Finite Group Databasebeta

← #26225760

Subgroups — #26225760

Order 12032.

1302 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689176C211yesmaximal, characteristic
2#20245795C211yesmaximal, characteristic
3#20244350C214yesmaximal, complemented
4#20244341C214yesmaximal, complemented
5#20243474C214yesmaximal
6C22×C4⋊Q8×C47C211yesmaximal, characteristic
7C2×#13093983—21no—
8#15689176—24no—
9C2×C4⋊(C47⋊C8)C2212yes—
10C2×C22⋊(C47⋊C8)C2212yes—
11C2×C4×C47⋊C8C2212yes—
12#15688993C2212yes—
13#15688989—21no—
14#15688983—21no—
15#15688983C2212yes—
16C4×C4⋊Dic47—21no—
17C4×C4⋊Dic47C2212yes—
18#15688885C22116yes—
19C23×Q8×C47—21no—
20C2×C4⋊Q8×C47—21no—
21C2×C4⋊Q8×C47—21no—
22C2×C4⋊Q8×C47—24no—
23C2×C4⋊Q8×C47C418yescomplemented
24C2×C4⋊Q8×C47C2212yes—
25C2×C4⋊Q8×C47C2212yes—
26C22×C4⋊C4×C47—21no—
27C22×C4⋊C4×C47C2212yes—
28C22×C42×C47C2211yescharacteristic
29C22×C47⋊C8C2314yes—
30#13093983—44no—
31C2×C4⋊Dic47—22no—
32C2×C4⋊Dic47—22no—
33C2×C4⋊Dic47C2314yes—
34C2×C22⋊Dic47—22no—
35C2×C4×Dic47—22no—
36C4⋊(C47⋊C8)—24no—
37C22⋊(C47⋊C8)—24no—
38C4×C47⋊C8—24no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C22×Q8×C47—28no—
42C22×Q8×C47—22no—
43C22×Q8×C47—22no—
44C22×Q8×C47—44no—
45C22×Q8×C47—44no—
46C23×C4×C47—22no—
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C4⋊Q8×C47—28no—
51C4⋊Q8×C47—28no—
52C4⋊Q8×C47—24no—
53C4⋊Q8×C47—24no—
54C4⋊Q8×C47C2×C418yes—
55C4⋊Q8×C47C2×C418yes—
56C2×C4⋊C4×C47—28no—
57C2×C4⋊C4×C47—22no—
58C2×C4⋊C4×C47—22no—
59C2×C4⋊C4×C47—22no—
60C2×C4⋊C4×C47—24no—
61C2×C4⋊C4×C47C2×C418yes—
62C2×C4⋊C4×C47C2314yes—
63C2×C42×C47C2×C414yes—
64C2×C42×C47D411yescharacteristic
65C2×C42×C47D411yescharacteristic
66C2×C42×C47D414yes—
67C2×C42×C47C2311yescharacteristic
68C2×C42×C47C2311yescharacteristic
69C24×C47C2×D411yescharacteristic
70C2×Q8×C47—28no—
71C2×Q8×C47—28no—
72C2×Q8×C47—432no—
73C2×Q8×C47—44no—
74C2×Q8×C47—44no—
75C2×Q8×C47—44no—
76C2×Q8×C47—44no—
77C22×C4×C47—24no—
78C22×C4×C47—24no—
79C22×C4×C47—28no—
80C22×C4×C47—24no—
81C22×C4×C47—44no—
82C22×C4×C47—44no—
83C22×C4×C47C22⋊C414yes—
84C22×C4×C47C22⋊C414yes—
85C22×C4×C47D814yes—
86C22×C4×C47SD1614yes—
87C22×C4×C47C22×C412yes—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2411yescharacteristic
95C22×Dic47—24no—
96C4⋊C4×C47—28no—
97C4⋊C4×C47—28no—
98C4⋊C4×C47—28no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47C22×C418yes—
101C42×C47C22⋊C414yes—
102C42×C47C22⋊C414yes—
103C42×C47C22×C412yes—
104C42×C47C22×C412yes—
105C42×C47C2×D412yes—
106C42×C47C2×D412yes—
107C22⋊Dic47—44no—
108C4⋊Dic47—28no—
109C4⋊Dic47—44no—
110C4⋊Dic47—44no—
111C4×Dic47—44no—
112C2×C47⋊C8—24no—
113C2×C47⋊C8—28no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47C2×C22⋊C412yes—
