Finite Group Databasebeta

← #26119423

Subgroups — #26119423

Order 12032.

508 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 101 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#20244224C211yesmaximal, characteristic
2#20241955C211yesmaximal, characteristic
3C2×#15688856C211yesmaximal, characteristic
4#20241267C211yesmaximal, characteristic, complemented
5#20241255C211yesmaximal, characteristic, complemented
6#20241236C211yesmaximal, characteristic, complemented
7C47×#2549C211yesmaximal, characteristic, complemented
8C2×#13093998C2211yescharacteristic
9C2×#13093938—21no—
10C2×#13093938—21no—
11C2×C22⋊(C47⋊C8)C2211yescharacteristic
12C2×C22⋊(C47⋊C8)C2211yescharacteristic
13#15688983C2211yescharacteristic
14#15688856—21no—
15#15688856—21no—
16#15688856—21no—
17#15688856—21no—
18C2×C22⋊Q8×C47—21no—
19C47×#398—21no—
20C47×#395—21no—
21C47×#391—21no—
22C47×#389—21no—
23C47×#385—21no—
24C47×#380C2211yescharacteristic
25C4×C22⋊C4×C47C2211yescharacteristic
26C2×#95×C47C2211yescharacteristic
27C22×D188Q811yescharacteristic, complemented
28C22×C22⋊D47—21no—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—21no—
31#13093998—21no—
32#13093998—21no—
33#13093998—21no—
34#13093998—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47Q811yescharacteristic, complemented
37C2×C22⋊Dic47—21no—
38C2×(C2×D47)⋊C4—21no—
39#13093938—42no—
40#13093938—42no—
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C47⋊#95—21no—
46C22×Q8×C47—21no—
47C23×C4×C47—21no—
48C23×C4×C47C2311yescharacteristic
49C22⋊Q8×C47—42no—
50C22⋊Q8×C47—42no—
51C2×C4⋊C4×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C22⋊C4×C47—21no—
58C2×C22⋊C4×C47D411yescharacteristic
59C2×C22⋊C4×C47D411yescharacteristic
60C2×C42×C47—21no—
61#95×C47—21no—
62#95×C47—21no—
63#95×C47—21no—
64#95×C47—21no—
65#95×C47D411yescharacteristic
66#95×C47D411yescharacteristic
67#95×C47D411yescharacteristic
68#95×C47D411yescharacteristic
69C24×C47C2×D411yescharacteristic
70C23×D47—21no—
71C2×Q8×C47—42no—
72C2×Q8×C47—42no—
73C2×Q8×C47—81no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—42no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47—81no—
84C22×C4×C47—81no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×Q811yescharacteristic
88C22×C4×C47#4111yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C22×C4×C47#4111yescharacteristic
91C2×C22⋊D47—41no—
92C2×C22⋊D47—41no—
93C2×C22⋊D47—41no—
94C2×C22⋊D47—41no—
95C2×C22⋊D47—41no—
96C2×C22⋊D47—41no—
97C22×Dic47—21no—
98C22×Dic47—41no—
99C2×D188—21no—
100C2×D188—21no—
101C2×D188—81no—
102C4⋊C4×C47—42no—
103C4⋊C4×C47—42no—
104C4⋊C4×C47—42no—
105C4⋊C4×C47—42no—
106C4⋊C4×C47—81no—
107C4⋊C4×C47—81no—
108C22⋊C4×C47—21no—
109C22⋊C4×C47—21no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47D812yes—
113C22⋊C4×C47SD1612yes—
114C42×C47—81no—
115C22⋊Dic47—41no—
116C22⋊Dic47—41no—
117(C2×D47)⋊C4—41no—
118(C2×D47)⋊C4—41no—
119C4⋊Dic47—21no—
120C4⋊Dic47—21no—
121C4⋊Dic47—81no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—81no—
125C2×C47⋊C8—81no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47#12011yescharacteristic
129C23×C47C22⋊Q811yescharacteristic
130C23×C47#12411yescharacteristic
131C23×C47C2×D811yescharacteristic
132C23×C47C2×SD1611yescharacteristic
133C23×C47AGL(1,ℤ/8)11yescharacteristic
134C23×C47#13711yescharacteristic
135C22×D47—41no—
136C22×D47—41no—
137C22×D47—41no—
138C22×D47—81no—
139C22×D47—81no—
140Q8×C47—84no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
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—82no—
166C2×C4×C47—82no—
167C2×C4×C47C22⋊Q811yescharacteristic
168C2×C4×C47C22⋊Q811yescharacteristic
169C2×C4×C47#12411yescharacteristic
170C2×C4×C47#12411yescharacteristic
171C22⋊D47—81no—
172C22⋊D47—81no—
173C22⋊D47—81no—
174C22⋊D47—81no—
175C2×Dic47—41no—
176C2×Dic47—41no—
177C2×Dic47—41no—
178C2×Dic47—81no—
179C2×Dic47—81no—
180C2×Dic47—81no—
181D188—81no—
182D188—81no—
183C47⋊C8—81no—
184C47⋊C8—81no—
185#10374—471nomaximal, Sylow
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—41no—
193C22×C47—41no—
194C22×C47—41no—
195C22×C47—41no—
196C22×C47#39311yescharacteristic
197C22×C47#44711yescharacteristic
198C22×C47#44811yescharacteristic
199C22×C47#47411yescharacteristic
200C22×C47#47611yescharacteristic
201C22×C47#48611yescharacteristic
202C22×C47#48911yescharacteristic
203C2×D47—81no—
204C2×D47—81no—
205C2×D47—81no—
206C2×D47—161no—
207C2×D47—161no—
208C4×C47—21no—
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—42no—
213C4×C47—42no—
214C4×C47—42no—
215C4×C47—81no—
216C4×C47—81no—
217C4×C47—81no—
218Dic47—81no—
219Dic47—161no—
220#2549D4711yescharacteristic, complemented
