Finite Group Databasebeta

← #26126563

Subgroups — #26126563

Order 12032.

592 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#15688890C211yesmaximal, characteristic
2C2×#15688847C211yesmaximal, characteristic
3#20241412C212yesmaximal, complemented
4#20241409C212yesmaximal, complemented
5C47×#2626C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic, complemented
13C22×C22⋊C4×C47C2211yescharacteristic
14C47×#399C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#396—21no—
17C47×#396—21no—
18C47×#393C411yescharacteristic, complemented
19C47×#393C2211yescharacteristic
20C47×#385—21no—
21C47×#380C411yescharacteristic, complemented
22C47×#379—21no—
23C23×Dic47—41no—
24C22×C47⋊C8—41no—
25C2×C22⋊Dic47—24no—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—41no—
28C2×C22⋊Dic47—42no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—41no—
31C25×C47D411yescharacteristic
32C22×Q8×C47C2×C411yescharacteristic, complemented
33C23×C4×C47C2×C411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C22⋊Q8×C47—41no—
36C22⋊Q8×C47—41no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
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×C47D411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47C2311yescharacteristic
54#95×C47—21no—
55#95×C47—21no—
56#95×C47C2×C411yescharacteristic, complemented
57#95×C47C2×C411yescharacteristic, complemented
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—22no—
61C24×C47—81no—
62C24×C47—81no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C22⋊C411yescharacteristic
65C24×C47SD1612yes—
66C24×C47Q1612yes—
67C24×C47C2×D411yescharacteristic
68C2×Q8×C47—41no—
69C2×Q8×C47—81no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—81no—
79C22×C4×C47—81no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×Dic47—44no—
85C22×Dic47—42no—
86C22×Dic47—82no—
87C22×Dic47—82no—
88C4⋊C4×C47—42no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—81no—
91C4⋊C4×C47—81no—
92C4⋊C4×C47—81no—
93C22⋊C4×C47—41no—
94C22⋊C4×C47—41no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—81no—
101C22⋊C4×C47—81no—
102C22⋊C4×C47—81no—
103C22⋊C4×C47—81no—
104C22⋊Dic47—24no—
105C22⋊Dic47—44no—
106C22⋊Dic47—42no—
107C2×C47⋊C8—42no—
108C2×C47⋊C8—82no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—81no—
126C23×C47—81no—
127C23×C47—82no—
128C23×C47—82no—
129C23×C47—84no—
130C23×C47#9911yescharacteristic
131C23×C47#9911yescharacteristic
132C23×C47#10312yes—
133C23×C47#10312yes—
134C23×C47#10412yes—
135C23×C47C2×C22⋊C411yescharacteristic
136C23×C47C2×SD1611yescharacteristic
137C23×C47C2×Q1611yescharacteristic
138Q8×C47—82no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—42no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—81no—
161C2×C4×C47—81no—
162C2×C4×C47—81no—
163C2×Dic47—44no—
164C2×Dic47—88no—
165C2×Dic47—82no—
166C2×Dic47—82no—
167C47⋊C8—82no—
168#11716—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—22no—
177C22×C47—21no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47—41no—
182C22×C47—41no—
183C22×C47—44no—
184C22×C47—44no—
185C22×C47—41no—
186C22×C47—41no—
187C22×C47—41no—
188C22×C47—41no—
189C22×C47—81no—
190C22×C47—81no—
191C22×C47—82no—
192C22×C47—82no—
193C22×C47#32812yes—
194C22×C47#32812yes—
195C22×C47C2×#9911yescharacteristic
196C22×C47C2×#10311yescharacteristic
197C22×C47C2×#10411yescharacteristic
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—41no—
201C4×C47—41no—
202C4×C47—81no—
203C4×C47—81no—
204C4×C47—81no—
205C4×C47—81no—
206Dic47—84no—
207#2626D4711yescharacteristic
208C2×#342—471no—
209C2×#323—471no—
210#1352—472no—
211#1351—472no—
212C2×C47—21no—
213C2×C47—22no—
214C2×C47—21no—
215C2×C47—22no—
216C2×C47—41no—
