Finite Group Databasebeta

← #26190193

Subgroups — #26190193

Order 12032.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 187 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×#15689754C211yesmaximal, characteristic, complemented
2#20254145C212yesmaximal
3#20242395C214yesmaximal, complemented
4#20242327C214yesmaximal, complemented
5#20241959C212yesmaximal, complemented
6C2×#15688849C211yesmaximal, characteristic, complemented
7C47×#2618C211yesmaximal, characteristic
8#15689754—24no—
9#15689754—22no—
10#15689754C2214yes—
11C2×#13094023—22no—
12C2×#13094015—22no—
13C2×#13094003—21no—
14C2×#13094003C2212yes—
15C2×#13093983—21no—
16#15689453C2214yes—
17#15689439—22no—
18#15689350—24no—
19#15689274C2214yes—
20#15689261—22no—
21C2×C22⋊(C47⋊C8)C2212yes—
22#15688859C2218yes—
23#15688849C2214yes—
24C2×#123×C47—21no—
25C2×#121×C47C2211yescharacteristic
26C22×C4⋊C4×C47C2211yescharacteristic
27C47×#398C2212yes—
28C47×#396—22no—
29C47×#394C2212yes—
30C47×#386—21no—
31C47×#380C2211yescharacteristic
32C22×Dic94—21no—
33C22×Dic94D412yes—
34C2×#10625335—24no—
35C22×C47⋊C8—22no—
36#13094023—28no—
37#13094023—28no—
38#13094015—28no—
39#13094015—28no—
40#13094003—24no—
41#13094003—22no—
42#13094003—22no—
43#13094003—24no—
44#13094003D414yes—
45#13094003C2314yes—
46#13093983—24no—
47#13093983—24no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47D412yes—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—22no—
54C2×C22⋊Dic47—22no—
55C2×C4×Dic47—22no—
56#13093955—24no—
57#13093947—24no—
58#13093945—24no—
59C22⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)C2314yes—
61C22×D4×C47—21no—
62C22×D4×C47C2311yescharacteristic
63C23×C4×C47D411yescharacteristic
64C23×C4×C47D411yescharacteristic
65C23×C4×C47C2311yescharacteristic
66#123×C47—44no—
67#121×C47—22no—
68#121×C47—22no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47—22no—
71C2×C4⋊C4×C47—22no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47D412yes—
74C2×C4⋊C4×C47C2311yescharacteristic
75C2×C4⋊C4×C47C2312yes—
76C2×C22⋊C4×C47—21no—
77C2×C22⋊C4×C47—21no—
78C2×C22⋊C4×C47—22no—
79C2×C22⋊C4×C47—22no—
80#95×C47—22no—
81#95×C47C2312yes—
82C24×C47—21no—
83C24×C47C2×D411yescharacteristic
84C2×D4×C47—22no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—42no—
88C2×D4×C47—42no—
89C2×D4×C47—81no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47—81no—
98C22×C4×C47—81no—
99C22×C4×C47SD1614yescomplemented
100C22×C4×C47C2×D412yes—
101C22×C4×C47C2×D412yes—
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D412yes—
104C22×C4×C47C2×D412yes—
105C22×C4×C47C2411yescharacteristic
106C22×Dic47—22no—
107C22×Dic47—22no—
108C2×Dic94—24no—
109C2×Dic94—22no—
110C2×Dic94—82no—
111C2×Dic94—82no—
112C2×Dic94C2×D414yes—
113C4⋊C4×C47—24no—
114C4⋊C4×C47—24no—
115C4⋊C4×C47—22no—
116C4⋊C4×C47—44no—
117C4⋊C4×C47—81no—
118C22⋊C4×C47—42no—
119C22⋊C4×C47—42no—
120C22⋊C4×C47—44no—
121C22⋊C4×C47—82no—
122C22⋊Dic47—24no—
123C22⋊Dic47—24no—
124#10625335—84no—
125C4⋊Dic47—24no—
126C4⋊Dic47—24no—
127C4⋊Dic47—24no—
128C4⋊Dic47—22no—
129C4⋊Dic47C2×D414yes—
130Dic47⋊C4—28no—
131Dic47⋊C4—24no—
132Dic47⋊C4—24no—
133C4×Dic47—28no—
134C2×C47⋊C8—24no—
135C2×C47⋊C8—82no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47—42no—
139C23×C47—42no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47#12011yescharacteristic
144C23×C47#12011yescharacteristic
145C23×C47C2×SD1612yes—
146C23×C47#13712yes—
147C23×C47C22×D411yescharacteristic
148D4×C47—44no—
149D4×C47—82no—
150D4×C47—82no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—24no—
154C2×C4×C47—24no—
155C2×C4×C47—22no—
156C2×C4×C47—42no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47—41no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—81no—
169C2×C4×C47#12012yes—
170C2×C4×C47C2×SD1614yes—
171C2×C4×C47C22×D412yes—
172C2×Dic47—24no—
173C2×Dic47—24no—
174C2×Dic47—44no—
175C2×Dic47—44no—
176Dic94—44no—
177Dic94—82no—
178Dic94—84no—
179C47⋊C8—44no—
180#18294—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—42no—
188C22×C47—42no—
189C22×C47—82no—
190C22×C47—82no—
191C22×C47—81no—
192C22×C47—81no—
193C22×C47—81no—
194C22×C47#44812yes—
195C22×C47#44812yes—
196C22×C47C2×#12011yescharacteristic
