Finite Group Databasebeta

← #26224186

Subgroups — #26224186

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C47⋊C8C211yesmaximal, characteristic, complemented
2C2×#15689201C211yesmaximal, characteristic
3#20244325C211yesmaximal, characteristic, complemented
4#20244183C211yesmaximal, characteristic
5#20244003C214yesmaximal, complemented
6#20243505C211yesmaximal, characteristic, complemented
7#20243172C214yesmaximal, complemented
8C4×#13093935C211yesmaximal, characteristic, complemented
9C4×#124×C47C211yesmaximal, characteristic
10C22×#10625329C2212yes—
11Q8×C47⋊C8—24no—
12#15689201—24no—
13C2×C4⋊(C47⋊C8)C2212yes—
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C22⋊(C47⋊C8)C2212yes—
17C2×C22⋊(C47⋊C8)C2214yes—
18C2×#13093935C2212yes—
19C2×#13093935C2212yes—
20C2×C4×C47⋊C8C2211yescharacteristic
21C2×C4×C47⋊C8C2212yes—
22C2×C4×C47⋊C8C2211yescharacteristic
23#15688878C2214yes—
24#15688878C2214yes—
25C2×#124×C47C411yescharacteristic
26C2×C4×Q8×C47C2211yescharacteristic
27C2×C4×D4×C47C2211yescharacteristic
28C47×#391C411yescharacteristic
29C47×#390C411yescharacteristic
30C47×#388C414yes—
31C47×#382C411yescharacteristic
32C4×C22⋊C4×C47C2214yes—
33C43×C47C2211yescharacteristic
34C2×#10625329—24no—
35C2×#10625329—28no—
36C22×C47⋊C8C2312yes—
37C22×C47⋊C8C2312yes—
38C22×C47⋊C8C2311yescharacteristic
39C22×C47⋊C8C2312yes—
40C22×C47⋊C8C2311yescharacteristic
41C4⋊(C47⋊C8)—22no—
42C4⋊(C47⋊C8)—22no—
43C4⋊(C47⋊C8)—24no—
44C22⋊(C47⋊C8)—24no—
45C22⋊(C47⋊C8)—28no—
46#13093935—24no—
47#13093935—24no—
48C4×C47⋊C8—22no—
49C4×C47⋊C8—24no—
50C4×C47⋊C8D414yes—
51C22×Q8×C47C2×C411yescharacteristic
52C22×D4×C47C2×C411yescharacteristic
53C23×C4×C47C2312yes—
54#124×C47—24no—
55C4×Q8×C47—22no—
56C4×Q8×C47—22no—
57C4×D4×C47—22no—
58C4×D4×C47—22no—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C4⋊C4×C47C2×C412yes—
62C2×C22⋊C4×C47C2×C412yes—
63C2×C22⋊C4×C47C2×C414yes—
64C2×C22⋊C4×C47C2×C414yes—
65C2×C42×C47C2×C411yescharacteristic
66C2×C42×C47C2×C411yescharacteristic
67C2×C42×C47C2×C412yes—
68C2×C42×C47C2312yes—
69C2×C42×C47C2312yes—
70C2×C42×C47C2311yescharacteristic
71#95×C47C2×C414yes—
72#95×C47C2×C414yes—
73C24×C47C22×C412yes—
74C2×Q8×C47—22no—
75C2×Q8×C47—22no—
76C2×Q8×C47M1614yescomplemented
77C2×D4×C47—22no—
78C2×D4×C47—22no—
79C2×D4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—24no—
83C22×C4×C47—24no—
84C22×C4×C47C22×C412yes—
85C22×C4×C47C22×C412yes—
86C22×C4×C47C22×C412yes—
87C22×C4×C47C22×C412yes—
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C22×C412yes—
90C22×C4×C47C22×C411yescharacteristic
91C22×C4×C47C2411yescharacteristic
92C4⋊C4×C47—22no—
93C4⋊C4×C47—24no—
94C4⋊C4×C47—22no—
95C22⋊C4×C47—24no—
96C22⋊C4×C47—28no—
97C22⋊C4×C47—44no—
98C42×C47—22no—
99C42×C47—22no—
100C42×C47—22no—
101C42×C47—22no—
