Finite Group Databasebeta

← #26185499

Subgroups — #26185499

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#10625350C211yesmaximal, characteristic
2#20251953C212yesmaximal
3#20242158C212yesmaximal, complemented
4#20242157C214yesmaximal, complemented
5#20242144C211yesmaximal, characteristic, complemented
6#20242142C212yesmaximal, complemented
7#334×D47C212yesmaximal, complemented
8C2×C47×#334C211yesmaximal, characteristic, complemented
9C2×#13094030—22no—
10C4×#10625350C2211yescharacteristic
11C8×Dic94C411yescharacteristic
12C8×Dic94C411yescharacteristic
13C8×D188C411yescharacteristic
14C8×D188C411yescharacteristic
15C8×C22⋊D47C412yes—
16C8×C22⋊D47C412yes—
17#15689176C2211yescharacteristic
18#15689174C2212yes—
19#15689149—22no—
20#15689148—24no—
21#15689147C2212yes—
22#15689145C2214yes—
23C4⋊C8×D47C2214yes—
24C4×C8×D47C2212yes—
25#15688874C2212yes—
26#15688874C2212yes—
27#15688873C2214yes—
28#15688873C2214yes—
29C2×C4⋊C8×C47C2212yes—
30C2×C4×C8×C47C2211yescharacteristic
31C47×#334C2211yescharacteristic
32C47×#334C2211yescharacteristic
33C47×#334C2212yes—
34C2×#10625350D411yescharacteristic
35#13094030—48no—
36C2×#10625329—22no—
37C2×#10625324—24no—
38C2×C8×D47—24no—
39C2×C8×D47C2×C412yes—
40C2×C8×D47C2×C412yes—
41C4×Dic94C2×C411yescharacteristic
42C4×Dic94Q811yescharacteristic
43C4×D188C2×C411yescharacteristic
44C4×D188Q811yescharacteristic
45C4×C22⋊D47C2×C412yes—
46C4×C22⋊D47Q812yes—
47#13093983D411yescharacteristic
48#13093981D412yes—
49C42×D47C2312yes—
50C4⋊(C47⋊C8)—22no—
51C4⋊(C47⋊C8)C2×C411yescharacteristic
52C4⋊(C47⋊C8)C2×C411yescharacteristic
53C4⋊(C47⋊C8)C2314yes—
54Dic47⋊C8—24no—
55Dic47⋊C8—24no—
56Dic47⋊C8C2×C412yes—
57Dic47⋊C8C2×C412yes—
58C22⋊(C47⋊C8)C2×C412yes—
59(C2×D47)⋊C8C2×C412yes—
60(C2×D47)⋊C8C2×C412yes—
61C4×C47⋊C8C2312yes—
62C8×Dic47C2×C412yes—
63C8×Dic47C2×C412yes—
64C22×C8×C47—22no—
65C22×C8×C47C2×C411yescharacteristic
66C22×C8×C47C2×C411yescharacteristic
67C2×C42×C47C2311yescharacteristic
68C4⋊C8×C47—22no—
69C4⋊C8×C47C2312yes—
70C4⋊C8×C47C2312yes—
71C4×C8×C47C2×C411yescharacteristic
72C4×C8×C47C2×C411yescharacteristic
73C4×C8×C47C2311yescharacteristic
74C4×C8×C47C2311yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×Q811yescharacteristic
78C2×C22⋊D47C4⋊C412yes—
79#10625350SD1618yes—
80C2×D188C4⋊C411yescharacteristic
81C2×C4×D47C22×C412yes—
82C2×C4×D47C2×D412yes—
83C2×C4×D47C2×Q812yes—
84C2×Dic94C4⋊C411yescharacteristic
85C2×C8×C47—22no—
86C2×C8×C47—22no—
87C2×C8×C47—22no—
88C2×C8×C47—41no—
89C2×C8×C47—44no—
90C2×C8×C47C22×C411yescharacteristic
91C2×C8×C47C22×C411yescharacteristic
92C2×C8×C47C22×C411yescharacteristic
93C2×C8×C47C22×C411yescharacteristic
94C42×C47C22×C411yescharacteristic
95C42×C47C2×D411yescharacteristic
96C42×C47C2×Q811yescharacteristic
97C42×C47C2411yescharacteristic
98C22⋊Dic47C4⋊C412yes—
99(C2×D47)⋊C4C4⋊C412yes—
100(C2×D47)⋊C4C4⋊C412yes—
