Finite Group Databasebeta

← #26133335

Subgroups — #26133335

Order 12032.

590 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×#13093965C211yesmaximal, characteristic
2C2×#15688924C211yesmaximal, characteristic, complemented
3C2×#15688923C211yesmaximal, characteristic, complemented
4#20244254C211yesmaximal, characteristic, complemented
5#20241479C211yesmaximal, characteristic, complemented
6#20241466C211yesmaximal, characteristic, complemented
7C2×C4×D8×C47C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C2×#13093965—22no—
10C2×#13093965—22no—
11C2×#13093965C414yescomplemented
12C22×C47⋊C16C2211yescharacteristic
13C22×C47⋊C16C2211yescharacteristic
14C2×#13093958C2211yescharacteristic
15C2×#13093946—21no—
16#15689025—21no—
17#15688924—22no—
18#15688923—22no—
19C22×D8×C47C2211yescharacteristic
20C2×C4×D4×C47—21no—
21C4×D8×C47—22no—
22C4×D8×C47—22no—
23C2×#107×C47C2211yescharacteristic
24C2×#102×C47—21no—
25C2×C4×C8×C47C2211yescharacteristic
26C22×D188—21no—
27C2×D376—21no—
28C2×D376—21no—
29C2×D376—21no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376C2×C412yescomplemented
33C2×C4⋊Dic47—21no—
34C2×(C2×D47)⋊C4—41no—
35C2×(C2×D47)⋊C4—41no—
36#13093965—28no—
37C2×C47⋊C16—21no—
38C2×C47⋊C16—21no—
39C2×C47⋊C16—21no—
40C2×C47⋊C16—21no—
41C2×C47⋊C16—21no—
42C2×C47⋊C16C2×C412yescomplemented
43#13093958—21no—
44#13093958—21no—
45#13093946—42no—
46C22×D4×C47—21no—
47C23×C4×C47—41no—
48C2×D8×C47—21no—
49C2×D8×C47—21no—
50C2×D8×C47—21no—
51C2×D8×C47C2×C412yes—
52C2×D8×C47D412yes—
53C2×D8×C47D412yes—
54C22×C8×C47D411yescharacteristic
55C22×C8×C47C2311yescharacteristic
56C4×D4×C47—24no—
57C4×D4×C47—42no—
58C2×C4⋊C4×C47—21no—
59C2×C22⋊C4×C47—41no—
60C2×C42×C47D411yescharacteristic
61#107×C47—21no—
62#107×C47D412yes—
63#102×C47—24no—
64C4×C8×C47—21no—
65C4×C8×C47—21no—
66C24×C47—41no—
67C23×D47—41no—
68C2×D4×C47—22no—
69C2×D4×C47—22no—
70C2×D4×C47—22no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C22×C4×C47—44no—
75C22×C4×C47—41no—
76C22×C4×C47—82no—
77C22×C4×C47—82no—
78C22×C4×C47D811yescharacteristic
79C22×C4×C47D811yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×Dic47—41no—
82C2×D188—22no—
83C2×D188—41no—
84C2×D188—41no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—41no—
88D8×C47—22no—
89D8×C47—22no—
90D8×C47—22no—
91D8×C47C22⋊C414yescomplemented
92C2×C8×C47—21no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47C22×C411yescharacteristic
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47#4111yescharacteristic
100C2×C8×C47#4111yescharacteristic
101C4⋊C4×C47—22no—
102C4⋊C4×C47—41no—
103C22⋊C4×C47—44no—
104C42×C47—21no—
105C42×C47—21no—
106(C2×D47)⋊C4—82no—
107(C2×D47)⋊C4—82no—
108C4⋊Dic47—41no—
109C4⋊Dic47—41no—
110D376—22no—
111D376—22no—
112D376—41no—
113D376—41no—
114C47⋊C16—22no—
115C47⋊C16—41no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—81no—
120C23×C47—81no—
121C23×C47—81no—
122C23×C47—81no—
123C23×C47C2×D811yescharacteristic
124C22×D47—42no—
125C22×D47—81no—
126C22×D47—81no—
127C22×D47—81no—
128C22×D47—81no—
129C22×D47—81no—
130C22×D47—81no—
131D4×C47—24no—
132D4×C47—42no—
133D4×C47—42no—
134D4×C47—42no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—42no—
