Finite Group Databasebeta

← #26188693

Subgroups — #26188693

Order 12032.

1054 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 167 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×#15689749C211yesmaximal, characteristic
2#20253336C212yesmaximal, complemented
3#20242412C214yesmaximal, complemented
4#20242378C214yesmaximal, complemented
5#20241920C212yesmaximal, complemented
6#20241715C211yesmaximal, characteristic
7C47×#2482C211yesmaximal, characteristic
8#15689749—24no—
9#15689749—22no—
10#15689749C2214yes—
11C2×#13094003—21no—
12C2×#13094003C2212yes—
13C2×#13094001—22no—
14C2×#13094001—22no—
15C22×C22⋊Dic47—21no—
16#15689464—24no—
17#15689458C2214yescomplemented
18#15689453C2214yescomplemented
19#15689166—22no—
20#15689131—22no—
21#15689087C2212yes—
22#15688859C2218yescomplemented
23#15688847C2214yes—
24C2×#117×C47C2211yescharacteristic
25C22×C22⋊C4×C47C2211yescharacteristic
26C47×#399C2212yes—
27C47×#397C2212yes—
28C47×#388—22no—
29C47×#385—21no—
30C47×#381C2211yescharacteristic
31C47×#379—21no—
32C22×Dic94—21no—
33C22×Dic94D412yes—
34C23×Dic47—21no—
35C23×Dic47—21no—
36C2×#10625329—22no—
37#13094003—24no—
38#13094003—22no—
39#13094003—22no—
40#13094003—24no—
41#13094003D414yes—
42#13094003C2314yes—
43#13094001—28no—
44#13094001—28no—
45#13094001—28no—
46#13094001—28no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47D412yes—
49C2×Dic47⋊C4—22no—
50C2×Dic47⋊C4—22no—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47—22no—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47—22no—
55#13093957—24no—
56#13093955—24no—
57#13093947—24no—
58#13093947—24no—
59C22⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)C2314yes—
61C25×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47C2311yescharacteristic
64#117×C47—22no—
65#117×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47C2311yescharacteristic
68C2×C4⋊C4×C47C2311yescharacteristic
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—22no—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47D412yes—
76C2×C22⋊C4×C47C2312yes—
77C2×C42×C47—21no—
78#95×C47—21no—
79#95×C47—22no—
80#95×C47C2312yes—
81C24×C47—21no—
82C24×C47—21no—
83C24×C47—24no—
84C24×C47—42no—
85C24×C47—81no—
86C24×C47C2×D411yescharacteristic
87C24×C47C2×D412yes—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—42no—
94C22×C4×C47—81no—
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D412yes—
99C22×C4×C47C2411yescharacteristic
100C22×Dic47—22no—
101C22×Dic47—22no—
102C22×Dic47—24no—
103C22×Dic47—24no—
104C22×Dic47—82no—
105C22×Dic47—82no—
106C2×Dic94—24no—
107C2×Dic94—22no—
108C2×Dic94—44no—
109C2×Dic94—82no—
110C2×Dic94C2×D414yes—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—82no—
114C22⋊C4×C47—24no—
115C22⋊C4×C47—24no—
116C22⋊C4×C47—42no—
117C22⋊C4×C47—41no—
118C22⋊C4×C47—41no—
119C22⋊C4×C47—44no—
120C22⋊C4×C47—82no—
121C42×C47—42no—
122C22⋊Dic47—28no—
123C22⋊Dic47—24no—
124C22⋊Dic47—24no—
125C4⋊Dic47—24no—
126C4⋊Dic47—22no—
127C4⋊Dic47C2×D414yes—
128Dic47⋊C4—28no—
129Dic47⋊C4—24no—
130Dic47⋊C4—24no—
131#10625329—44no—
132C2×C47⋊C8—24no—
133C23×C47—24no—
134C23×C47—24no—
135C23×C47—22no—
136C23×C47—22no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—41no—
140C23×C47—42no—
141C23×C47—42no—
142C23×C47—42no—
143C23×C47—42no—
144C23×C47—81no—
145C23×C47—81no—
146C23×C47—88no—
147C23×C47#12011yescharacteristic
148C23×C47#12011yescharacteristic
149C23×C47#13714yes—
150C23×C47C22×D411yescharacteristic
151C2×C4×C47—24no—
152C2×C4×C47—22no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—42no—
159C2×C4×C47—42no—
160C2×C4×C47—42no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47—81no—
166C2×C4×C47#12012yes—
167C2×C4×C47C22×D412yes—
168C2×Dic47—24no—
169C2×Dic47—24no—
170C2×Dic47—44no—
171C2×Dic47—44no—
172C2×Dic47—84no—
173C2×Dic47—84no—
174Dic94—44no—
175Dic94—82no—
176Dic94—84no—
177C47⋊C8—44no—
178#18138—471nomaximal, Sylow
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—24no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—44no—
188C22×C47—44no—
189C22×C47—48no—
190C22×C47—42no—
191C22×C47—42no—
192C22×C47—82no—
