Finite Group Databasebeta

← #27079730

Subgroups — #27079730

Order 12032.

5134 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×#15689890C212yesmaximal, complemented
2#20251837C2116yesmaximal, complemented
3#20251833C216yesmaximal, complemented
4C2×#15689087C216yesmaximal, complemented
5C23×#41×C47C211yesmaximal, characteristic
6#15689890—216no—
7C2×#13094031C22116yescomplemented
8C22×#10625329C22112yescomplemented
9#15689090C22148yescomplemented
10#15689087C2214yes—
11#15689087C22124yescomplemented
12#15689087C22112yescomplemented
13C2×C22⋊(C47⋊C8)C22124yescomplemented
14C22×#41×C47—216no—
15C22×#41×C47C418yes—
16C22×#41×C47C411yescharacteristic
17C22×#41×C47C413yes—
18C22×#41×C47C2213yes—
19C22×#41×C47C2218yes—
20C22×#41×C47C2211yescharacteristic
21C23×Q8×C47C411yescharacteristic
22C23×D4×C47C413yes—
23C24×C4×C47C2213yes—
24#13094031—264no—
25C2×#10625329—248no—
26C2×#10625329C2318yes—
27C2×#10625329C23124yescomplemented
28C22×C47⋊C8C23124yescomplemented
29C22⋊(C47⋊C8)C23116yes—
30C22⋊(C47⋊C8)C23148yescomplemented
31C25×C47C2×C413yes—
32C2×#41×C47—264no—
33C2×#41×C47—264no—
34C2×#41×C47—28no—
35C2×#41×C47—224no—
36C2×#41×C47—224no—
37C2×#41×C47C2×C418yes—
38C2×#41×C47C2×C4124yes—
39C2×#41×C47C2×C4112yes—
40C2×#41×C47C2×C414yes—
41C2×#41×C47D4116yes—
42C2×#41×C47C23112yes—
43C2×#41×C47C2314yes—
44C22×Q8×C47—28no—
45C22×Q8×C47C2×C418yes—
46C22×Q8×C47C2×C413yes—
47C22×Q8×C47C2×C411yescharacteristic
48C22×D4×C47—224no—
49C22×D4×C47C2×C416yes—
50C22×D4×C47C2×C4124yes—
51C22×D4×C47C2×C413yes—
52C22×D4×C47C2×C413yes—
53C23×C4×C47—224no—
54C23×C4×C47C2×C413yes—
55C23×C4×C47C2×C413yes—
56C23×C4×C47C2×C4112yes—
57C23×C4×C47C2×C413yes—
58C23×C4×C47C23112yes—
59C23×C4×C47C2311yescharacteristic
60C23×C4×C47C2313yes—
61C23×C4×C47C2313yes—
62C24×C47—212no—
63C24×C47—412no—
64C24×C47C22×C411yescharacteristic
65C24×C47C22×C4112yes—
66C24×C47C22×C413yes—
67C24×C47C22×C413yes—
68#41×C47—232no—
69#41×C47—296no—
70#41×C47—248no—
71#41×C47—248no—
72#41×C47C22⋊C4132yes—
73#41×C47C2×D4132yes—
74C2×Q8×C47—232no—
75C2×Q8×C47—212no—
76C2×Q8×C47C22⋊C418yes—
77C2×Q8×C47C22×C4112yes—
78C2×Q8×C47C22×C414yes—
79C2×D4×C47—224no—
80C2×D4×C47—296no—
81C2×D4×C47—212no—
82C2×D4×C47C22⋊C4124yes—
83C2×D4×C47C22×C4124yes—
84C2×D4×C47C22×C4112yes—
85C2×D4×C47C22×C4112yes—
86C22×C4×C47—212no—
87C22×C4×C47—212no—
88C22×C4×C47—212no—
89C22×C4×C47—248no—
90C22×C4×C47—24no—
91C22×C4×C47—212no—
92C22×C4×C47—248no—
93C22×C4×C47C22×C411yescharacteristic
94C22×C4×C47C22×C412yes—
95C22×C4×C47C22×C413yes—
96C22×C4×C47C22×C4112yes—
97C22×C4×C47C22×C4112yes—
98C22×C4×C47C22×C4112yes—
99C22×C4×C47C22×C416yes—
100C22×C4×C47C22×C413yes—
101C22×C4×C47C22×C416yes—
102C22×C4×C47C2×D4124yes—
103C22×C4×C47C2412yes—
104C22×C4×C47C2416yes—
105C22×C4×C47C2411yescharacteristic
106C22×C4×C47C2416yes—
107#10625329—216no—
108#10625329—248no—
109C2×C47⋊C8—248no—
110C2×C47⋊C8C24116yes—
111C23×C47—22no—
112C23×C47—24no—