141C2×C4×C47—88no—
142C2×C4×C47—81no—
143C2×C4×C47—81no—
144C2×C4×C47D1612yescomplemented
145C2×C4×C47C4×D411yescharacteristic
146C2×C4×C47#12011yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#13511yescharacteristic
149C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
151C2×Dic47—81no—
152C2×Dic47—81no—
153C2×Dic47—81no—
154D188—42no—
155D188—42no—
156D188—81no—
157D188—81no—
158C8×C47—41no—
159C8×C47C2×C22⋊C411yescharacteristic
160C8×C47C4×D411yescharacteristic
161C8×C47#12111yescharacteristic
162C8×C47#12311yescharacteristic
163#13012—471nomaximal, Sylow
164C22×C47—44no—
165C22×C47—82no—
166C22×C47—82no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47—82no—
170C22×C47—82no—
171C22×C47C4×D811yescharacteristic
172C22×C47#44711yescharacteristic
173C22×C47#45911yescharacteristic
174C22×C47C2×D1611yescharacteristic
175C22×C47#50811yescharacteristic
176C22×C47#50911yescharacteristic
177C22×C47#50911yescharacteristic
178C2×D47—82no—
179C2×D47—82no—
180C2×D47—82no—
181C2×D47—161no—
182C2×D47—161no—
183C2×D47—161no—
184C2×D47—161no—
185C4×C47—22no—
186C4×C47—41no—
187C4×C47—82no—
188C4×C47#38611yescharacteristic
189C4×C47#44211yescharacteristic
190C4×C47#44911yescharacteristic
191C4×C47#46311yescharacteristic
192Dic47—161no—
193C22×D16—471no—
194C2×C4×D8D4711yescharacteristic, complemented
195C2×#357—471no—
196C2×#357—471no—
197#1970—471no—
198#1380—471no—
199#1374—471no—
200C2×C47—84no—
201C2×C47#187911yescharacteristic
202C2×C47C4×D1611yescharacteristic
203C2×C47#217711yescharacteristic
204C2×C47#218411yescharacteristic
205C2×C47#218611yescharacteristic
206C2×C47#220611yescharacteristic
207C2×C47#221011yescharacteristic
208D47—162no—
209C22×D8—471no—
210C22×D8C2×D4711yescharacteristic
211C2×C4×D4—21no—
212C2×D16—474no—
213C2×D16—942no—
214C2×D16—942no—
215C22×C16—471no—
216C22×C16—471no—
217C4×D8—22no—
218C4×D8—22no—
219C2×#107—471no—
220C2×#107C2×D4711yescharacteristic
221C2×#102—21no—
222C2×#102—941no—
223C2×C4×C8C2×D4711yescharacteristic
224#390—941no—
225#357—942no—
226#357—942no—
227C47#1301211yescharacteristic, complemented, Sylow
228C22×D4—21no—
229C22×D4—941no—
230C23×C4—41no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—472no—
235C2×D8—941no—
236C2×D8—941no—
237C2×D8—941no—
238C2×D8—941no—
239C2×D8—941no—
240C2×D8C4×D4712yes—
241C2×D8D18812yes—
242C2×D8C22⋊D4712yes—
243C22×C8C22⋊D4711yescharacteristic
244C22×C8C22×D4711yescharacteristic
245C4×D4—24no—
246C4×D4—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—941no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—1881no—
251C2×C22⋊C4—1881no—
252C2×C42C22⋊D4711yescharacteristic
253D16—948no—
254C2×C16—472no—
255C2×C16—941no—
256C2×C16—941no—
257C2×C16—941no—
258C2×C16—941no—
259C2×C16—941no—
260#107—21no—
261#107—941no—
262#107—941no—
263#107C22⋊D4712yes—
264#102—24no—
265#102—1882no—
266C4×C8—21no—
267C4×C8—21no—
268C24—41no—
269C24—1881no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—942no—
277C2×D4—1881no—
278C2×D4—1881no—
279C2×D4—1881no—
280C2×D4—1881no—
281C2×D4—1881no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—1881no—
287C22×C4#1062533411yescharacteristic
