Finite Group Databasebeta

← #26376334

Subgroups — #26376334

Order 12032.

2806 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D47C211yesmaximal, characteristic
2#20246261C218yesmaximal, complemented
3#20246253C218yesmaximal, complemented
4#20246004C212yesmaximal, complemented
5#20245988C214yesmaximal, complemented
6#20245769C214yesmaximal
7#20245767C212yesmaximal
8#20245745C211yesmaximal, characteristic, complemented
9C47×#2274C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C23×C4×D47C2212yes—
12C2×#13093983C2212yes—
13C2×#13093981C2218yes—
14C22×C4⋊Dic47C2211yescharacteristic
15C22×Dic47⋊C4C2212yes—
16C2×C4⋊C4×D47C4116yescomplemented
17C2×C4⋊C4×D47C2214yes—
18C2×C4⋊C4×D47C2214yes—
19#15689021C2218yescomplemented
20#15689019C2218yescomplemented
21#15689007C22116yescomplemented
22#15689005C22116yescomplemented
23#15688985C2218yes—
24#15688984C2214yes—
25#15688982C2218yes—
26#15688981C2218yes—
27#15688980C2214yes—
28#15688980C2214yes—
29#15688978C2214yes—
30C4⋊C4×Dic47C2218yescomplemented
31C4×(C2×D47)⋊C4C2214yes—
32C4×(C2×D47)⋊C4C2214yes—
33C4×(C2×D47)⋊C4C2218yes—
34C4×C22⋊Dic47C2214yes—
35C2×#117×C47C2212yes—
36C22×C4⋊C4×C47C2211yescharacteristic
37C4×C4⋊C4×C47C2218yescomplemented
38C4×C22⋊C4×C47C2214yes—
39C24×D47C2311yescharacteristic
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2312yes—
42C22×C4×D47—28no—
43C22×C4×D47—48no—
44C22×C4×D47C2×C418yes—
45C22×C4×D47C2×C4116yes—
46C22×C4×D47C2312yes—
47C22×C4×D47C2312yes—
48C22×C4×D47C2314yes—
49C22×C4×D47C2314yes—
50#13093983—28no—
51#13093981—416no—
52C2×C4⋊Dic47C2×C418yes—
53C2×C4⋊Dic47C2314yes—
54C2×C4⋊Dic47C2314yes—
55C2×Dic47⋊C4C2×C4116yes—
56C2×Dic47⋊C4C23116yes—
57C2×Dic47⋊C4C2318yes—
58C4⋊C4×D47C2×C4132yes—
59C4⋊C4×D47C2×C4132yes—
60C2×C22⋊Dic47C2314yes—
61C2×(C2×D47)⋊C4C2318yes—
62C2×(C2×D47)⋊C4C2318yes—
63C2×C4×Dic47C2314yes—
64C2×C4×Dic47C2318yes—
65C47⋊#95C2318yes—
66C47⋊#95C2318yes—
67C47⋊#95C2318yes—
68C47⋊#95C23116yes—
69C23×C4×C47C2312yes—
70C23×C4×C47C2311yescharacteristic
71#117×C47—28no—
72C2×C4⋊C4×C47C2×C418yes—
73C2×C4⋊C4×C47C2314yes—
74C2×C4⋊C4×C47C2314yes—
75C2×C22⋊C4×C47C2314yes—
76C2×C42×C47C2314yes—
77C2×C42×C47C2318yes—
78#95×C47C2318yes—
79C24×C47C2411yescharacteristic
80C23×D47—24no—
81C23×D47—24no—
82C23×D47—24no—
83C23×D47C22×C414yes—
84C23×D47C2411yescharacteristic
85C23×D47C2411yescharacteristic
86C22×C4×C47—44no—
87C22×C4×C47C22×C418yes—
88C22×C4×C47C22×C414yes—
89C22×C4×C47#4118yes—
90C22×C4×C47C2418yes—
91C22×C4×C47C2414yes—
92C22×C4×C47C2412yes—
93C22×Dic47—24no—
94C22×Dic47—44no—
95C22×Dic47C22×C414yes—
96C22×Dic47C22×C418yes—
97C22×Dic47C2418yes—
98C22×Dic47C2412yes—
99C22×Dic47C2414yes—
100C2×C4×D47—216no—
101C2×C4×D47—232no—
102C2×C4×D47—24no—
103C2×C4×D47—24no—
104C2×C4×D47—24no—
105C2×C4×D47—24no—
106C2×C4×D47—432no—
107C2×C4×D47—48no—
108C2×C4×D47C22×C4116yes—
109C2×C4×D47C22×C418yes—
110C2×C4×D47C22×C418yes—
111C2×C4×D47C22×C4116yes—
112C4⋊C4×C47—28no—
113C4⋊C4×C47C22×C4116yes—
114C22⋊C4×C47—28no—
115C42×C47—28no—
116C42×C47—48no—
117C22⋊Dic47—28no—
118(C2×D47)⋊C4—216no—
119(C2×D47)⋊C4—216no—
120C4⋊Dic47—28no—
