Finite Group Databasebeta

← C2×#20252101 #27080153

Subgroups — C2×#20252101 #27080153

Order 12032.

2894 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094030C212yesmaximal, complemented
2C2×#15689673C211yesmaximal, characteristic
3#20252101C2116yesmaximal, direct product factor
4C2×#15689202C214yesmaximal, complemented
5C2×C8×D188C214yesmaximal, complemented
6C2×#15689147C211yesmaximal, characteristic, complemented
7C2×#15689089C212yesmaximal, complemented
8C22×C4×C8×C47C211yesmaximal, characteristic, complemented
9C22×#10625350C2212yes—
10C2×#13094030—216no—
11C2×#13094030C22116yescomplemented
12C22×#10625329C2212yes—
13C22×#10625324C2214yes—
14C22×C8×D47C2214yes—
15#15689673C418yes—
16#15689673C2218yes—
17C2×#13094024C411yescharacteristic
18C2×#13094021C411yescharacteristic
19C2×#13094012C412yes—
20C2×#13093998C414yes—
21C2×C4×D188C2214yes—
22#15689202C22132yescomplemented
23C8×D188C22132yescomplemented
24#15689147C2218yescomplemented
25C2×C4⋊(C47⋊C8)C2214yes—
26#15689089C22116yescomplemented
27C2×(C2×D47)⋊C8C2218yes—
28C23×C8×C47C2211yescharacteristic
29C23×C8×C47C2211yescharacteristic
30C22×C42×C47C2211yescharacteristic
31C2×C4×C8×C47C2212yes—
32C2×C4×C8×C47C2212yes—
33C2×C4×C8×C47C2218yescomplemented
34C2×#10625350—28no—
35C2×#10625350—28no—
36C2×#10625350C2×C418yes—
37C2×#10625350C2318yes—
38C22×Dic94C2×C412yes—
39C22×D188C2×C414yes—
40C22×D188C2×C412yes—
41C22×C22⋊D47C2×C414yes—
42C22×C4×D47C2314yes—
43#13094030—264no—
44C2×#10625329—28no—
45C2×#10625329C2318yes—
46C2×#10625324—216no—
47C2×#10625324C23116yes—
48C22×C47⋊C8C2314yes—
49C2×C8×D47—216no—
50C2×C8×D47C23116yes—
51#13094024C2×C418yes—
52#13094021C2×C418yes—
53#13094012C2×C4116yes—
54#13093998C2×C4132yes—
55C4×D188C2×C4116yes—
56C4×D188C23116yes—
57C2×C4⋊Dic47C2×C414yes—
58C2×(C2×D47)⋊C4C2×C418yes—
59C4⋊(C47⋊C8)C23116yes—
60(C2×D47)⋊C8C23132yes—
61C23×C4×C47C2×C412yes—
62C23×C4×C47C2311yescharacteristic
63C22×C8×C47—24no—
64C22×C8×C47D418yescomplemented
65C22×C8×C47C2312yes—
66C22×C8×C47C2314yes—
67C22×C8×C47C2311yescharacteristic
68C22×C8×C47C2311yescharacteristic
69C22×C8×C47C2314yes—
70C2×C42×C47C2×C411yescharacteristic
71C2×C42×C47C2×C411yescharacteristic
72C2×C42×C47C2×C414yes—
73C2×C42×C47C2312yes—
74C2×C42×C47C2314yes—
75C4×C8×C47C2318yes—
76C4×C8×C47C2318yes—
77C24×C47C22×C411yescharacteristic
78C23×D47C22×C414yes—
79C22×C4×C47—28no—
80C22×C4×C47C22×C412yes—
81C22×C4×C47C22×C418yes—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C22×C412yes—
84C22×C4×C47C22×C412yes—
85C22×C4×C47C2×D414yes—
86C22×C4×C47#4114yes—
87C22×C4×C47C2411yescharacteristic
88C22×C4×C47C2412yes—
89C2×C22⋊D47—216no—
90C2×C22⋊D47C22×C4116yes—
91C22×Dic47C22×C414yes—
92#10625350—232no—
93#10625350—216no—
94#10625350—216no—
95C2×D188—28no—
96C2×D188—216no—
97C2×D188C22×C4116yes—
98C2×D188C22×C418yes—
99C2×C4×D47—28no—
100C2×C4×D47—28no—
101C2×C4×D47C22×C418yes—
102C2×C4×D47C2418yes—
103C2×Dic94—28no—
104C2×Dic94C22×C418yes—
105C2×C8×C47—24no—
106C2×C8×C47—28no—
107C2×C8×C47C2×D414yes—
108C2×C8×C47C2×D414yes—
109C2×C8×C47C2×D4116yescomplemented
110C2×C8×C47C2414yes—
111C2×C8×C47C2412yes—
112C2×C8×C47C2412yes—
113C42×C47C22×C414yes—
114C42×C47C22×C414yes—
115C42×C47C22×C414yes—
116C42×C47C2414yes—
