Finite Group Databasebeta

← #26135124

Subgroups — #26135124

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093967C211yesmaximal, characteristic, complemented
2C2×#15689498C211yesmaximal, characteristic
3C2×#15688925C211yesmaximal, characteristic, complemented
4C2×#15688924C211yesmaximal, characteristic, complemented
5#20244250C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C22×SD32×C47C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C22×#10625334—21no—
10C2×#13094020—21no—
11C2×#13093967—22no—
12C2×#13093967—22no—
13C2×#13093967—22no—
14C22×C47⋊C16C2211yescharacteristic
15#15689498—22no—
16#15689498—22no—
17C2×#13093958C2211yescharacteristic
18C2×#13093948—21no—
19C2×C8×Dic47C2211yescharacteristic
20#15689030—21no—
21#15688925—22no—
22#15688924—22no—
23C22×Q16×C47C2211yescharacteristic
24C22×D8×C47C2211yescharacteristic
25C2×SD32×C47—22no—
26C2×SD32×C47—22no—
27C2×SD32×C47—22no—
28C22×C16×C47C2211yescharacteristic
29C22×D188—21no—
30C22×C22⋊D47—42no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376—21no—
36C2×D376D412yes—
37C2×#10625334—24no—
38C2×#10625334—42no—
39C2×#10625334—42no—
40C22×C47⋊C8D411yescharacteristic
41#13094020—24no—
42#13094020—42no—
43C2×C4⋊Dic47—21no—
44C2×C4×Dic47D411yescharacteristic
45#13093967—28no—
46C2×C47⋊C16—21no—
47C2×C47⋊C16—21no—
48C2×C47⋊C16D412yescomplemented
49#13093958—21no—
50#13093958—21no—
51#13093948—42no—
52C8×Dic47—21no—
53C8×Dic47—21no—
54C47⋊#95—42no—
55C22×Q8×C47—21no—
56C22×D4×C47—21no—
57C2×Q16×C47—21no—
58C2×Q16×C47—21no—
59C2×Q16×C47—21no—
60C2×Q16×C47—21no—
61C2×Q16×C47—21no—
62C2×Q16×C47D412yescomplemented
63C2×D8×C47—21no—
64C2×D8×C47—21no—
65C2×D8×C47—21no—
66C2×D8×C47—21no—
67C2×D8×C47—21no—
68C2×D8×C47D412yes—
69C22×C8×C47C2311yescharacteristic
70SD32×C47—28no—
71C2×C16×C47—21no—
72C2×C16×C47—21no—
73C2×C16×C47D412yescomplemented
74C24×C47—41no—
75C23×D47—41no—
76C2×Q8×C47—22no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C2×Q8×C47—41no—
80C2×Q8×C47—41no—
81C2×Q8×C47—41no—
82C2×D4×C47—22no—
83C2×D4×C47—41no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47C2×D411yescharacteristic
90C2×C22⋊D47—48no—
91C2×C22⋊D47—84no—
92C2×C22⋊D47—84no—
93C22×Dic47—41no—
94C22×Dic47D812yes—
95C2×D188—22no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188—41no—
101Q16×C47—22no—
102Q16×C47—22no—
103Q16×C47—41no—
104Q16×C47—41no—
105D8×C47—22no—
106D8×C47—22no—
107D8×C47—41no—
108D8×C47—41no—
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47C2×D411yescharacteristic
113C2×C8×C47C2×D411yescharacteristic
114C2×C8×C47#4111yescharacteristic
115#10625334—48no—
116C4⋊Dic47—41no—
117C4⋊Dic47—41no—
118C4×Dic47—21no—
119C4×Dic47—21no—
120C2×C47⋊C8—21no—
121C2×C47⋊C8—21no—
122C2×C47⋊C8D812yes—
123D376—22no—
124D376—22no—
125D376—41no—
126D376—41no—
127C16×C47—22no—
128C47⋊C16—22no—
129C23×C47—42no—
130C23×C47—81no—
131C23×C47—81no—
132C23×C47—82no—
133C23×C47—82no—
134C23×C47C2×D811yescharacteristic
135C22×D47—42no—
136C22×D47—81no—
