Finite Group Databasebeta

← C2×#20252107 #27080160

Subgroups — C2×#20252107 #27080160

Order 12032.

2806 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094030C212yesmaximal, complemented
2C2×#15689673C211yesmaximal, characteristic
3#20252107C2116yesmaximal, direct product factor
4C2×#15689196C218yesmaximal, complemented
5C2×#15689147C211yesmaximal, characteristic, complemented
6C2×#15689089C212yesmaximal, complemented
7C22×#97×C47C211yesmaximal, characteristic, complemented
8C22×#10625350C2212yes—
9C2×#13094030—216no—
10C2×#13094030C22116yescomplemented
11C22×#10625329C2212yes—
12C22×#10625324C2214yes—
13C22×C8×D47C2214yes—
14#15689673C418yes—
15#15689673C2218yes—
16C2×#13094024C411yescharacteristic
17C2×#13094021C411yescharacteristic
18C2×#13094012C412yes—
19C2×#13093998C414yes—
20C2×C4×D188C2214yes—
21#15689196C22164yescomplemented
22#15689147C2218yescomplemented
23C2×C4⋊(C47⋊C8)C2214yes—
24#15689089C22116yescomplemented
25C2×(C2×D47)⋊C8C2218yes—
26C23×C8×C47C2211yescharacteristic
27C23×C8×C47C2211yescharacteristic
28C22×C42×C47C2211yescharacteristic
29C2×#97×C47C2214yes—
30C2×#97×C47C2218yescomplemented
31C2×#10625350—28no—
32C2×#10625350—28no—
33C2×#10625350C2×C418yes—
34C2×#10625350C2318yes—
35C22×Dic94C2×C412yes—
36C22×D188C2×C412yes—
37C22×D188C2×C414yes—
38C22×C22⋊D47C2×C414yes—
39C22×C4×D47C2314yes—
40#13094030—264no—
41C2×#10625329—28no—
42C2×#10625329C2318yes—
43C2×#10625324—216no—
44C2×#10625324C23116yes—
45C22×C47⋊C8C2314yes—
46C2×C8×D47—216no—
47C2×C8×D47C23116yes—
48#13094024C2×C418yes—
49#13094021C2×C418yes—
50#13094012C2×C4116yes—
51#13093998C2×C4132yes—
52C4×D188C2×C4116yes—
53C4×D188C23116yes—
54C2×C4⋊Dic47C2×C414yes—
55C2×(C2×D47)⋊C4C2×C418yes—
56C4⋊(C47⋊C8)C23116yes—
57(C2×D47)⋊C8C23132yes—
58C23×C4×C47C2×C412yes—
59C23×C4×C47C2311yescharacteristic
60C22×C8×C47—24no—
61C22×C8×C47D418yescomplemented
62C22×C8×C47C2312yes—
63C22×C8×C47C2314yes—
64C22×C8×C47C2312yes—
65C22×C8×C47C2314yes—
66C2×C42×C47C2×C411yescharacteristic
67C2×C42×C47C2×C411yescharacteristic
68C2×C42×C47C2×C414yes—
69C2×C42×C47C2312yes—
70C2×C42×C47C2314yes—
71#97×C47C23116yes—
72C24×C47C22×C411yescharacteristic
73C23×D47C22×C414yes—
74C22×C4×C47—44no—
75C22×C4×C47C22×C412yes—
76C22×C4×C47C22×C412yes—
77C22×C4×C47C22×C418yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C22×C412yes—
80C22×C4×C47C2×D414yes—
81C22×C4×C47#4114yes—
82C22×C4×C47C2412yes—
83C22×C4×C47C2411yescharacteristic
84C2×C22⋊D47—216no—
85C2×C22⋊D47C22×C4116yes—
86C22×Dic47C22×C414yes—
87#10625350—232no—
88#10625350—216no—
89#10625350—216no—
90C2×D188—28no—
91C2×D188—48no—
92C2×D188C22×C418yes—
93C2×D188C22×C4116yes—
94C2×C4×D47—28no—
95C2×C4×D47—28no—
96C2×C4×D47C22×C418yes—
97C2×C4×D47C2418yes—
98C2×Dic94—28no—
99C2×Dic94C22×C418yes—
100C2×C8×C47—24no—
101C2×C8×C47—28no—
102C2×C8×C47C2×D418yes—
103C2×C8×C47C2×D4116yescomplemented
104C2×C8×C47C2414yes—
105C2×C8×C47C2414yes—
106C42×C47C22×C414yes—
107C42×C47C22×C414yes—