121C4⋊Dic47C22×C4116yes—
122Dic47⋊C4—416no—
123Dic47⋊C4C22×C4132yes—
124C4×Dic47—28no—
125C4×Dic47—48no—
126C23×C47—22no—
127C23×C47—22no—
128C23×C47C23×C412yes—
129C23×C47C2×#4114yes—
130C23×C47C2511yescharacteristic
131C22×D47—232no—
132C22×D47—28no—
133C22×D47—22no—
134C22×D47—22no—
135C22×D47—22no—
136C22×D47—22no—
137C22×D47—28no—
138C22×D47—28no—
139C22×D47C23×C412yes—
140C22×D47C23×C412yes—
141C22×D47H(5,𝔽2)14yes—
142C22×D472−1+414yes—
143C2×C4×C47—28no—
144C2×C4×C47—28no—
145C2×C4×C47—28no—
146C2×C4×C47—28no—
147C2×C4×C47—48no—
148C2×C4×C47#117116yescomplemented
149C2×C4×C47C23×C418yes—
150C2×C4×C47C23×C414yes—
151C2×C4×C47C2×#4114yes—
152C2×C4×C47C2×#4114yes—
153C2×Dic47—28no—
154C2×Dic47—22no—
155C2×Dic47—22no—
156C2×Dic47—28no—
157C2×Dic47—28no—
158C2×Dic47—28no—
159C2×Dic47—28no—
160C2×Dic47—48no—
161C2×Dic47C23×C414yes—
162C2×Dic47C23×C418yes—
163C4×D47—232no—
164C4×D47—216no—
165C4×D47—216no—
166C4×D47—416no—
167#34178—471nomaximal, Sylow
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—28no—
171C22×C47C2×#11714yes—
172C22×C47C24×C411yescharacteristic
173C22×C47C22×#4112yes—
174C22×C47C2×H(5,𝔽2)12yes—
175C22×C47C2×2−1+412yes—
176C2×D47—232no—
177C2×D47—22no—
178C2×D47—22no—
179C2×D47—22no—
180C2×D47—22no—
181C2×D47—28no—
182C2×D47—28no—
183C2×D47#51814yes—
184C2×D47#51914yes—
185C4×C47—28no—
186C4×C47—48no—
187C4×C47C2×#11718yes—
188Dic47—24no—
189Dic47—28no—
190Dic47—48no—
191C23×C4⋊C4—471no—
192#2345—478no—
193#2344—478no—
194#2316—472no—
195#2313—474no—
196#2280—474no—
197#2280—472no—
198#2274—471no—
199#2274D4711yescharacteristic, complemented
200C2×C47—24no—
201C2×C47C22×#11711yescharacteristic
202C2×C47C2×#51811yescharacteristic
203C2×C47C2×#51911yescharacteristic
204C2×C47#349214yes—
205D47—24no—
206D47—24no—
207C24×C4—471no—
208C24×C4—472no—
209C2×#117—472no—
210C2×#117—478no—
211C2×#117C2×D4712yes—
212C22×C4⋊C4—472no—
213C22×C4⋊C4—471no—
214C22×C4⋊C4—4716no—
215C22×C4⋊C4—474no—
216C22×C4⋊C4—474no—
217C22×C4⋊C4C2×D4711yescharacteristic
218#389—478no—
219#389—478no—
220#387—4716no—
221#387—4716no—
222#381—474no—
223#381—478no—
224#380—474no—
225#380—474no—
226#380—478no—
227#380—478no—
228#379—474no—
229C4×C4⋊C4—478no—
230C4×C4⋊C4C2×D4718yescomplemented
231C4×C22⋊C4—474no—
232C4×C22⋊C4—474no—
233C4×C22⋊C4—474no—
234C4×C22⋊C4—478no—
235C4×C22⋊C4C2×D4714yes—
236C47#3417811yescharacteristic, complemented, Sylow
237C25—471no—
238C23×C4—471no—
239C23×C4—472no—
240C23×C4—478no—
241C23×C4—474no—
242C23×C4—472no—
243C23×C4—472no—
244C23×C4—4716no—
245C23×C4—474no—
246C23×C4—948no—
247C23×C4—1888no—
248C23×C4C22×D4712yes—
249C23×C4C22×D4711yescharacteristic
250#117—28no—
251#117—948no—
252#117—18816no—
253C2×C4⋊C4—4716no—
254C2×C4⋊C4—478no—
255C2×C4⋊C4—478no—
256C2×C4⋊C4—474no—
257C2×C4⋊C4—4732no—
258C2×C4⋊C4—4732no—
259C2×C4⋊C4—474no—
260C2×C4⋊C4—4716no—
261C2×C4⋊C4C4×D4718yes—
262C2×C4⋊C4C22×D4714yes—
263C2×C4⋊C4C22×D4714yes—
264C2×C22⋊C4—474no—
265C2×C22⋊C4—478no—
266C2×C22⋊C4—478no—
267C2×C22⋊C4C22×D4714yes—
268C2×C42—474no—
269C2×C42—478no—
270C2×C42C22×D4714yes—
271C2×C42C22×D4718yes—
272#95—478no—