288C22×C4#1062533411yescharacteristic
289C22×C4C2×C22⋊D4711yescharacteristic
290D8—22no—
291D8—22no—
292D8—22no—
293D8—942no—
294D8—942no—
295D8—1881no—
296D8—1881no—
297D8(C2×D47)⋊C414yescomplemented
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8C2×C4×D4711yescharacteristic
302C2×C8C2×D18811yescharacteristic
303C2×C8#1062535011yescharacteristic
304C2×C8#1062535011yescharacteristic
305C2×C8C2×C22⋊D4711yescharacteristic
306C2×C8C2×C22⋊D4711yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—1881no—
310C4⋊C4—1881no—
311C22⋊C4—44no—
312C22⋊C4—3762no—
313C22⋊C4—3762no—
314C42—21no—
315C42—21no—
316C16—942no—
317C16—1881no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—3761no—
326C23—3761no—
327C23—3761no—
328C23—3761no—
329C23—3761no—
330C23—3761no—
331C23—1882no—
332C23C2×#1062533411yescharacteristic
333D4—24no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—3761no—
338D4—3761no—
339D4—1882no—
340D4—1882no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—3761no—
351C2×C4—3761no—
352C2×C4—3761no—
353C2×C4#1309396512yescomplemented
354C2×C4C4×C22⋊D4711yescharacteristic
355C2×C4#1309399311yescharacteristic
356C2×C4#1309399811yescharacteristic
357C2×C4#1309404411yescharacteristic
358C2×C4#1309404911yescharacteristic
359C2×C4#1309405011yescharacteristic
360C8—41no—
361C8C2×(C2×D47)⋊C411yescharacteristic
362C8C4×C22⋊D4711yescharacteristic
363C8#1309399811yescharacteristic
364C8#1309400711yescharacteristic
365C22—44no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—3762no—
373C22—3762no—
374C22—3762no—
375C22—7521no—
376C22—7521no—
377C22—7521no—
378C22—7521no—
379C22C4×#1062533411yescharacteristic
380C22#1568925711yescharacteristic
381C22#1568931811yescharacteristic
382C22C2×#1309396511yescharacteristic
383C22#1568955311yescharacteristic
384C22#1568955811yescharacteristic
385C22#1568955911yescharacteristic
386C4—22no—
387C4—41no—
388C4—82no—
389C4—7521no—
390C4#1568900211yescharacteristic
391C4#1568923311yescharacteristic
392C4#1568926811yescharacteristic
393C4#1568933811yescharacteristic
394C2—84no—
395C2—7522no—
396C2#2024373611yescharacteristic
397C2C4×#1309396511yescharacteristic
398C2#2024526211yescharacteristic
399C2#2024530011yescharacteristic
400C2#2024531111yescharacteristic
401C2#2024542311yescharacteristic
402C2#2024544311yescharacteristic
subgroupG/H
centerC23C2×#10625334
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C4×D8×C47C2
socleC23×C47C2×D8
prime radicalC22×#13093965C2
cosocleC22×C8×C47C23
hypercenterC2×C4×D8D47
hypocenterC47#13012
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. C2×#15688924C2semidirect
3. C2×#15688923C2semidirect
4. #20244254C2semidirect
5. #20241479C2semidirect
6. #20241466C2semidirect
7. C2×C4×D8×C47C2semidirect
11. C2×#13093965C4semidirect
32. C2×D376C2×C4semidirect
42. C2×C47⋊C16C2×C4semidirect
91. D8×C47C22⋊C4semidirect
144. C2×C4×C47D16semidirect
194. C2×C4×D8D47semidirect
227. C47#13012semidirect
297. D8(C2×D47)⋊C4semidirect
353. C2×C4#13093965semidirect

Maximal subgroups

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

subgroupindex
C22×#130939652
C2×#156889242
C2×#156889232
#202442542
#202414792
#202414662
C2×C4×D8×C472
#1301247