Finite Group Databasebeta

← #26113299

Subgroups — #26113299

Order 12032.

534 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 99 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#20254194C211yesmaximal, characteristic
2#20244372C211yesmaximal, characteristic, complemented
3#20241920C211yesmaximal, characteristic, complemented
4#20241904C211yesmaximal, characteristic
5#20241294C211yesmaximal, characteristic, complemented
6#20241240C211yesmaximal, characteristic, complemented
7C47×#2377C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689458—21no—
10#15689434C412yescomplemented
11#15689367—22no—
12#15689274—21no—
13#15689261C412yescomplemented
14#15689087C2211yescharacteristic
15#15689087C2211yescharacteristic
16C4×C22⋊Dic47C2211yescharacteristic
17#15688889—21no—
18#15688889—21no—
19#15688859—21no—
20#15688859—21no—
21#15688857—22no—
22C2×#121×C47C2211yescharacteristic
23C47×#390—21no—
24C47×#388—21no—
25C47×#387—21no—
26C47×#386—21no—
27C47×#386—21no—
28C47×#385—21no—
29C47×#381C2211yescharacteristic
30C47×#380C2211yescharacteristic
31C22×Dic94C2×C411yescharacteristic, complemented
32C2×#10625335—22no—
33C2×#10625329—21no—
34C2×#10625329—21no—
35#13094003—21no—
36#13094003—21no—
37#13094003—21no—
38#13094003—21no—
39C2×C4⋊Dic47C2×C411yescharacteristic, complemented
40C2×Dic47⋊C4—21no—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47—21no—
43C2×C4×Dic47—21no—
44#13093957—22no—
45#13093947—22no—
46#13093945—22no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)C2×C412yescomplemented
51C47⋊#95—21no—
52C22×D4×C47—21no—
53C22×D4×C47D411yescharacteristic
54C23×C4×C47—21no—
55C23×C4×C47C2311yescharacteristic
56#121×C47—21no—
57#121×C47—21no—
58#121×C47C2×C412yes—
59#121×C47D412yes—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47D411yescharacteristic
69C2×C22⋊C4×C47D411yescharacteristic
70C2×C42×C47—21no—
71#95×C47—21no—
72#95×C47—21no—
73#95×C47—21no—
74#95×C47—21no—
75#95×C47D411yescharacteristic
76#95×C47D411yescharacteristic
77C24×C47—21no—
78C24×C47C2×D411yescharacteristic
79C2×D4×C47—22no—
80C2×D4×C47—22no—
81C2×D4×C47—21no—
82C2×D4×C47—81no—
83C2×D4×C47—81no—
84C2×D4×C47—81no—
85C2×D4×C47C22⋊C412yes—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—81no—
92C22×C4×C47—81no—
93C22×C4×C47—81no—
94C22×C4×C47—81no—
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47#4111yescharacteristic
100C22×C4×C47#4111yescharacteristic
101C22×Dic47—21no—
102C22×Dic47—21no—
103C22×Dic47—41no—
104C2×Dic94—21no—
105C2×Dic94—21no—
106C2×Dic94—81no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—81no—
109C4⋊C4×C47—81no—
110C4⋊C4×C47C22⋊C412yes—
111C22⋊C4×C47—22no—
112C22⋊C4×C47—22no—
113C22⋊C4×C47—41no—
114C22⋊C4×C47—41no—
115C22⋊C4×C47—81no—
116C22⋊C4×C47—81no—
117C22⋊C4×C47—81no—
118C22⋊C4×C47—81no—
119C42×C47—81no—
120C22⋊Dic47—22no—
121C22⋊Dic47—41no—
122C22⋊Dic47—41no—
123C22⋊Dic47—41no—
124#10625335—82no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127Dic47⋊C4—41no—
128Dic47⋊C4—41no—
129C4×Dic47—81no—
130#10625329—81no—
131#10625329—81no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—41no—
134C23×C47—21no—
135C23×C47—21no—
136C23×C47—22no—
137C23×C47—41no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47C4×D411yescharacteristic
146C23×C47#12011yescharacteristic
147C23×C47#12111yescharacteristic
148D4×C47—42no—
149D4×C47—81no—
150D4×C47—82no—
151D4×C47—82no—
152C2×C4×C47—22no—
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—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—84no—
174C2×C4×C47C2×C22⋊C411yescharacteristic
175C2×C4×C47C4×D411yescharacteristic
176C2×C4×C47#12111yescharacteristic
177C2×C4×C47#12311yescharacteristic
178C2×Dic47—42no—
179C2×Dic47—41no—
180C2×Dic47—41no—
181C2×Dic47—81no—
182C2×Dic47—81no—
183C2×Dic47—81no—
184C2×Dic47—81no—
185Dic94—42no—
186Dic94—81no—
187C47⋊C8—42no—
188C47⋊C8—81no—
189#9341—471nomaximal, Sylow
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—42no—
199C22×C47—42no—
200C22×C47—42no—
201C22×C47—81no—
202C22×C47—81no—
203C22×C47—81no—
204C22×C47—81no—
205C22×C47—81no—
206C22×C47—81no—
207C22×C47#38611yescharacteristic
208C22×C47#45511yescharacteristic
209C22×C47#45611yescharacteristic
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—21no—
213C4×C47—21no—
214C4×C47—42no—
