Finite Group Databasebeta

← C22×#15689171 #27079970

Subgroups — C22×#15689171 #27079970

Order 12032.

2718 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C211yesmaximal, characteristic
2C23×#10625329C211yesmaximal, characteristic
3C2×#15689171C2124yesmaximal, direct product factor
4C22×#13093959C214yesmaximal, complemented
5C23×M16×C47C211yesmaximal, characteristic
6C22×#10625329—21no—
7C22×#10625329—26no—
8C22×#10625329C412yes—
9C22×#10625329C412yes—
10C22×#10625329C4112yes—
11C22×#10625329C2212yes—
12C22×#10625329C22112yes—
13C22×#10625329C22112yes—
14C23×C47⋊C8—21no—
15C23×C47⋊C8C2212yes—
16#15689171C22164yesdirect product factor
17C2×#13093959C22148yescomplemented
18C24×C4×C47C2211yescharacteristic
19C22×M16×C47—21no—
20C22×M16×C47—26no—
21C22×M16×C47C2212yes—
22C22×M16×C47C22112yes—
23C23×C8×C47—21no—
24C2×#10625329—212no—
25C2×#10625329—26no—
26C2×#10625329—224no—
27C2×#10625329C2×C4124yes—
28C2×#10625329C2×C4124yes—
29C2×#10625329C2×C4148yes—
30C2×#10625329C23112yes—
31C2×#10625329C23116yes—
32C2×#10625329C23116yes—
33C22×C47⋊C8—212no—
34C22×C47⋊C8—22no—
35C22×C47⋊C8C2×C4112yes—
36C22×C47⋊C8C2×C414yes—
37C22×C47⋊C8C23112yes—
38#13093959C23164yescomplemented
39C25×C47Q811yescharacteristic
40C23×C4×C47C2×C411yescharacteristic
41C23×C4×C47C2×C411yescharacteristic
42C23×C4×C47C2×C416yes—
43C23×C4×C47D411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D416yes—
46C23×C4×C47Q811yescharacteristic
47C23×C4×C47Q816yes—
48C23×C4×C47C2311yescharacteristic
49C23×C4×C47C2316yes—
50C2×M16×C47—212no—
51C2×M16×C47—26no—
52C2×M16×C47—224no—
53C2×M16×C47C23112yes—
54C2×M16×C47C23116yes—
55C22×C8×C47—212no—
56C22×C8×C47—22no—
57C24×C47—44no—
58C24×C47C4⋊C411yescharacteristic
59C24×C47C4⋊C411yescharacteristic
60C24×C47C4⋊C416yes—
61C24×C47C2×Q811yescharacteristic
62C24×C47C2×Q816yes—
63C22×C4×C47C4⋊C416yes—
64C22×C4×C47C4⋊C416yes—
65C22×C4×C47C4⋊C416yes—
66C22×C4×C47C4⋊C416yes—
67C22×C4×C47C4⋊C416yes—
68C22×C4×C47C4⋊C4124yes—
69C22×C4×C47C4⋊C411yescharacteristic
70C22×C4×C47C4⋊C411yescharacteristic
71C22×C4×C47C22×C413yes—
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C22×C416yes—
74C22×C4×C47C22×C416yes—
75C22×C4×C47C22×C4112yes—
76C22×C4×C47C2×D416yes—
77C22×C4×C47C2×D416yes—
78C22×C4×C47C2×D413yes—
79C22×C4×C47C2×D4112yes—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×Q813yes—
82C22×C4×C47C2×Q816yes—
83C22×C4×C47C2×Q8112yes—
84C22×C4×C47C2413yes—
85C22×C4×C47C2414yes—
86M16×C47—216no—
87M16×C47—212no—
88M16×C47C2418yes—
89C2×C8×C47—212no—
90C2×C8×C47—216no—
91#10625329—216no—
92#10625329—212no—
93#10625329C22×C4132yes—
94#10625329C22×C4132yes—
95#10625329C2418yes—
96C2×C47⋊C8—212no—
97C2×C47⋊C8—216no—
98C2×C47⋊C8C22×C4124yes—
99C2×C47⋊C8C22×C4124yes—
100C2×C47⋊C8C2418yes—
101C23×C47—24no—
102C23×C47—24no—
103C23×C47—24no—
104C23×C47—412no—
105C23×C47—412no—
