Finite Group Databasebeta

← C2×#20243496 #26216133

Subgroups — C2×#20243496 #26216133

Order 12032.

1554 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093939C212yesmaximal
2#20243496C218yesmaximal, direct product factor
3C2×#15688889C214yesmaximal, complemented
4C2×C47×#518C211yesmaximal, characteristic
5C22×C22⋊Dic47C2214yes—
6C2×#13093939—24no—
7C2×#13093939—24no—
8C2×#13093939C2218yes—
9#15688889C22116yescomplemented
10C23×D4×C47C2211yescharacteristic
11C47×#518—24no—
12C47×#518C414yescomplemented
13C47×#518C2214yes—
14C2×C4×D4×C47—24no—
15C2×#117×C47C412yescomplemented
16C22×C22⋊C4×C47C412yescomplemented
17C22×C22⋊C4×C47C2212yes—
18C23×Dic47—24no—
19C2×C22⋊Dic47—24no—
20C2×C22⋊Dic47—24no—
21C2×C22⋊Dic47—24no—
22C2×C22⋊Dic47—24no—
23C2×C22⋊Dic47—24no—
24C2×C22⋊Dic47C2318yes—
25#13093939—216no—
26C25×C47C2×C411yescharacteristic
27C25×C47C2311yescharacteristic
28C22×D4×C47—24no—
29C22×D4×C47—24no—
30C22×D4×C47C2×C412yes—
31C22×D4×C47D414yes—
32C22×D4×C47D414yes—
33C22×D4×C47C2312yes—
34C23×C4×C47—22no—
35C23×C4×C47—22no—
36C23×C4×C47C2×C411yescharacteristic
37C4×D4×C47—216no—
38C4×D4×C47—216no—
39#117×C47—24no—
40#117×C47C2×C418yescomplemented
41C2×C4⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—24no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C414yes—
48C2×C22⋊C4×C47C2×C414yes—
49C2×C22⋊C4×C47C2×C418yescomplemented
50C2×C22⋊C4×C47C2314yes—
51C2×C42×C47—22no—
52C24×C47—22no—
53C24×C47—22no—
54C24×C47—82no—
55C24×C47—82no—
56C24×C47C22⋊C412yes—
57C24×C47C22⋊C412yes—
58C24×C47C22×C412yes—
59C24×C47C22×C411yescharacteristic
60C24×C47C22×C411yescharacteristic
61C24×C47C2×D412yes—
62C24×C47C2×D412yes—
63C24×C47#4114yes—
64C24×C47#4114yes—
65C24×C47C2411yescharacteristic
66C2×D4×C47—28no—
67C2×D4×C47—28no—
68C2×D4×C47—28no—
69C2×D4×C47—28no—
70C2×D4×C47—28no—
71C2×D4×C47—28no—
72C2×D4×C47C22⋊C418yes—
73C2×D4×C47C2×D418yes—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—24no—
77C22×C4×C47—24no—
78C22×C4×C47—28no—
79C22×C4×C47—24no—
80C22×C4×C47—84no—
81C22×C4×C47—84no—
82C22×C4×C47C22⋊C412yes—
83C22×C4×C47C22⋊C412yes—
84C22×C4×C47C22×C412yes—
85C22×Dic47—28no—
86C22×Dic47—44no—
87C22×Dic47—44no—
88C22×Dic47—88no—
89C4⋊C4×C47—28no—
90C22⋊C4×C47—24no—
91C22⋊C4×C47—24no—
92C22⋊C4×C47—28no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47C22×C418yes—
95C42×C47—28no—
96C22⋊Dic47—28no—
97C22⋊Dic47—28no—
98C22⋊Dic47—44no—
99C22⋊Dic47—44no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—24no—
103C23×C47—24no—
104C23×C47—48no—
105C23×C47—44no—
106C23×C47—42no—
107C23×C47—44no—
108C23×C47—84no—
109C23×C47—81no—
110C23×C47—82no—
111C23×C47—84no—
112C23×C47—81no—
113C23×C47—82no—
114C23×C47—82no—
115C23×C47—84no—
116C23×C47C2×C22⋊C411yescharacteristic