113C23×C47—224no—
114C23×C47—26no—
115C23×C47—224no—
116C23×C47—46no—
117C23×C47—412no—
118C23×C47—46no—
119C23×C47—448no—
120C23×C47C2×C22⋊C4112yes—
121C23×C47C23×C416yes—
122C23×C47C23×C411yescharacteristic
123C23×C47C23×C416yes—
124C23×C47C23×C412yes—
125Q8×C47—224no—
126Q8×C47C2×C22⋊C4116yes—
127D4×C47—248no—
128D4×C47—224no—
129D4×C47C2×C22⋊C4148yes—
130C2×C4×C47—22no—
131C2×C4×C47—26no—
132C2×C4×C47—224no—
133C2×C4×C47—24no—
134C2×C4×C47—224no—
135C2×C4×C47—224no—
136C2×C4×C47—212no—
137C2×C4×C47—24no—
138C2×C4×C47—212no—
139C2×C4×C47C2×C22⋊C4112yes—
140C2×C4×C47C2×C22⋊C4124yes—
141C2×C4×C47C23×C412yes—
142C2×C4×C47C23×C416yes—
143C2×C4×C47C23×C411yescharacteristic
144C2×C4×C47C23×C416yes—
145C2×C4×C47C22×D4124yes—
146C2×C4×C47C22×D414yes—
147C2×C4×C47C2511yescharacteristic
148C47⋊C8—216no—
149#60064—471nomaximal, Sylow
150C22×C47—28no—
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—212no—
155C22×C47—26no—
156C22×C47—212no—
157C22×C47—412no—
158C22×C47—448no—
159C22×C47—412no—
160C22×C47C22×C22⋊C412yes—
161C22×C47C22×C22⋊C4112yes—
162C22×C47#56814yescomplemented
163C22×C47C24×C411yescharacteristic
164C4×C47—22no—
165C4×C47—26no—
166C4×C47C22×C22⋊C412yes—
167C4×C47C22×C22⋊C4112yes—
168C4×C47C23×D412yes—
169C23×#41D4711yescharacteristic
170C2×#568—472no—
171#2880—4716no—
172#2879—476no—
173C2×#407—476no—
174C2×C47—22no—
175C2×C47—24no—
176C2×C47—212no—
177C2×C47—46no—
178C2×C47C23×C22⋊C411yescharacteristic
179C2×C47C2×#56812yes—
180C22×#41—216no—
181C22×#41Dic4718yes—
182C22×#41Dic4711yescharacteristic
183C22×#41Dic4713yes—
184C22×#41C2×D4713yes—
185C22×#41C2×D4718yes—
186C22×#41C2×D4711yescharacteristic
187C23×Q8Dic4711yescharacteristic
188C23×D4Dic4713yes—
189C24×C4C2×D4713yes—
190#568—9416no—
191C2×#131—4716no—
192C22×M16—4712no—
193#408—4748no—
194#407—474no—
195#407—4712no—
196#407—4724no—
197C2×C22⋊C8—4724no—
198C47#6006411yescharacteristic, complemented, Sylow
199C25C2×Dic4713yes—
200C2×#41—264no—
201C2×#41—264no—
202C2×#41—28no—
203C2×#41—224no—
204C2×#41—224no—
205C2×#41C2×Dic4718yes—
206C2×#41C2×Dic47124yes—
207C2×#41C2×Dic47112yes—
208C2×#41C2×Dic4714yes—
209C2×#41C22⋊D47116yes—
210C2×#41C22×D47112yes—
211C2×#41C22×D4714yes—
212C22×Q8—28no—
213C22×Q8C2×Dic4718yes—
214C22×Q8C2×Dic4713yes—
215C22×Q8C2×Dic4711yescharacteristic
216C22×D4—224no—
217C22×D4C2×Dic4716yes—
218C22×D4C2×Dic47124yes—
219C22×D4C2×Dic4713yes—
220C22×D4C2×Dic4713yes—
221C23×C4—224no—
222C23×C4C2×Dic4713yes—
223C23×C4C2×Dic4713yes—
224C23×C4C2×Dic47112yes—
225C23×C4C2×Dic4713yes—
226C23×C4C22×D4711yescharacteristic
227C23×C4C22×D47112yes—
228C23×C4C22×D4713yes—
229C23×C4C22×D4713yes—
230#131—9464no—
231C2×M16—478no—
232C2×M16—4724no—
233C2×M16—9448no—
234C22×C8—4724no—
235C22⋊C8—4716no—
236C22⋊C8—4748no—
237C24—212no—
238C24—412no—
239C24C22×Dic4711yescharacteristic
240C24C22×Dic47112yes—
241C24C22×Dic4713yes—
242C24C22×Dic4713yes—
243#41—232no—
244#41—296no—