108C42×C47C22×C414yes—
109C42×C47C2414yes—
110(C2×D47)⋊C4C22×C4132yes—
111C4⋊Dic47C22×C4116yes—
112#10625329—216no—
113C2×C47⋊C8—28no—
114C2×C47⋊C8C2418yes—
115#10625324—232no—
116C8×D47—232no—
117C23×C47—24no—
118C23×C47C4×D414yes—
119C23×C47C23×C412yes—
120C23×C47C23×C411yescharacteristic
121C22×D47—28no—
122C22×D47—216no—
123C22×D47C23×C418yes—
124C2×C4×C47—24no—
125C2×C4×C47—24no—
126C2×C4×C47—48no—
127C2×C4×C47C4×D414yes—
128C2×C4×C47C4×D418yes—
129C2×C4×C47C23×C414yes—
130C2×C4×C47C23×C414yes—
131C2×C4×C47C23×C412yes—
132C2×C4×C47C23×C411yescharacteristic
133C2×C4×C47C22×D412yes—
134C2×C4×C47C22×D414yes—
135C2×C4×C47C2×#4112yes—
136C2×C4×C47C2×#4114yes—
137C2×C4×C47C2511yescharacteristic
138C22⋊D47—232no—
139C2×Dic47—28no—
140C2×Dic47C23×C418yes—
141D188—216no—
142D188—416no—
143C4×D47—216no—
144C4×D47—28no—
145C4×D47—28no—
146Dic94—216no—
147C8×C47—24no—
148C8×C47C22×D418yes—
149C47⋊C8—28no—
150C2×#2932—471nomaximal, Sylow
151C22×C47—28no—
152C22×C47—24no—
153C22×C47C2×C4×D414yes—
154C22×C47C2×C4×D412yes—
155C22×C47#56814yes—
156C22×C47C24×C411yescharacteristic
157C2×D47—28no—
158C2×D47—216no—
159C2×D47—416no—
160C4×C47—24no—
161C4×C47—24no—
162C4×C47C2×C4×D414yes—
163C4×C47C2×C4×D412yes—
164C4×C47C23×D411yescharacteristic
165C4×C47C22×#4111yescharacteristic
166Dic47—28no—
167C22×#131—472no—
168C2×#532—471no—
169#2932—4716no—
170C2×#435—478no—
171C2×#423—471no—
172C2×#408—472no—
173C22×#97D4711yescharacteristic, complemented
174C2×C47—24no—
175C2×C47#293214yescomplemented
176C2×C47C22×C4×D411yescharacteristic
177C2×C47C2×#56812yes—
178D47—48no—
179C22×#41—472no—
180C2×#131—4716no—
181C2×#131—9416no—
182C22×M16—472no—
183C22×M16—474no—
184C23×C8—474no—
185C23×C8C2×D4711yescharacteristic
186C23×C8C2×D4711yescharacteristic
187#532—478no—
188#532—478no—
189C2×C4⋊Q8—471no—
190C2×#127—471no—
191C2×#124—472no—
192C2×#121—474no—
193C2×C4×D4—474no—
194C22×C42C2×D4711yescharacteristic
195#435—4764no—
196#423—478no—
197C2×C4⋊C8—474no—
198#408—4716no—
199C2×C22⋊C8—478no—
200C2×#97C2×D4714yes—
201C2×#97C2×D4718yescomplemented
202C47C2×#293211yescharacteristic, complemented, Sylow
203C2×#41—478no—
204C2×#41—478no—
205C2×#41—948no—
206C2×#41—948no—
207C22×Q8—472no—
208C22×D4—474no—
209C22×D4—472no—
210C22×D4—474no—
211C23×C4—474no—
212C23×C4C4×D4712yes—
213C23×C4C22×D4711yescharacteristic
214#131—9464no—
215C2×M16—478no—
216C2×M16—4716no—
217C2×M16—948no—
218C2×M16—9416no—
219C22×C8—24no—
220C22×C8—474no—
221C22×C8—4716no—
222C22×C8—9416no—
223C22×C8D18818yescomplemented
224C22×C8C22×D4714yes—
225C22×C8C22×D4712yes—
226C22×C8C22×D4712yes—
227C22×C8C22×D4714yes—
228C4⋊Q8—478no—
229#127—478no—
230#124—4716no—
231#121—4732no—
232C4×D4—4716no—
233C4×D4—4716no—
234C2×C4⋊C4—474no—
235C2×C22⋊C4—478no—
236C2×C42C4×D4711yescharacteristic
237C2×C42C4×D4711yescharacteristic
238C2×C42C4×D4714yes—
239C2×C42C22×D4714yes—
240C2×C42C22×D4712yes—
241C4⋊C8—4716no—
242C22⋊C8—4732no—
243#97C22×D47116yes—
244C24—474no—
