Finite Group Databasebeta

← C2×#20242396 #26188507

Subgroups — C2×#20242396 #26188507

Order 12032.

1028 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689451C212yesmaximal
2C2×#15689053C211yesmaximal, characteristic
3#20242396C218yesmaximal, direct product factor
4C2×#15688860C212yesmaximal
5C2×#15688849C211yesmaximal, characteristic
6C2×C47×#544C211yesmaximal, characteristic
7C22×C4⋊Dic47C2211yescharacteristic
8C22×Dic47⋊C4—21no—
9#15689451—24no—
10#15689451C2218yes—
11C2×#13093955—22no—
12C2×#13093948—22no—
13C2×C22⋊(C47⋊C8)C2212yes—
14#15689053—22no—
15#15689053C2214yes—
16C2×C47⋊#95—21no—
17C2×C47⋊#95C2211yescharacteristic
18#15688860C2218yes—
19#15688849C2214yes—
20C47×#544—24no—
21C47×#544—22no—
22C47×#544C2214yes—
23C2×C4⋊Q8×C47—22no—
24C2×#125×C47—22no—
25C2×C22⋊Q8×C47—21no—
26C2×C22⋊Q8×C47C2212yes—
27C2×#117×C47—21no—
28C22×C4⋊C4×C47C2211yescharacteristic
29C23×Dic47—21no—
30C23×Dic47—21no—
31C22×C47⋊C8—22no—
32C2×C4⋊Dic47—22no—
33C2×C4⋊Dic47—22no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2312yes—
36C2×Dic47⋊C4—24no—
37C2×Dic47⋊C4—44no—
38#13093955—28no—
39#13093948—28no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)C2314yes—
42C47⋊#95—24no—
43C47⋊#95—21no—
44C47⋊#95C2312yes—
45C22×Q8×C47—21no—
46C22×Q8×C47D412yes—
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C4⋊Q8×C47—28no—
51C4⋊Q8×C47—28no—
52#125×C47—28no—
53#125×C47—28no—
54C22⋊Q8×C47—24no—
55C22⋊Q8×C47—24no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47—22no—
58C22⋊Q8×C47D414yes—
59C22⋊Q8×C47C2314yes—
60#117×C47—24no—
61#117×C47—24no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47D412yes—
71C2×C4⋊C4×C47C2312yes—
72C2×C22⋊C4×C47—22no—
73C2×C42×C47—22no—
74C24×C47C2×D411yescharacteristic
75C2×Q8×C47—24no—
76C2×Q8×C47—22no—
77C2×Q8×C47—44no—
78C2×Q8×C47—82no—
79C2×Q8×C47C2×D414yes—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—42no—
88C22×C4×C47—42no—
89C22×C4×C47SD1614yes—
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D412yes—
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D412yes—
95C22×C4×C47C2411yescharacteristic
96C22×Dic47—24no—
97C22×Dic47—22no—
98C22×Dic47—22no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C22×Dic47—41no—
102C22×Dic47—41no—
103C22×Dic47—42no—
104C22×Dic47—42no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—24no—
107C4⋊C4×C47—24no—
108C4⋊C4×C47—28no—
109C4⋊C4×C47—24no—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—24no—
113C4⋊C4×C47—24no—
114C4⋊C4×C47C2×D414yes—
115C22⋊C4×C47—24no—
116C22⋊C4×C47—24no—
117C42×C47—28no—
118C4⋊Dic47—24no—
119C4⋊Dic47—42no—
120Dic47⋊C4—48no—
121C2×C47⋊C8—24no—
122C2×C47⋊C8—44no—
123C23×C47—22no—
124C23×C47—22no—
125C23×C47#12011yescharacteristic
126C23×C47#12011yescharacteristic
127C23×C47C2×SD1612yes—
128C23×C47#13712yes—
129C23×C47C22×D411yescharacteristic
130Q8×C47—44no—
131Q8×C47—82no—
132Q8×C47—84no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—24no—
136C2×C4×C47—24no—
137C2×C4×C47—24no—
138C2×C4×C47—24no—
139C2×C4×C47—24no—
140C2×C4×C47—22no—
141C2×C4×C47—42no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—44no—
145C2×C4×C47—42no—
146C2×C4×C47—44no—
147C2×C4×C47—42no—
148C2×C4×C47—81no—
149C2×C4×C47#12012yes—
150C2×C4×C47C2×SD1614yes—
151C2×C4×C47C22×D412yes—
152C2×Dic47—24no—
153C2×Dic47—48no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C2×Dic47—82no—
160C2×Dic47—82no—
161C47⋊C8—44no—
162C2×#1611—471nomaximal, Sylow
163C22×C47—24no—
164C22×C47—24no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47#44812yes—
170C22×C47#44812yes—
171C22×C47#45612yes—
172C22×C47#45812yes—
173C22×C47C2×#12011yescharacteristic
174C22×C47C22×SD1611yescharacteristic
175C22×C47C2×#13711yescharacteristic
176C4×C47—24no—
177C4×C47—22no—
178C4×C47—22no—
179C4×C47—44no—
180C4×C47—42no—
181C4×C47—44no—
182Dic47—44no—
