Finite Group Databasebeta

← #27079736

Subgroups — #27079736

Order 12032.

4874 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689959C211yesmaximal, characteristic
2C22×#13094031C211yesmaximal, characteristic, complemented
3#20251835C2116yesmaximal, complemented
4#20251832C216yesmaximal, complemented
5C2×#15689091C216yesmaximal, complemented
6C2×C47×#568C211yesmaximal, characteristic, complemented
7#15689959—28no—
8#15689959—28no—
9C2×#13094072C412yes—
10C2×#13094071C414yes—
11C2×#13094070C414yes—
12C2×#13094069C416yes—
13C22×#10625350C2216yes—
14C2×#41×D47C2218yes—
15C2×#13094031—28no—
16C2×#13094031C2218yescomplemented
17C22×#10625329C2213yescomplemented
18C23×C47⋊C8C2213yescomplemented
19#15689091C2214yescomplemented
20#15689091C22112yescomplemented
21#15689091C22124yescomplemented
22#15689089C22124yescomplemented
23#15689086C22124yescomplemented
24C2×(C2×D47)⋊C8C22124yescomplemented
25C22×#41×C47C2211yescharacteristic
26C47×#568—28no—
27C2×#131×C47C2218yescomplemented
28C22×M16×C47C2216yescomplemented
29#13094072—216no—
30#13094071—232no—
31#13094070—232no—
32#13094069—248no—
33C2×#10625354C2×C418yes—
34C2×#10625350—224no—
35C2×#10625350—224no—
36C2×#10625350C2×C4112yes—
37C2×#10625350C2×C4112yes—
38C2×#10625350C23112yes—
39C2×#10625350C2314yes—
40C2×#10625352C2×C4124yes—
41#41×D47—232no—
42#41×D47—232no—
43C22×Dic94C2×C416yes—
44C2×Q8×D47C2×C418yes—
45C22×D188C2×C416yes—
46C22×C22⋊D47C2×C4112yes—
47C2×D4×D47C2×C4124yes—
48C22×C4×D47C23112yes—
49#13094031—232no—
50C2×#10625329—212no—
51C2×#10625329C23112yes—
52C22×C47⋊C8—212no—
53C22×C47⋊C8C23112yes—
54C22×C47⋊C8C2311yescharacteristic
55C22×C47⋊C8C2313yes—
56(C2×D47)⋊C8C23116yes—
57(C2×D47)⋊C8C23148yes—
58C2×#41×C47C2×C414yes—
59C2×#41×C47D418yes—
60C2×#41×C47D418yes—
61C2×#41×C47C2314yes—
62C22×Q8×C47C2×C411yescharacteristic
63C22×D4×C47C2×C413yes—
64C23×C4×C47C2313yes—
65#131×C47—232no—
66C2×M16×C47—224no—
67C2×M16×C47C2314yes—
68C2×M16×C47C23112yes—
69C22×C8×C47C23112yes—
70C24×C47C22×C413yes—
71C23×D47C22×C4112yes—
72#41×C47C22⋊C4132yes—
73#41×C47C2×D4116yes—
74#41×C47C2×D4116yes—
75C2×Q8×C47C22⋊C418yes—
76C2×Q8×C47C22×C414yes—
77C2×D4×C47C22⋊C4124yes—
78C2×D4×C47C22×C4112yes—
79C22×C4×C47C22×C413yes—
80C22×C4×C47C22×C416yes—
81C22×C4×C47C2×D4112yes—
82C22×C4×C47C2×D4112yes—
83C22×C4×C47C2416yes—
84C22×C4×C47C2411yescharacteristic
85C2×C22⋊D47—248no—
86C2×C22⋊D47C22×C4124yes—
87C2×C22⋊D47C22×C418yes—
88C22×Dic47C22×C4112yes—
89#10625354—232no—
90Q8×D47—232no—
91#10625352—296no—
92D4×D47—296no—
93#10625350—248no—
94#10625350—248no—
95#10625350—224no—
96#10625350—224no—
97#10625350—28no—
98#10625350—28no—
99C2×D188—224no—
100C2×D188C22×C4112yes—
101C2×D188C22×C414yes—
102C2×C4×D47—224no—
103C2×C4×D47—224no—
104C2×C4×D47C22×C4124yes—
105C2×C4×D47C2418yes—
106C2×Dic94—224no—
107C2×Dic94C22×C4112yes—
108C2×Dic94C22×C414yes—
109M16×C47—28no—