117C23×C47C2×C22⋊C414yes—
118C23×C47C2×C22⋊C411yescharacteristic
119C23×C47C2×C22⋊C412yes—
120C23×C47#11712yes—
121C23×C47#11712yes—
122C23×C47#11714yes—
123C23×C47#12314yes—
124C23×C47#12314yes—
125C23×C47#12314yes—
126C23×C47#12314yes—
127C23×C47C23×C411yescharacteristic
128C23×C47C22×D412yes—
129C23×C47C2×#4114yes—
130D4×C47—216no—
131D4×C47—216no—
132C2×C4×C47—24no—
133C2×C4×C47—22no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—28no—
137C2×C4×C47—48no—
138C2×C4×C47—84no—
139C2×C4×C47—84no—
140C2×C4×C47—88no—
141C2×C4×C47C2×C22⋊C414yes—
142C2×Dic47—48no—
143C2×Dic47—88no—
144C2×Dic47—88no—
145C2×Dic47—88no—
146C2×#1828—471nomaximal, Sylow
147C22×C47—28no—
148C22×C47—22no—
149C22×C47—24no—
150C22×C47—24no—
151C22×C47—44no—
152C22×C47—44no—
153C22×C47—42no—
154C22×C47—42no—
155C22×C47—41no—
156C22×C47—42no—
157C22×C47—41no—
158C22×C47—44no—
159C22×C47—44no—
160C22×C47—44no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—44no—
164C22×C47—84no—
165C22×C47—84no—
166C22×C47#38111yescharacteristic
167C22×C47#38112yes—
168C22×C47#38114yes—
169C22×C47#38111yescharacteristic
170C22×C47C22×C22⋊C411yescharacteristic
171C22×C47C2×#11712yes—
172C22×C47C2×#12314yes—
173C4×C47—24no—
174C4×C47—48no—
175Dic47—88no—
176C2×#518D4711yescharacteristic
177C22×#99—472no—
178#1828—478no—
179C2×#342—474no—
180C2×C47—24no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47—22no—
184C2×C47—22no—
185C2×C47—24no—
186C2×C47—44no—
187C2×C47#182812yescomplemented
188C2×C47C2×#38111yescharacteristic
189C23×D4C2×D4711yescharacteristic
190#518—24no—
191#518Dic4714yescomplemented
192#518C2×D4714yes—
193C2×C4×D4—24no—
194C2×#117Dic4712yescomplemented
195C22×C22⋊C4—474no—
196C22×C22⋊C4Dic4712yescomplemented
197C22×C22⋊C4C2×D4712yes—
198C2×#99—478no—
199C2×#99—944no—
200C2×#99—944no—
201#342—4716no—
202C47C2×#182811yescharacteristic, complemented, Sylow
203C25C2×Dic4711yescharacteristic
204C25C22×D4711yescharacteristic
205C22×D4—24no—
206C22×D4—24no—
207C22×D4C2×Dic4712yes—
208C22×D4C22⋊D4714yes—
209C22×D4C22⋊D4714yes—
210C22×D4C22×D4712yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—944no—
214C23×C4C2×Dic4711yescharacteristic
215C4×D4—216no—
216C4×D4—216no—
217#117—24no—
218#117C2×Dic4718yescomplemented
219C2×C4⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—478no—
226C2×C22⋊C4—944no—
227C2×C22⋊C4—944no—
228C2×C22⋊C4—944no—
229C2×C22⋊C4—944no—
230C2×C22⋊C4—944no—
231C2×C22⋊C4C2×Dic4714yes—
232C2×C22⋊C4C2×Dic4714yes—
233C2×C22⋊C4C2×Dic4718yescomplemented
234C2×C22⋊C4C22×D4714yes—
235C2×C42—22no—
236#99—9416no—
237C24—22no—
238C24—22no—
239C24—82no—
240C24—82no—
241C24C22⋊Dic4712yes—
242C24C22⋊Dic4712yes—
243C24#1062535014yes—
244C24#1062535014yes—
245C24C22×Dic4712yes—
246C24C22×Dic4711yescharacteristic
