Finite Group Databasebeta

← #26125814

Subgroups — #26125814

Order 12032.

700 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20241715C211yesmaximal, characteristic, complemented
2C2×#15688847C211yesmaximal, characteristic, complemented
3#20241336C212yesmaximal, complemented
4#20241316C212yesmaximal, complemented
5C47×#2405C211yesmaximal, characteristic
6#15689087—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688847—21no—
9#15688847—21no—
10#15688847C2212yes—
11#15688847C2212yes—
12C23×D4×C47C411yescharacteristic
13C2×#123×C47—21no—
14C22×C22⋊C4×C47C2211yescharacteristic
15C47×#394—21no—
16C47×#394—21no—
17C47×#386—21no—
18C47×#385C411yescharacteristic
19C47×#385C2211yescharacteristic
20C47×#379C411yescharacteristic
21C47×#379C411yescharacteristic
22C47×#379C2211yescharacteristic
23C2×#10625329—41no—
24C22×C47⋊C8—41no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—41no—
28C22⋊(C47⋊C8)—41no—
29C25×C47C2×C411yescharacteristic
30C25×C47C2×C411yescharacteristic
31C22×D4×C47—21no—
32C22×D4×C47—21no—
33C22×D4×C47—44no—
34C22×D4×C47—41no—
35C22×D4×C47—41no—
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39#123×C47—44no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—41no—
49C2×C22⋊C4×C47—41no—
50C2×C22⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C22⋊C4×C47C2311yescharacteristic
56#95×C47—21no—
57#95×C47—21no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—21no—
61C24×C47—22no—
62C24×C47—22no—
63C24×C47—41no—
64C24×C47—42no—
65C24×C47—41no—
66C24×C47—42no—
67C24×C47—41no—
68C24×C47—81no—
69C24×C47—81no—
70C24×C47C22⋊C411yescharacteristic
71C24×C47C22⋊C411yescharacteristic
72C24×C47C22×C411yescharacteristic
73C2×D4×C47—24no—
74C2×D4×C47—42no—
75C2×D4×C47—42no—
76C2×D4×C47—42no—
77C2×D4×C47—42no—
78C2×D4×C47—44no—
79C2×D4×C47—44no—
80C2×D4×C47—44no—
81C2×D4×C47—41no—
82C2×D4×C47—41no—
83C2×D4×C47—44no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—42no—
92C22×C4×C47—41no—
93C22×C4×C47—42no—
94C22×C4×C47—81no—
95C22×C4×C47C22⋊C411yescharacteristic
96C22×C4×C47C22⋊C411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C4⋊C4×C47—42no—
100C4⋊C4×C47—42no—
101C22⋊C4×C47—22no—
102C22⋊C4×C47—41no—
103C22⋊C4×C47—41no—
104C22⋊C4×C47—41no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—42no—
107C22⋊C4×C47—42no—
108C22⋊C4×C47—42no—
109C22⋊C4×C47—41no—
110C22⋊C4×C47—81no—
111C22⋊C4×C47—81no—
112C22⋊C4×C47—81no—
113#10625329—82no—
114C2×C47⋊C8—42no—
115C2×C47⋊C8—42no—
116C2×C47⋊C8—82no—
117C23×C47—22no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—21no—
121C23×C47—21no—
122C23×C47—22no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—41no—
127C23×C47—42no—
128C23×C47—42no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47—42no—
135C23×C47—41no—
136C23×C47—41no—
137C23×C47—42no—
138C23×C47—42no—
139C23×C47—42no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47—81no—
144C23×C47—81no—
145C23×C47—82no—
146C23×C47—84no—
147C23×C47—84no—
148C23×C47—84no—
149C23×C47—84no—
150C23×C47—82no—
151C23×C47#9911yescharacteristic
152C23×C47#9911yescharacteristic
153C23×C47C2×C22⋊C411yescharacteristic
154C23×C47#13112yes—
155D4×C47—44no—
156D4×C47—48no—
157D4×C47—44no—
158C2×C4×C47—22no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—42no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—81no—
180C2×C4×C47—81no—
181C47⋊C8—82no—
182C47⋊C8—82no—
183#11545—471nomaximal, Sylow
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—22no—
191C22×C47—21no—
192C22×C47—42no—
193C22×C47—42no—
194C22×C47—42no—
195C22×C47—42no—
196C22×C47—42no—
197C22×C47—42no—
198C22×C47—41no—
199C22×C47—41no—
200C22×C47—42no—
201C22×C47—41no—
202C22×C47—41no—
203C22×C47—41no—
204C22×C47—84no—
205C22×C47—81no—
206C22×C47—81no—
207C22×C47—81no—
208C22×C47—81no—
209C22×C47—84no—
210C22×C47—84no—
211C22×C47—84no—
212C22×C47—81no—