197C22×C47C22×SD1611yescharacteristic
198C22×C47C2×#13711yescharacteristic
199C4×C47—24no—
200C4×C47—22no—
201C4×C47—22no—
202C4×C47—42no—
203C4×C47—82no—
204C4×C47—81no—
205Dic47—44no—
206Dic47—44no—
207C2×#544—471no—
208#3187—472no—
209#2618D4711yescharacteristic
210#1611—474no—
211#1599—474no—
212#1520—472no—
213C2×#324—471no—
214C2×C47—21no—
215C2×C47—21no—
216C2×C47—22no—
217C2×C47—22no—
218C2×C47—161no—
219C2×C47C2×#44811yescharacteristic
220C2×C47#302011yescharacteristic
221C2×C47#302611yescharacteristic
222#544—474no—
223#544—944no—
224#544—942no—
225C2×C4⋊Q8—942no—
226C2×#125—942no—
227C2×#123—21no—
228C2×C22⋊Q8—472no—
229C2×C22⋊Q8—941no—
230C2×#121C2×D4711yescharacteristic
231C2×#117—941no—
232C22×C4⋊C4C2×D4711yescharacteristic
233#483—474no—
234#481—942no—
235#465—944no—
236#450—474no—
237#448—942no—
238C2×C22⋊C8—472no—
239#398C2×D4712yes—
240#396—22no—
241#394C2×D4712yes—
242#386—21no—
243#380C2×D4711yescharacteristic
244#328—478no—
245#324—474no—
246C47#1829411yescharacteristic, complemented, Sylow
247C22×Q8—472no—
248C22×Q8—941no—
249C22×D4—21no—
250C22×D4C22×D4711yescharacteristic
251C23×C4C22⋊D4711yescharacteristic
252C23×C4C22⋊D4711yescharacteristic
253C23×C4C22×D4711yescharacteristic
254C2×SD16—944no—
255C22×C8—942no—
256C4⋊Q8—948no—
257C4⋊Q8—948no—
258#125—948no—
259#125—948no—
260#123—44no—
261C22⋊Q8—474no—
262C22⋊Q8—474no—
263C22⋊Q8—944no—
264C22⋊Q8—942no—
265C22⋊Q8—942no—
266C22⋊Q8—944no—
267#121—22no—
268#121—22no—
269#117—944no—
270#117—944no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—472no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4—941no—
278C2×C4⋊C4—941no—
279C2×C4⋊C4—942no—
280C2×C4⋊C4—942no—
281C2×C4⋊C4C22⋊D4712yes—
282C2×C4⋊C4C22×D4711yescharacteristic
283C2×C4⋊C4C22×D4712yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—942no—
289C2×C42—942no—
290#106—944no—
291#103—944no—
292#102—944no—
293C22⋊C8—474no—
294C22⋊C8—942no—
295#95—22no—
296#95C22×D4712yes—
297C24—21no—
298C24C2×C22⋊D4711yescharacteristic
299C2×Q8—474no—
300C2×Q8—944no—
301C2×Q8—942no—
302C2×Q8—3762no—
303C2×Q8—3762no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—942no—
320C22×C4—942no—
321C22×C4#1062533514yescomplemented
322C22×C4D4×D4712yes—
323C22×C4D4×D4712yes—
324C22×C4C2×C22⋊D4712yes—
325C22×C4C2×C22⋊D4712yes—
326C22×C4C2×C22⋊D4711yescharacteristic
327C22×C4C23×D4711yescharacteristic
328SD16—3764no—
329C2×C8—944no—
330C2×C8—3762no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—81no—
336C4⋊C4—474no—
337C4⋊C4—948no—
338C4⋊C4—944no—
339C4⋊C4—944no—
340C4⋊C4—944no—
341C4⋊C4—944no—
342C4⋊C4—942no—
343C4⋊C4—944no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—944no—
349C22⋊C4—944no—
350C42—948no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#1309399211yescharacteristic
359C23#1309399211yescharacteristic
360C23C2×#1062533512yes—
361C23#1309405412yes—
362C23C22×C22⋊D4711yescharacteristic
363Q8—3762no—
364Q8—3764no—
365Q8—1884no—
366D4—44no—
367D4—82no—
368D4—82no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—944no—
388C2×C4—944no—
389C2×C4—1884no—
390C2×C4—1884no—
391C2×C4#1309399212yes—
392C2×C4C2×#1062533514yes—
393C2×C4C2×D4×D4712yes—
394C8—1884no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#1568926112yes—
409C22#1568926112yes—
410C22C2×#1309399211yescharacteristic
411C22C22×#1062533511yescharacteristic
412C22C2×#1309405411yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—1884no—
420C4—1884no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×#1568926111yescharacteristic
427C2#2025275111yescharacteristic
428C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2618C2
socleC22×C47C2×#120
prime radicalC2×#121×C47C22
hypercenter#2618D47
hypocenterC47#18294
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#15689754C2semidirect
3. #20242395C2semidirect
4. #20242327C2semidirect
5. #20241959C2semidirect
6. C2×#15688849C2semidirect
99. C22×C4×C47SD16semidirect
246. C47#18294semidirect
321. C22×C4#10625335semidirect

Maximal subgroups

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

subgroupindex
C2×#156897542
#202541452
#202423952
#202423272
#202419592
C2×#156888492
C47×#26182
#1829447