Finite Group Databasebeta

← #26108885

Subgroups — #26108885

Order 12032.

1578 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 155 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#20243462C212yesmaximal, complemented
2#20243303C212yesmaximal, complemented
3#20243303C212yesmaximal, complemented
4C22×#121×C47C211yesmaximal, characteristic
5C2×#13093939—24no—
6C2×#13093939—24no—
7#15688979C2214yes—
8#15688890—24no—
9C23×D4×C47C411yescharacteristic, complemented
10C23×D4×C47C2211yescharacteristic
11C23×D4×C47C2211yescharacteristic
12C24×C4×C47C411yescharacteristic, complemented
13C2×#121×C47—24no—
14C2×#121×C47—24no—
15C2×#121×C47—24no—
16C22×C4⋊C4×C47C411yescharacteristic, complemented
17C22×C22⋊C4×C47C411yescharacteristic, complemented
18C22×C22⋊C4×C47C2211yescharacteristic
19C2×C22⋊Dic47—24no—
20C2×C22⋊Dic47—24no—
21C2×C22⋊Dic47—24no—
22#13093939—48no—
23#13093939—48no—
24C25×C47C2×C411yescharacteristic
25C25×C47C2×C411yescharacteristic
26C25×C47C2311yescharacteristic
27C22×D4×C47—22no—
28C22×D4×C47—22no—
29C22×D4×C47—22no—
30C22×D4×C47—22no—
31C22×D4×C47—22no—
32C22×D4×C47—22no—
33C22×D4×C47—22no—
34C22×D4×C47—44no—
35C22×D4×C47—44no—
36C22×D4×C47—44no—
37C22×D4×C47D414yes—
38C22×D4×C47D414yes—
39C23×C4×C47—22no—
40C23×C4×C47—22no—
41C23×C4×C47—22no—
42C23×C4×C47—44no—
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47#121×C47—232no—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47—22no—
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47—22no—
53C2×C22⋊C4×C47—22no—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47—22no—
56C2×C22⋊C4×C47D414yes—
57C24×C47—24no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—21no—
61C24×C47—21no—
62C24×C47—42no—
63C24×C47—42no—
64C24×C47—42no—
65C24×C47—42no—
66C24×C47—42no—
67C24×C47—42no—
68C24×C47—42no—
69C24×C47—42no—
70C24×C47C22⋊C412yes—
71C24×C47C22⋊C412yes—
72C24×C47C22×C411yescharacteristic
73C24×C47C2×D412yes—
74C24×C47C2×D412yes—
75C24×C47C2×D412yes—
76C2×D4×C47—28no—
77C2×D4×C47—28no—
78C2×D4×C47—28no—
79C2×D4×C47—28no—
80C2×D4×C47—28no—
81C2×D4×C47—44no—
82C2×D4×C47—44no—
83C2×D4×C47—48no—
84C2×D4×C47—48no—
85C2×D4×C47—48no—
86C2×D4×C47—48no—
87C2×D4×C47—48no—
88C2×D4×C47—44no—
89C2×D4×C47—48no—
90C2×D4×C47—44no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—24no—
94C22×C4×C47—28no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—42no—
102C22×C4×C47—48no—
103C22×C4×C47—48no—
104C22×C4×C47—44no—
105C22×C4×C47—44no—
106C22×C4×C47—42no—
107C22×C4×C47—42no—
108C22×C4×C47C22⋊C412yes—
109C22×C4×C47C22⋊C412yes—
110C22×C4×C47C22⋊C412yes—
111C22×C4×C47C22⋊C412yes—
112C22×Dic47—44no—
113C4⋊C4×C47—28no—
114C22⋊C4×C47—28no—
115C22⋊C4×C47—28no—
116C22⋊Dic47—48no—
117C22⋊Dic47—48no—
118C22⋊Dic47—48no—
119C23×C47—22no—
120C23×C47—22no—
121C23×C47—22no—
122C23×C47—24no—
123C23×C47—24no—
124C23×C47—22no—
125C23×C47—22no—
126C23×C47—24no—
127C23×C47—24no—
128C23×C47—24no—
129C23×C47—22no—
130C23×C47—24no—
131C23×C47—42no—
132C23×C47—42no—
133C23×C47—42no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—42no—
137C23×C47—42no—
138C23×C47—42no—
139C23×C47—42no—
140C23×C47—416no—
141C23×C47—42no—
142C23×C47—42no—
143C23×C47—42no—
144C23×C47—42no—
145C23×C47—42no—
146C23×C47—42no—
147C23×C47—42no—
148C23×C47—42no—
149C23×C47—42no—
150C23×C47—42no—
151C23×C47—42no—
152C23×C47—88no—
153C23×C47—88no—
154C23×C47#9914yescomplemented
155C23×C47C2×C22⋊C411yescharacteristic
156C23×C47C2×C22⋊C411yescharacteristic
157C23×C47C2×C22⋊C411yescharacteristic
158C23×C47#12014yes—
159D4×C47—416no—
160D4×C47—416no—
161D4×C47—416no—
162C2×C4×C47—24no—
163C2×C4×C47—24no—
164C2×C4×C47—22no—
165C2×C4×C47—22no—
166C2×C4×C47—22no—
167C2×C4×C47—22no—
168C2×C4×C47—22no—
169C2×C4×C47—24no—
170C2×C4×C47—416no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47—42no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
179C2×C4×C47#9914yescomplemented
180C2×Dic47—84no—
181C2×Dic47—84no—
182C2×Dic47—84no—
183#8542—471nomaximal, Sylow
184C22×C47—28no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—24no—
192C22×C47—24no—
193C22×C47—24no—
194C22×C47—24no—
195C22×C47—44no—
196C22×C47—48no—
197C22×C47—44no—
198C22×C47—48no—
199C22×C47—42no—
200C22×C47—42no—
201C22×C47—42no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—84no—
