Finite Group Databasebeta

← #26107705

Subgroups — #26107705

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689603C211yesmaximal, characteristic
2#20241782C212yesmaximal, complemented
3#20241780C212yesmaximal, complemented
4#20241702C211yesmaximal, characteristic, complemented
5C47×#1458C211yesmaximal, characteristic, complemented
6C22×#10625350C411yescharacteristic
7#15689603—24no—
8#15689603—24no—
9C2×C4×Dic94C411yescharacteristic
10C2×C4×Dic94C2211yescharacteristic
11C2×C4×D188C411yescharacteristic
12C2×C4×D188C2211yescharacteristic
13C2×(C22⋊D47)⋊C4—21no—
14C2×(C22⋊D47)⋊C4—21no—
15C2×#13093982—21no—
16C22×C4⋊Dic47C411yescharacteristic
17C2×C4⋊C4×D47—21no—
18#15689176C2212yes—
19#15688853—24no—
20#15688852—24no—
21#15688849—21no—
22#15688847—21no—
23C2×#117×C47C2211yescharacteristic
24C47×#431C2212yes—
25C47×#324—21no—
26C47×#323—21no—
27C2×#10625350—22no—
28C2×#10625350—22no—
29C2×#10625350—22no—
30C2×#10625350—22no—
31C2×#10625350—22no—
32C2×#10625350C814yescomplemented
33C22×Dic94C2×C411yescharacteristic
34C22×D188C2×C411yescharacteristic
35C22×C22⋊D47—21no—
36C23×Dic47—21no—
37C22×C4×D47—21no—
38C22×C4×D47—21no—
39C22×C4×D47—21no—
40C4×Dic94—22no—
41C4×Dic94—22no—
42C4×Dic94—22no—
43C4×Dic94D414yes—
44C4×D188—22no—
45C4×D188—22no—
46C4×D188—22no—
47C4×D188D414yes—
48(C22⋊D47)⋊C4—28no—
49(C22⋊D47)⋊C4—28no—
50#13093982—28no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47C2×C411yescharacteristic
55C2×C4⋊Dic47C2×C411yescharacteristic
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C4⋊C4×D47—28no—
59C2×(C2×D47)⋊C4—21no—
60C2×(C2×D47)⋊C4—21no—
61C2×C4×Dic47—21no—
62C2×C4×Dic47—21no—
63C4⋊(C47⋊C8)—22no—
64C22⋊(C47⋊C8)—22no—
65C4×C47⋊C8—22no—
66C23×C4×C47C2×C411yescharacteristic
67#117×C47—22no—
68#117×C47—22no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C4⋊C4×C47D411yescharacteristic
71C2×C22⋊C4×C47D411yescharacteristic
72C2×C22⋊C4×C47D411yescharacteristic
73C2×C42×C47C2×C411yescharacteristic
74C2×C42×C47C2311yescharacteristic
75C4⋊C8×C47—22no—
76C22⋊C8×C47—22no—
77C4×C8×C47—22no—
78C24×C47C22⋊C411yescharacteristic
79C23×D47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47C2×C812yes—
88C22×C4×C47M1612yes—
89C22×C4×C47C22×C411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C2×C22⋊D47—24no—
93C2×C22⋊D47—24no—
94C2×C22⋊D47—24no—
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—21no—
98C22×Dic47—24no—
99C22×Dic47—82no—
100#10625350—28no—
101#10625350—28no—
102#10625350—28no—
103#10625350—28no—
104C2×D188—21no—
105C2×D188—21no—
106C2×D188—21no—
107C2×D188C22⋊C412yes—
108C2×D188C2×C812yes—
109C2×D188M1612yes—
110C2×C4×D47—24no—
111C2×C4×D47—24no—
112C2×C4×D47—24no—
113C2×C4×D47—24no—
114C2×C4×D47—24no—
115C2×C4×D47—82no—
116C2×C4×D47—82no—
117C2×Dic94—21no—
118C2×Dic94—21no—
119C2×Dic94—21no—
120C2×Dic94C22⋊C412yes—
121C2×Dic94C2×C812yes—
122C2×Dic94M1612yes—
123C2×C8×C47—42no—
124C4⋊C4×C47—22no—
125C4⋊C4×C47—22no—
126C22⋊C4×C47—22no—
127C22⋊C4×C47—22no—
128C42×C47—21no—
129C42×C47—21no—
130C42×C47C2×D412yes—
131C42×C47C2×D412yes—
132(C2×D47)⋊C4—24no—
133(C2×D47)⋊C4—24no—
134C4⋊Dic47—24no—
135C4⋊Dic47—21no—
136C4⋊Dic47—21no—
137C4⋊Dic47C22⋊C412yes—
138C4⋊Dic47C22⋊C412yes—
139Dic47⋊C4—24no—
140Dic47⋊C4—24no—
141C4×Dic47—24no—
142C4×Dic47—24no—