102C42×C47—22no—
103C42×C47—22no—
104C42×C47—42no—
105C42×C47C2×D412yes—
106C42×C47#4112yes—
107#10625329—48no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—24no—
114C2×C47⋊C8—24no—
115C2×C47⋊C8C2×D412yes—
116C2×C47⋊C8C2×D412yes—
117C2×C47⋊C8#4114yes—
118C2×C47⋊C8#4114yes—
119C23×C47—22no—
120C23×C47—24no—
121C23×C47—22no—
122C23×C47—22no—
123C23×C47—24no—
124C23×C47C23×C411yescharacteristic
125Q8×C47—28no—
126D4×C47—44no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—24no—
140C2×C4×C47—42no—
141C2×C4×C47—44no—
142C2×C4×C47—42no—
143C2×C4×C47C4×D412yes—
144C2×C4×C47C4×D412yes—
145C2×C4×C47C2×M1614yes—
146C2×C4×C47C2×M1612yes—
147C2×C4×C47#13112yes—
148C2×C4×C47C22×D411yescharacteristic
149C2×C4×C47C2×#4112yes—
150C2×C4×C47C2×#4112yes—
151C2×C4×C47C2×#4111yescharacteristic
152C47⋊C8—24no—
153C47⋊C8—22no—
154C47⋊C8—42no—
155C47⋊C8—42no—
156C47⋊C8#12414yescomplemented
157#21634—471nomaximal, Sylow
158C22×C47—42no—
159C22×C47—42no—
160C22×C47—44no—
161C22×C47—44no—
162C22×C47—42no—
163C22×C47C2×C4×D411yescharacteristic
164C22×C47C4×#4112yes—
165C22×C47C22×M1611yescharacteristic
166C22×C47C2×#13111yescharacteristic
167C22×C47#56812yes—
168C4×C47—24no—
169C4×C47—22no—
170C4×C47—42no—
171C4×C47—42no—
172C4×C47—42no—
173C4×C47—42no—
174C4×C47#43514yes—
175C4×C47C2×#12411yescharacteristic
176C4×C47#52912yes—
177C4×C47#53211yescharacteristic
178C2×C8×Q8—471no—
179C2×#436—471no—
180C4×#124D4711yescharacteristic
181#1984—471no—
182#1954—471no—
183#1924—474no—
184#1831—471no—
185#1753—474no—
186C4×#97—471no—
187C2×C47—82no—
188C2×C47C4×#12411yescharacteristic
189C2×C47C2×#43511yescharacteristic
190C2×C47#293211yescharacteristic
191C2×C47#296912yes—
192C2×C47#297112yes—
193C22×M16—472no—
194C2×#124Dic4711yescharacteristic
195C2×C4×Q8C2×D4711yescharacteristic
196C2×C4×D4C2×D4711yescharacteristic
197C8×Q8—944no—
198#436—944no—
199C2×C4⋊C8—471no—
200C2×C4⋊C8—471no—
201C2×C4⋊C8—472no—
202C2×C22⋊C8—472no—
203C2×C22⋊C8—474no—
204C2×#97—472no—
205C2×#97—472no—
206C2×C4×C8—471no—
207C2×C4×C8—472no—
208C2×C4×C8—471no—
209#391Dic4711yescharacteristic
210#390Dic4711yescharacteristic
211#388Dic4714yes—
212#382Dic4711yescharacteristic
213C4×C22⋊C4C2×D4714yes—
214C43C2×D4711yescharacteristic
215#336—474no—
216#336—474no—
217C47#2163411yescharacteristic, complemented, Sylow
218C22×Q8C2×Dic4711yescharacteristic
219C22×D4C2×Dic4711yescharacteristic
220C23×C4C22×D4712yes—
221C2×M16—948no—
222C2×M16—944no—
223C22×C8—471no—
224C22×C8—472no—
225C22×C8—471no—
226C22×C8—472no—
227C22×C8—472no—
228#124—24no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4C2×Dic4711yescharacteristic
234C2×C4⋊C4C2×Dic4711yescharacteristic
235C2×C4⋊C4C2×Dic4712yes—
236C2×C22⋊C4C2×Dic4712yes—
237C2×C22⋊C4C2×Dic4714yes—
238C2×C22⋊C4C2×Dic4714yes—
