Finite Group Databasebeta

← #26112117

Subgroups — #26112117

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252423C211yesmaximal, characteristic
2#20243213C211yesmaximal, characteristic
3#20242003C211yesmaximal, characteristic, complemented
4#20241887C211yesmaximal, characteristic, complemented
5#20241286C211yesmaximal, characteristic, complemented
6#20241245C211yesmaximal, characteristic, complemented
7C47×#3110C211yesmaximal, characteristic
8#15689201—21no—
9#15689195—22no—
10D4×C47⋊C8—21no—
11#15689180C2211yescharacteristic
12#15689176C2211yescharacteristic
13C2×#13093945—21no—
14#15689087—21no—
15C2×C22⋊(C47⋊C8)—21no—
16C2×#13093935C2211yescharacteristic
17C4×C4⋊Dic47C2211yescharacteristic
18#15688885—21no—
19#15688865—22no—
20#15688860—21no—
21#15688860—21no—
22C47×#533C2211yescharacteristic
23C2×C4×D4×C47C2211yescharacteristic
24C47×#484—21no—
25C47×#474C411yescharacteristic, complemented
26C47×#474C411yescharacteristic, complemented
27C47×#461C411yescharacteristic, complemented
28C47×#461C411yescharacteristic, complemented
29C47×#450—21no—
30C47×#440—21no—
31C47×#440—21no—
32C47×#432C2211yescharacteristic
33C2×#10625329—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—22no—
36C2×C4⋊Dic47—21no—
37C2×C4×Dic47—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40#13093945—24no—
41C22⋊(C47⋊C8)—22no—
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)D411yescharacteristic
45C22⋊(C47⋊C8)D411yescharacteristic
46#13093935—21no—
47#13093935—21no—
48#13093935D412yes—
49C4×C47⋊C8—21no—
50C47⋊#95—21no—
51C22×D4×C47D411yescharacteristic
52C23×C4×C47—21no—
53C2×SD16×C47—21no—
54C2×SD16×C47—21no—
55C4⋊Q8×C47C2×C411yescharacteristic
56C4⋊Q8×C47C2×C411yescharacteristic
57#125×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47—21no—
62C4×D4×C47—21no—
63C4×D4×C47—21no—
64C4×D4×C47C2×C411yescharacteristic
65C4×D4×C47C2×C411yescharacteristic
66C4×D4×C47D411yescharacteristic
67C4×D4×C47D411yescharacteristic
68#117×C47—21no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C22⋊C4×C47—21no—
71C2×C42×C47C2311yescharacteristic
72#107×C47—21no—
73#107×C47—21no—
74C4⋊C8×C47C2×C411yescharacteristic
75C4⋊C8×C47C2×C411yescharacteristic
76#103×C47—21no—
77#103×C47—21no—
78#102×C47—21no—
79#102×C47—21no—
80C22⋊C8×C47—21no—
81#97×C47—21no—
82C24×C47—21no—
83C2×Q8×C47—21no—
84C2×Q8×C47—21no—
85C2×D4×C47—22no—
86C2×D4×C47—21no—
87C2×D4×C47—41no—
88C2×D4×C47C22⋊C411yescharacteristic
89C2×D4×C47C22⋊C411yescharacteristic
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—82no—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47#4111yescharacteristic
99C22×Dic47—41no—
100SD16×C47—42no—
101SD16×C47—42no—
102C2×C8×C47—21no—
103C2×C8×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47C22⋊C411yescharacteristic
112C4⋊C4×C47C22⋊C411yescharacteristic
113C22⋊C4×C47—21no—
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C22⋊C4×C47—41no—
118C42×C47—41no—
119C42×C47C22×C411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C4⋊Dic47—22no—
124C4⋊Dic47—41no—
125C4×Dic47—81no—
126#10625329—82no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—82no—
134C23×C47—21no—
135C23×C47—21no—
136C23×C47—42no—
137C23×C47—41no—
138C23×C47—41no—
139C23×C47—81no—
140C23×C47#12111yescharacteristic
141Q8×C47—41no—
142Q8×C47—41no—
143Q8×C47—81no—
144D4×C47—21no—
145D4×C47—21no—
146D4×C47—21no—
147D4×C47—42no—
148D4×C47#10412yes—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—41no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—81no—
161C2×C4×C47—81no—
162C2×C4×C47—81no—
163C2×C4×C47C2×C22⋊C411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C2×C4×C47#12011yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12311yescharacteristic
169C2×Dic47—42no—
170C2×Dic47—81no—
171C2×Dic47—81no—
172C8×C47—41no—
173C8×C47—41no—
174C47⋊C8—42no—
175C47⋊C8—41no—
176C47⋊C8—42no—
177C47⋊C8—41no—
178#9165—471nomaximal, Sylow
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—22no—
182C22×C47—41no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47—41no—
186C22×C47—41no—
187C22×C47—82no—
188C22×C47—82no—
189C22×C47#38611yescharacteristic
190C22×C47#47111yescharacteristic
191C22×C47#47311yescharacteristic
192C4×C47—21no—
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—81no—
200C4×C47—81no—
201C4×C47C2×#10411yescharacteristic
202C4×C47#42111yescharacteristic
203Dic47—82no—
204#3110D4711yescharacteristic
