Finite Group Databasebeta

← #26133225

Subgroups — #26133225

Order 12032.

574 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689365C211yesmaximal, characteristic
2C2×#107×D47C211yesmaximal, characteristic
3C2×#15688928C211yesmaximal, characteristic, complemented
4C2×#15688926C211yesmaximal, characteristic, complemented
5C2×(C2×D47)⋊C16C211yesmaximal, characteristic
6#20241479C211yesmaximal, characteristic, complemented
7C47×#1380C211yesmaximal, characteristic, complemented
8C22×Dic188C2211yescharacteristic
9C22×C8×D47C2211yescharacteristic
10C2×#13094004—21no—
11C2×C4⋊C4×D47—21no—
12C22×C47⋊C16C2211yescharacteristic
13#15689365—22no—
14#15689365C414yescomplemented
15C2×#13093958C2211yescharacteristic
16C2×#13093957—21no—
17#107×D47—22no—
18#107×D47—22no—
19C2×#13093947—21no—
20#15688928—22no—
21#15688926—22no—
22(C2×D47)⋊C16—22no—
23C22×C16×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C2×#107×C47C2211yescharacteristic
26C22×Dic94—21no—
27C22×C4×D47—41no—
28C22×C4×D47D411yescharacteristic
29C2×Dic188—21no—
30C2×Dic188—21no—
31C2×Dic188—21no—
32C2×Dic188—21no—
33C2×Dic188—21no—
34C2×Dic188C2×C412yescomplemented
35C22×C47⋊C8D411yescharacteristic
36C2×C8×D47—21no—
37C2×C8×D47—21no—
38C2×C8×D47—21no—
39C2×C8×D47—21no—
40C2×C8×D47C2×C412yes—
41C2×C8×D47D412yes—
42#13094004—24no—
43#13094004—42no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—41no—
46C2×Dic47⋊C4—41no—
47C4⋊C4×D47—24no—
48C4⋊C4×D47—42no—
49C2×(C2×D47)⋊C4—41no—
50C2×C47⋊C16—21no—
51C2×C47⋊C16—21no—
52C2×C47⋊C16—21no—
53#13093958—21no—
54#13093958D412yes—
55#13093957—24no—
56#13093947—24no—
57C22×C8×C47C2311yescharacteristic
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C16×C47—21no—
61C2×C16×C47—21no—
62C2×C16×C47—21no—
63#107×C47—21no—
64#107×C47—21no—
65#107×C47C2×C412yescomplemented
66#107×C47D412yes—
67C23×D47D811yescharacteristic
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—41no—
72C22×Dic47—41no—
73C22×Dic47D811yescharacteristic
74C2×C4×D47—21no—
75C2×C4×D47—21no—
76C2×C4×D47—21no—
77C2×C4×D47—21no—
78C2×C4×D47—44no—
79C2×C4×D47—82no—
80C2×C4×D47—82no—
81C2×C4×D47C22⋊C412yes—
82C2×C4×D47SD1612yes—
83C2×Dic94—22no—
84C2×Dic94—41no—
85C2×Dic94—41no—
86C2×Dic94—41no—
87C2×Dic94—41no—
88C2×Dic94—41no—
89C2×C8×C47C22×C411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C4⋊C4×C47—22no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99(C2×D47)⋊C4—44no—
100C4⋊Dic47—22no—
101C4⋊Dic47—41no—
102C4⋊Dic47—44no—
103Dic47⋊C4—44no—
104C2×C47⋊C8—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8C22⋊C412yes—
107Dic188—22no—
108Dic188—22no—
109Dic188—41no—
110Dic188—41no—
111C8×D47—22no—
112C8×D47—22no—
113C8×D47—22no—
114C8×D47—22no—
115C16×C47—41no—
116C47⋊C16—41no—
117C23×C47C2×D811yescharacteristic
118C22×D47—21no—
119C22×D47—21no—
120C22×D47—21no—
121C22×D47—21no—
122C22×D47#10212yes—
123C22×D47SD3212yes—
124C22×D47Q3212yes—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—81no—
128C2×C4×C47—81no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47C2×C22⋊C411yescharacteristic
