Finite Group Databasebeta

← #26112446

Subgroups — #26112446

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689404C211yesmaximal, characteristic
2C2×#15689188C211yesmaximal, characteristic
3#20244189C211yesmaximal, characteristic, complemented
4#20243385C211yesmaximal, characteristic
5C2×#15688870C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C47×#1296C211yesmaximal, characteristic, complemented
8C22×C8×D47—21no—
9C2×#13094024C2211yescharacteristic
10C2×C4×D188C2211yescharacteristic
11#15689404—22no—
12#15689404C414yescomplemented
13#15689188—22no—
14#15689188—22no—
15C2×#13093958—21no—
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093947—21no—
18C2×#13093947—21no—
19C2×#13093946—21no—
20C2×#13093944—21no—
21C2×(C2×D47)⋊C8—21no—
22C2×C4×C47⋊C8C2211yescharacteristic
23#15688885—21no—
24#15688870—22no—
25C2×C4⋊C8×C47C2211yescharacteristic
26C2×C4⋊C8×C47C2211yescharacteristic
27C47×#384C2211yescharacteristic
28C22×Dic94—21no—
29C22×D188D411yescharacteristic
30C22×C4×D47—21no—
31C22×C47⋊C8—21no—
32C22×C47⋊C8D411yescharacteristic
33C22×C47⋊C8D411yescharacteristic
34C2×C8×D47—24no—
35C2×C8×D47—42no—
36C2×C8×D47—42no—
37#13094024—21no—
38#13094024—21no—
39#13094024—21no—
40#13094024C2×C412yes—
41C4×D188—21no—
42C4×D188—21no—
43C4×D188C2×C412yes—
44C4×D188D412yes—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47D411yescharacteristic
48C2×(C2×D47)⋊C4—21no—
49#13093958—24no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)—21no—
52#13093947—42no—
53#13093947—42no—
54#13093946—24no—
55#13093944—42no—
56(C2×D47)⋊C8—24no—
57C4×C47⋊C8—21no—
58C4×C47⋊C8—21no—
59C22×C8×C47—21no—
60C22×C8×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C42×C47C2311yescharacteristic
65C4⋊C8×C47—21no—
66C4⋊C8×C47—21no—
67C4⋊C8×C47C2×C412yes—
68C4⋊C8×C47D412yes—
69C23×D47—21no—
70C22×C4×C47—41no—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4111yescharacteristic
74C22×Dic47—21no—
75C22×Dic47—41no—
76C2×D188—21no—
77C2×D188—21no—
78C2×D188—21no—
79C2×D188C22⋊C412yes—
80C2×D188SD1612yes—
81C2×D188Q1612yes—
82C2×C4×D47—22no—
83C2×C4×D47—22no—
84C2×C4×D47—41no—
85C2×C4×D47—41no—
86C2×C4×D47—41no—
87C2×C4×D47—41no—
88C2×Dic94—22no—
89C2×Dic94—41no—
90C2×Dic94—41no—
91C2×Dic94—41no—
92C2×Dic94—41no—
93C2×Dic94—41no—
94C2×C8×C47—22no—
95C2×C8×C47—22no—
96C2×C8×C47—41no—
97C2×C8×C47—41no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—81no—
105C42×C47C22×C411yescharacteristic
106C42×C47C2×D411yescharacteristic
107C42×C47C2×D411yescharacteristic
108C42×C47#4111yescharacteristic
109(C2×D47)⋊C4—22no—
110(C2×D47)⋊C4—22no—
111C4⋊Dic47—21no—
112C4⋊Dic47—22no—
113C4⋊Dic47—22no—
114C4⋊Dic47—41no—
115C4⋊Dic47—41no—
116C4⋊Dic47C22⋊C412yes—
117C2×C47⋊C8—21no—
118C2×C47⋊C8—21no—
119C2×C47⋊C8—21no—
120C2×C47⋊C8—21no—
121C2×C47⋊C8—21no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8D812yes—
126C8×D47—48no—
127C23×C47#12111yescharacteristic
128C22×D47—22no—
129C22×D47—22no—
130C22×D47—22no—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47C4×D411yescharacteristic
143C2×C4×C47#12011yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12311yescharacteristic
146C2×C4×C47C2×D811yescharacteristic
147C2×C4×C47C2×SD1611yescharacteristic
148C2×C4×C47C2×Q1611yescharacteristic
