Finite Group Databasebeta

← #26112867

Subgroups — #26112867

Order 12032.

422 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 103 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#20252464C211yesmaximal, characteristic
2#20241900C211yesmaximal, characteristic, complemented
3#20241896C211yesmaximal, characteristic
4#20241861C211yesmaximal, characteristic, complemented
5#20241250C211yesmaximal, characteristic, complemented
6#20241216C211yesmaximal, characteristic, complemented
7C47×#3228C211yesmaximal, characteristic
8#15689201—21no—
9#15689195—22no—
10D4×C47⋊C8—21no—
11#15689180C2211yescharacteristic
12#15689180C2211yescharacteristic
13#15689176C2211yescharacteristic
14#15689090—22no—
15#15688969C2211yescharacteristic
16#15688860—22no—
17#15688857—22no—
18Q8⋊(C47⋊C8)—21no—
19Q8⋊(C47⋊C8)—21no—
20C47×#533C2211yescharacteristic
21C47×#532C2211yescharacteristic
22C47×#500C411yescharacteristic, complemented
23C47×#498C411yescharacteristic, complemented
24C47×#486C411yescharacteristic, complemented
25C47×#486C411yescharacteristic, complemented
26C47×#482—22no—
27C47×#478—22no—
28C47×#431C2211yescharacteristic
29C2×#10625329—22no—
30C22×C47⋊C8—22no—
31C2×C4×Dic47—21no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35C22⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—21no—
38C22⋊(C47⋊C8)—21no—
39C22⋊(C47⋊C8)D412yes—
40#13093935—21no—
41#13093935—21no—
42C4×C47⋊C8D411yescharacteristic
43C4×C47⋊C8D411yescharacteristic
44C47⋊#95—21no—
45C47⋊#95—21no—
46C2×#41×C47—21no—
47C4⋊Q8×C47C2×C411yescharacteristic
48C4⋊Q8×C47C2×C411yescharacteristic
49C4⋊Q8×C47C2×C411yescharacteristic
50#127×C47C2×C411yescharacteristic
51#125×C47—21no—
52#124×C47—21no—
53C22⋊Q8×C47—21no—
54C22⋊Q8×C47—21no—
55#121×C47—21no—
56#121×C47D412yes—
57C4×Q8×C47—21no—
58C4×Q8×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47D412yes—
61#117×C47—21no—
62C2×C42×C47C2311yescharacteristic
63#107×C47—22no—
64#106×C47—22no—
65C4⋊C8×C47—21no—
66#102×C47—22no—
67#102×C47—22no—
68C22⋊C8×C47—21no—
69C4×C8×C47C2×C411yescharacteristic
70C4×C8×C47C2×C411yescharacteristic
71#41×C47—82no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×D4×C47—22no—
76C2×D4×C47—21no—
77C2×D4×C47—21no—
78C2×D4×C47C22⋊C412yes—
79C22×C4×C47—22no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×Dic47—41no—
84C2×C8×C47—21no—
85C2×C8×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47C22⋊C412yes—
94C22⋊C4×C47—22no—
95C22⋊C4×C47—22no—
96C22⋊C4×C47—41no—
97C42×C47—41no—
98C42×C47C22×C411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102C4×Dic47—22no—
103C4×Dic47—41no—
104#10625329—82no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—22no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—82no—
110C23×C47—22no—
111C23×C47#12111yescharacteristic
112Q8×C47—41no—
113Q8×C47—41no—
114Q8×C47—81no—
115Q8×C47—81no—
116D4×C47—42no—
117D4×C47—41no—
118D4×C47—41no—
119D4×C47—81no—
120D4×C47—82no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—22no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47C2×C22⋊C411yescharacteristic
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47#12011yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#12311yescharacteristic
136C2×Dic47—42no—
137C2×Dic47—81no—
138C2×Dic47—81no—
139C8×C47—41no—
140C8×C47—41no—
141C47⋊C8—41no—
142C47⋊C8—42no—
143C47⋊C8—41no—
144C47⋊C8—81no—
145#9279—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—42no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47#38611yescharacteristic
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—42no—
159C4×C47—81no—
160C4×C47—81no—
161C4×C47#44311yescharacteristic
162C4×C47#44411yescharacteristic
163Dic47—82no—
164#3228D4711yescharacteristic
165#2981—471no—
166#1506—471no—
167#1505—471no—
168#1497—471no—
169#1287—471no—
170#1274—471no—
171C2×C47—41no—
172C2×C47—82no—