143C2×C47⋊C8—42no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47C22⋊C812yes—
149C23×C47C2×C22⋊C411yescharacteristic
150C22×D47—22no—
151C22×D47—22no—
152C22×D47—22no—
153C22×D47—42no—
154C22×D47—42no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—22no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47C22⋊C812yes—
168C2×C4×C47C2×C22⋊C411yescharacteristic
169C2×C4×C47C2×C22⋊C411yescharacteristic
170C2×C4×C47#12012yes—
171C2×C4×C47#12012yes—
172C2×C4×C47C22×C811yescharacteristic
173C2×C4×C47C2×M1611yescharacteristic
174C2×C4×C47C2×M1611yescharacteristic
175C2×C4×C47C2×M1611yescharacteristic
176C22⋊D47—216no—
177C2×Dic47—22no—
178C2×Dic47—22no—
179C2×Dic47—22no—
180C2×Dic47—22no—
181C2×Dic47—22no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47—84no—
185D188—22no—
186D188—22no—
187D188—22no—
188D188C22⋊C814yes—
189C4×D47—216no—
190C4×D47—84no—
191C4×D47—84no—
192Dic94—22no—
193Dic94—22no—
194Dic94—22no—
195Dic94C22⋊C814yes—
196C8×C47—82no—
197C47⋊C8—82no—
198#8324—471nomaximal, Sylow
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—22no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—41no—
205C22×C47—41no—
206C22×C47#37911yescharacteristic
207C22×C47C2×C22⋊C811yescharacteristic
208C22×C47#40711yescharacteristic
209C2×D47—24no—
210C2×D47—44no—
211C2×D47—44no—
212C2×D47—42no—
213C2×D47—42no—
214C4×C47—22no—
215C4×C47—22no—
216C4×C47—22no—
217C4×C47—41no—
218C4×C47—41no—
219C4×C47C2×C22⋊C811yescharacteristic
220C4×C47C2×C22⋊C811yescharacteristic
221C4×C47#40711yescharacteristic
222C4×C47#40711yescharacteristic
223Dic47—24no—
224Dic47—42no—
225Dic47—42no—
226C2×#520—471no—
227#1479—472no—
228#1478—472no—
229#1458—471no—
230#1458D4711yescharacteristic, complemented
231C2×C47—21no—
232C2×C47—21no—
233C2×C47—42no—
234C2×C47#177911yescharacteristic
235C2×C47#179511yescharacteristic
236C2×C47#179611yescharacteristic
237D47—44no—
238C22×#41—471no—
239#520—944no—
240#520—944no—
241C2×C4×Q8—471no—
242C2×C4×Q8—471no—
243C2×C4×D4—471no—
244C2×C4×D4—471no—
245C2×C4×D4—941no—
246C2×C4×D4—941no—
247C2×#117—941no—
248C2×#117C2×D4711yescharacteristic
249C22×C4⋊C4—471no—
250C22×C4⋊C4—941no—
251#431—472no—
252#431C2×D4712yes—
253Q8⋊C8—944no—
254D4⋊C8—944no—
255#324—21no—
256#324—941no—
257#323—21no—
258#323—941no—
259C47#832411yescharacteristic, complemented, Sylow
260C2×#41—474no—
261C2×#41—942no—
262C2×#41—942no—
263C2×#41—942no—
264C2×#41—942no—
265C2×#41—942no—
266C22×Q8—471no—
267C22×D4—471no—
268C22×D4—941no—
269C23×C4—941no—
270C23×C4—941no—
271C23×C4—941no—
272C23×C4—941no—
273C23×C4C4×D4711yescharacteristic
274C4×Q8—474no—
275C4×Q8—942no—
276C4×Q8—942no—
277C4×Q8—942no—
278C4×D4—474no—
279C4×D4—942no—
280C4×D4—942no—
281C4×D4—948no—
282C4×D4—948no—
283C4×D4—942no—
284#117—22no—
285#117—22no—
286#117—948no—
287C2×C4⋊C4—471no—
288C2×C4⋊C4—471no—
289C2×C4⋊C4—948no—
290C2×C4⋊C4—942no—
291C2×C4⋊C4—942no—
292C2×C4⋊C4—941no—
293C2×C4⋊C4—941no—
294C2×C4⋊C4—941no—
295C2×C4⋊C4D18811yescharacteristic
296C2×C4⋊C4C22⋊D4711yescharacteristic
297C2×C22⋊C4—941no—
298C2×C22⋊C4—941no—
299C2×C22⋊C4D18811yescharacteristic
300C2×C22⋊C4C22⋊D4711yescharacteristic
301C2×C42—941no—
302C2×C42—941no—
303C2×C42C4×D4711yescharacteristic
304C2×C42C22×D4711yescharacteristic
305C4⋊C8—22no—
306C4⋊C8—942no—
307C22⋊C8—22no—
308C22⋊C8—942no—