101C4⋊Dic47C4⋊C411yescharacteristic
102C4⋊Dic47C4⋊C411yescharacteristic
103Dic47⋊C4C4⋊C412yes—
104Dic47⋊C4C4⋊C412yes—
105C4×Dic47C22×C412yes—
106C4×Dic47C2×D412yes—
107C4×Dic47C2×Q812yes—
108#10625329—44no—
109C2×C47⋊C8—24no—
110C2×C47⋊C8C22×C412yes—
111C2×C47⋊C8C22×C412yes—
112#10625324—48no—
113C8×D47—24no—
114C8×D47—42no—
115C8×D47—48no—
116C23×C47C2×C4⋊C411yescharacteristic
117C22×D47C2×C4⋊C412yes—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—41no—
121C2×C4×C47C2×C4⋊C411yescharacteristic
122C2×C4×C47C2×C4⋊C411yescharacteristic
123C2×C4×C47C2×C4⋊C411yescharacteristic
124C2×C4×C47C2×SD1614yes—
125C2×C4×C47C23×C411yescharacteristic
126C2×C4×C47C22×D411yescharacteristic
127C2×C4×C47C22×Q811yescharacteristic
128C22⋊D47#10618yes—
129C2×Dic47C2×C4⋊C412yes—
130C2×Dic47C2×C4⋊C412yes—
131C2×Dic47C2×C4⋊C412yes—
132D188#10614yes—
133C4×D47—22no—
134C4×D47—22no—
135C4×D47—42no—
136C4×D47C2×SD1618yes—
137Dic94#10614yes—
138C8×C47—21no—
139C8×C47—21no—
140C8×C47—21no—
141C8×C47—42no—
142C8×C47—42no—
143C8×C47#13112yescomplemented
144C47⋊C8—22no—
145C47⋊C8—22no—
146C47⋊C8—44no—
147#17826—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47C2×#10612yes—
151C22×C47C22×C4⋊C411yescharacteristic
152C2×D47—22no—
153C2×D47—22no—
154C2×D47C2×#10614yes—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47C2×#10612yes—
159C4×C47C2×#13111yescharacteristic
160C4×C47#56811yescharacteristic
161C4×C47C22×SD1612yes—
162Dic47—22no—
163Dic47—22no—
164Dic47C2×#10614yes—
165C8×#41—471no—
166#2906—472no—
167#1570—472no—
168#1570—474no—
169#1566—471no—
170#1566—472no—
171C2×#334—472no—
172C2×#334D4711yescharacteristic, complemented
173C2×C47—22no—
174C2×C47#290611yescharacteristic
175C2×C47C22×#10611yescharacteristic
176C2×C47#291611yescharacteristic
177D47—24no—
178C2×#131—942no—
179C4×#41—471no—
180C8×Q8—471no—
181C8×Q8—471no—
182C8×D4—472no—
183C8×D4—472no—
184C8×D4—471no—
185C8×D4—471no—
186#431—471no—
187#431—472no—
188#424—944no—
189#424—942no—
190#423—472no—
191#423—474no—
192C2×C4⋊C8—474no—
193C2×C4⋊C8C2×D4712yes—
194C2×C4×C8—472no—
195C2×C4×C8C2×D4711yescharacteristic
196#334—474no—
197#334—474no—
198#334—472no—
199#334—472no—
200#334C2×D4711yescharacteristic
201#334C2×D4711yescharacteristic
202#334C2×D4712yes—
203C47#1782611yescharacteristic, complemented, Sylow
204C2×#41—471no—
205#131—1888no—
206C2×M16—944no—
207C2×M16—942no—
208C22×C8—22no—
209C22×C8—472no—
210C22×C8—472no—
211C22×C8—944no—
212C22×C8C4×D4711yescharacteristic
213C22×C8C4×D4711yescharacteristic
214C4×Q8—471no—
215C4×Q8—471no—
216C4×D4—472no—
217C4×D4—472no—
218C4×D4—471no—
219C4×D4—471no—
220#117—471no—
221#117—472no—
222C2×C42—472no—
223C2×C42C22×D4711yescharacteristic
224C4⋊C8—22no—
225C4⋊C8—472no—
226C4⋊C8—472no—
227C4⋊C8—471no—
228C4⋊C8—471no—
229C4⋊C8—474no—
230C4⋊C8—944no—
231C4⋊C8—944no—
232C4⋊C8—942no—