117(C2×D47)⋊C4C22×C4132yes—
118C4⋊Dic47C22×C4116yes—
119#10625329—216no—
120C2×C47⋊C8—28no—
121C2×C47⋊C8C2418yes—
122#10625324—232no—
123C8×D47—232no—
124C23×C47—24no—
125C23×C47C4×D414yes—
126C23×C47C23×C412yes—
127C23×C47C23×C411yescharacteristic
128C22×D47—28no—
129C22×D47—216no—
130C22×D47C23×C418yes—
131C2×C4×C47—24no—
132C2×C4×C47—216no—
133C2×C4×C47C4×D414yes—
134C2×C4×C47C4×D418yes—
135C2×C4×C47#13118yescomplemented
136C2×C4×C47C23×C414yes—
137C2×C4×C47C23×C412yes—
138C2×C4×C47C23×C411yescharacteristic
139C2×C4×C47C23×C414yes—
140C2×C4×C47C22×D412yes—
141C2×C4×C47C22×D414yes—
142C2×C4×C47C2×#4112yes—
143C2×C4×C47C2×#4114yes—
144C2×C4×C47C2511yescharacteristic
145C22⋊D47—232no—
146C2×Dic47—28no—
147C2×Dic47C23×C418yes—
148D188—216no—
149D188—232no—
150C4×D47—216no—
151C4×D47—28no—
152C4×D47—28no—
153Dic94—216no—
154C8×C47—24no—
155C8×C47C22×D414yes—
156C8×C47C22×D414yes—
157C47⋊C8—28no—
158C2×#2931—471nomaximal, Sylow
159C22×C47—28no—
160C22×C47—24no—
161C22×C47C2×C4×D414yes—
162C22×C47C2×C4×D412yes—
163C22×C47C2×#13114yes—
164C22×C47C24×C411yescharacteristic
165C2×D47—28no—
166C2×D47—216no—
167C2×D47—416no—
168C4×C47—24no—
169C4×C47C2×C4×D414yes—
170C4×C47C2×C4×D412yes—
171C4×C47C2×#13118yescomplemented
172C4×C47C23×D411yescharacteristic
173C4×C47C22×#4111yescharacteristic
174Dic47—28no—
175C22×#131—472no—
176C2×#532—471no—
177#2931—4716no—
178C2×#436—474no—
179C2×C8×D4—474no—
180C2×#423—471no—
181C2×#408—472no—
182C22×C4×C8D4711yescharacteristic, complemented
183C2×C47—24no—
184C2×C47#293114yescomplemented
185C2×C47C22×C4×D411yescharacteristic
186C2×C47C22×#13112yes—
187D47—48no—
188C22×#41—472no—
189C2×#131—4716no—
190C2×#131—9416no—
191C22×M16—472no—
192C22×M16—474no—
193C23×C8—474no—
194C23×C8C2×D4711yescharacteristic
195C23×C8C2×D4711yescharacteristic
196#532—478no—
197#532—478no—
198C2×C4⋊Q8—471no—
199C2×#127—471no—
200C2×#124—472no—
201C2×#121—474no—
202C2×C4×D4—474no—
203C22×C42C2×D4711yescharacteristic
204#436—4732no—
205C8×D4—4732no—
206#423—478no—
207C2×C4⋊C8—474no—
208#408—4716no—
209C2×C22⋊C8—478no—
210C2×C4×C8C2×D4712yes—
211C2×C4×C8C2×D4712yes—
212C2×C4×C8C2×D4718yescomplemented
213C47C2×#293111yescharacteristic, complemented, Sylow
214C2×#41—478no—
215C2×#41—478no—
216C2×#41—948no—
217C2×#41—948no—
218C22×Q8—472no—
219C22×D4—474no—
220C22×D4—474no—
221C22×D4—472no—
222C23×C4—474no—
223C23×C4C4×D4712yes—
224C23×C4C22×D4711yescharacteristic
225#131—9464no—
226C2×M16—478no—
227C2×M16—4716no—
228C2×M16—948no—
229C2×M16—9416no—
230C22×C8—24no—
231C22×C8—474no—
232C22×C8—4716no—
233C22×C8—9416no—
234C22×C8D18818yescomplemented
235C22×C8C22×D4712yes—
236C22×C8C22×D4714yes—
237C22×C8C22×D4711yescharacteristic
238C22×C8C22×D4714yes—
239C22×C8C22×D4711yescharacteristic
240C4⋊Q8—478no—
241#127—478no—
242#124—4716no—
243#121—4732no—
244C4×D4—4716no—
245C4×D4—4716no—
246C2×C4⋊C4—474no—
247C2×C22⋊C4—478no—
248C2×C42C4×D4711yescharacteristic
249C2×C42C4×D4711yescharacteristic
250C2×C42C4×D4714yes—
251C2×C42C22×D4712yes—
252C2×C42C22×D4714yes—
253C4⋊C8—4716no—
254C22⋊C8—4732no—
255C4×C8C22×D4718yes—
256C4×C8C22×D4718yes—
257C24—474no—
258C24C2×C4×D4711yescharacteristic
259#41—9432no—
260#41—9416no—
261#41—9416no—
262C2×Q8—478no—
263C2×Q8—948no—
264C2×D4—4716no—