149C2×C4×C47#13511yescharacteristic
150C2×C4×C47#13511yescharacteristic
151C2×C4×C47#13511yescharacteristic
152C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
154C2×Dic47—22no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—81no—
159C2×Dic47—81no—
160D188—22no—
161D188—22no—
162D188—22no—
163D188#10314yescomplemented
164C4×D47—42no—
165C4×D47—42no—
166C4×D47—42no—
167C4×D47—42no—
168Dic94—42no—
169Dic94—42no—
170Dic94—81no—
171Dic94—81no—
172C8×C47—42no—
173C8×C47—42no—
174C47⋊C8—22no—
175C47⋊C8—41no—
176C47⋊C8—81no—
177#9219—471nomaximal, Sylow
178C22×C47#38611yescharacteristic
179C22×C47#45911yescharacteristic
180C22×C47#46311yescharacteristic
181C22×C47#46511yescharacteristic
182C22×C47#46711yescharacteristic
183C22×C47#47111yescharacteristic
184C22×C47#47311yescharacteristic
185C2×D47—24no—
186C2×D47—42no—
187C2×D47—42no—
188C2×D47—42no—
189C2×D47—42no—
190C2×D47—42no—
191C2×D47—42no—
192C4×C47—21no—
193C4×C47—21no—
194C4×C47—161no—
195C4×C47C2×#10311yescharacteristic
196C4×C47#41611yescharacteristic
197C4×C47C4×D811yescharacteristic
198C4×C47#44211yescharacteristic
199C4×C47#44711yescharacteristic
200C4×C47#44911yescharacteristic
201C4×C47#46511yescharacteristic
202C4×C47#46711yescharacteristic
203Dic47—42no—
204Dic47—82no—
205C2×#474—471no—
206C2×C8×D4—471no—
207#1956—471no—
208#1802—471no—
209C2×#332—471no—
210#1296D4711yescharacteristic, complemented
211#1277—471no—
212C2×C47#166511yescharacteristic
213C2×C47#166911yescharacteristic
214C2×C47#167911yescharacteristic
215C2×C47#168311yescharacteristic
216C2×C47#187911yescharacteristic
217C2×C47#189211yescharacteristic
218C2×C47#190811yescharacteristic
219D47—44no—
220C23×C8—941no—
221C2×C4⋊Q8—471no—
222C2×C4×D4—471no—
223#474—474no—
224#474—942no—
225C8×D4—942no—
226C8×D4—942no—
227C2×#107—941no—
228C2×C4⋊C8—471no—
229C2×C4⋊C8C2×D4711yescharacteristic
230C2×C4⋊C8C2×D4711yescharacteristic
231C2×#103—941no—
232C2×#103—941no—
233C2×#102—941no—
234C2×#102—941no—
235C2×C22⋊C8—941no—
236C2×C4×C8—471no—
237#384C2×D4711yescharacteristic
238#340—941no—
239#332—942no—
240C47#921911yescharacteristic, complemented, Sylow
241C22×Q8—941no—
242C22×D4—471no—
243C23×C4—941no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—471no—
247C22×C8—471no—
248C22×C8—944no—
249C22×C8—941no—
250C22×C8—1882no—
251C22×C8—1882no—
252C4⋊Q8—472no—
253C4⋊Q8—941no—
254C4⋊Q8—941no—
255C4⋊Q8—941no—
256C4×D4—472no—
257C4×D4—472no—
258C4×D4—941no—
259C4×D4—941no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—471no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4—941no—
266C2×C22⋊C4—941no—
267C2×C42C22×D4711yescharacteristic
268#107—944no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—941no—
272C4⋊C8—941no—
273C4⋊C8C4×D4712yes—
274C4⋊C8D18812yes—
275#103—1882no—
276#103—1882no—
277#102—944no—
278#102—1882no—
279C22⋊C8—944no—
280C4×C8—941no—
281C4×C8—941no—
282C24—941no—
283C2×Q8—942no—
284C2×Q8—1881no—
285C2×Q8—1881no—
286C2×Q8—1881no—
287C2×Q8—1881no—
288C2×Q8—1881no—
289C2×D4—472no—
290C2×D4—472no—
291C2×D4—472no—
292C2×D4—941no—
293C2×D4—941no—
294C2×D4—941no—
295C22×C4—41no—
296C22×C4—941no—
297C22×C4—942no—
298C22×C4—942no—
299C22×C4—1881no—
300C22×C4—1881no—
301C22×C4—1881no—
302C22×C4—1881no—