245#41—248no—
246#41—248no—
247#41C22⋊Dic47132yes—
248#41C2×C22⋊D47132yes—
249C2×Q8—232no—
250C2×Q8—212no—
251C2×Q8C22⋊Dic4718yes—
252C2×Q8C22×Dic47112yes—
253C2×Q8C22×Dic4714yes—
254C2×D4—224no—
255C2×D4—296no—
256C2×D4—212no—
257C2×D4C22⋊Dic47124yes—
258C2×D4C22×Dic47124yes—
259C2×D4C22×Dic47112yes—
260C2×D4C22×Dic47112yes—
261C22×C4—212no—
262C22×C4—212no—
263C22×C4—212no—
264C22×C4—248no—
265C22×C4—24no—
266C22×C4—212no—
267C22×C4—248no—
268C22×C4C22×Dic4711yescharacteristic
269C22×C4C22×Dic4712yes—
270C22×C4C22×Dic4713yes—
271C22×C4C22×Dic47112yes—
272C22×C4C22×Dic47112yes—
273C22×C4C22×Dic47112yes—
274C22×C4C22×Dic4716yes—
275C22×C4C22×Dic4713yes—
276C22×C4C22×Dic4716yes—
277C22×C4C2×C22⋊D47124yes—
278C22×C4C23×D4712yes—
279C22×C4C23×D4716yes—
280C22×C4C23×D4716yes—
281C22×C4C23×D4711yescharacteristic
282M16—9416no—
283M16—9448no—
284C2×C8—4716no—
285C2×C8—9448no—
286C23—22no—
287C23—24no—
288C23—224no—
289C23—26no—
290C23—224no—
291C23—46no—
292C23—412no—
293C23—46no—
294C23—448no—
295C23C2×C22⋊Dic47112yes—
296C23C23×Dic4716yes—
297C23C23×Dic4711yescharacteristic
298C23C23×Dic4716yes—
299C23C23×Dic4712yes—
300Q8—224no—
301Q8C2×C22⋊Dic47116yes—
302D4—248no—
303D4—224no—
304D4C2×C22⋊Dic47148yes—
305C2×C4—22no—
306C2×C4—26no—
307C2×C4—224no—
308C2×C4—24no—
309C2×C4—224no—
310C2×C4—224no—
311C2×C4—212no—
312C2×C4—24no—
313C2×C4—212no—
314C2×C4C2×C22⋊Dic47112yes—
315C2×C4C2×C22⋊Dic47124yes—
316C2×C4C23×Dic4712yes—
317C2×C4C23×Dic4716yes—
318C2×C4C23×Dic4711yescharacteristic
319C2×C4C23×Dic4716yes—
320C2×C4C22×C22⋊D47124yes—
321C2×C4C22×C22⋊D4714yes—
322C2×C4C24×D4711yescharacteristic
323C8—9416no—
324C22—28no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—212no—
329C22—26no—
330C22—212no—
331C22—412no—
332C22—448no—
333C22—412no—
334C22C22×C22⋊Dic4712yes—
335C22C22×C22⋊Dic47112yes—
336C22#1568989014yescomplemented
337C22C24×Dic4711yescharacteristic
338C4—22no—
339C4—26no—
340C4C22×C22⋊Dic4712yes—
341C4C22×C22⋊Dic47112yes—
342C4C23×C22⋊D4712yes—
343C2—22no—
344C2—24no—
345C2—212no—
346C2—46no—
347C2C23×C22⋊Dic4711yescharacteristic
348C2C2×#1568989012yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
Fitting, prime radicalC23×#41×C47C2
cosocleC2×C4×C47C25
hypercenterC23×#41D47
hypocenterC47#60064
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15689890C2semidirect
2. #20251837C2semidirect
3. #20251833C2semidirect
4. C2×#15689087C2semidirect
7. C2×#13094031C22semidirect
8. C22×#10625329C22semidirect
9. #15689090C22semidirect
11. #15689087C22semidirect
12. #15689087C22semidirect
13. C2×C22⋊(C47⋊C8)C22semidirect
27. C2×#10625329C23semidirect
28. C22×C47⋊C8C23semidirect
30. C22⋊(C47⋊C8)C23semidirect
162. C22×C47#568semidirect
198. C47#60064semidirect
336. C22#15689890semidirect

Maximal subgroups

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

subgroupindex
C2×#156898902
#202518372
#202518332
C2×#156890872
C23×#41×C472
#6006447