247C24C22×Dic4711yescharacteristic
248C24C2×C22⋊D4712yes—
249C24C2×C22⋊D4712yes—
250C24C23×D4711yescharacteristic
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4—28no—
257C2×D4C22⋊Dic4718yes—
258C2×D4C2×C22⋊D4718yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—948no—
268C22×C4—3768no—
269C22×C4—1884no—
270C22×C4—1884no—
271C22×C4C22⋊Dic4712yes—
272C22×C4C22⋊Dic4712yes—
273C22×C4C22×Dic4712yes—
274C4⋊C4—28no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—28no—
278C22⋊C4—24no—
279C22⋊C4—948no—
280C22⋊C4—948no—
281C22⋊C4—1884no—
282C22⋊C4—1884no—
283C22⋊C4C22×Dic4718yes—
284C42—28no—
285C23—22no—
286C23—22no—
287C23—24no—
288C23—24no—
289C23—44no—
290C23—48no—
291C23—42no—
292C23—44no—
293C23—84no—
294C23—84no—
295C23—81no—
296C23—82no—
297C23—81no—
298C23—82no—
299C23—82no—
300C23—84no—
301C23C2×C22⋊Dic4711yescharacteristic
302C23C2×C22⋊Dic4714yes—
303C23C2×C22⋊Dic4711yescharacteristic
304C23C2×C22⋊Dic4712yes—
305C23#1309398312yes—
306C23#1309398312yes—
307C23#1309398314yes—
308C23#1309400714yes—
309C23#1309400714yes—
310C23#1309400714yes—
311C23#1309400714yes—
312C23C23×Dic4711yescharacteristic
313C23C22×C22⋊D4712yes—
314C23C2×#1062535014yes—
315D4—216no—
316D4—216no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—48no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—3768no—
327C2×C4—3768no—
328C2×C4—3768no—
329C2×C4—1888no—
330C2×C4C2×C22⋊Dic4714yes—
331C22—22no—
332C22—28no—
333C22—24no—
334C22—24no—
335C22—42no—
336C22—42no—
337C22—44no—
338C22—44no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—84no—
350C22#1568898611yescharacteristic
351C22#1568898612yes—
352C22#1568898614yes—
353C22#1568898611yescharacteristic
354C22C22×C22⋊Dic4711yescharacteristic
355C22C2×#1309398312yes—
356C22C2×#1309400714yes—
357C4—24no—
358C4—48no—
359C4—3768no—
360C2—21no—
361C2—24no—
362C2—21no—
363C2—22no—
364C2—22no—
365C2—24no—
366C2—44no—
367C2#2024349612yesdirect product factor
368C2C2×#1568898611yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#518C2
socleC22×C47#381
prime radicalC23×D4×C47C22
cosocleC24×C47C24
hypercenterC2×#518D47
hypocenterC47C2×#1828
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
2. #20243496C2direct (N × H)
3. C2×#15688889C2semidirect
9. #15688889C22semidirect
12. C47×#518C4semidirect
15. C2×#117×C47C4semidirect
16. C22×C22⋊C4×C47C4semidirect
40. #117×C47C2×C4semidirect
49. C2×C22⋊C4×C47C2×C4semidirect
187. C2×C47#1828semidirect
191. #518Dic47semidirect
194. C2×#117Dic47semidirect
196. C22×C22⋊C4Dic47semidirect
202. C47C2×#1828semidirect
218. #117C2×Dic47semidirect
233. C2×C22⋊C4C2×Dic47semidirect
367. C2#20243496direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939392
#202434962
C2×#156888892
C2×C47×#5182
C2×#182847