Finite Group Databasebeta

← #26104732

Subgroups — #26104732

Order 12032.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#20251729C211yesmaximal, characteristic
2#20243080C211yesmaximal, characteristic
3#336×D47C211yesmaximal, characteristic
4#20241214C211yesmaximal, characteristic, complemented
5#20241212C211yesmaximal, characteristic, complemented
6#20241210C211yesmaximal, characteristic, complemented
7C47×#1272C211yesmaximal, characteristic, complemented
8C22×#10625324—21no—
9C22×C8×D47—21no—
10C2×#13094000C411yescharacteristic
11C2×C42×D47C2211yescharacteristic
12#15689030C411yescharacteristic
13#15689022C411yescharacteristic
14#15689021C2211yescharacteristic
15#15689006—21no—
16#15689005—21no—
17#15688997—21no—
18#15688985—21no—
19#15688980C411yescharacteristic
20#15688878C2211yescharacteristic
21#15688878C2211yescharacteristic
22#15688877—21no—
23#15688877—21no—
24C47×#391C2211yescharacteristic
25C47×#336C2211yescharacteristic
26C47×#336C2211yescharacteristic
27C22×C4×D47C2×C411yescharacteristic
28C22×C4×D47D411yescharacteristic
29C22×C4×D47D411yescharacteristic
30C2×#10625324—24no—
31C2×#10625324—42no—
32C2×#10625324—42no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35C2×C8×D47—24no—
36C2×C8×D47—42no—
37C2×C8×D47—42no—
38#13094000—42no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×C4⋊Dic47C2×C411yescharacteristic
42C2×Dic47⋊C4—21no—
43C2×Dic47⋊C4—21no—
44C2×(C2×D47)⋊C4—21no—
45C2×(C2×D47)⋊C4—21no—
46C2×C4×Dic47C2×C411yescharacteristic
47C2×C4×Dic47D411yescharacteristic
48C2×C4×Dic47D411yescharacteristic
49C42×D47—42no—
50C47⋊#95—21no—
51C47⋊#95—21no—
52C47⋊#95—21no—
53C47⋊#95—21no—
54C47⋊#95—21no—
55C22×Q8×C47C2×C411yescharacteristic
56C22×C8×C47—21no—
57C22×C8×C47—21no—
58C2×C4⋊C4×C47C2×C411yescharacteristic
59C2×C42×C47C2311yescharacteristic
60#95×C47—21no—
61#95×C47—21no—
62C23×D47C22⋊C411yescharacteristic
63C2×Q8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47C22⋊C411yescharacteristic
75C22×Dic47C22⋊C411yescharacteristic
76C22×Dic47C22⋊C411yescharacteristic
77C2×C4×D47—22no—
78C2×C4×D47—22no—
79C2×C4×D47—22no—
80C2×C4×D47—21no—
81C2×C4×D47—21no—
82C2×C4×D47—22no—
83C2×C4×D47—22no—
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86C2×C4×D47—41no—
87C2×C4×D47SD1612yes—
88C2×C4×D47Q1612yes—
89C2×C8×C47—22no—
90C2×C8×C47—22no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C2×C8×C47—41no—
94C2×C8×C47—41no—
95C4⋊C4×C47—41no—
96C42×C47—41no—
97(C2×D47)⋊C4—42no—
98(C2×D47)⋊C4—42no—
99C4⋊Dic47—21no—
100C4⋊Dic47—21no—
101C4⋊Dic47—42no—
102C4⋊Dic47—41no—
103Dic47⋊C4—42no—
104Dic47⋊C4—41no—
105Dic47⋊C4—41no—
106C4×Dic47—41no—
107C4×Dic47—41no—
108C4×Dic47—41no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115#10625324—48no—
116C8×D47—48no—
117C23×C47C2×C22⋊C411yescharacteristic
118C22×D47—21no—
119C22×D47—21no—
120C22×D47—21no—
121C22×D47—21no—
122C22×D47#9912yes—
123C22×D47#10312yes—
124C22×D47#10412yes—
125Q8×C47—81no—
126Q8×C47—81no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47#13111yescharacteristic
140C2×C4×C47#13111yescharacteristic
141C2×C4×C47C2×SD1611yescharacteristic
142C2×C4×C47C2×Q1611yescharacteristic
143C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×Dic47—21no—
146C2×Dic47—21no—
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—21no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47#10312yes—
159C2×Dic47#10412yes—
160C4×D47—24no—
161C4×D47—24no—
162C4×D47—24no—
163C4×D47—24no—
164C4×D47—42no—
165C4×D47—42no—
166C8×C47—42no—
167C8×C47—42no—
168C47⋊C8—42no—
169C47⋊C8—42no—
170#7681—471nomaximal, Sylow
171C22×C47#40811yescharacteristic
172C22×C47C2×#9911yescharacteristic
173C22×C47#41311yescharacteristic
174C22×C47C2×#10311yescharacteristic
175C22×C47#41711yescharacteristic
176C22×C47C2×#10411yescharacteristic
177C22×C47C2×#10411yescharacteristic
178C2×D47—22no—
179C2×D47—22no—
180C2×D47—22no—
181C2×D47—22no—
182C2×D47—22no—
183C2×D47—22no—
184C2×D47#32814yes—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—81no—
192C4×C47—81no—
193Dic47—22no—
194Dic47—22no—
195Dic47—41no—
196Dic47—81no—
197Dic47—81no—
198#2855—471no—
199#1728—471no—
200C2×#336—471no—
201#1272—471no—