193C22×C47—84no—
194C22×C47#45812yes—
195C22×C47C2×#12011yescharacteristic
196C22×C47C2×#13712yes—
197C4×C47—22no—
198C4×C47—22no—
199C4×C47—42no—
200C4×C47—82no—
201C4×C47—81no—
202C4×C47—81no—
203Dic47—44no—
204Dic47—44no—
205C2×#543—471no—
206#3095—472no—
207#2482D4711yescharacteristic
208#1614—474no—
209#1608—474no—
210#1511—472no—
211#1462—471no—
212C2×C47—22no—
213C2×C47—22no—
214C2×C47—41no—
215C2×C47—44no—
216C2×C47#300411yescharacteristic
217C2×C47#301911yescharacteristic
218C2×C47C2×#45811yescharacteristic
219#543—474no—
220#543—944no—
221#543—942no—
222C2×C22⋊Q8—472no—
223C2×C22⋊Q8—942no—
224C2×C22⋊Q8—942no—
225C2×C22⋊Q8—941no—
226C2×#117C2×D4711yescharacteristic
227C22×C22⋊C4—941no—
228C22×C22⋊C4C2×D4711yescharacteristic
229#485—944no—
230#484—474no—
231#483—474no—
232#428—942no—
233#419—942no—
234#407—472no—
235#399C2×D4712yes—
236#397C2×D4712yes—
237#388—22no—
238#385—21no—
239#381C2×D4711yescharacteristic
240#379—21no—
241#328—478no—
242#323—474no—
243C47#1813811yescharacteristic, complemented, Sylow
244C25C22⋊D4711yescharacteristic
245C22×Q8—472no—
246C22×Q8—941no—
247C23×C4—941no—
248C23×C4—941no—
249C23×C4C22⋊D4711yescharacteristic
250C23×C4C22×D4711yescharacteristic
251C2×M16—942no—
252C22⋊Q8—474no—
253C22⋊Q8—474no—
254C22⋊Q8—948no—
255C22⋊Q8—948no—
256C22⋊Q8—948no—
257C22⋊Q8—948no—
258C22⋊Q8—944no—
259C22⋊Q8—942no—
260C22⋊Q8—942no—
261C22⋊Q8—944no—
262#117—22no—
263#117—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—472no—
266C2×C4⋊C4—942no—
267C2×C4⋊C4—941no—
268C2×C4⋊C4—942no—
269C2×C4⋊C4C22×D4711yescharacteristic
270C2×C4⋊C4C22×D4711yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—942no—
278C2×C22⋊C4—944no—
279C2×C22⋊C4—942no—
280C2×C22⋊C4—944no—
281C2×C22⋊C4C22⋊D4712yes—
282C2×C22⋊C4C22×D4712yes—
283C2×C42—21no—
284#107—944no—
285#106—944no—
286#103—944no—
287#103—944no—
288C22⋊C8—474no—
289C22⋊C8—942no—
290#95—21no—
291#95—22no—
292#95C22×D4712yes—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—81no—
298C24C2×C22⋊D4711yescharacteristic
299C24C2×C22⋊D4712yes—
300C2×Q8—474no—
301C2×Q8—944no—
302C2×Q8—942no—
303C2×Q8—3762no—
304C2×Q8—1884no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—81no—
312C22×C4—944no—
313C22×C4—944no—
314C22×C4—942no—
315C22×C4—942no—
316C22×C4—3762no—
317C22×C4—3762no—
318C22×C4D4×D4712yes—
319C22×C4D4×D4712yes—
320C22×C4C2×C22⋊D4712yes—
321C22×C4C2×C22⋊D4711yescharacteristic
322C22×C4C23×D4711yescharacteristic
323M16—1884no—
324C2×C8—944no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—82no—
328C4⋊C4—474no—
329C4⋊C4—948no—
330C4⋊C4—944no—
331C4⋊C4—944no—
332C4⋊C4—942no—
333C4⋊C4—944no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—948no—
342C22⋊C4—944no—
343C22⋊C4—944no—
344C42—42no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—88no—
359C23#1309399211yescharacteristic
360C23#1309399211yescharacteristic
361C23#1309405414yes—
362C23C22×C22⋊D4711yescharacteristic
363Q8—3762no—
364Q8—3764no—
365Q8—1884no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—944no—
382C2×C4—944no—
383C2×C4—3764no—
384C2×C4—3764no—
385C2×C4—1884no—
386C2×C4—1884no—
387C2×C4#1309399212yes—
388C2×C4C2×D4×D4712yes—
389C8—1884no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1568931212yes—
406C22C2×#1309399211yescharacteristic
407C22C2×#1309405412yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—1884no—
415C4—1884no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#2025262111yescharacteristic
421C2#2025274411yescharacteristic
422C2C2×#1568931211yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2482C2
socleC22×C47C2×#120
prime radicalC25×C47D4
hypercenter#2482D47
hypocenterC47#18138
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #20253336C2semidirect
3. #20242412C2semidirect
4. #20242378C2semidirect
5. #20241920C2semidirect
17. #15689458C22semidirect
18. #15689453C22semidirect
22. #15688859C22semidirect
243. C47#18138semidirect

Maximal subgroups

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

subgroupindex
C2×#156897492
#202533362
#202424122
#202423782
#202419202
#202417152
C47×#24822
#1813847