110M16×C47—224no—
111C2×C8×C47—224no—
112C2×C8×C47C2418yes—
113#10625329—224no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—26no—
116C2×C47⋊C8—224no—
117C2×C47⋊C8C2412yes—
118C2×C47⋊C8C2416yes—
119C23×C47—46no—
120C23×C47C2×C22⋊C4112yes—
121C23×C47C23×C416yes—
122C23×C47C23×C411yescharacteristic
123C22×D47—224no—
124C22×D47—424no—
125C22×D47C23×C418yes—
126Q8×C47C2×C22⋊C4116yes—
127D4×C47C2×C22⋊C4148yes—
128C2×C4×C47C2×C22⋊C4124yes—
129C2×C4×C47C2×C22⋊C4112yes—
130C2×C4×C47C23×C411yescharacteristic
131C2×C4×C47C23×C416yes—
132C2×C4×C47C22×D4112yes—
133C2×C4×C47C22×D412yes—
134C2×C4×C47C22×D412yes—
135C2×C4×C47C22×D4112yes—
136C2×C4×C47C2511yescharacteristic
137C22⋊D47—216no—
138C22⋊D47—248no—
139C2×Dic47—224no—
140C2×Dic47C23×C418yes—
141D188—28no—
142D188—224no—
143C4×D47—248no—
144C4×D47—28no—
145C4×D47—28no—
146Dic94—28no—
147Dic94—224no—
148C8×C47—28no—
149C47⋊C8—22no—
150C47⋊C8—26no—
151#60065—471nomaximal, Sylow
152C22×C47—212no—
153C22×C47—22no—
154C22×C47—412no—
155C22×C47C22×C22⋊C4112yes—
156C22×C47C22×C22⋊C412yes—
157C22×C47C24×C411yescharacteristic
158C2×D47—28no—
159C2×D47—216no—
160C2×D47—824no—
161C4×C47C22×C22⋊C412yes—
162C4×C47C22×C22⋊C4112yes—
163C4×C47C23×D411yescharacteristic
164C4×C47C23×D411yescharacteristic
165Dic47—28no—
166C2×#585—471no—
167C2×#568D4711yescharacteristic, complemented
168C22×#131—471no—
169#2880—4716no—
170#2879—476no—
171C2×#408—476no—
172C2×C47—212no—
173C2×C47—22no—
174C2×C47C23×C22⋊C411yescharacteristic
175C2×C47#357312yes—
176D47—48no—
177#585—948no—
178#585—948no—
179C2×2−1+4—474no—
180C2×2−1+4—472no—
181C2×H(5,𝔽2)—474no—
182C2×H(5,𝔽2)—476no—
183C22×#41—478no—
184C22×#41—476no—
185C22×#41C2×D4711yescharacteristic
186#568—28no—
187C2×#131—478no—
188C2×#131—948no—
189C2×#131C2×D4718yescomplemented
190C22×M16—473no—
191C22×M16C2×D4716yescomplemented
192C23×C8—473no—
193#408—474no—
194#408—4712no—
195#408—4724no—
196#408—4724no—
197#407—4724no—
198C2×C22⋊C8—4724no—
199C47#6006511yescharacteristic, complemented, Sylow
2002−1+4—9432no—
2012−1+4—9416no—
202H(5,𝔽2)—9432no—
203H(5,𝔽2)—9448no—
204C2×#41—478no—
205C2×#41—4724no—
206C2×#41—4712no—
207C2×#41—4712no—
208C2×#41—474no—
209C2×#41—4712no—
210C2×#41—9432no—
211C2×#41—9432no—
212C2×#41—9424no—
213C2×#41—9424no—
214C2×#41C4×D4714yes—
215C2×#41D18818yes—
216C2×#41C22⋊D4718yes—
217C2×#41C22×D4714yes—
218C22×Q8—478no—
219C22×Q8—476no—
220C22×Q8C4×D4711yescharacteristic
221C22×D4—4712no—
222C22×D4—4724no—
223C22×D4—476no—
224C22×D4C4×D4713yes—
225C23×C4—4712no—
226C23×C4C22×D4713yes—
227#131—232no—
228#131—9432no—
229C2×M16—224no—
230C2×M16—4712no—
231C2×M16—9412no—
232C2×M16C22×D4714yes—
233C2×M16C22×D47112yes—
234C22×C8—471no—
235C22×C8—473no—
236C22×C8—4712no—
237C22×C8—9412no—
238C22×C8C22×D47112yes—
239C22⋊C8—4716no—
240C22⋊C8—4748no—
241C24—4712no—
242C24C2×C4×D4713yes—
243#41—9432no—
244#41—9496no—
245#41—9448no—