215C4×C47—81no—
216C4×C47—81no—
217C4×C47—81no—
218C4×C47—81no—
219Dic47—82no—
220Dic47—81no—
221Dic47—81no—
222#3193—471no—
223#2377D4711yescharacteristic
224#1996—471no—
225#1511—471no—
226#1507—471no—
227#1305—471no—
228#1284—471no—
229C2×C47—21no—
230C2×C47—21no—
231C2×C47—41no—
232C2×C47—41no—
233C2×C47—82no—
234C2×C47#188211yescharacteristic
235C2×C47#190411yescharacteristic
236C2×C47#191311yescharacteristic
237C2×C22⋊Q8—471no—
238C2×#121C2×D4711yescharacteristic
239#484—941no—
240#480—472no—
241#468—942no—
242#450—941no—
243#448—472no—
244#407—471no—
245#407—471no—
246#390—21no—
247#388—21no—
248#387—21no—
249#386—21no—
250#386—21no—
251#385—21no—
252#381C2×D4711yescharacteristic
253#380C2×D4711yescharacteristic
254C4×C22⋊C4—471no—
255#342—941no—
256#342—941no—
257#328—941no—
258#328—941no—
259#327—942no—
260C47#934111yescharacteristic, complemented, Sylow
261C22×Q8—471no—
262C22×D4—21no—
263C22×D4C22⋊D4711yescharacteristic
264C23×C4—21no—
265C23×C4C22×D4711yescharacteristic
266C2×SD16—942no—
267C2×M16—941no—
268C2×M16—941no—
269C22⋊Q8—941no—
270C22⋊Q8—941no—
271C22⋊Q8—941no—
272C22⋊Q8—941no—
273#121—21no—
274#121—21no—
275#121C4×D4712yes—
276#121D18812yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—471no—
280C2×C4⋊C4—941no—
281C2×C4⋊C4C22⋊D4711yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—941no—
288C2×C22⋊C4—941no—
289C2×C22⋊C4C22⋊D4711yescharacteristic
290C2×C22⋊C4C22⋊D4711yescharacteristic
291C2×C42—21no—
292C2×C42—941no—
293#107—942no—
294#103—942no—
295#102—942no—
296C22⋊C8—472no—
297C22⋊C8—941no—
298C22⋊C8—941no—
299C22⋊C8—941no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—941no—
305#95C22⋊D4711yescharacteristic
306#95C22⋊D4711yescharacteristic
307C24—21no—
308C24C2×C22⋊D4711yescharacteristic
309C2×Q8—941no—
310C2×Q8—941no—
311C2×Q8—3761no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4(C2×D47)⋊C412yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—941no—
329C22×C4—941no—
330C22×C4—1881no—
331C22×C4C2×C4×D4711yescharacteristic
332C22×C4C2×D18811yescharacteristic
333C22×C4#1062535011yescharacteristic
334C22×C4#1062535011yescharacteristic
335C22×C4C2×C22⋊D4711yescharacteristic
336C22×C4C2×C22⋊D4711yescharacteristic
337SD16—3762no—
338M16—3761no—
339M16—3761no—
340C2×C8—942no—
341C2×C8—1881no—
342C4⋊C4—21no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—941no—
346C4⋊C4—941no—
347C4⋊C4—1881no—
348C4⋊C4—1881no—
349C4⋊C4(C2×D47)⋊C412yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—81no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C22⋊C4—942no—
359C22⋊C4—1881no—
360C22⋊C4—1881no—
361C22⋊C4—1881no—
362C42—81no—
363C42—3761no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C4×C22⋊D4711yescharacteristic
376C23#1309399311yescharacteristic
377C23#1309399811yescharacteristic
378Q8—3761no—
379Q8—1882no—
380D4—42no—
381D4—81no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—84no—
406C2×C4—3761no—
407C2×C4—3761no—
408C2×C4—3761no—
409C2×C4—3761no—
410C2×C4—1882no—
411C2×C4—1881no—
412C2×C4—1881no—
413C2×C4C2×(C2×D47)⋊C411yescharacteristic
414C2×C4C4×C22⋊D4711yescharacteristic
415C2×C4#1309399811yescharacteristic
416C2×C4#1309400711yescharacteristic
417C8—3761no—
418C8—1882no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#1568900211yescharacteristic
437C22#1568930111yescharacteristic
438C22#1568930611yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—3762no—
449C4—3761no—
450C4—3761no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2#2024375711yescharacteristic
457C2#2024387511yescharacteristic
458C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2377C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2377D47
hypocenterC47#9341
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2377

Complements

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

NH ≅ G/Ntype
2. #20244372C2semidirect
3. #20241920C2semidirect
5. #20241294C2semidirect
6. #20241240C2semidirect
10. #15689434C4semidirect
13. #15689261C4semidirect
31. C22×Dic94C2×C4semidirect
39. C2×C4⋊Dic47C2×C4semidirect
50. C22⋊(C47⋊C8)C2×C4semidirect
260. C47#9341semidirect

Maximal subgroups

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

subgroupindex
#202541942
#202443722
#202419202
#202419042
#202412942
#202412402
C47×#23772
#934147