217C2×C47—42no—
218C2×C47—41no—
219C2×C47C2×#32811yescharacteristic
220C2×C47#212111yescharacteristic
221C2×C47#212711yescharacteristic
222C2×C22⋊Q8Dic4711yescharacteristic, complemented
223C22×C22⋊C4—941no—
224C22×C22⋊C4C2×D4711yescharacteristic
225C2×C22⋊C8—941no—
226#399Dic4711yescharacteristic, complemented
227#399C2×D4711yescharacteristic
228#396—21no—
229#396—21no—
230#393Dic4711yescharacteristic, complemented
231#393C2×D4711yescharacteristic
232#385—21no—
233#380Dic4711yescharacteristic, complemented
234#379—21no—
235#342—472no—
236#342—941no—
237#323—472no—
238#323—941no—
239C47#1171611yescharacteristic, complemented, Sylow
240C25C22⋊D4711yescharacteristic
241C22×Q8C2×Dic4711yescharacteristic, complemented
242C23×C4—1881no—
243C23×C4C2×Dic4711yescharacteristic
244C23×C4C22⋊D4711yescharacteristic
245C22×C8—1881no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C2×Dic4711yescharacteristic, complemented
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—944no—
262C2×C22⋊C4—942no—
263C2×C22⋊C4—1881no—
264C2×C22⋊C4—1882no—
265C2×C22⋊C4C2×Dic4711yescharacteristic
266C2×C22⋊C4C22⋊D4711yescharacteristic
267C2×C22⋊C4C22⋊D4711yescharacteristic
268C2×C22⋊C4C22×D4711yescharacteristic
269C22⋊C8—942no—
270C22⋊C8—1881no—
271#95—21no—
272#95—21no—
273#95C2×Dic4711yescharacteristic, complemented
274#95C2×Dic4711yescharacteristic, complemented
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24#1062533612yes—
281C24#1062533712yes—
282C24C22⋊Dic4711yescharacteristic
283C24C22⋊Dic4711yescharacteristic
284C24C2×C22⋊D4711yescharacteristic
285C2×Q8—41no—
286C2×Q8—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—3762no—
298C22×C4—3762no—
299C22×C4—1882no—
300C22×C4—1884no—
301C22×C4C22⋊Dic4711yescharacteristic
302C22×C4C22⋊Dic4711yescharacteristic
303C22×C4C22×Dic4711yescharacteristic
304C22×C4C2×C22⋊D4711yescharacteristic
305C2×C8—3762no—
306C2×C8—1882no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—81no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—944no—
324C22⋊C4—1884no—
325C22⋊C4—1882no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1309393911yescharacteristic
348C23#1309393911yescharacteristic
349C23#1309394812yes—
350C23#1309394812yes—
351C23#1309395112yes—
352C23C2×C22⋊Dic4711yescharacteristic
353C23C2×#1062533611yescharacteristic
354C23C2×#1062533711yescharacteristic
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—3768no—
381C2×C4—3762no—
382C2×C4—3762no—
383C2×C4—1884no—
384C8—3762no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1568886012yes—
410C22#1568886012yes—
411C22C2×#1309393911yescharacteristic
412C22C2×#1309394811yescharacteristic
413C22C2×#1309395111yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—3764no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#1568886011yescharacteristic
431C2#2024503411yescharacteristic
432C2#2024505711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2626C2
socleC22×C47C2×#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2626D47
hypocenterC47#11716
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #20241412C2semidirect
4. #20241409C2semidirect
12. C2×C22⋊Q8×C47C4semidirect
14. C47×#399C4semidirect
18. C47×#393C4semidirect
21. C47×#380C4semidirect
32. C22×Q8×C47C2×C4semidirect
41. C2×C4⋊C4×C47C2×C4semidirect
56. #95×C47C2×C4semidirect
57. #95×C47C2×C4semidirect
222. C2×C22⋊Q8Dic47semidirect
226. #399Dic47semidirect
230. #393Dic47semidirect
233. #380Dic47semidirect
239. C47#11716semidirect
241. C22×Q8C2×Dic47semidirect
252. C2×C4⋊C4C2×Dic47semidirect
273. #95C2×Dic47semidirect
274. #95C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156888902
C2×#156888472
#202414122
#202414092
C47×#26262
#1171647