Finite Group Databasebeta

← #26118927

Subgroups — #26118927

Order 12032.

626 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 113 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#20254516C211yesmaximal, characteristic
2#20254313C211yesmaximal, characteristic
3#20244216C211yesmaximal, characteristic
4#20242073C211yesmaximal, characteristic, complemented
5#20242069C211yesmaximal, characteristic, complemented
6#20241221C211yesmaximal, characteristic, complemented
7C47×#2868C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689519—21no—
10#15689497—21no—
11#15689473—21no—
12#15689471—21no—
13#15689441—22no—
14#15689410—22no—
15#15689325—22no—
16#15689262—22no—
17#15689176C2211yescharacteristic
18#15689176C2211yescharacteristic
19#15689147C2211yescharacteristic
20C2×#13093952—22no—
21#15688872—22no—
22#15688869—22no—
23C2×C4⋊Q8×C47—22no—
24C2×C4⋊Q8×C47C2211yescharacteristic
25C2×C4⋊Q8×C47C2211yescharacteristic
26C47×#384—22no—
27C47×#384—22no—
28C43×C47C2211yescharacteristic
29C2×#10625350—21no—
30C2×#10625336—24no—
31C2×#10625329—21no—
32#13094024D411yescharacteristic
33#13094021D411yescharacteristic
34#13094012—21no—
35#13093998—21no—
36#13093998—21no—
37C4×D188—21no—
38C4×D188—21no—
39#13093955—24no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)D412yes—
43C4⋊(C47⋊C8)D412yes—
44#13093952—44no—
45#13093948—24no—
46#13093944—24no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8—21no—
51C22×Q8×C47—21no—
52C22×Q8×C47D412yes—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47—21no—
61C4⋊Q8×C47—48no—
62C2×C4⋊C4×C47—28no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47D412yes—
67C2×C42×C47—21no—
68C2×C42×C47—21no—
69C2×C42×C47D412yes—
70C2×C42×C47C2311yescharacteristic
71C2×Q8×C47—22no—
72C2×Q8×C47—22no—
73C2×Q8×C47—44no—
74C2×Q8×C47—44no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C2×C22⋊D47—41no—
84#10625350—42no—
85#10625350—42no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×C4×D47—41no—
90C2×Dic94—21no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—416no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C42×C47—24no—
101C42×C47—24no—
102C42×C47—22no—
103C42×C47SD1614yescomplemented
104C42×C47C2×D411yescharacteristic
105C42×C47C2×D411yescharacteristic
106C42×C47C2×D411yescharacteristic
107C42×C47#4111yescharacteristic
108#10625336—84no—
109(C2×D47)⋊C4—41no—
110(C2×D47)⋊C4—41no—
111C4⋊Dic47—21no—
112C4⋊Dic47—21no—
113#10625329—42no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—22no—
116C2×C47⋊C8—41no—
117C23×C47#12011yescharacteristic
118C22×D47—41no—
119Q8×C47—44no—
120Q8×C47—82no—
121Q8×C47—82no—
122Q8×C47—82no—
123C2×C4×C47—24no—
124C2×C4×C47—24no—
125C2×C4×C47—24no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—44no—
129C2×C4×C47—44no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47#12011yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12711yescharacteristic
138C2×C4×C47C2×SD1612yes—
139C2×C4×C47#13712yes—
140C22⋊D47—82no—
141C2×Dic47—41no—
142D188—42no—
143D188—42no—
144D188—42no—
145C4×D47—81no—
146C4×D47—81no—
147Dic94—42no—
148C47⋊C8—42no—
149C47⋊C8—42no—
150C47⋊C8—81no—
151#10283—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47#39211yescharacteristic
155C22×C47#44812yes—
156C2×D47—81no—
157C2×D47—81no—
158C2×D47—81no—
159C4×C47—28no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—82no—
163C4×C47—82no—
164C4×C47#45312yes—
165C4×C47#45612yes—
166C4×C47#46112yes—
167C4×C47#49211yescharacteristic
168C4×C47#49711yescharacteristic
169Dic47—81no—
170#3227—471no—
171#3206—471no—
172#2868D4711yescharacteristic, complemented
173#1962—471no—