265C2×D4—4716no—
266C2×D4—478no—
267C2×D4—9416no—
268C2×D4—948no—
269C2×D4—9416no—
270C22×C4—28no—
271C22×C4—474no—
272C22×C4—478no—
273C22×C4—478no—
274C22×C4—948no—
275C22×C4—948no—
276C22×C4C2×C4×D4712yes—
277C22×C4C2×C4×D4718yes—
278C22×C4C2×C4×D4711yescharacteristic
279C22×C4C2×C4×D4712yes—
280C22×C4C2×C4×D4712yes—
281C22×C4C2×D18814yes—
282C22×C4#1062535014yes—
283C22×C4C23×D4711yescharacteristic
284C22×C4C23×D4712yes—
285M16—9416no—
286M16—9432no—
287C2×C8—24no—
288C2×C8—28no—
289C2×C8—478no—
290C2×C8—948no—
291C2×C8—9432no—
292C2×C8C2×D18814yes—
293C2×C8C2×D18814yes—
294C2×C8C2×D188116yescomplemented
295C2×C8C23×D4714yes—
296C2×C8C23×D4712yes—
297C2×C8C23×D4712yes—
298C4⋊C4—4716no—
299C22⋊C4—4732no—
300C42C2×C4×D4714yes—
301C42C2×C4×D4714yes—
302C42C2×C4×D4714yes—
303C42C23×D4714yes—
304C23—24no—
305C23—478no—
306C23—948no—
307C23—9416no—
308C23C4×D18814yes—
309C23C22×C4×D4712yes—
310C23C22×C4×D4711yescharacteristic
311Q8—9416no—
312D4—9432no—
313D4—9416no—
314D4—9432no—
315C2×C4—24no—
316C2×C4—216no—
317C2×C4—478no—
318C2×C4—948no—
319C2×C4—9416no—
320C2×C4—948no—
321C2×C4—948no—
322C2×C4C4×D18814yes—
323C2×C4C4×D18818yes—
324C2×C4#1309403018yescomplemented
325C2×C4C22×C4×D4714yes—
326C2×C4C22×C4×D4711yescharacteristic
327C2×C4C22×C4×D4712yes—
328C2×C4C22×C4×D4714yes—
329C2×C4C22×D18812yes—
330C2×C4C22×D18814yes—
331C2×C4C2×#1062535012yes—
332C2×C4C2×#1062535014yes—
333C2×C4C24×D4711yescharacteristic
334C8—24no—
335C8—948no—
336C8C22×D18814yes—
337C8C22×D18814yes—
338C22—24no—
339C22—28no—
340C22—9416no—
341C22—948no—
342C22—18816no—
343C22C2×C4×D18814yes—
344C22C2×C4×D18812yes—
345C22C2×#1309403014yes—
346C22C23×C4×D4711yescharacteristic
347C4—24no—
348C4—948no—
349C4C2×C4×D18812yes—
350C4C2×C4×D18814yes—
351C4C2×#1309403018yescomplemented
352C4C23×D18811yescharacteristic
353C4C22×#1062535011yescharacteristic
354C2—24no—
355C2—1888no—
356C2#2025210114yesdirect product factor
357C2C22×C4×D18811yescharacteristic
358C2C22×#1309403012yes—
subgroupG/H
centerC22×C8C22×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC22×C4×C8×C47C2
socleC23×C47C23×C4
prime radicalC2×#15689673C2
cosocleC2×C4×C47C25
hypercenterC22×C4×C8D47
hypocenterC47C2×#2931
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C8 ◁ C22×C4×C8

Complements

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

NH ≅ G/Ntype
1. C22×#13094030C2semidirect
3. #20252101C2direct (N × H)
4. C2×#15689202C2semidirect
5. C2×C8×D188C2semidirect
6. C2×#15689147C2semidirect
7. C2×#15689089C2semidirect
8. C22×C4×C8×C47C2semidirect
11. C2×#13094030C22semidirect
22. #15689202C22semidirect
23. C8×D188C22semidirect
24. #15689147C22semidirect
26. #15689089C22semidirect
33. C2×C4×C8×C47C22semidirect
64. C22×C8×C47D4semidirect
109. C2×C8×C47C2×D4semidirect
135. C2×C4×C47#131semidirect
171. C4×C47C2×#131semidirect
182. C22×C4×C8D47semidirect
184. C2×C47#2931semidirect
212. C2×C4×C8C2×D47semidirect
213. C47C2×#2931semidirect
234. C22×C8D188semidirect
294. C2×C8C2×D188semidirect
324. C2×C4#13094030semidirect
351. C4C2×#13094030semidirect
356. C2#20252101direct (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
#202521012
C2×#156892022
C2×C8×D1882
C2×#156891472
C2×#156890892
C22×C4×C8×C472
C2×#293147