117C23×C47C2×D812yes—
118C23×C47C2×SD1612yes—
119C23×C47C22×D411yescharacteristic
120Q8×C47—416no—
121Q8×C47—416no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—24no—
126C2×C4×C47—28no—
127C2×C4×C47—28no—
128C2×C4×C47—44no—
129C2×C4×C47—44no—
130C2×C4×C47—416no—
131C2×C4×C47#102116yes—
132C2×C4×C47C2×C22⋊C412yes—
133C2×C4×C47C2×C22⋊C412yes—
134C2×C4×C47C2×C22⋊C412yes—
135C2×C4×C47C2×C22⋊C412yes—
136C2×C4×C47C2×C22⋊C412yes—
137C2×C4×C47C2×D812yes—
138C2×C4×C47C2×D812yes—
139C2×C4×C47C2×SD1612yes—
140C2×C4×C47C2×SD1612yes—
141C2×C4×C47#13714yes—
142C2×C4×C47C23×C411yescharacteristic
143C2×C4×C47C22×D411yescharacteristic
144C2×C4×C472−1+414yes—
145C2×Dic47—44no—
146C2×Dic47—44no—
147C2×Dic47—44no—
148C47⋊C8—44no—
149#21833—471nomaximal, Sylow
150C22×C47—22no—
151C22×C47—22no—
152C22×C47—24no—
153C22×C47—24no—
154C22×C47C2×#10214yes—
155C22×C47C22×C22⋊C411yescharacteristic
156C22×C47#53612yes—
157C22×C47C22×D811yescharacteristic
158C22×C47C22×SD1611yescharacteristic
159C22×C47C2×#13712yes—
160C4×C47—22no—
161C4×C47—22no—
162C4×C47—48no—
163C4×C47C2×#10214yes—
164C4×C47C2×#10214yes—
165C4×C47#41914yes—
166C4×C47#51912yes—
167C4×C47#53612yes—
168Dic47—84no—
169C22×C4⋊Q8D4711yescharacteristic
170C2×#431—471no—
171#2285—471no—
172#1990—474no—
173#1988—474no—
174#1823—474no—
175C2×C47—24no—
176C2×C47#231011yescharacteristic
177C2×C47C22×#10211yescharacteristic
178C2×C47C2×#41911yescharacteristic
179C2×C47#322612yes—
180C2×C47#322912yes—
181C23×Q8—21no—
182C2×C4⋊Q8—21no—
183C2×C4⋊Q8—21no—
184C2×C4⋊Q8—24no—
185C2×C4⋊Q8Dic4718yescomplemented
186C2×C4⋊Q8C2×D4712yes—
187C2×C4⋊Q8C2×D4712yes—
188C2×#117—941no—
189C22×C4⋊C4—21no—
190C22×C4⋊C4C2×D4712yes—
191C22×C42C2×D4711yescharacteristic
192#431—944no—
193C2×C4⋊C8—472no—
194C2×C22⋊C8—472no—
195C2×C4×C8—472no—
196#384—472no—
197#382—941no—
198#380—472no—
199#380—941no—
200C4×C4⋊C4—472no—
201C4×C4⋊C4—941no—
202#340—4716no—
203C47#2183311yescharacteristic, complemented, Sylow
204C22×Q8—28no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—44no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4C22⋊D4711yescharacteristic
211C23×C4C22⋊D4711yescharacteristic
212C23×C4C22×D4711yescharacteristic
213C22×C8—474no—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8C2×Dic4718yes—
219C4⋊Q8C2×Dic4718yes—
220#117—1884no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—474no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—942no—
229C2×C4⋊C4C2×Dic4718yes—
230C2×C4⋊C4C22×D4714yes—
231C2×C22⋊C4—942no—
232C2×C42—942no—
233C2×C42C2×Dic4714yes—
234C2×C42C22⋊D4711yescharacteristic
235C2×C42C22⋊D4711yescharacteristic
236C2×C42C22⋊D4714yes—
237C2×C42C22×D4711yescharacteristic