106C23×C47C2×C4⋊C413yes—
107C23×C47C2×C4⋊C416yes—
108C23×C47C2×C4⋊C416yes—
109C23×C47C2×C4⋊C4112yes—
110C23×C47C2×C4⋊C411yescharacteristic
111C23×C47C22×Q813yes—
112C23×C47C22×Q814yes—
113C2×C4×C47C2×C4⋊C416yes—
114C2×C4×C47C2×C4⋊C4112yes—
115C2×C4×C47C2×C4⋊C4112yes—
116C2×C4×C47C2×C4⋊C4112yes—
117C2×C4×C47C2×C4⋊C4112yes—
118C2×C4×C47C2×C4⋊C4112yes—
119C2×C4×C47C2×C4⋊C413yes—
120C2×C4×C47C2×C4⋊C416yes—
121C2×C4×C47C2×C4⋊C416yes—
122C2×C4×C47C2×C4⋊C413yes—
123C2×C4×C47C23×C413yes—
124C2×C4×C47C23×C413yes—
125C2×C4×C47C23×C414yes—
126C2×C4×C47C23×C414yes—
127C2×C4×C47C22×D413yes—
128C2×C4×C47C22×D414yes—
129C2×C4×C47C22×D414yes—
130C2×C4×C47C22×D413yes—
131C2×C4×C47C22×Q813yes—
132C2×C4×C47C22×Q814yes—
133C2×C4×C47C2511yescharacteristic
134C8×C47—28no—
135C47⋊C8—28no—
136C47⋊C8C23×C4116yes—
137C22×#430—471nomaximal, Sylow
138C22×C47—212no—
139C22×C47—212no—
140C22×C47—26no—
141C22×C47—26no—
142C22×C47—412no—
143C22×C47—44no—
144C22×C47#43014yescomplemented
145C22×C47C22×C4⋊C413yes—
146C22×C47C22×C4⋊C414yes—
147C22×C47C22×C4⋊C414yes—
148C22×C47C22×C4⋊C413yes—
149C22×C47C23×Q811yescharacteristic
150C4×C47C22×C4⋊C413yes—
151C4×C47C22×C4⋊C413yes—
152C4×C47C22×C4⋊C414yes—
153C4×C47C22×C4⋊C414yes—
154C4×C47C24×C411yescharacteristic
155C4×C47C23×D411yescharacteristic
156C23×M16—471no—
157C23×M16—471no—
158C23×M16D4711yescharacteristic
159C2×#430—4724no—
160C22×#108—474no—
161C2×C47—24no—
162C2×C47—24no—
163C2×C47—23no—
164C2×C47—21no—
165C2×C47C2×#43016yescomplemented
166C2×C47C23×C4⋊C411yescharacteristic
167C24×C4C2×D4711yescharacteristic
168C22×M16—26no—
169C22×M16—21no—
170C22×M16—472no—
171C22×M16—472no—
172C22×M16—4712no—
173C22×M16—4712no—
174C22×M16—472no—
175C22×M16—4712no—
176C22×M16—946no—
177C22×M16—941no—
178C22×M16C2×D4712yes—
179C22×M16C2×D47112yes—
180C23×C8—21no—
181C23×C8—472no—
182C23×C8—941no—
183#430—4764no—
184C2×#108—4748no—
185C47C22×#43011yescharacteristic, complemented, Sylow
186C25Dic9411yescharacteristic
187C23×C4Dic9411yescharacteristic
188C23×C4Dic9416yes—
189C23×C4C4×D4711yescharacteristic
190C23×C4C4×D4711yescharacteristic
191C23×C4C4×D4716yes—
192C23×C4C22⋊D4716yes—
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22⋊D4711yescharacteristic
195C23×C4C22×D4716yes—
196C23×C4C22×D4711yescharacteristic
197C2×M16—212no—
198C2×M16—26no—
199C2×M16—224no—
200C2×M16—4724no—
201C2×M16—4724no—
202C2×M16—4748no—
203C2×M16—4716no—
204C2×M16—4712no—
205C2×M16—4716no—
206C2×M16—9412no—
207C2×M16—946no—
208C2×M16—9424no—
209C2×M16C22×D47112yes—
210C2×M16C22×D47116yes—
211C22×C8—212no—
212C22×C8—22no—
213C22×C8—474no—
214C22×C8—4712no—
215C22×C8—4712no—
216C22×C8—9412no—
217C22×C8—942no—
218#108—4764no—
219C24—44no—
220C24Dic47⋊C411yescharacteristic
221C24Dic47⋊C411yescharacteristic
222C24Dic47⋊C416yes—
223C24C2×Dic9411yescharacteristic
224C24C2×Dic9416yes—
225C22×C4Dic47⋊C416yes—