239C2×C42C2×Dic4711yescharacteristic
240C2×C42C2×Dic4711yescharacteristic
241C2×C42C2×Dic4712yes—
242C2×C42C22×D4712yes—
243C2×C42C22×D4712yes—
244C2×C42C22×D4711yescharacteristic
245C4⋊C8—944no—
246C4⋊C8—942no—
247C4⋊C8—942no—
248C22⋊C8—948no—
249C22⋊C8—944no—
250#97—944no—
251#97—944no—
252C4×C8—474no—
253C4×C8—944no—
254C4×C8—942no—
255#95C2×Dic4714yes—
256#95C2×Dic4714yes—
257C24C22×Dic4712yes—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8#1062532914yescomplemented
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4C22×Dic4712yes—
269C22×C4C22×Dic4712yes—
270C22×C4C22×Dic4712yes—
271C22×C4C22×Dic4712yes—
272C22×C4C22×Dic4711yescharacteristic
273C22×C4C22×Dic4712yes—
274C22×C4C22×Dic4711yescharacteristic
275C22×C4C23×D4711yescharacteristic
276M16—1888no—
277C2×C8—472no—
278C2×C8—474no—
279C2×C8—472no—
280C2×C8—474no—
281C2×C8—942no—
282C2×C8—942no—
283C2×C8—942no—
284C2×C8—942no—
285C2×C8—942no—
286C2×C8—944no—
287C2×C8—944no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—28no—
293C22⋊C4—44no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42D4×D4712yes—
302C42#1062535212yes—
303C23—22no—
304C23—24no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23C23×Dic4711yescharacteristic
309Q8—28no—
310D4—44no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4D4×Dic4712yes—
328C2×C4D4×Dic4712yes—
329C2×C4C2×#1062532914yes—
330C2×C4C2×#1062532912yes—
331C2×C4#1309403112yes—
332C2×C4C2×D4×D4711yescharacteristic
333C2×C4#41×D4712yes—
334C2×C4#41×D4712yes—
335C2×C4C2×#1062535211yescharacteristic
336C8—474no—
337C8—944no—
338C8—942no—
339C8—1882no—
340C8—1882no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22C2×D4×Dic4711yescharacteristic
347C22#41×Dic4712yes—
348C22C22×#1062532911yescharacteristic
349C22C2×#1309403111yescharacteristic
350C22#1568989012yes—
351C4—24no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4#1568919514yes—
358C4#124×D4711yescharacteristic
359C4#1568965212yes—
360C4#1568967211yescharacteristic
361C2—82no—
362C2#124×Dic4711yescharacteristic
363C2C2×#1568919511yescharacteristic
364C2#2025210611yescharacteristic
365C2#2025239312yes—
366C2#2025240812yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC4×#124×C47C2
prime radicalC22×D4×C47C2×C4
cosocleC22×C4×C47C24
hypercenterC4×#124D47
hypocenterC47#21634
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C47⋊C8C2semidirect
3. #20244325C2semidirect
5. #20244003C2semidirect
6. #20243505C2semidirect
7. #20243172C2semidirect
8. C4×#13093935C2semidirect
76. C2×Q8×C47M16semidirect
156. C47⋊C8#124semidirect
217. C47#21634semidirect
260. C2×Q8#10625329semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C47⋊C82
C2×#156892012
#202443252
#202441832
#202440032
#202435052
#202431722
C4×#130939352
C4×#124×C472
#2163447