238C2×C42C22×D4711yescharacteristic
239C4⋊C8—944no—
240C22⋊C8—944no—
241C4×C8—944no—
242#95—942no—
243#95—942no—
244C24C2×C22⋊D4711yescharacteristic
245C2×Q8—28no—
246C2×Q8—28no—
247C2×Q8—432no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—944no—
259C22×C4#1062533414yes—
260C22×C4#1062533514yes—
261C22×C4C22⋊Dic4714yes—
262C22×C4C22⋊Dic4714yes—
263C22×C4C22×Dic4712yes—
264C22×C4C2×C22⋊D4711yescharacteristic
265C22×C4C2×C22⋊D4711yescharacteristic
266C22×C4C2×C22⋊D4712yes—
267C22×C4C2×C22⋊D4711yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270C22×C4C23×D4711yescharacteristic
271C2×C8—944no—
272C2×C8—948no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—24no—
277C4⋊C4—948no—
278C4⋊C4—1884no—
279C4⋊C4—1884no—
280C4⋊C4C22×Dic4718yes—
281C22⋊C4—1884no—
282C42—1884no—
283C42C22⋊Dic4714yes—
284C42C22⋊Dic4714yes—
285C42C22×Dic4712yes—
286C42C22×Dic4712yes—
287C42C2×C22⋊D4712yes—
288C42C2×C22⋊D4712yes—
289C23—22no—
290C23—22no—
291C23C2×C22⋊Dic4712yes—
292C23C2×#1062533412yes—
293C23C2×#1062533512yes—
294C23C22×C22⋊D4711yescharacteristic
295Q8—416no—
296Q8—416no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—1884no—
307C2×C4—1884no—
308C2×C4—1884no—
309C2×C4#13093945116yes—
310C2×C4C2×C22⋊Dic4712yes—
311C2×C4C2×C22⋊Dic4712yes—
312C2×C4C2×C22⋊Dic4712yes—
313C2×C4C2×C22⋊Dic4712yes—
314C2×C4C2×C22⋊Dic4712yes—
315C2×C4C2×#1062533412yes—
316C2×C4C2×#1062533412yes—
317C2×C4C2×#1062533512yes—
318C2×C4C2×#1062533512yes—
319C2×C4#1309405514yes—
320C2×C4C23×Dic4711yescharacteristic
321C2×C4C22×C22⋊D4711yescharacteristic
322C2×C4#1309407214yes—
323C8—1884no—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—22no—
328C22C2×#1309394514yes—
329C22C22×C22⋊Dic4711yescharacteristic
330C22#1568969712yes—
331C22C22×#1062533411yescharacteristic
332C22C22×#1062533511yescharacteristic
333C22C2×#1309405512yes—
334C4—22no—
335C4—22no—
336C4—48no—
337C4—3764no—
338C4C2×#1309394514yes—
339C4C2×#1309394514yes—
340C4#1568913214yes—
341C4#1568960112yes—
342C4#1568969712yes—
343C2—24no—
344C2#2024597411yescharacteristic
345C2C22×#1309394511yescharacteristic
346C2C2×#1568913211yescharacteristic
347C2#2025450912yes—
348C2#2025453212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C4⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC22×C4⋊Q8D47
hypocenterC47#21833
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20244350C2semidirect
4. #20244341C2semidirect
23. C2×C4⋊Q8×C47C4semidirect
185. C2×C4⋊Q8Dic47semidirect
203. C47#21833semidirect

Maximal subgroups

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

subgroupindex
C2×#156891762
#202457952
#202443502
#202443412
#202434742
C22×C4⋊Q8×C472
#2183347