226C22×C4Dic47⋊C416yes—
227C22×C4Dic47⋊C416yes—
228C22×C4Dic47⋊C4124yes—
229C22×C4Dic47⋊C411yescharacteristic
230C22×C4Dic47⋊C411yescharacteristic
231C22×C4Dic47⋊C416yes—
232C22×C4Dic47⋊C416yes—
233C22×C4C2×Dic9416yes—
234C22×C4C2×Dic94112yes—
235C22×C4C2×Dic9413yes—
236C22×C4C2×C4×D4711yescharacteristic
237C22×C4C2×C4×D4713yes—
238C22×C4C2×C4×D4716yes—
239C22×C4C2×C4×D4716yes—
240C22×C4C2×C4×D47112yes—
241C22×C4C2×C22⋊D4716yes—
242C22×C4C2×C22⋊D4716yes—
243C22×C4C2×C22⋊D4713yes—
244C22×C4C2×C22⋊D47112yes—
245C22×C4C2×C22⋊D4711yescharacteristic
246C22×C4C23×D4713yes—
247C22×C4C23×D4714yes—
248M16—216no—
249M16—212no—
250M16—4732no—
251M16—4732no—
252M16—478no—
253M16—9416no—
254M16—9412no—
255M16C23×D4718yes—
256C2×C8—212no—
257C2×C8—216no—
258C2×C8—4724no—
259C2×C8—4724no—
260C2×C8—478no—
261C2×C8—9412no—
262C2×C8—9416no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—412no—
267C23—412no—
268C23C2×Dic47⋊C413yes—
269C23C2×Dic47⋊C416yes—
270C23C2×Dic47⋊C416yes—
271C23C2×Dic47⋊C4112yes—
272C23C2×Dic47⋊C411yescharacteristic
273C23C22×Dic9413yes—
274C23C22×Dic9414yes—
275C2×C4C2×Dic47⋊C416yes—
276C2×C4C2×Dic47⋊C4112yes—
277C2×C4C2×Dic47⋊C4112yes—
278C2×C4C2×Dic47⋊C4112yes—
279C2×C4C2×Dic47⋊C4112yes—
280C2×C4C2×Dic47⋊C4112yes—
281C2×C4C2×Dic47⋊C413yes—
282C2×C4C2×Dic47⋊C416yes—
283C2×C4C2×Dic47⋊C416yes—
284C2×C4C2×Dic47⋊C413yes—
285C2×C4C22×C4×D4713yes—
286C2×C4C22×C4×D4713yes—
287C2×C4C22×C4×D4714yes—
288C2×C4C22×C4×D4714yes—
289C2×C4C22×C22⋊D4713yes—
290C2×C4C22×C22⋊D4714yes—
291C2×C4C22×C22⋊D4714yes—
292C2×C4C22×C22⋊D4713yes—
293C2×C4C22×Dic9413yes—
294C2×C4C22×Dic9414yes—
295C2×C4C24×D4711yescharacteristic
296C8—28no—
297C8—4716no—
298C8—948no—
299C22—212no—
300C22—212no—
301C22—26no—
302C22—26no—
303C22—412no—
304C22—44no—
305C22#1568917114yesdirect product factor
306C22C22×Dic47⋊C413yes—
307C22C22×Dic47⋊C414yes—
308C22C22×Dic47⋊C414yes—
309C22C22×Dic47⋊C413yes—
310C22C23×Dic9411yescharacteristic
311C4C22×Dic47⋊C413yes—
312C4C22×Dic47⋊C414yes—
313C4C22×Dic47⋊C414yes—
314C4C22×Dic47⋊C413yes—
315C4C23×C4×D4711yescharacteristic
316C4C23×C22⋊D4711yescharacteristic
317C2—24no—
318C2—24no—
319C2—23no—
320C2—21no—
321C2C2×#1568917116yesdirect product factor
322C2C23×Dic47⋊C411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC23×M16×C47C2
socleC23×C47C2×C4⋊C4
prime radicalC25×C47Q8
cosocleC2×C4×C47C25
hypercenterC23×M16D47
hypocenterC47C22×#430
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#15689171C2direct (N × H)
4. C22×#13093959C2semidirect
16. #15689171C22direct (N × H)
17. C2×#13093959C22semidirect
38. #13093959C23semidirect
144. C22×C47#430semidirect
165. C2×C47C2×#430semidirect
185. C47C22×#430semidirect
305. C22#15689171direct (N × H)
321. C2C2×#15689171direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253292
C2×#156891712
C22×#130939592
C23×M16×C472
C22×#43047