309C4×C8—22no—
310C4×C8—942no—
311C24—941no—
312C24(C2×D47)⋊C411yescharacteristic
313#41—948no—
314#41—948no—
315#41—948no—
316#41—948no—
317C2×Q8—472no—
318C2×Q8—472no—
319C2×Q8—472no—
320C2×Q8—941no—
321C2×Q8—941no—
322C2×Q8—941no—
323C2×D4—472no—
324C2×D4—472no—
325C2×D4—472no—
326C2×D4—944no—
327C2×D4—944no—
328C2×D4—944no—
329C2×D4—941no—
330C2×D4—941no—
331C2×D4—941no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—944no—
337C22×C4—944no—
338C22×C4—944no—
339C22×C4—941no—
340C22×C4—944no—
341C22×C4—944no—
342C22×C4—941no—
343C22×C4—941no—
344C22×C4—944no—
345C22×C4—3762no—
346C22×C4—3762no—
347C22×C4—3762no—
348C22×C4C8×D4712yes—
349C22×C4#1062532412yes—
350C22×C4(C2×D47)⋊C411yescharacteristic
351C22×C4(C2×D47)⋊C411yescharacteristic
352C22×C4(C2×D47)⋊C411yescharacteristic
353C22×C4C2×C4×D4711yescharacteristic
354C22×C4C2×D18811yescharacteristic
355C22×C4C2×C22⋊D4711yescharacteristic
356C2×C8—42no—
357C2×C8—1882no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—472no—
361C4⋊C4—472no—
362C4⋊C4—944no—
363C4⋊C4—944no—
364C4⋊C4—941no—
365C4⋊C4—941no—
366C4⋊C4—944no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—944no—
370C22⋊C4—944no—
371C42—21no—
372C42—21no—
373C42—944no—
374C42—944no—
375C42D4×D4712yes—
376C42D4×D4712yes—
377C23—21no—
378C23—21no—
379C23—41no—
380C23—41no—
381C23—942no—
382C23—942no—
383C23—942no—
384C23—1882no—
385C23—1882no—
386C23(C2×D47)⋊C812yes—
387C23C2×(C2×D47)⋊C411yescharacteristic
388Q8—474no—
389Q8—942no—
390Q8—942no—
391Q8—942no—
392D4—474no—
393D4—9416no—
394D4—942no—
395D4—942no—
396D4—942no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—9416no—
410C2×C4—942no—
411C2×C4—942no—
412C2×C4—942no—
413C2×C4—942no—
414C2×C4—942no—
415C2×C4—3764no—
416C2×C4—3764no—
417C2×C4—3764no—
418C2×C4—1882no—
419C2×C4—1882no—
420C2×C4(C2×D47)⋊C812yes—
421C2×C4C22⋊C4×D4711yescharacteristic
422C2×C4C22⋊C4×D4711yescharacteristic
423C2×C4#1309399212yes—
424C2×C4#1309399412yes—
425C2×C4C2×C8×D4711yescharacteristic
426C2×C4M16×D4711yescharacteristic
427C2×C4M16×D4711yescharacteristic
428C2×C4C2×#1062532411yescharacteristic
429C8—82no—
430C8—3762no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—42no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—944no—
439C22—1884no—
440C22—1884no—
441C22—1882no—
442C22—1882no—
443C22#1568897811yescharacteristic
444C22C2×(C2×D47)⋊C811yescharacteristic
445C22#1568908611yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—944no—
452C4—1882no—
453C4—1882no—
454C4C22⋊C8×D4711yescharacteristic
455C4C22⋊C8×D4711yescharacteristic
456C4#1568908511yescharacteristic
457C4#1568908511yescharacteristic
458C2—21no—
459C2—21no—
460C2—42no—
461C2—1884no—
462C2#2024329711yescharacteristic
463C2#2024335911yescharacteristic
464C2#2024336111yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#379
prime radicalC22×#10625350C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#8324
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20241782C2semidirect
3. #20241780C2semidirect
4. #20241702C2semidirect
5. C47×#1458C2semidirect
32. C2×#10625350C8semidirect
230. #1458D47semidirect
259. C47#8324semidirect

Maximal subgroups

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

subgroupindex
C2×#156896032
#202417822
#202417802
#202417022
C47×#14582
#832447