137C22×D47—81no—
138C22×D47—82no—
139C22×D47—82no—
140Q8×C47—42no—
141Q8×C47—42no—
142Q8×C47—81no—
143Q8×C47—81no—
144D4×C47—42no—
145D4×C47—42no—
146D4×C47—81no—
147D4×C47—81no—
148C2×C4×C47—42no—
149C2×C4×C47—81no—
150C2×C4×C47—81no—
151C2×C4×C47#12011yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12711yescharacteristic
154C2×C4×C47C2×D811yescharacteristic
155C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C22⋊D47—816no—
158C2×Dic47—22no—
159C2×Dic47—22no—
160C2×Dic47—81no—
161C2×Dic47—82no—
162C2×Dic47SD3214yescomplemented
163D188—42no—
164D188—42no—
165D188—81no—
166D188—81no—
167C8×C47#12011yescharacteristic
168C8×C47#12011yescharacteristic
169C8×C47#12111yescharacteristic
170C8×C47#12111yescharacteristic
171C47⋊C8—22no—
172#13328—471nomaximal, Sylow
173C22×C47—84no—
174C22×C47—82no—
175C22×C47—162no—
176C22×C47—161no—
177C22×C47—161no—
178C22×C47#44711yescharacteristic
179C22×C47#45911yescharacteristic
180C22×C47#49311yescharacteristic
181C22×C47C2×SD3212yes—
182C22×C47#50912yes—
183C2×D47—84no—
184C2×D47—82no—
185C2×D47—162no—
186C2×D47—161no—
187C2×D47—161no—
188C4×C47—82no—
189C4×C47#39211yescharacteristic
190C4×C47#44711yescharacteristic
191C4×C47#45911yescharacteristic
192C4×C47#49611yescharacteristic
193Dic47—24no—
194Dic47—161no—
195C22×SD32—471no—
196C22×SD32D4711yescharacteristic, complemented
197C2×#493—471no—
198C2×#357—471no—
199C2×#357—471no—
200#1969—471no—
201#1374—471no—
202C2×C47—162no—
203C2×C47#201111yescharacteristic
204C2×C47#218712yes—
205C2×C47#220712yes—
206C2×C47#224811yescharacteristic
207C2×C47#225011yescharacteristic
208D47—162no—
209C22×Q16C2×D4711yescharacteristic
210C22×D8—471no—
211C22×D8—941no—
212C22×D8C2×D4711yescharacteristic
213C2×#127—941no—
214C2×SD32—22no—
215C2×SD32—22no—
216C2×SD32—22no—
217C2×SD32—942no—
218C2×SD32—942no—
219C2×SD32—942no—
220C22×C16—471no—
221C22×C16C2×D4711yescharacteristic
222#493—942no—
223#493—942no—
224C2×#107—471no—
225C2×#103—941no—
226C2×C4×C8—471no—
227#391—941no—
228#357—942no—
229#357—942no—
230C47#1332811yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×D4—21no—
233C22×D4—941no—
234C22×D4—1882no—
235C2×Q16—21no—
236C2×Q16—21no—
237C2×Q16—21no—
238C2×Q16—21no—
239C2×Q16—21no—
240C2×Q16C22⋊D4712yescomplemented
241C2×D8—21no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—21no—
245C2×D8—21no—
246C2×D8—472no—
247C2×D8—944no—
248C2×D8—941no—
249C2×D8—941no—
250C2×D8—941no—
251C2×D8—941no—
252C2×D8—941no—
253C2×D8—1882no—
254C2×D8—1882no—
255C2×D8C22⋊D4712yes—
256C22×C8—471no—
257C22×C8C22×D4711yescharacteristic
258#127—944no—
259#127—1882no—
260C2×C4⋊C4—941no—
261C2×C42—471no—
262SD32—28no—
263SD32—948no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—472no—
267C2×C16—941no—
268C2×C16—941no—
269C2×C16C22⋊D4712yescomplemented
270#107—941no—
271#107—941no—
272#103—1882no—
273C4×C8—941no—
274C4×C8—941no—
275#95—1882no—
276C24—41no—
277C24—1881no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—942no—
