Finite Group Databasebeta

← #26133340

Subgroups — #26133340

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093967C211yesmaximal, characteristic
2C2×Q16×Dic47C211yesmaximal, characteristic, complemented
3C2×#15688925C211yesmaximal, characteristic
4#20244252C211yesmaximal, characteristic
5#20241479C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C2×C47×#358C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C2×Q8×Dic47—21no—
10C2×#13093967—22no—
11C2×#13093967—22no—
12C2×#13093967C414yescomplemented
13C22×C47⋊C16C2211yescharacteristic
14Q16×Dic47—22no—
15Q16×Dic47—22no—
16C2×#13093958C2211yescharacteristic
17C2×#13093948—21no—
18C2×#13093944—21no—
19C2×C8×Dic47C2211yescharacteristic
20#15689024—21no—
21#15688925—22no—
22C22×Q16×C47C2211yescharacteristic
23C22×C16×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C47×#358—22no—
26C22×D188—21no—
27C2×D376—21no—
28C2×D376—21no—
29C2×D376—21no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376C2×C412yes—
33C22×C47⋊C8D411yescharacteristic
34Q8×Dic47—24no—
35Q8×Dic47—42no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—41no—
38C2×(C2×D47)⋊C4—41no—
39C2×(C2×D47)⋊C4—41no—
40C2×C4×Dic47—41no—
41C2×C4×Dic47D411yescharacteristic
42#13093967—28no—
43C2×C47⋊C16—21no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16C2×C412yes—
46#13093958—21no—
47#13093958D412yes—
48#13093948—24no—
49#13093944—42no—
50C8×Dic47—21no—
51C8×Dic47—21no—
52C22×Q8×C47—21no—
53C2×Q16×C47—21no—
54C2×Q16×C47—21no—
55C2×Q16×C47—21no—
56C2×Q16×C47C2×C412yes—
57C2×Q16×C47D412yes—
58C2×Q16×C47D412yes—
59C22×C8×C47C2311yescharacteristic
60C2×C4⋊C4×C47—21no—
61C2×C16×C47—21no—
62C2×C16×C47—21no—
63C2×C16×C47—21no—
64#107×C47—21no—
65#107×C47—21no—
66C23×D47—41no—
67C2×Q8×C47—22no—
68C2×Q8×C47—22no—
69C2×Q8×C47—22no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47—41no—
77C22×Dic47D811yescharacteristic
78C22×Dic47D811yescharacteristic
79C2×D188—22no—
80C2×D188—41no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—41no—
84C2×D188—41no—
85Q16×C47—22no—
86Q16×C47—22no—
87Q16×C47—22no—
88Q16×C47C22⋊C414yescomplemented
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—41no—
96C4⋊C4×C47—41no—
97(C2×D47)⋊C4—82no—
98(C2×D47)⋊C4—82no—
99C4⋊Dic47—22no—
100C4⋊Dic47—44no—
101C4⋊Dic47—41no—
102C4×Dic47—21no—
103C4×Dic47—21no—
104C4×Dic47—44no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108D376—22no—
109D376—22no—
110D376—41no—
111D376—41no—
112C16×C47—41no—
113C47⋊C16—22no—
114C23×C47C2×D811yescharacteristic
115C22×D47—42no—
116C22×D47—81no—
117C22×D47—81no—
118C22×D47—81no—
119C22×D47—81no—
120C22×D47—81no—
121C22×D47—81no—
122Q8×C47—24no—
123Q8×C47—42no—
124Q8×C47—42no—
125Q8×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47#12011yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#13511yescharacteristic
136C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
138C2×Dic47—21no—
139C2×Dic47—21no—
140C2×Dic47—21no—
141C2×Dic47—21no—
142C2×Dic47—21no—
143C2×Dic47—42no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146C2×Dic47SD3212yescomplemented
147D188—42no—
148D188—42no—
149D188—81no—
150D188—81no—
151C8×C47C2×C22⋊C411yescharacteristic
152C8×C47C4×D411yescharacteristic
153C8×C47#12111yescharacteristic
154C8×C47#12311yescharacteristic
155C47⋊C8—41no—
156#13013—471nomaximal, Sylow
157C22×C47C4×D811yescharacteristic
158C22×C47#44711yescharacteristic
159C22×C47#45911yescharacteristic
160C22×C47C2×SD3211yescharacteristic
161C22×C47#50811yescharacteristic
162C22×C47#50911yescharacteristic
163C22×C47#51011yescharacteristic
164C2×D47—82no—
165C2×D47—82no—
166C2×D47—82no—
167C2×D47—161no—
168C2×D47—161no—
169C2×D47—161no—
170C2×D47—161no—
171C4×C47—44no—
172C4×C47—161no—
173C4×C47#38611yescharacteristic
174C4×C47#44211yescharacteristic
175C4×C47#44911yescharacteristic
176C4×C47#46311yescharacteristic
177Dic47—22no—
178Dic47—41no—
179Dic47—82no—
180C22×SD32—471no—
181C2×C4×Q16—471no—
182C2×#358D4711yescharacteristic, complemented
183C2×#357—471no—
184#1970—471no—
185#1380—471no—
186#1374—471no—
187C2×C47#187911yescharacteristic
188C2×C47C4×SD3211yescharacteristic
189C2×C47#217511yescharacteristic
190C2×C47#218511yescharacteristic
191C2×C47#218611yescharacteristic
192C2×C47#220711yescharacteristic
