Finite Group Databasebeta

← #26109181

Subgroups — #26109181

Order 12032.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689194C211yesmaximal, characteristic
2#20243413C211yesmaximal, characteristic, complemented
3#20243373C212yesmaximal
4#20241255C212yesmaximal, complemented
5C47×#1808C211yesmaximal, characteristic, complemented
6C22×#10625324C411yescharacteristic, complemented
7C2×C4×C22⋊D47C2211yescharacteristic
8#15689194—21no—
9#15689194—21no—
10#15689194—21no—
11#15689194—21no—
12C2×#13093956—21no—
13C2×Dic47⋊C8C411yescharacteristic, complemented
14C2×#13093938—22no—
15C2×C22⋊(C47⋊C8)C2211yescharacteristic
16C2×(C2×D47)⋊C8C411yescharacteristic, complemented
17C2×#13093934C411yescharacteristic, complemented
18#15688983C2212yes—
19#15688889—22no—
20#15688886—21no—
21C23×C8×C47C2211yescharacteristic
22C2×#107×C47—21no—
23C47×#380C2212yes—
24C47×#340—21no—
25C22×C22⋊D47D411yescharacteristic
26C22×C4×D47C2×C411yescharacteristic
27C2×#10625324—21no—
28C2×#10625324—21no—
29C2×#10625324—42no—
30C22×C47⋊C8C2×C411yescharacteristic
31C22×C47⋊C8C2×C411yescharacteristic
32C4×C22⋊D47—21no—
33C4×C22⋊D47—21no—
34C4×C22⋊D47—21no—
35C4×C22⋊D47—21no—
36C2×C4⋊Dic47—22no—
37C2×Dic47⋊C4Q811yescharacteristic
38C2×C22⋊Dic47—22no—
39C2×C22⋊Dic47D411yescharacteristic
40C2×(C2×D47)⋊C4Q811yescharacteristic
41C2×C4×Dic47C2×C411yescharacteristic
42#13093956—42no—
43Dic47⋊C8—21no—
44Dic47⋊C8—21no—
45#13093938—44no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48(C2×D47)⋊C8—21no—
49(C2×D47)⋊C8—21no—
50#13093934—21no—
51#13093934—21no—
52C47⋊#95—22no—
53C23×C4×C47C2311yescharacteristic
54C22×C8×C47—42no—
55C22×C8×C47C2×C411yescharacteristic
56C22×C8×C47C2×C411yescharacteristic
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C22×C8×C47D411yescharacteristic
61C2×C4⋊C4×C47—22no—
62C2×C22⋊C4×C47—22no—
63#107×C47—42no—
64#95×C47—22no—
65C24×C47C2×D411yescharacteristic
66C23×D47C4⋊C411yescharacteristic
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47—42no—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×Q811yescharacteristic
74C22×C4×C47#4111yescharacteristic
75C22×C4×C47#4111yescharacteristic
76C2×C22⋊D47—21no—
77C2×C22⋊D47—21no—
78C2×C22⋊D47—22no—
79C2×C22⋊D47—22no—
80C22×Dic47—42no—
81C22×Dic47C4⋊C411yescharacteristic
82C22×Dic47C4⋊C411yescharacteristic
83C22×Dic47C4⋊C411yescharacteristic
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86C2×C4×D47—41no—
87C2×C4×D47—41no—
88C2×C8×C47—22no—
89C2×C8×C47—22no—
90C2×C8×C47—22no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C2×C8×C47—41no—
94C2×C8×C47C22⋊C411yescharacteristic
95C2×C8×C47C22⋊C411yescharacteristic
96C2×C8×C47C22⋊C411yescharacteristic
97C2×C8×C47C22⋊C411yescharacteristic
98C4⋊C4×C47—42no—
99C4⋊C4×C47—42no—
100C22⋊C4×C47—42no—
101C22⋊C4×C47—42no—
102C22⋊Dic47—21no—
103C22⋊Dic47—21no—
104C22⋊Dic47—42no—
105C22⋊Dic47—42no—
106(C2×D47)⋊C4—21no—
107(C2×D47)⋊C4—21no—
108C4⋊Dic47—42no—
109C4⋊Dic47—42no—
110Dic47⋊C4—21no—
111Dic47⋊C4—21no—
112C4×Dic47—21no—