303C22×C4—1881no—
304C22×C4#1062535011yescharacteristic
305C22×C4D4×D4711yescharacteristic
306C22×C4D4×D4711yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—472no—
314C2×C8—941no—
315C2×C8—941no—
316C2×C8—941no—
317C2×C8—941no—
318C2×C8—941no—
319C2×C8—1888no—
320C2×C8—1881no—
321C2×C8—1881no—
322C2×C8—1881no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—81no—
328C4⋊C4—472no—
329C4⋊C4—941no—
330C4⋊C4—942no—
331C4⋊C4—942no—
332C4⋊C4—1881no—
333C4⋊C4—1881no—
334C22⋊C4—942no—
335C22⋊C4—942no—
336C42C2×C4×D4711yescharacteristic
337C42C2×D18811yescharacteristic
338C42D4×D4711yescharacteristic
339C42#1062535411yescharacteristic
340C23—942no—
341C23—942no—
342C23—942no—
343C23—1881no—
344C23—1881no—
345C23—1881no—
346C23—1881no—
347C23#1309399711yescharacteristic
348Q8—3761no—
349Q8—3761no—
350Q8—1882no—
351Q8—1882no—
352D4—474no—
353D4—942no—
354D4—942no—
355D4—942no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—942no—
362C2×C4—3761no—
363C2×C4—3761no—
364C2×C4—1882no—
365C2×C4—1881no—
366C2×C4—1881no—
367C2×C4—1882no—
368C2×C4—1882no—
369C2×C4—1882no—
370C2×C4—1882no—
371C2×C4C22⋊C4×D4711yescharacteristic
372C2×C4#1309398611yescharacteristic
373C2×C4C4×D18811yescharacteristic
374C2×C4#1309399411yescharacteristic
375C2×C4#1309399911yescharacteristic
376C2×C4#1309400911yescharacteristic
377C2×C4D8×D4711yescharacteristic
378C2×C4SD16×D4711yescharacteristic
379C2×C4Q16×D4711yescharacteristic
380C2×C4#1309404311yescharacteristic
381C2×C4#1309404511yescharacteristic
382C2×C4#1309404711yescharacteristic
383C2×C4#1309405111yescharacteristic
384C2×C4#1309405111yescharacteristic
385C8—42no—
386C8—42no—
387C8—942no—
388C8—3761no—
389C8—1881no—
390C22—944no—
391C22—1882no—
392C22—1882no—
393C22—1882no—
394C22—1882no—
395C22—1882no—
396C22—1882no—
397C22#1568900111yescharacteristic
398C22#1568931711yescharacteristic
399C22#1568934111yescharacteristic
400C22#1568935111yescharacteristic
401C22#1568936311yescharacteristic
402C22#1568938511yescharacteristic
403C22#1568939711yescharacteristic
404C4—21no—
405C4—21no—
406C4—161no—
407C4—3762no—
408C4—1882no—
409C4#103×D4711yescharacteristic
410C4#1568911611yescharacteristic
411C4#1568920511yescharacteristic
412C4#1568923411yescharacteristic
413C4#1568925811yescharacteristic
414C4#1568927111yescharacteristic
415C4#1568935311yescharacteristic
416C4#1568936511yescharacteristic
417C2—1884no—
418C2#2024272811yescharacteristic
419C2#2024275211yescharacteristic
420C2#2024281411yescharacteristic
421C2#2024284211yescharacteristic
422C2#2024373711yescharacteristic
423C2#2024381011yescharacteristic
424C2#2024390411yescharacteristic
subgroupG/H
centerC23#13093997
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47#121
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1296D47
hypocenterC47#9219
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20244189C2semidirect
5. C2×#15688870C2semidirect
6. #20241222C2semidirect
7. C47×#1296C2semidirect
12. #15689404C4semidirect
163. D188#103semidirect
210. #1296D47semidirect
240. C47#9219semidirect

Maximal subgroups

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

subgroupindex
C2×#156894042
C2×#156891882
#202441892
#202433852
C2×#156888702
#202412222
C47×#12962
#921947