202#1272D4711yescharacteristic, complemented
203#1271—471no—
204#1271—471no—
205C2×C47#146811yescharacteristic
206C2×C47#148311yescharacteristic
207C2×C47#149311yescharacteristic
208C2×C47C2×#32811yescharacteristic
209C2×C47#150411yescharacteristic
210C2×C47#152411yescharacteristic
211C2×C47#155011yescharacteristic
212D47—24no—
213C22×M16—941no—
214C23×C8—941no—
215C2×C22⋊Q8—471no—
216C22×C42—471no—
217#391—471no—
218#391C2×D4711yescharacteristic
219#389—471no—
220#389—471no—
221#387—941no—
222#387—941no—
223#385—941no—
224#381—941no—
225#380—471no—
226#336—471no—
227#336—471no—
228#336—941no—
229#336—941no—
230#336C2×D4711yescharacteristic
231#336C2×D4711yescharacteristic
232C47#768111yescharacteristic, complemented, Sylow
233C22×Q8C4×D4711yescharacteristic
234C23×C4—471no—
235C23×C4—471no—
236C23×C4—471no—
237C2×M16—944no—
238C2×M16—1882no—
239C2×M16—1882no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—944no—
243C22×C8—941no—
244C22×C8—941no—
245C22×C8—1882no—
246C22×C8—1882no—
247C22⋊Q8—1882no—
248C2×C4⋊C4—471no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4C4×D4711yescharacteristic
254C2×C22⋊C4—941no—
255C2×C22⋊C4—941no—
256C2×C42—471no—
257C2×C42—471no—
258C2×C42—471no—
259C2×C42—1882no—
260C2×C42C22×D4711yescharacteristic
261#95—21no—
262#95—21no—
263#95—941no—
264#95—941no—
265#95—941no—
266#95—941no—
267#95—941no—
268C24—471no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—41no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—471no—
276C22×C4—471no—
277C22×C4—472no—
278C22×C4—472no—
279C22×C4—471no—
280C22×C4—942no—
281C22×C4—942no—
282C22×C4—942no—
283C22×C4—942no—
284C22×C4—942no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—1881no—
292C22×C4C2×C4×D4711yescharacteristic
293C22×C4D4×D4711yescharacteristic
294C22×C4D4×D4711yescharacteristic
295M16—1888no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—942no—
303C2×C8—942no—
304C2×C8—1888no—
305C2×C8—1881no—
306C2×C8—1881no—
307C2×C8—1881no—
308C2×C8—1881no—
309C4⋊C4—41no—
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—1881no—
313C4⋊C4—1882no—
314C4⋊C4—1882no—
315C4⋊C4—1881no—
316C4⋊C4—1881no—
317C22⋊C4—1882no—
318C22⋊C4—1882no—
319C42—41no—
320C42—1881no—
321C42—1881no—
322C42—1881no—
323C23—472no—
324C23—472no—
325C23—472no—
326C23—941no—
327C23—941no—
328C23—941no—
329C23—941no—
330C23C22⋊C4×D4711yescharacteristic
331Q8—81no—
332Q8—81no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—472no—
346C2×C4—472no—
347C2×C4—944no—
348C2×C4—944no—
349C2×C4—944no—
350C2×C4—944no—
351C2×C4—941no—
352C2×C4—941no—
353C2×C4—941no—
354C2×C4—941no—
355C2×C4—941no—
356C2×C4—941no—
357C2×C4—941no—
358C2×C4—1882no—
359C2×C4—1882no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4—1881no—
363C2×C4—1881no—
364C2×C4—1881no—
365C2×C4—1881no—
366C2×C4#1309403011yescharacteristic
367C2×C4#1309403211yescharacteristic
368C2×C4SD16×D4711yescharacteristic
369C2×C4Q16×D4711yescharacteristic
370C2×C4#1309405111yescharacteristic
371C2×C4#1309405611yescharacteristic
372C8—42no—
373C8—42no—
374C8—1882no—
375C8—1882no—
376C22—474no—
377C22—942no—
378C22—942no—
379C22—942no—
380C22—942no—
381C22—942no—
382C22—942no—
383C22#1568908811yescharacteristic
384C22#99×D4711yescharacteristic
385C22#1568910411yescharacteristic
386C22#103×D4711yescharacteristic
387C22#1568912311yescharacteristic
388C22#104×D4711yescharacteristic
389C22#104×D4711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—942no—
399C4—942no—
400C4—3761no—
401C4—3761no—
402C4—1881no—
403C2—944no—
404C2#2024173311yescharacteristic
405C2#2024180011yescharacteristic
406C2#2024184111yescharacteristic
407C2#328×D4711yescharacteristic
408C2#2024189111yescharacteristic
409C2#2024197311yescharacteristic
410C2#2024207511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7681
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #20241214C2semidirect
5. #20241212C2semidirect
6. #20241210C2semidirect
7. C47×#1272C2semidirect
202. #1272D47semidirect
232. C47#7681semidirect

Maximal subgroups

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

subgroupindex
#202517292
#202430802
#336×D472
#202412142
#202412122
#202412102
C47×#12722
#768147