193C2×C47#221111yescharacteristic
194D47—162no—
195C22×Q16C2×D4711yescharacteristic
196C22×D8—471no—
197C2×C4×Q8—941no—
198C2×SD32—474no—
199C2×SD32—942no—
200C2×SD32—942no—
201C22×C16—471no—
202C22×C16C2×D4711yescharacteristic
203C4×Q16—942no—
204C4×Q16—942no—
205C2×#107—471no—
206C2×#107C2×D4711yescharacteristic
207C2×#103—941no—
208C2×#102—941no—
209C2×C4×C8—471no—
210#390—941no—
211#358—22no—
212#357—942no—
213C47#1301311yescharacteristic, complemented, Sylow
214C22×Q8—21no—
215C22×D4—941no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D4712yes—
220C2×Q16D18812yes—
221C2×Q16C22⋊D4712yes—
222C2×D8—472no—
223C2×D8—941no—
224C2×D8—941no—
225C2×D8—941no—
226C2×D8—941no—
227C2×D8—941no—
228C22×C8—471no—
229C22×C8C22×D4711yescharacteristic
230C4×Q8—944no—
231C4×Q8—1882no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—1881no—
235C2×C22⋊C4—1881no—
236C2×C22⋊C4—1881no—
237C2×C42—471no—
238C2×C42—1881no—
239SD32—948no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—472no—
244C2×C16—941no—
245C2×C16—941no—
246#107—21no—
247#107—21no—
248#107—472no—
249#107—941no—
250#103—944no—
251#102—1882no—
252C4×C8—941no—
253C4×C8—941no—
254C24—1881no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—942no—
262C2×D4—1881no—
263C2×D4—1881no—
264C2×D4—1881no—
265C2×D4—1881no—
266C2×D4—1881no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—471no—
270C22×C4—471no—
271C22×C4—1881no—
272C22×C4D4×D4711yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16(C2×D47)⋊C414yescomplemented
277D8—942no—
278D8—942no—
279D8—1881no—
280D8—1881no—
281C2×C8—941no—
282C2×C8—941no—
283C2×C8—941no—
284C2×C8C2×C4×D4711yescharacteristic
285C2×C8C2×D18811yescharacteristic
286C2×C8D4×D4711yescharacteristic
287C2×C8#1062535211yescharacteristic
288C2×C8#1062535211yescharacteristic
289C2×C8C2×C22⋊D4711yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—942no—
293C4⋊C4—1884no—
294C4⋊C4—1881no—
295C22⋊C4—3762no—
296C22⋊C4—3762no—
297C42—941no—
298C42—941no—
299C42—1884no—
300C16—41no—
301C16—942no—
302C23—3761no—
303C23—3761no—
304C23—3761no—
305C23—3761no—
306C23—3761no—
307C23—3761no—
308C23—1882no—
309C23D8×D4711yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—3761no—
315D4—3761no—
316D4—1882no—
317D4—1882no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—472no—
325C2×C4—941no—
326C2×C4—941no—
327C2×C4—941no—
328C2×C4—941no—
329C2×C4—941no—
330C2×C4—3761no—
331C2×C4—3761no—
332C2×C4—1882no—
333C2×C4(C22⋊D47)⋊C411yescharacteristic
334C2×C4#1309399411yescharacteristic
335C2×C4#1309399611yescharacteristic
336C2×C4#1309404511yescharacteristic
337C2×C4#1309404811yescharacteristic
338C2×C4#1309405111yescharacteristic
339C8—1881no—
340C8C2×(C2×D47)⋊C411yescharacteristic
341C8(C22⋊D47)⋊C411yescharacteristic
342C8#1309399611yescharacteristic
343C8#1309400811yescharacteristic
344C22—3762no—
345C22—3762no—
346C22—3762no—
347C22—7521no—
348C22—7521no—
349C22—7521no—
350C22—7521no—
351C22#1568920311yescharacteristic
352C22#1568925811yescharacteristic
353C22#1568931611yescharacteristic
354C22SD32×D4711yescharacteristic
355C22#1568955611yescharacteristic
356C22#1568956011yescharacteristic
357C22#1568956511yescharacteristic
358C4—44no—
359C4—161no—
360C4—942no—
361C4—3762no—
362C4—1881no—
363C4#1568900011yescharacteristic
364C4#1568923211yescharacteristic
365C4#1568926911yescharacteristic
366C4#1568934011yescharacteristic
367C2—7522no—
368C2#2024373411yescharacteristic
369C2#2024523711yescharacteristic
370C2#2024525511yescharacteristic
371C2#2024530611yescharacteristic
372C2#2024531411yescharacteristic
373C2#2024542611yescharacteristic
374C2#2024545211yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#358C2
socleC23×C47C2×D8
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenterC2×#358D47
hypocenterC47#13013
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic47C2semidirect
5. #20241479C2semidirect
6. #20241465C2semidirect
7. C2×C47×#358C2semidirect
12. C2×#13093967C4semidirect
88. Q16×C47C22⋊C4semidirect
146. C2×Dic47SD32semidirect
182. C2×#358D47semidirect
213. C47#13013semidirect
276. Q16(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130939672
C2×Q16×Dic472
C2×#156889252
#202442522
#202414792
#202414652
C2×C47×#3582
#1301347