213C22×C47—81no—
214C22×C47AΣL(1,𝔽16)12yes—
215C22×C47#40811yescharacteristic
216C22×C47C2×#9911yescharacteristic
217C22×C47#41311yescharacteristic
218C4×C47—41no—
219C4×C47—42no—
220C4×C47—41no—
221C4×C47—41no—
222C4×C47—41no—
223C4×C47—81no—
224C4×C47—81no—
225#2405D4711yescharacteristic
226#1462—471no—
227C2×#323—471no—
228#1326—472no—
229#1316—472no—
230C2×C47—21no—
231C2×C47—21no—
232C2×C47—22no—
233C2×C47—41no—
234C2×C47—42no—
235C2×C47—82no—
236C2×C47—81no—
237C2×C47#146811yescharacteristic
238C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
239C2×C47#212111yescharacteristic
240C23×D4Dic4711yescharacteristic
241C2×#123—21no—
242C22×C22⋊C4C2×D4711yescharacteristic
243#407—941no—
244C2×C22⋊C8—941no—
245#394—21no—
246#394—21no—
247#386—21no—
248#385Dic4711yescharacteristic
249#385C2×D4711yescharacteristic
250#379Dic4711yescharacteristic
251#379Dic4711yescharacteristic
252#379C2×D4711yescharacteristic
253#323—472no—
254#323—472no—
255#323—941no—
256#323—941no—
257C47#1154511yescharacteristic, complemented, Sylow
258C25C2×Dic4711yescharacteristic
259C25C2×Dic4711yescharacteristic
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—44no—
263C22×D4—41no—
264C22×D4—41no—
265C23×C4C2×Dic4711yescharacteristic
266C23×C4C22⋊D4711yescharacteristic
267C23×C4C22⋊D4711yescharacteristic
268C2×M16—1881no—
269C22×C8—1881no—
270#123—44no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4C2×Dic4711yescharacteristic
282C2×C22⋊C4C2×Dic4711yescharacteristic
283C2×C22⋊C4C2×Dic4711yescharacteristic
284C2×C22⋊C4C22⋊D4711yescharacteristic
285C2×C22⋊C4C22⋊D4711yescharacteristic
286C2×C22⋊C4C22×D4711yescharacteristic
287C22⋊C8—942no—
288C22⋊C8—942no—
289C22⋊C8—1881no—
290C22⋊C8—1881no—
291#95—21no—
292#95—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—41no—
303C24—81no—
304C24—81no—
305C24C22⋊Dic4711yescharacteristic
306C24C22⋊Dic4711yescharacteristic
307C24C22×Dic4711yescharacteristic
308C2×D4—24no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—44no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—81no—
330C22×C4C22⋊Dic4711yescharacteristic
331C22×C4C22⋊Dic4711yescharacteristic
332C22×C4C2×C22⋊D4711yescharacteristic
333C22×C4C2×C22⋊D4711yescharacteristic
334M16—3762no—
335C2×C8—3762no—
336C2×C8—1882no—
337C2×C8—1882no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C22⋊C4—22no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—41no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—82no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23—82no—
386C23#1309393911yescharacteristic
387C23#1309393911yescharacteristic
388C23C2×C22⋊Dic4711yescharacteristic
389C23#1309403112yes—
390D4—44no—
391D4—48no—
392D4—44no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—81no—
416C8—3762no—
417C8—3762no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—81no—
447C22—81no—
448C22#1568891012yes—
449C22#1568909011yescharacteristic
450C22C2×#1309393911yescharacteristic
451C22#1568910711yescharacteristic
452C4—41no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—81no—
458C4—81no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—41no—
463C2—42no—
464C2—82no—
465C2—81no—
466C2#2024173611yescharacteristic
467C2C2×#1568891011yescharacteristic
468C2#2024503411yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2405C2
socleC22×C47C2×#99
prime radicalC23×D4×C47C4
cosocleC2×C22⋊C4×C47C23
hypercenter#2405D47
hypocenterC47#11545
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2405

Complements

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

NH ≅ G/Ntype
1. #20241715C2semidirect
2. C2×#15688847C2semidirect
3. #20241336C2semidirect
4. #20241316C2semidirect
257. C47#11545semidirect

Maximal subgroups

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

subgroupindex
#202417152
C2×#156888472
#202413362
#202413162
C47×#24052
#1154547