221#1964—471no—
222#1519—471no—
223C2×#327—471no—
224#1294—471no—
225#1289—471no—
226#1282—471no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47—21no—
230C2×C47—21no—
231C2×C47—21no—
232C2×C47—21no—
233C2×C47#202311yescharacteristic
234C2×C47#203111yescharacteristic
235C2×C47#203211yescharacteristic
236D47—161no—
237C2×C22⋊Q8—21no—
238C2×#121—471no—
239C2×#99—941no—
240C2×#99—941no—
241C2×C22⋊C8—471no—
242C2×C22⋊C8—471no—
243#398—21no—
244#395—21no—
245#391—21no—
246#389—21no—
247#385—21no—
248#380—471no—
249#380C2×D4711yescharacteristic
250C4×C22⋊C4C2×D4711yescharacteristic
251C2×#95C2×D4711yescharacteristic
252#327—941no—
253#327—941no—
254#327—941no—
255#327—941no—
256C47#1037411yescharacteristic, complemented, Sylow
257C22×Q8—21no—
258C22×D4—471no—
259C22×D4—941no—
260C23×C4—21no—
261C23×C4C22×D4711yescharacteristic
262C22×C8—941no—
263C22×C8—941no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266#121—941no—
267#121—941no—
268#121—941no—
269#121—941no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—471no—
277C2×C4⋊C4—941no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—941no—
280C2×C22⋊C4—941no—
281C2×C22⋊C4C22⋊D4711yescharacteristic
282C2×C22⋊C4C22⋊D4711yescharacteristic
283C2×C42—21no—
284#99—1882no—
285#99—1882no—
286C22⋊C8—941no—
287C22⋊C8—941no—
288C22⋊C8—941no—
289C22⋊C8—941no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—941no—
295#95C22⋊D4711yescharacteristic
296#95C22⋊D4711yescharacteristic
297#95C22⋊D4711yescharacteristic
298#95C22⋊D4711yescharacteristic
299C24—941no—
300C24C2×C22⋊D4711yescharacteristic
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—81no—
304C2×D4—941no—
305C2×D4—941no—
306C2×D4—3761no—
307C2×D4—1881no—
308C2×D4—1881no—
309C2×D4—1881no—
310C2×D4—1881no—
311C2×D4—1881no—
312C2×D4—1881no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—941no—
325C22×C4—1881no—
326C22×C4Q8×D4711yescharacteristic
327C22×C4#1062535411yescharacteristic
328C22×C4#1062535411yescharacteristic
329C22×C4#1062535411yescharacteristic
330C22×C4C2×C22⋊D4711yescharacteristic
331C22×C4C2×C22⋊D4711yescharacteristic
332C2×C8—3761no—
333C2×C8—3761no—
334C2×C8—1881no—
335C2×C8—1881no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—941no—
343C4⋊C4—941no—
344C4⋊C4—3761no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—1881no—
350C22⋊C4—1881no—
351C22⋊C4—1881no—
352C22⋊C4—1881no—
353C22⋊C4#1062533412yes—
354C22⋊C4#1062533612yes—
355C42—81no—
356C23—41no—
357C23—41no—
358C23—3761no—
359C23—3761no—
360C23—1881no—
361C23—1881no—
362C23—1881no—
363C23#1309399311yescharacteristic
364C23#1309400011yescharacteristic
365C23#1309401311yescharacteristic
366C23C2×#1062533411yescharacteristic
367C23C2×#1062533611yescharacteristic
368C23#1309405011yescharacteristic
369C23#1309405511yescharacteristic
370Q8—84no—
371D4—3761no—
372D4—3761no—
373D4—3761no—
374D4—3761no—
375D4—3761no—
376D4—3761no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
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—82no—
402C2×C4—82no—
403C2×C4—3761no—
404C2×C4—3761no—
405C2×C4—3761no—
406C2×C4—1881no—
407C2×C4—1881no—
408C2×C4—1881no—
409C2×C4#1309400011yescharacteristic
410C2×C4#1309400011yescharacteristic
411C2×C4#1309401311yescharacteristic
412C2×C4#1309401311yescharacteristic
413C8—3761no—
414C8—3761no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—3761no—
426C22—3761no—
427C22—3761no—
428C22—7521no—
429C22—7521no—
430C22#1568903411yescharacteristic
431C22#1568925711yescharacteristic
432C22#1568926211yescharacteristic
433C22#1568940011yescharacteristic
434C22#1568941011yescharacteristic
435C22#1568947111yescharacteristic
436C22#1568948511yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—3761no—
448C4—7521no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—7521no—
456C2#2024452111yescharacteristic
457C2#2024455711yescharacteristic
458C2#2024456411yescharacteristic
subgroupG/H
centerC22#15689034
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2549C2
socleC22×C47#393
prime radicalC2×#13093998C22
hypercenter#2549D47
hypocenterC47#10374
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2549

Complements

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

NH ≅ G/Ntype
4. #20241267C2semidirect
5. #20241255C2semidirect
6. #20241236C2semidirect
7. C47×#2549C2semidirect
27. C22×D188Q8semidirect
36. C2×C4⋊Dic47Q8semidirect
220. #2549D47semidirect
256. C47#10374semidirect

Maximal subgroups

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

subgroupindex
#202442242
#202419552
C2×#156888562
#202412672
#202412552
#202412362
C47×#25492
#1037447