273#95—478no—
274#95—478no—
275#95—4716no—
276#95C22×D4718yes—
277C24—471no—
278C24—471no—
279C24—474no—
280C24—944no—
281C24—944no—
282C24—944no—
283C24C23×D4711yescharacteristic
284C22×C4—44no—
285C22×C4—474no—
286C22×C4—472no—
287C22×C4—474no—
288C22×C4—478no—
289C22×C4—4716no—
290C22×C4—478no—
291C22×C4—478no—
292C22×C4—4716no—
293C22×C4—478no—
294C22×C4—9416no—
295C22×C4—944no—
296C22×C4—9432no—
297C22×C4—944no—
298C22×C4—944no—
299C22×C4—944no—
300C22×C4—944no—
301C22×C4—1888no—
302C22×C4—18832no—
303C22×C4—1884no—
304C22×C4C2×C4×D4718yes—
305C22×C4C2×C4×D4714yes—
306C22×C4#1062535018yes—
307C22×C4C23×D4718yes—
308C22×C4C23×D4714yes—
309C22×C4C23×D4712yes—
310C4⋊C4—28no—
311C4⋊C4—4732no—
312C4⋊C4—4716no—
313C4⋊C4—948no—
314C4⋊C4—18816no—
315C4⋊C4C2×C4×D47116yes—
316C22⋊C4—28no—
317C22⋊C4—9416no—
318C22⋊C4—9416no—
319C22⋊C4—948no—
320C42—28no—
321C42—48no—
322C42—948no—
323C42—1888no—
324C23—22no—
325C23—22no—
326C23—472no—
327C23—472no—
328C23—474no—
329C23—474no—
330C23—948no—
331C23—948no—
332C23—9432no—
333C23—948no—
334C23—942no—
335C23—942no—
336C23—942no—
337C23—942no—
338C23C22×C4×D4712yes—
339C23C2×#1062535014yes—
340C23C24×D4711yescharacteristic
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—48no—
346C2×C4—474no—
347C2×C4—478no—
348C2×C4—9432no—
349C2×C4—948no—
350C2×C4—942no—
351C2×C4—942no—
352C2×C4—948no—
353C2×C4—9416no—
354C2×C4—9416no—
355C2×C4—948no—
356C2×C4—948no—
357C2×C4—948no—
358C2×C4—18816no—
359C2×C4—1888no—
360C2×C4#13093981116yescomplemented
361C2×C4C22×C4×D4718yes—
362C2×C4C22×C4×D4714yes—
363C2×C4C2×#1062535014yes—
364C2×C4C2×#1062535014yes—
365C22—22no—
366C22—22no—
367C22—28no—
368C22—474no—
369C22—474no—
370C22—9432no—
371C22—942no—
372C22—942no—
373C22—942no—
374C22—942no—
375C22—948no—
376C22—948no—
377C22C2×#1309398114yes—
378C22C23×C4×D4711yescharacteristic
379C22C22×#1062535012yes—
380C22H(5,𝔽2)×D4712yes—
381C222−1+4×D4712yes—
382C4—28no—
383C4—48no—
384C4—944no—
385C4—948no—
386C4—1888no—
387C4C2×#1309398118yes—
388C2—24no—
389C2—944no—
390C2—944no—
391C2C22×#1309398111yescharacteristic
392C2#518×D4711yescharacteristic
393C2#519×D4711yescharacteristic
394C2#2025709114yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2274C2
socle, cosocleC23×C47C25
prime radicalC24×D47C23
hypercenter#2274D47
hypocenterC47#34178
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2274

Complements

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

NH ≅ G/Ntype
2. #20246261C2semidirect
3. #20246253C2semidirect
4. #20246004C2semidirect
5. #20245988C2semidirect
8. #20245745C2semidirect
9. C47×#2274C2semidirect
16. C2×C4⋊C4×D47C4semidirect
19. #15689021C22semidirect
20. #15689019C22semidirect
21. #15689007C22semidirect
22. #15689005C22semidirect
30. C4⋊C4×Dic47C22semidirect
37. C4×C4⋊C4×C47C22semidirect
148. C2×C4×C47#117semidirect
199. #2274D47semidirect
230. C4×C4⋊C4C2×D47semidirect
236. C47#34178semidirect
360. C2×C4#13093981semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D472
#202462612
#202462532
#202460042
#202459882
#202457692
#202457672
#202457452
C47×#22742
#3417847