245C24C2×C4×D4711yescharacteristic
246#41—9432no—
247#41—9416no—
248#41—9416no—
249C2×Q8—478no—
250C2×Q8—948no—
251C2×D4—4716no—
252C2×D4—478no—
253C2×D4—4716no—
254C2×D4—9416no—
255C2×D4—948no—
256C2×D4—1888no—
257C22×C4—44no—
258C22×C4—474no—
259C22×C4—478no—
260C22×C4—478no—
261C22×C4—948no—
262C22×C4—948no—
263C22×C4C2×C4×D4712yes—
264C22×C4C2×C4×D4712yes—
265C22×C4C2×C4×D4718yes—
266C22×C4C2×C4×D4711yescharacteristic
267C22×C4C2×C4×D4712yes—
268C22×C4C2×D18814yes—
269C22×C4#1062535014yes—
270C22×C4C23×D4712yes—
271C22×C4C23×D4711yescharacteristic
272M16—9416no—
273M16—9432no—
274C2×C8—24no—
275C2×C8—28no—
276C2×C8—478no—
277C2×C8—948no—
278C2×C8—9432no—
279C2×C8C2×D18818yes—
280C2×C8C2×D188116yescomplemented
281C2×C8C23×D4714yes—
282C2×C8C23×D4714yes—
283C4⋊C4—4716no—
284C22⋊C4—4732no—
285C42C2×C4×D4714yes—
286C42C2×C4×D4714yes—
287C42C2×C4×D4714yes—
288C42C23×D4714yes—
289C23—24no—
290C23—478no—
291C23—948no—
292C23—9416no—
293C23C4×D18814yes—
294C23C22×C4×D4712yes—
295C23C22×C4×D4711yescharacteristic
296Q8—9416no—
297D4—9432no—
298D4—9416no—
299D4—18816no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—48no—
303C2×C4—478no—
304C2×C4—948no—
305C2×C4—9416no—
306C2×C4—948no—
307C2×C4—948no—
308C2×C4C4×D18814yes—
309C2×C4C4×D18818yes—
310C2×C4C22×C4×D4714yes—
311C2×C4C22×C4×D4714yes—
312C2×C4C22×C4×D4711yescharacteristic
313C2×C4C22×C4×D4712yes—
314C2×C4C22×D18812yes—
315C2×C4C22×D18814yes—
316C2×C4C2×#1062535012yes—
317C2×C4C2×#1062535014yes—
318C2×C4C24×D4711yescharacteristic
319C8—24no—
320C8—948no—
321C8C22×D18818yes—
322C22—24no—
323C22—28no—
324C22—948no—
325C22—9416no—
326C22—18816no—
327C22C2×C4×D18814yes—
328C22C2×C4×D18812yes—
329C22#1568988914yes—
330C22C23×C4×D4711yescharacteristic
331C4—24no—
332C4—24no—
333C4—948no—
334C4C2×C4×D18812yes—
335C4C2×C4×D18814yes—
336C4C23×D18811yescharacteristic
337C4C22×#1062535011yescharacteristic
338C2—24no—
339C2—1888no—
340C2#2025210714yesdirect product factor
341C2C22×C4×D18811yescharacteristic
342C2C2×#1568988912yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC22×#97×C47C2
socleC23×C47C23×C4
prime radicalC2×#15689673C2
cosocleC2×C4×C47C25
hypercenterC22×#97D47
hypocenterC47C2×#2932
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
1. C22×#13094030C2semidirect
3. #20252107C2direct (N × H)
4. C2×#15689196C2semidirect
5. C2×#15689147C2semidirect
6. C2×#15689089C2semidirect
7. C22×#97×C47C2semidirect
10. C2×#13094030C22semidirect
21. #15689196C22semidirect
22. #15689147C22semidirect
24. #15689089C22semidirect
30. C2×#97×C47C22semidirect
61. C22×C8×C47D4semidirect
103. C2×C8×C47C2×D4semidirect
173. C22×#97D47semidirect
175. C2×C47#2932semidirect
201. C2×#97C2×D47semidirect
202. C47C2×#2932semidirect
223. C22×C8D188semidirect
280. C2×C8C2×D188semidirect
340. C2#20252107direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940302
C2×#156896732
#202521072
C2×#156891962
C2×#156891472
C2×#156890892
C22×#97×C472
C2×#293247