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47#12311yescharacteristic
134C2×C4×C47C2×SD1611yescharacteristic
135C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C47#13711yescharacteristic
137C2×Dic47—21no—
138C2×Dic47—21no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—81no—
142C2×Dic47—81no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C2×Dic47#10212yes—
146C4×D47—22no—
147C4×D47—22no—
148C4×D47—22no—
149C4×D47—22no—
150C4×D47—88no—
151Dic94—42no—
152Dic94—42no—
153Dic94—81no—
154Dic94—81no—
155C8×C47C4×D411yescharacteristic
156C8×C47C4×D411yescharacteristic
157C8×C47#12111yescharacteristic
158C8×C47#12111yescharacteristic
159C47⋊C8—22no—
160#12993—471nomaximal, Sylow
161C22×C47C2×#10211yescharacteristic
162C22×C47#44711yescharacteristic
163C22×C47#48011yescharacteristic
164C22×C47C2×SD3211yescharacteristic
165C22×C47C2×Q3211yescharacteristic
166C22×C47#50911yescharacteristic
167C22×C47#51011yescharacteristic
168C2×D47—22no—
169C2×D47—22no—
170C2×D47—22no—
171C2×D47—22no—
172C2×D47—22no—
173C2×D47—22no—
174C2×D47#35814yescomplemented
175C4×C47—82no—
176C4×C47—82no—
177C4×C47#38611yescharacteristic
178C4×C47#41711yescharacteristic
179C4×C47#44811yescharacteristic
180C4×C47#48111yescharacteristic
181Dic47—22no—
182Dic47—82no—
183Dic47—82no—
184C2×#467—471no—
185C22×#107—471no—
186C2×#358—471no—
187C2×#358—471no—
188C2×C22⋊C16—471no—
189#1380—471no—
190#1380D4711yescharacteristic, complemented
191C2×C47#187411yescharacteristic
192C2×C47C2×#35811yescharacteristic
193C2×C47#213911yescharacteristic
194C2×C47#218711yescharacteristic
195C2×C47#218811yescharacteristic
196C2×C47#223511yescharacteristic
197C2×C47#223611yescharacteristic
198D47—24no—
199C22×Q16—471no—
200C23×C8—471no—
201C2×C22⋊Q8—941no—
202C22×C4⋊C4—941no—
203C22×C16—471no—
204C22×C16C2×D4711yescharacteristic
205#467—474no—
206#467—942no—
207C2×#107—471no—
208C2×#107—942no—
209C2×#107—941no—
210C2×#107—942no—
211C2×#107C2×D4711yescharacteristic
212C2×#107C2×D4711yescharacteristic
213C2×#103—941no—
214#358—942no—
215#358—942no—
216C22⋊C16—942no—
217C47#1299311yescharacteristic, complemented, Sylow
218C22×Q8—941no—
219C23×C4—471no—
220C23×C4—1881no—
221C2×Q16—472no—
222C2×Q16—941no—
223C2×Q16—941no—
224C2×Q16—941no—
225C2×Q16—941no—
226C2×Q16—941no—
227C22×C8—471no—
228C22×C8—472no—
229C22×C8—472no—
230C22×C8—941no—
231C22×C8—941no—
232C22×C8—941no—
233C22×C8—941no—
234C22×C8C22×D4711yescharacteristic
235C22⋊Q8—944no—
236C22⋊Q8—1882no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—944no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—1882no—
242C2×C4⋊C4—1881no—
243C2×C4⋊C4—1881no—
244C2×C22⋊C4—1881no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—941no—
249C2×C16—941no—
250C2×C16—941no—
251#107—21no—
252#107—21no—
253#107—472no—
254#107—944no—
255#107—941no—
256#107C4×D4712yescomplemented
257#107D18812yes—
258#103—944no—
259C24—471no—
260C2×Q8—942no—
261C2×Q8—1881no—
262C2×Q8—1881no—
263C2×Q8—1881no—
264C2×Q8—1881no—
265C2×Q8—1881no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—471no—