113C4×Dic47—21no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118#10625324—42no—
119#10625324—42no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47C2×C4⋊C411yescharacteristic
123C23×C47#12111yescharacteristic
124C23×C47#12111yescharacteristic
125C23×C47AGL(1,ℤ/8)12yes—
126C23×C47#13712yes—
127C22×D47—21no—
128C22×D47—22no—
129C22×D47—41no—
130C22×D47—41no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—82no—
141C2×C4×C47—82no—
142C2×C4×C47—82no—
143C2×C4×C47C2×C22⋊C411yescharacteristic
144C2×C4×C47#11711yescharacteristic
145C2×C4×C47C22⋊Q811yescharacteristic
146C2×C4×C47C22⋊Q811yescharacteristic
147C22⋊D47—42no—
148C22⋊D47—41no—
149C22⋊D47—41no—
150C2×Dic47—21no—
151C2×Dic47—21no—
152C2×Dic47—21no—
153C2×Dic47—22no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—82no—
157C2×Dic47—82no—
158C2×Dic47—82no—
159C4×D47—42no—
160C4×D47—41no—
161C4×D47—41no—
162C8×C47—21no—
163C8×C47—21no—
164C8×C47—21no—
165C8×C47#9912yescomplemented
166C47⋊C8—41no—
167C47⋊C8—41no—
168#8610—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—42no—
176C22×C47—42no—
177C22×C47#38011yescharacteristic
178C22×C47#42812yes—
179C22×C47#46812yes—
180C22×C47#46911yescharacteristic
181C22×C47#47011yescharacteristic
182C2×D47—41no—
183C2×D47—41no—
184C2×D47—41no—
185C2×D47—82no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—162no—
190C4×C47C2×#9911yescharacteristic
191C4×C47#41011yescharacteristic
192Dic47—41no—
193Dic47—41no—
194Dic47—41no—
195Dic47—162no—
196C2×#435—471no—
197#1808D4711yescharacteristic, complemented
198#1807—471no—
199#1799—472no—
200#1289—472no—
201C2×C47—22no—
202C2×C47—21no—
203C2×C47—22no—
204C2×C47—21no—
205C2×C47#179911yescharacteristic
206C2×C47#180911yescharacteristic
207C2×C47#181211yescharacteristic
208D47—81no—
209C22×M16—471no—
210C23×C8C2×D4711yescharacteristic
211C2×C4×D4—471no—
212#435—941no—
213#435—941no—
214#435—941no—
215#435—941no—
216C2×#107—21no—
217C2×#106—941no—
218C2×C4⋊C8—471no—
219C2×#99—942no—
220C2×C22⋊C8—471no—
221C2×C22⋊C8—471no—
222C2×#97—471no—
223#380—472no—
224#380C2×D4712yes—
225#342—942no—
226#340—21no—
227#340—941no—
228C47#861011yescharacteristic, complemented, Sylow
229C22×D4—471no—
230C23×C4—471no—
231C23×C4C22×D4711yescharacteristic
232C2×M16—941no—
233C2×M16—941no—
234C2×M16—1882no—
235C22×C8—42no—
236C22×C8—471no—
237C22×C8—471no—
238C22×C8C4×D4711yescharacteristic
239C22×C8C4×D4711yescharacteristic
240C22×C8D18811yescharacteristic
241C22×C8D18811yescharacteristic
242C22×C8C22⋊D4711yescharacteristic
243C22×C8C22⋊D4711yescharacteristic
244C4×D4—941no—
245C4×D4—941no—
246C4×D4—941no—
247C4×D4—941no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—942no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—471no—
253C2×C22⋊C4—471no—
254C2×C22⋊C4—942no—
255C2×C42—471no—
256#107—42no—
257#106—1882no—
258C4⋊C8—941no—
259C4⋊C8—941no—
260#99—1884no—
261C22⋊C8—941no—
262C22⋊C8—941no—
263C22⋊C8—941no—
264C22⋊C8—941no—
265#97—941no—
266#97—941no—
267#95—22no—
268#95—942no—
269C24—471no—
270C24D4×D4711yescharacteristic
271C2×D4—941no—
272C2×D4—941no—