233C4⋊C8C22×D4712yes—
234C4⋊C8C22×D4712yes—
235C22⋊C8—472no—
236C22⋊C8—472no—
237C22⋊C8—472no—
238C4×C8—472no—
239C4×C8—472no—
240C4×C8—472no—
241C4×C8C4×D4711yescharacteristic
242C4×C8C4×D4711yescharacteristic
243C4×C8C22×D4711yescharacteristic
244C4×C8C22×D4711yescharacteristic
245#41—478no—
246C2×Q8—471no—
247C2×D4—472no—
248C2×D4—471no—
249C22×C4—472no—
250C22×C4—472no—
251C22×C4—472no—
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4D4×D4711yescharacteristic
254C22×C4Q8×D4711yescharacteristic
255M16—1888no—
256M16—1884no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—44no—
262C2×C8—472no—
263C2×C8—472no—
264C2×C8—944no—
265C2×C8—944no—
266C2×C8—1882no—
267C2×C8—1888no—
268C2×C8C2×C4×D4711yescharacteristic
269C2×C8C2×C4×D4711yescharacteristic
270C2×C8C2×C4×D4711yescharacteristic
271C2×C8C2×C4×D4711yescharacteristic
272C4⋊C4—472no—
273C4⋊C4—472no—
274C4⋊C4—471no—
275C4⋊C4—471no—
276C22⋊C4—472no—
277C22⋊C4—472no—
278C22⋊C4—472no—
279C42—472no—
280C42—472no—
281C42—472no—
282C42C2×C4×D4711yescharacteristic
283C42D4×D4711yescharacteristic
284C42Q8×D4711yescharacteristic
285C42C23×D4711yescharacteristic
286C23—472no—
287C23C4⋊C4×D4711yescharacteristic
288Q8—474no—
289D4—478no—
290D4—474no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—472no—
295C2×C4—478no—
296C2×C4—472no—
297C2×C4—472no—
298C2×C4—942no—
299C2×C4—942no—
300C2×C4—1882no—
301C2×C4C4⋊C4×D4711yescharacteristic
302C2×C4C4⋊C4×D4711yescharacteristic
303C2×C4C4⋊C4×D4711yescharacteristic
304C2×C4SD16×D4714yes—
305C2×C4C22×C4×D4711yescharacteristic
306C2×C4C2×D4×D4711yescharacteristic
307C2×C4C2×Q8×D4711yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8—42no—
312C8—42no—
313C8—942no—
314C8—942no—
315C8—1884no—
316C8#1309403012yescomplemented
317C22—21no—
318C22—21no—
319C22—474no—
320C22—942no—
321C22—942no—
322C22#106×D4712yes—
323C22C2×C4⋊C4×D4711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—474no—
328C4—942no—
329C4—942no—
330C4#106×D4712yes—
331C4C2×#1309403011yescharacteristic
332C4#1568988911yescharacteristic
333C4C2×SD16×D4712yes—
334C2—22no—
335C2—944no—
336C2#2025195311yescharacteristic
337C2C2×#106×D4711yescharacteristic
338C2#2025199611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC2×C47×#334C2
socleC22×C47C22×C4⋊C4
prime radicalC2×#10625350D4
cosocleC42×C47C24
hypercenterC2×#334D47
hypocenterC47#17826
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
3. #20242158C2semidirect
4. #20242157C2semidirect
5. #20242144C2semidirect
6. #20242142C2semidirect
7. #334×D47C2semidirect
8. C2×C47×#334C2semidirect
143. C8×C47#131semidirect
172. C2×#334D47semidirect
203. C47#17826semidirect
316. C8#13094030semidirect

Maximal subgroups

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

subgroupindex
C8×#106253502
#202519532
#202421582
#202421572
#202421442
#202421422
#334×D472
C2×C47×#3342
#1782647