291C2×D4—3764no—
292C2×D4—3764no—
293C2×D4—1888no—
294C2×D4—1881no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1881no—
298C2×D4—1881no—
299C22×C4—41no—
300C22×C4—472no—
301C22×C4—1881no—
302C22×C4D4×D4711yescharacteristic
303Q16—22no—
304Q16—22no—
305Q16—41no—
306Q16—41no—
307D8—22no—
308D8—22no—
309D8—41no—
310D8—41no—
311D8—942no—
312D8—942no—
313D8—1888no—
314D8—1881no—
315D8—1881no—
316C2×C8—472no—
317C2×C8—941no—
318C2×C8—941no—
319C2×C8D4×D4711yescharacteristic
320C2×C8D4×D4711yescharacteristic
321C2×C8#1062535211yescharacteristic
322C2×C8C2×C22⋊D4711yescharacteristic
323C2×C8C2×C22⋊D4711yescharacteristic
324C2×C8C2×C22⋊D4711yescharacteristic
325C4⋊C4—1881no—
326C4⋊C4—1881no—
327C42—941no—
328C42—941no—
329C16—22no—
330C16—942no—
331C23—42no—
332C23—81no—
333C23—82no—
334C23—82no—
335C23—81no—
336C23—3761no—
337C23—3762no—
338C23—3762no—
339C23—3761no—
340C23—1882no—
341C23D8×D4711yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346D4—42no—
347D4—42no—
348D4—81no—
349D4—81no—
350D4—37616no—
351D4—3761no—
352D4—3761no—
353D4—1882no—
354D4—1882no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—474no—
359C2×C4—942no—
360C2×C4—942no—
361C2×C4—3761no—
362C2×C4—3762no—
363C2×C4#1309399211yescharacteristic
364C2×C4#1309399611yescharacteristic
365C2×C4#1309402011yescharacteristic
366C2×C4D8×D4711yescharacteristic
367C2×C4#1309404811yescharacteristic
368C2×C4#1309404811yescharacteristic
369C8—942no—
370C8#1309399211yescharacteristic
371C8#1309399311yescharacteristic
372C8#1309399511yescharacteristic
373C8#1309399611yescharacteristic
374C22—84no—
375C22—82no—
376C22—161no—
377C22—161no—
378C22—162no—
379C22—3764no—
380C22—3762no—
381C22—7521no—
382C22—7521no—
383C22—7522no—
384C22#1568925511yescharacteristic
385C22#1568931611yescharacteristic
386C22#1568949811yescharacteristic
387C22SD32×D4712yes—
388C22#1568956012yes—
389C4—82no—
390C4—944no—
391C4—7521no—
392C4#1568903111yescharacteristic
393C4#1568925511yescharacteristic
394C4#1568931611yescharacteristic
395C4#1568951111yescharacteristic
396C2—162no—
397C2—7522no—
398C2#2024445711yescharacteristic
399C2#2024531512yes—
400C2#2024542612yes—
401C2#2024564211yescharacteristic
402C2#2024565311yescharacteristic
subgroupG/H
centerC23D8×D47
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC22×SD32×C47C2
socleC23×C47C2×D8
prime radicalC2×#15689498C2
hypercenterC22×SD32D47
hypocenterC47#13328
LCSG ▷ C22×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C22×#13093967C2semidirect
3. C2×#15688925C2semidirect
4. C2×#15688924C2semidirect
5. #20244250C2semidirect
6. #20241465C2semidirect
7. C22×SD32×C47C2semidirect
48. C2×C47⋊C16D4semidirect
62. C2×Q16×C47D4semidirect
73. C2×C16×C47D4semidirect
162. C2×Dic47SD32semidirect
196. C22×SD32D47semidirect
230. C47#13328semidirect
240. C2×Q16C22⋊D47semidirect
269. C2×C16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939672
C2×#156894982
C2×#156889252
C2×#156889242
#202442502
#202414652
C22×SD32×C472
#1332847