174#1549—471no—
175#1548—471no—
176#1276—471no—
177C2×C47—22no—
178C2×C47#200211yescharacteristic
179C2×C47#200511yescharacteristic
180C2×C47#201311yescharacteristic
181D47—161no—
182#532—471no—
183C2×C4⋊Q8—22no—
184C2×C4⋊Q8C2×D4711yescharacteristic
185C2×C4⋊Q8C2×D4711yescharacteristic
186#498—941no—
187#492—941no—
188#487—941no—
189#486—941no—
190#481—942no—
191#476—942no—
192#461—942no—
193#448—942no—
194#431—471no—
195#431—471no—
196#423—471no—
197C2×#104—942no—
198#384—22no—
199#384—22no—
200C43C2×D4711yescharacteristic
201#333—942no—
202#332—942no—
203C47#1028311yescharacteristic, complemented, Sylow
204C2×#41—941no—
205C22×Q8—21no—
206C22×Q8C22⋊D4712yes—
207C2×SD16—944no—
208C2×M16—941no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—48no—
218C4⋊Q8—471no—
219#127—471no—
220#124—941no—
221#121—941no—
222#121—941no—
223C4×D4—941no—
224C4×D4—941no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C22⋊D4712yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C22⋊D4712yes—
233C2×C42C22×D4711yescharacteristic
234#106—944no—
235C4⋊C8—472no—
236C4⋊C8—472no—
237C4⋊C8—941no—
238C4⋊C8—941no—
239#104—1884no—
240#103—944no—
241#102—944no—
242C22⋊C8—941no—
243C22⋊C8—941no—
244C4×C8—941no—
245C4×C8—941no—
246#41—1882no—
247#41—1882no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—941no—
255C2×D4—941no—
256C2×D4—941no—
257C2×D4—941no—
258C2×D4—1881no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—1881no—
263C22×C4C2×C22⋊D4711yescharacteristic
264C22×C4C2×C22⋊D4711yescharacteristic
265C22×C4C2×C22⋊D4711yescharacteristic
266SD16—3764no—
267M16—1882no—
268C2×C8—942no—
269C2×C8—942no—
270C2×C8—1881no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—416no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—941no—
281C4⋊C4—941no—
282C22⋊C4—1881no—
283C22⋊C4—1881no—
284C42—24no—
285C42—24no—
286C42—22no—
287C42#1062533614yescomplemented
288C42D4×D4711yescharacteristic
289C42D4×D4711yescharacteristic
290C42D4×D4711yescharacteristic
291C42#1062535211yescharacteristic
292C23—1881no—
293C23#1309399311yescharacteristic
294Q8—44no—
295Q8—82no—
296Q8—82no—
297Q8—82no—
298Q8—1882no—
299D4—3762no—
300D4—1882no—
301D4—1882no—
302D4—1882no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—3761no—
313C2×C4—3761no—
314C2×C4—1881no—
315C2×C4#1309399211yescharacteristic
316C2×C4#1309399211yescharacteristic
317C2×C4#1309399511yescharacteristic
318C2×C4#1309399611yescharacteristic
319C2×C4#1309399611yescharacteristic
320C2×C4#1309402011yescharacteristic
321C2×C4C2×#1062533612yes—
322C2×C4#1309405512yes—
323C8—3761no—
324C8—1882no—
325C8—1882no—
326C22—21no—
327C22—21no—
328C22—3761no—
329C22—3761no—
330C22—3761no—
331C22#1568903111yescharacteristic
332C22#1568926212yes—
333C4—28no—
334C4—21no—
335C4—21no—
336C4—82no—
337C4—82no—
338C4—3761no—
339C4#1568929012yes—
340C4#1568930512yes—
341C4#1568932512yes—
342C4#1568949711yescharacteristic
343C4#1568951511yescharacteristic
344C2—22no—
345C2—7521no—
346C2#2024440311yescharacteristic
347C2#2024442411yescharacteristic
348C2#2024447111yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2868C2
socleC22×C47#392
prime radical#15689673C22
hypercenter#2868D47
hypocenterC47#10283
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
4. #20242073C2semidirect
5. #20242069C2semidirect
6. #20241221C2semidirect
7. C47×#2868C2semidirect
103. C42×C47SD16semidirect
172. #2868D47semidirect
203. C47#10283semidirect
287. C42#10625336semidirect

Maximal subgroups

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

subgroupindex
#202545162
#202543132
#202442162
#202420732
#202420692
#202412212
C47×#28682
#1028347