273C2×D4—942no—
274C2×D4—942no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—471no—
279C22×C4—471no—
280C22×C4—471no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—1881no—
284C22×C4—1881no—
285C22×C4—1882no—
286C22×C4C2×C4×D4711yescharacteristic
287C22×C4C2×D18811yescharacteristic
288C22×C4#1062535211yescharacteristic
289C22×C4Q8×D4711yescharacteristic
290C22×C4#1062535411yescharacteristic
291C22×C4C2×C22⋊D4711yescharacteristic
292M16—1882no—
293M16—1882no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—941no—
301C2×C8—941no—
302C2×C8—1881no—
303C2×C8—1881no—
304C2×C8(C2×D47)⋊C411yescharacteristic
305C2×C8(C2×D47)⋊C411yescharacteristic
306C2×C8(C2×D47)⋊C411yescharacteristic
307C2×C8(C2×D47)⋊C411yescharacteristic
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—1882no—
313C4⋊C4—1882no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—941no—
317C22⋊C4—941no—
318C22⋊C4—941no—
319C22⋊C4—941no—
320C22⋊C4—1882no—
321C22⋊C4—1882no—
322C42—941no—
323C42—941no—
324C23—41no—
325C23—41no—
326C23—942no—
327C23—941no—
328C23—1881no—
329C23—1881no—
330C23C4⋊C4×D4711yescharacteristic
331C23#1309399511yescharacteristic
332C23#1309399911yescharacteristic
333C23#1309404812yes—
334C23#1309405612yes—
335D4—1882no—
336D4—1881no—
337D4—1881no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—941no—
351C2×C4—941no—
352C2×C4—941no—
353C2×C4—942no—
354C2×C4—3762no—
355C2×C4—3762no—
356C2×C4—3762no—
357C2×C4—1882no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4C2×(C2×D47)⋊C411yescharacteristic
363C2×C4#1309398211yescharacteristic
364C2×C4#1309400011yescharacteristic
365C2×C4#1309400411yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—1881no—
370C8—1881no—
371C8#1309393812yescomplemented
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—3762no—
381C22—1881no—
382C22—1881no—
383C22—1881no—
384C22#1568898011yescharacteristic
385C22#1568916412yes—
386C22#1568937012yes—
387C22#1568937311yescharacteristic
388C22#1568937811yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—162no—
393C4—1881no—
394C4—1881no—
395C4—1881no—
396C4—7522no—
397C4C2×#1309393811yescharacteristic
398C4#1568909611yescharacteristic
399C2—21no—
400C2—22no—
401C2—22no—
402C2—21no—
403C2—3761no—
404C2#2024337311yescharacteristic
405C2#2024342211yescharacteristic
406C2#2024343111yescharacteristic
subgroupG/H
centerC22#15688980
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1808C2
socleC22×C47#380
prime radicalC22×C22⋊D47D4
cosocleC23×C4×C47C23
hypercenter#1808D47
hypocenterC47#8610
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. #20243413C2semidirect
4. #20241255C2semidirect
5. C47×#1808C2semidirect
6. C22×#10625324C4semidirect
13. C2×Dic47⋊C8C4semidirect
16. C2×(C2×D47)⋊C8C4semidirect
17. C2×#13093934C4semidirect
165. C8×C47#99semidirect
197. #1808D47semidirect
228. C47#8610semidirect
371. C8#13093938semidirect

Maximal subgroups

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

subgroupindex
C2×#156891942
#202434132
#202433732
#202412552
C47×#18082
#861047