183Dic47—82no—
184C2×#544D4711yescharacteristic
185C2×#483—472no—
186C2×#398—471no—
187#1611—478no—
188C2×#328—472no—
189C2×#324—471no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—22no—
193C2×C47#161114yescomplemented
194C2×C47C2×#44811yescharacteristic
195C2×C47C2×#45611yescharacteristic
196C2×C47C2×#45811yescharacteristic
197#544—24no—
198#544—22no—
199#544C2×D4714yes—
200C2×C4⋊Q8—22no—
201C2×#125—22no—
202C2×C22⋊Q8—21no—
203C2×C22⋊Q8C2×D4712yes—
204C2×#117—21no—
205C22×C4⋊C4—471no—
206C22×C4⋊C4—941no—
207C22×C4⋊C4C2×D4711yescharacteristic
208#483—478no—
209#483—944no—
210C2×#106—942no—
211C2×#103—942no—
212C2×C22⋊C8—472no—
213#398—474no—
214#398—942no—
215C2×#95—471no—
216C2×#95—941no—
217#328—478no—
218#324—474no—
219C47C2×#161111yescharacteristic, complemented, Sylow
220C22×Q8—21no—
221C22×Q8C22⋊D4712yes—
222C23×C4—941no—
223C23×C4—941no—
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22⋊D4711yescharacteristic
226C23×C4C22×D4711yescharacteristic
227C22×C8—942no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230#125—28no—
231#125—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8C22⋊D4714yes—
237C22⋊Q8C22×D4714yes—
238#117—24no—
239#117—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—472no—
248C2×C4⋊C4—942no—
249C2×C4⋊C4—942no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4—944no—
252C2×C4⋊C4—1884no—
253C2×C4⋊C4C22⋊D4712yes—
254C2×C4⋊C4C22⋊D4712yes—
255C2×C4⋊C4C22×D4712yes—
256C2×C22⋊C4—22no—
257C2×C42—22no—
258#106—948no—
259#103—948no—
260C22⋊C8—474no—
261C22⋊C8—942no—
262#95—472no—
263#95—941no—
264#95—944no—
265C24C2×C22⋊D4711yescharacteristic
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8C2×C22⋊D4714yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—944no—
281C22×C4—942no—
282C22×C4—942no—
283C22×C4—1881no—
284C22×C4—1881no—
285C22×C4—1882no—
286C22×C4—1882no—
287C22×C4—1882no—
288C22×C4—1882no—
289C22×C4#1062533614yes—
290C22×C4C2×C22⋊D4712yes—
291C22×C4C2×C22⋊D4712yes—
292C22×C4C2×C22⋊D4712yes—
293C22×C4C2×C22⋊D4711yescharacteristic
294C22×C4C2×C22⋊D4712yes—
295C22×C4C23×D4711yescharacteristic
296C2×C8—944no—
297C2×C8—1884no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—944no—
308C4⋊C4—1882no—
309C4⋊C4—1888no—
310C4⋊C4C2×C22⋊D4714yes—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C42—28no—
314C23—22no—
315C23—22no—
316C23#1309399311yescharacteristic
317C23#1309399311yescharacteristic
318C23C2×#1062533612yes—
319C23#1309405512yes—
320C23C22×C22⋊D4711yescharacteristic
321Q8—44no—
322Q8—82no—
323Q8—84no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—944no—
341C2×C4—3762no—
342C2×C4—3762no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4—1882no—
346C2×C4—1882no—
347C2×C4—1888no—
348C2×C4—1882no—
349C2×C4#1309399312yes—
350C2×C4C2×#1062533614yes—
351C2×C4C22×C22⋊D4712yes—
352C8—1884no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—24no—
359C22#1568926212yes—
360C22#1568926212yes—
361C22#1568930612yes—
362C22#1568931312yes—
363C22C2×#1309399311yescharacteristic
364C22C22×#1062533611yescharacteristic
365C22C2×#1309405511yescharacteristic
366C4—22no—
367C4—22no—
368C4—24no—
369C4—44no—
370C4—42no—
371C4—44no—
372C4—3762no—
373C4—1884no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2#2024239614yesdirect product factor
378C2C2×#1568926211yescharacteristic
379C2C2×#1568930611yescharacteristic
380C2C2×#1568931311yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#544C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenterC2×#544D47
hypocenterC47C2×#1611
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20242396C2direct (N × H)
193. C2×C47#1611semidirect
219. C47C2×#1611semidirect
377. C2#20242396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894512
C2×#156890532
#202423962
C2×#156888602
C2×#156888492
C2×C47×#5442
C2×#161147