173C2×C47#187611yescharacteristic
174C2×C47#190411yescharacteristic
175C2×C47#190911yescharacteristic
176#533C2×D4711yescharacteristic
177#532C2×D4711yescharacteristic
178#500Dic4711yescharacteristic, complemented
179#498Dic4711yescharacteristic, complemented
180#486Dic4711yescharacteristic, complemented
181#486Dic4711yescharacteristic, complemented
182#482—22no—
183#478—22no—
184#436—941no—
185#435—942no—
186C8×D4—941no—
187#432—471no—
188#432—471no—
189#431—471no—
190#431C2×D4711yescharacteristic
191#408—942no—
192#376—471no—
193#328—942no—
194#327—942no—
195Q8⋊C8—941no—
196Q8⋊C8—941no—
197C47#927911yescharacteristic, complemented, Sylow
198C2×#41—21no—
199C2×M16—942no—
200C22×C8—942no—
201C4⋊Q8C2×Dic4711yescharacteristic
202C4⋊Q8C2×Dic4711yescharacteristic
203C4⋊Q8C2×Dic4711yescharacteristic
204#127C2×Dic4711yescharacteristic
205#125—21no—
206#124—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209#121—21no—
210#121C22⋊D4712yes—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×D4—21no—
214C4×D4C22⋊D4712yes—
215#117—21no—
216C2×C42—941no—
217C2×C42C22×D4711yescharacteristic
218#107—22no—
219#106—22no—
220C4⋊C8—21no—
221C4⋊C8—941no—
222C4⋊C8—941no—
223C4⋊C8—941no—
224#102—22no—
225#102—22no—
226C22⋊C8—21no—
227C22⋊C8—472no—
228C22⋊C8—942no—
229C22⋊C8—942no—
230C22⋊C8—941no—
231C22⋊C8—941no—
232#97—941no—
233#97—941no—
234C4×C8—471no—
235C4×C8—471no—
236C4×C8C2×Dic4711yescharacteristic
237C4×C8C2×Dic4711yescharacteristic
238#95—941no—
239#95—941no—
240#41—82no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4C22⋊Dic4712yes—
248C22×C4—22no—
249C22×C4—1881no—
250C22×C4D4×D4711yescharacteristic
251C22×C4#1062535211yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253M16—3762no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—941no—
257C2×C8—941no—
258C2×C8—942no—
259C2×C8—3762no—
260C2×C8—1881no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4C22⋊Dic4712yes—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—41no—
272C42—41no—
273C42—942no—
274C42—1881no—
275C42D4×D4711yescharacteristic
276C42#1062535211yescharacteristic
277C42C22×Dic4711yescharacteristic
278C42C2×C22⋊D4711yescharacteristic
279C23—22no—
280C23#1309399611yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—81no—
284Q8—81no—
285D4—42no—
286D4—41no—
287D4—41no—
288D4—81no—
289D4—82no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—3761no—
300C2×C4—3761no—
301C2×C4—1882no—
302C2×C4C2×C22⋊Dic4711yescharacteristic
303C2×C4D4×Dic4711yescharacteristic
304C2×C4D4×Dic4711yescharacteristic
305C2×C4#1309399211yescharacteristic
306C2×C4#1309399611yescharacteristic
307C2×C4#1309400511yescharacteristic
308C8—41no—
309C8—41no—
310C8—3761no—
311C8—1881no—
312C8—1882no—
313C8—1881no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22#1568900311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—81no—
328C4—81no—
329C4—3762no—
330C4#1568924011yescharacteristic
331C4#1568924511yescharacteristic
332C2—41no—
333C2—82no—
334C2#2024371711yescharacteristic
335C2#2024387711yescharacteristic
336C2#2024391111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3228C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3228D47
hypocenterC47#9279
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

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

NH ≅ G/Ntype
2. #20241900C2semidirect
4. #20241861C2semidirect
5. #20241250C2semidirect
6. #20241216C2semidirect
22. C47×#500C4semidirect
23. C47×#498C4semidirect
24. C47×#486C4semidirect
25. C47×#486C4semidirect
178. #500Dic47semidirect
179. #498Dic47semidirect
180. #486Dic47semidirect
181. #486Dic47semidirect
197. C47#9279semidirect

Maximal subgroups

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

subgroupindex
#202524642
#202419002
#202418962
#202418612
#202412502
#202412162
C47×#32282
#927947