205#2974—471no—
206#1762—471no—
207#1531—471no—
208#1503—471no—
209#1302—471no—
210#1285—471no—
211C2×C47—21no—
212C2×C47—21no—
213C2×C47—42no—
214C2×C47—81no—
215C2×C47#187111yescharacteristic
216C2×C47#190411yescharacteristic
217C2×C47#190811yescharacteristic
218#533C2×D4711yescharacteristic
219C2×C4×D4C2×D4711yescharacteristic
220#484—21no—
221#474Dic4711yescharacteristic, complemented
222#474Dic4711yescharacteristic, complemented
223#461Dic4711yescharacteristic, complemented
224#461Dic4711yescharacteristic, complemented
225#450—21no—
226#440—21no—
227#440—21no—
228#436—941no—
229#435—942no—
230C8×D4—941no—
231#432—471no—
232#432C2×D4711yescharacteristic
233#431—471no—
234C2×#102—941no—
235#407—941no—
236C2×C22⋊C8—941no—
237C2×#97—471no—
238C4×C4⋊C4—471no—
239#340—941no—
240#330—942no—
241#328—941no—
242#328—941no—
243C47#916511yescharacteristic, complemented, Sylow
244C22×D4C22⋊D4711yescharacteristic
245C23×C4—21no—
246C2×SD16—21no—
247C2×SD16—21no—
248C2×M16—942no—
249C22×C8—942no—
250C22×C8—941no—
251C4⋊Q8C2×Dic4711yescharacteristic
252C4⋊Q8C2×Dic4711yescharacteristic
253#125—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×Dic4711yescharacteristic
261C4×D4C2×Dic4711yescharacteristic
262C4×D4C22⋊D4711yescharacteristic
263C4×D4C22⋊D4711yescharacteristic
264#117—21no—
265C2×C4⋊C4—941no—
266C2×C4⋊C4C22⋊D4711yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—941no—
269C2×C42C22×D4711yescharacteristic
270#107—21no—
271#107—21no—
272C4⋊C8—941no—
273C4⋊C8—941no—
274C4⋊C8C2×Dic4711yescharacteristic
275C4⋊C8C2×Dic4711yescharacteristic
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280#102—944no—
281C22⋊C8—21no—
282C22⋊C8—471no—
283C22⋊C8—471no—
284C22⋊C8—942no—
285C22⋊C8—942no—
286C22⋊C8—941no—
287#97—21no—
288#97—472no—
289#97—941no—
290#97—941no—
291C4×C8—941no—
292#95—941no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4C22⋊Dic4711yescharacteristic
300C2×D4C22⋊Dic4711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—82no—
307C22×C4—1881no—
308C22×C4D4×D4711yescharacteristic
309C22×C4#1062535211yescharacteristic
310C22×C4C2×C22⋊D4711yescharacteristic
311SD16—42no—
312SD16—42no—
313M16—3762no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—942no—
317C2×C8—941no—
318C2×C8—941no—
319C2×C8—942no—
320C2×C8—3762no—
321C2×C8—1881no—
322C2×C8—1881no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—942no—
331C4⋊C4—1881no—
332C4⋊C4C22⋊Dic4711yescharacteristic
333C4⋊C4C22⋊Dic4711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—41no—
339C42—41no—
340C42—3761no—
341C42D4×D4711yescharacteristic
342C42#1062535211yescharacteristic
343C42C22×Dic4711yescharacteristic
344C42C2×C22⋊D4711yescharacteristic
345C23—21no—
346C23—21no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23#1309399611yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—81no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4#1309395112yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—3761no—
375C2×C4—3761no—
376C2×C4—1882no—
377C2×C4C2×C22⋊Dic4711yescharacteristic
378C2×C4D4×Dic4711yescharacteristic
379C2×C4D4×Dic4711yescharacteristic
380C2×C4#1309399211yescharacteristic
381C2×C4#1309399611yescharacteristic
382C2×C4#1309400511yescharacteristic
383C8—41no—
384C8—41no—
385C8—1881no—
386C8—1882no—
387C8—1882no—
388C8—1881no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22#1568900311yescharacteristic
400C22#1568938811yescharacteristic
401C22#1568939611yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—3762no—
412C4C2×#1309395111yescharacteristic
413C4#1568914011yescharacteristic
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#2024369111yescharacteristic
419C2#2024387711yescharacteristic
420C2#2024390511yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3110C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#3110D47
hypocenterC47#9165
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #20242003C2semidirect
4. #20241887C2semidirect
5. #20241286C2semidirect
6. #20241245C2semidirect
25. C47×#474C4semidirect
26. C47×#474C4semidirect
27. C47×#461C4semidirect
28. C47×#461C4semidirect
221. #474Dic47semidirect
222. #474Dic47semidirect
223. #461Dic47semidirect
224. #461Dic47semidirect
243. C47#9165semidirect

Maximal subgroups

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

subgroupindex
#202524232
#202432132
#202420032
#202418872
#202412862
#202412452
C47×#31102
#916547