269C22×C4—472no—
270C22×C4—472no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—941no—
274C22×C4—941no—
275C22×C4—3762no—
276C22×C4—3762no—
277C22×C4—1884no—
278C22×C4—1881no—
279C22×C4—1881no—
280C22×C4D4×D4711yescharacteristic
281Q16—942no—
282Q16—942no—
283Q16—1881no—
284Q16—1881no—
285C2×C8—472no—
286C2×C8—941no—
287C2×C8—941no—
288C2×C8—942no—
289C2×C8—942no—
290C2×C8—942no—
291C2×C8—942no—
292C2×C8C2×C4×D4711yescharacteristic
293C2×C8C2×D18811yescharacteristic
294C2×C8#1062535011yescharacteristic
295C2×C8D4×D4711yescharacteristic
296C2×C8D4×D4711yescharacteristic
297C2×C8#1062535411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—942no—
303C4⋊C4—1884no—
304C4⋊C4—1884no—
305C4⋊C4—1881no—
306C22⋊C4—1884no—
307C16—41no—
308C16—1881no—
309C23—472no—
310C23—472no—
311C23—472no—
312C23—941no—
313C23—941no—
314C23—941no—
315C23—941no—
316C23D8×D4711yescharacteristic
317Q8—3761no—
318Q8—3761no—
319Q8—1882no—
320Q8—1882no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—472no—
328C2×C4—941no—
329C2×C4—941no—
330C2×C4—942no—
331C2×C4—942no—
332C2×C4—942no—
333C2×C4—942no—
334C2×C4—3768no—
335C2×C4—3761no—
336C2×C4—3761no—
337C2×C4—3761no—
338C2×C4—3761no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4C22⋊C4×D4711yescharacteristic
342C2×C4#1309399411yescharacteristic
343C2×C4#1309400911yescharacteristic
344C2×C4SD16×D4711yescharacteristic
345C2×C4#1309405111yescharacteristic
346C2×C4#1309405611yescharacteristic
347C8—942no—
348C8#1309398611yescharacteristic
349C8C4×D18811yescharacteristic
350C8#1309399711yescharacteristic
351C8#1309399911yescharacteristic
352C22—474no—
353C22—942no—
354C22—942no—
355C22—942no—
356C22—942no—
357C22—942no—
358C22—942no—
359C22#102×D4711yescharacteristic
360C22#1568925811yescharacteristic
361C22#1568943811yescharacteristic
362C22SD32×D4711yescharacteristic
363C22Q32×D4711yescharacteristic
364C22#1568955711yescharacteristic
365C22#1568956211yescharacteristic
366C4—82no—
367C4—82no—
368C4—942no—
369C4—3762no—
370C4—3762no—
371C4#1568900111yescharacteristic
372C4#1568912311yescharacteristic
373C4#1568926311yescharacteristic
374C4#1568944311yescharacteristic
375C2—944no—
376C2#2024370411yescharacteristic
377C2#358×D4711yescharacteristic
378C2#2024509711yescharacteristic
379C2#2024531811yescharacteristic
380C2#2024532311yescharacteristic
381C2#2024558711yescharacteristic
382C2#2024559211yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1380C2
socleC23×C47C2×D8
prime radicalC23×D47D8
cosocleC22×C8×C47C23
hypercenter#1380D47
hypocenterC47#12993
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#15688928C2semidirect
4. C2×#15688926C2semidirect
6. #20241479C2semidirect
7. C47×#1380C2semidirect
14. #15689365C4semidirect
34. C2×Dic188C2×C4semidirect
65. #107×C47C2×C4semidirect
174. C2×D47#358semidirect
190. #1380D47semidirect
217. C47#12993semidirect
256. #107C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156893652
C2×#107×D472
C2×#156889282
C2×#156889262
C2×(C2×D47)⋊C162
#202414792
C47×#13802
#1299347