246#41—9448no—
247#41—9424no—
248#41—948no—
249#41—948no—
250#41—9424no—
251#41(C2×D47)⋊C4132yes—
252#41C2×D188116yes—
253#41C2×C22⋊D47116yes—
254C2×Q8—4712no—
255C2×Q8—474no—
256C2×Q8—9432no—
257C2×Q8—9424no—
258C2×Q8(C2×D47)⋊C418yes—
259C2×Q8C2×C4×D4714yes—
260C2×D4—4724no—
261C2×D4—478no—
262C2×D4—4712no—
263C2×D4—474no—
264C2×D4—9448no—
265C2×D4—9496no—
266C2×D4—9424no—
267C2×D4(C2×D47)⋊C4124yes—
268C2×D4C2×C4×D47112yes—
269C22×C4—4712no—
270C22×C4—4724no—
271C22×C4—478no—
272C22×C4—9424no—
273C22×C4—9424no—
274C22×C4C2×C4×D4716yes—
275C22×C4C2×C4×D4713yes—
276C22×C4C2×D188112yes—
277C22×C4C2×C22⋊D47112yes—
278C22×C4C23×D4711yescharacteristic
279C22×C4C23×D4716yes—
280M16—28no—
281M16—224no—
282M16—9424no—
283C2×C8—224no—
284C2×C8—472no—
285C2×C8—476no—
286C2×C8—942no—
287C2×C8—946no—
288C2×C8—9424no—
289C2×C8C23×D4718yes—
290C23—46no—
291C23—478no—
292C23—9424no—
293C23—18824no—
294C23C2×(C2×D47)⋊C4112yes—
295C23C22×C4×D4716yes—
296C23C22×C4×D4711yescharacteristic
297Q8—9424no—
298Q8—948no—
299Q8C2×(C2×D47)⋊C4116yes—
300D4—9448no—
301D4—9416no—
302D4—9424no—
303D4—948no—
304D4C2×(C2×D47)⋊C4148yes—
305C2×C4—478no—
306C2×C4—9424no—
307C2×C4—9448no—
308C2×C4—948no—
309C2×C4—948no—
310C2×C4C2×(C2×D47)⋊C4124yes—
311C2×C4C2×(C2×D47)⋊C4112yes—
312C2×C4C22×C4×D4716yes—
313C2×C4C22×C4×D4711yescharacteristic
314C2×C4C22×C22⋊D47112yes—
315C2×C4C22×C22⋊D4712yes—
316C2×C4C22×D18812yes—
317C2×C4C22×D188112yes—
318C2×C4C24×D4711yescharacteristic
319C8—28no—
320C8—942no—
321C8—946no—
322C22—22no—
323C22—212no—
324C22—412no—
325C22—948no—
326C22—9416no—
327C22—37624no—
328C22C22×(C2×D47)⋊C4112yes—
329C22C22×(C2×D47)⋊C412yes—
330C22C23×C4×D4711yescharacteristic
331C4—948no—
332C4C22×(C2×D47)⋊C4112yes—
333C4C22×(C2×D47)⋊C412yes—
334C4C23×C22⋊D4711yescharacteristic
335C4C23×D18811yescharacteristic
336C2—22no—
337C2—212no—
338C2—1888no—
339C2C23×(C2×D47)⋊C411yescharacteristic
340C2#2025810112yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC2×C47×#568C2
prime radicalC2×#15689959C2
cosocleC2×C4×C47C25
hypercenterC2×#568D47
hypocenterC47#60065
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
2. C22×#13094031C2semidirect
3. #20251835C2semidirect
4. #20251832C2semidirect
5. C2×#15689091C2semidirect
6. C2×C47×#568C2semidirect
16. C2×#13094031C22semidirect
17. C22×#10625329C22semidirect
18. C23×C47⋊C8C22semidirect
19. #15689091C22semidirect
20. #15689091C22semidirect
21. #15689091C22semidirect
22. #15689089C22semidirect
23. #15689086C22semidirect
24. C2×(C2×D47)⋊C8C22semidirect
27. C2×#131×C47C22semidirect
28. C22×M16×C47C22semidirect
167. C2×#568D47semidirect
189. C2×#131C2×D47semidirect
191. C22×M16C2×D47semidirect
199. C47#60065semidirect

Maximal subgroups

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

subgroupindex
C2×#156899592
C22×#130940312
#202518352
#202518322
C2×#156890912
C2×C47×#5682
#6006547