205C22×C47—84no—
206C22×C47—84no—
207C22×C47—84no—
208C22×C47—84no—
209C22×C47—84no—
210C22×C47#37911yescharacteristic
211C22×C47C2×#9912yes—
212C22×C47C2×#9912yes—
213C22×C47#41012yes—
214C4×C47—24no—
215C4×C47—42no—
216C4×C47—42no—
217C4×C47—42no—
218Dic47—164no—
219C22×#121D4711yescharacteristic
220#1819—472no—
221#1781—472no—
222#1781—472no—
223C2×C47—22no—
224C2×C47—22no—
225C2×C47—22no—
226C2×C47—44no—
227C2×C47—44no—
228C2×C47—82no—
229C2×C47#178111yescharacteristic
230C2×C47#178111yescharacteristic
231C2×C47#178211yescharacteristic
232C23×D4Dic4711yescharacteristic, complemented
233C23×D4C2×D4711yescharacteristic
234C23×D4C2×D4711yescharacteristic
235C24×C4Dic4711yescharacteristic, complemented
236C2×#121—24no—
237C2×#121—24no—
238C2×#121—24no—
239C22×C4⋊C4Dic4711yescharacteristic, complemented
240C22×C22⋊C4Dic4711yescharacteristic, complemented
241C22×C22⋊C4C2×D4711yescharacteristic
242C2×#99—944no—
243C2×#99—944no—
244#379—474no—
245#342—944no—
246C47#854211yescharacteristic, complemented, Sylow
247C25C2×Dic4711yescharacteristic
248C25C2×Dic4711yescharacteristic
249C25C22×D4711yescharacteristic
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—44no—
258C22×D4—44no—
259C22×D4—44no—
260C22×D4C22⋊D4714yes—
261C22×D4C22⋊D4714yes—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—44no—
266C23×C4C2×Dic4711yescharacteristic
267C23×C4C2×Dic4711yescharacteristic
268C23×C4C2×Dic4711yescharacteristic
269C23×C4C2×Dic4711yescharacteristic
270#121—232no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—944no—
280C2×C22⋊C4—944no—
281C2×C22⋊C4—944no—
282C2×C22⋊C4C22⋊D4714yes—
283#99—1888no—
284#99—1888no—
285C24—24no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C22⋊Dic4712yes—
299C24C22⋊Dic4712yes—
300C24C22×Dic4711yescharacteristic
301C24C2×C22⋊D4712yes—
302C24C2×C22⋊D4712yes—
303C24C2×C22⋊D4712yes—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—28no—
307C2×D4—28no—
308C2×D4—28no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—48no—
318C2×D4—44no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—1884no—
337C22×C4C22⋊Dic4712yes—
338C22×C4C22⋊Dic4712yes—
339C22×C4C22⋊Dic4712yes—
340C22×C4C22⋊Dic4712yes—
341C4⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—1888no—
345C22⋊C4—1888no—
346C22⋊C4—1888no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—24no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—416no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—88no—
381C23—88no—
382C23#1309393914yescomplemented
383C23C2×C22⋊Dic4711yescharacteristic
384C23C2×C22⋊Dic4711yescharacteristic
385C23C2×C22⋊Dic4711yescharacteristic
386C23#1309399314yes—
387D4—416no—
388D4—416no—
389D4—416no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—416no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—3764no—
408C2×C4—3764no—
409C2×C4—3764no—
410C2×C4#1309393914yescomplemented
411C22—28no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—44no—
423C22—48no—
424C22—44no—
425C22—48no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—84no—
432C22—84no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22#1568897911yescharacteristic
438C22C2×#1309393912yes—
439C22C2×#1309393912yes—
440C22#1568909712yes—
441C4—24no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—7524no—
446C2—22no—
447C2—22no—
448C2—22no—
449C2—44no—
450C2—44no—
451C2—82no—
452C2#2024330311yescharacteristic
453C2#2024330311yescharacteristic
454C2#2024330611yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC24×C47C22×C4
FrattiniC25C22×D47
Fitting, prime radicalC22×#121×C47C2
socleC22×C47#379
cosocleC25×C47C23
hypercenterC22×#121D47
hypocenterC47#8542
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. #20243462C2semidirect
2. #20243303C2semidirect
3. #20243303C2semidirect
9. C23×D4×C47C4semidirect
12. C24×C4×C47C4semidirect
16. C22×C4⋊C4×C47C4semidirect
17. C22×C22⋊C4×C47C4semidirect
154. C23×C47#99semidirect
179. C2×C4×C47#99semidirect
232. C23×D4Dic47semidirect
235. C24×C4Dic47semidirect
239. C22×C4⋊C4Dic47semidirect
240. C22×C22⋊C4Dic47semidirect
246. C47#8542semidirect
382. C23#13093939semidirect
410. C2×C4#13093939semidirect

Maximal subgroups

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

subgroupindex
#202434622
#202433032
C22×#121×C472
#854247