
231C2×C47#145812yes—
232C2×C47#146412yes—
233C2×C47#177911yescharacteristic
234C2×C47#178211yescharacteristic
235C2×C47#178511yescharacteristic
236D47—44no—
237C24×C4—471no—
238C2×#117—944no—
239C2×#117—944no—
240C2×#117—942no—
241C2×#117—942no—
242C22×C4⋊C4—471no—
243C22×C4⋊C4C2×D4711yescharacteristic
244C22×C22⋊C4—471no—
245C22×C22⋊C4—471no—
246C22×C42—471no—
247C22×C42—471no—
248C2×C22⋊C8—472no—
249C2×C22⋊C8C2×D4712yes—
250#336—944no—
251#324—22no—
252#324—942no—
253#323—944no—
254C47#830011yescharacteristic, complemented, Sylow
255C25—471no—
256C23×C4—471no—
257C23×C4—471no—
258C23×C4—471no—
259C23×C4—474no—
260C23×C4—942no—
261C23×C4—942no—
262C23×C4—942no—
263C23×C4—942no—
264C23×C4—942no—
265C23×C4—942no—
266C23×C4C4×D4711yescharacteristic
267C23×C4C4×D4711yescharacteristic
268C23×C4C22×D4711yescharacteristic
269C22×C8—22no—
270C22×C8—942no—
271#117—9416no—
272#117—9416no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—942no—
276C2×C4⋊C4—942no—
277C2×C4⋊C4—942no—
278C2×C4⋊C4D18812yes—
279C2×C4⋊C4C22⋊D4712yes—
280C2×C22⋊C4—474no—
281C2×C22⋊C4—942no—
282C2×C22⋊C4—942no—
283C2×C22⋊C4—942no—
284C2×C22⋊C4—941no—
285C2×C22⋊C4—942no—
286C2×C22⋊C4—941no—
287C2×C42—474no—
288C2×C42—942no—
289C2×C42—942no—
290C2×C42—942no—
291C2×C42—941no—
292C2×C42—942no—
293C2×C42—941no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—942no—
297C22⋊C8—942no—
298C24—472no—
299C24—472no—
300C24—472no—
301C24—941no—
302C24—941no—
303C24—941no—
304C24—941no—
305C24—1882no—
306C24—1882no—
307C24C2×C4×D4711yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—472no—
317C22×C4—472no—
318C22×C4—472no—
319C22×C4—472no—
320C22×C4—472no—
321C22×C4—948no—
322C22×C4—948no—
323C22×C4—948no—
324C22×C4—948no—
325C22×C4—948no—
326C22×C4—948no—
327C22×C4—941no—
328C22×C4—941no—
329C22×C4—941no—
330C22×C4—941no—
331C22×C4—948no—
332C22×C4—941no—
333C22×C4—941no—
334C22×C4—941no—
335C22×C4—941no—
336C22×C4—1882no—
337C22×C4—1882no—
338C22×C4C8×D4712yes—
339C22×C4#1062532412yes—
340C22×C4(C2×D47)⋊C412yes—
341C22×C4(C2×D47)⋊C412yes—
342C22×C4C2×D18811yescharacteristic
343C22×C4D4×D4712yes—
344C22×C4D4×D4712yes—
345C22×C4C2×C22⋊D4711yescharacteristic
346C2×C8—44no—
347C2×C8—42no—
348C2×C8—1884no—
349C2×C8—1882no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—948no—
353C22⋊C4—948no—
354C22⋊C4—944no—
355C22⋊C4—944no—
356C42—948no—
357C42—944no—
358C42—944no—
359C23—22no—
360C23—22no—
361C23—474no—
362C23—942no—
363C23—942no—
364C23—942no—
365C23—942no—
366C23—942no—
367C23—942no—
368C23—1884no—
369C23—1884no—
370C23—1882no—
371C23—1882no—
372C23—1884no—
373C23—1884no—
374C23—1882no—
375C23—1882no—
376C23—1882no—
377C23—1882no—
378C23C22⋊C4×D4711yescharacteristic
379C23C22⋊C4×D4711yescharacteristic
380C23C2×(C2×D47)⋊C411yescharacteristic
381C23C2×C8×D4711yescharacteristic
382C23M16×D4711yescharacteristic
383C23M16×D4711yescharacteristic
384C23C2×#1062532411yescharacteristic
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—474no—
400C2×C4—9432no—
401C2×C4—942no—
402C2×C4—942no—
403C2×C4—942no—
404C2×C4—942no—
405C2×C4—942no—
406C2×C4—942no—
407C2×C4—942no—
408C2×C4—942no—
409C2×C4—944no—
410C2×C4—1882no—
411C2×C4—1882no—
412C2×C4—1882no—
413C2×C4—1882no—
414C2×C4—1882no—
415C2×C4(C2×D47)⋊C814yes—
416C2×C4#1309399212yes—
417C2×C4#1309399412yes—
418C8—82no—
419C8—3762no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—944no—
429C22—1884no—
430C22—1884no—
431C22—1884no—
432C22—1884no—
433C22—1884no—
434C22—1884no—
435C22—1882no—
436C22—1882no—
437C22#1568897811yescharacteristic
438C22C22⋊C8×D4711yescharacteristic
439C22C22⋊C8×D4711yescharacteristic
440C22C2×(C2×D47)⋊C811yescharacteristic
441C22#1568908511yescharacteristic
442C22#1568908511yescharacteristic
443C22#1568908611yescharacteristic
444C22#1568909512yes—
445C22#1568910612yes—
446C4—24no—
447C4—22no—
448C4—22no—
449C4—42no—
450C4—944no—
451C4—1882no—
452C4—1882no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—1884no—
458C2#2024170112yes—
459C2#2024172112yes—
460C2#2024329711yescharacteristic
461C2#2024330411yescharacteristic
462C2#2024331511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC23×C47C2×C22⋊C4
prime radicalC24×D47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#8300
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20243454C2semidirect
3. C2×#15688849C2semidirect
5. C2×C47×#324C2semidirect
33. C22×C4×D47C8semidirect
225. C2×#324D47semidirect
254. C47#8300semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
#202434542
C2×#156888492
C2×#156888462
C2×C47×#3242
#830047