Finite Group Databasebeta

← C2×#20242362 #26188462

Subgroups — C2×#20242362 #26188462

Order 12032.

1200 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689278C212yesmaximal
2C2×#15689043C211yesmaximal, characteristic
3#20242362C218yesmaximal, direct product factor
4C2×#15688860C212yesmaximal
5C2×#15688849C211yesmaximal, characteristic
6C2×C47×#544C211yesmaximal, characteristic
7C23×Dic94C2211yescharacteristic
8C22×#10625337—22no—
9C22×Dic47⋊C4—21no—
10#15689278—24no—
11#15689278C2218yes—
12C2×#13093947—22no—
13C2×C22⋊(C47⋊C8)C2212yes—
14#15689043—22no—
15#15689043C2214yes—
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
29C22×Dic94—22no—
30C22×Dic94—22no—
31C22×Dic94—21no—
32C22×Dic94—44no—
33C22×Dic94C2312yes—
34C23×Dic47—21no—
35C23×Dic47—21no—
36C2×#10625337—28no—
37C2×#10625337—48no—
38C22×C47⋊C8—22no—
39C2×Dic47⋊C4—24no—
40C2×Dic47⋊C4—44no—
41#13093947—28no—
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)C2314yes—
44C47⋊#95—24no—
45C47⋊#95—21no—
46C47⋊#95C2312yes—
47C22×Q8×C47—21no—
48C22×Q8×C47D412yes—
49C23×C4×C47D411yescharacteristic
50C23×C4×C47D411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52C4⋊Q8×C47—28no—
53C4⋊Q8×C47—28no—
54#125×C47—28no—
55#125×C47—28no—
56C22⋊Q8×C47—24no—
57C22⋊Q8×C47—24no—
58C22⋊Q8×C47—22no—
59C22⋊Q8×C47—22no—
60C22⋊Q8×C47D414yes—
61C22⋊Q8×C47C2314yes—
62#117×C47—24no—
63#117×C47—24no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47—22no—
71C2×C4⋊C4×C47D412yes—
72C2×C4⋊C4×C47D412yes—
73C2×C4⋊C4×C47C2312yes—
74C2×C22⋊C4×C47—22no—
75C2×C42×C47—22no—
76C24×C47C2×D411yescharacteristic
77C2×Q8×C47—24no—
78C2×Q8×C47—22no—
79C2×Q8×C47—44no—
80C2×Q8×C47—82no—
81C2×Q8×C47C2×D414yes—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—42no—
90C22×C4×C47—42no—
91C22×C4×C47Q1614yescomplemented
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2×D412yes—
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2411yescharacteristic
98C22×Dic47—24no—
99C22×Dic47—24no—
100C22×Dic47—22no—
101C22×Dic47—22no—
102C22×Dic47—42no—
103C22×Dic47—42no—
104C22×Dic47—41no—
105C22×Dic47—41no—
106C22×Dic47—42no—
107C2×Dic94—28no—
108C2×Dic94—24no—
109C2×Dic94—44no—
110C2×Dic94—48no—
111C2×Dic94—44no—
112C2×Dic94—44no—
113C2×Dic94—42no—
114C4⋊C4×C47—24no—
115C4⋊C4×C47—24no—
116C4⋊C4×C47—24no—
117C4⋊C4×C47—28no—
118C4⋊C4×C47—24no—
119C4⋊C4×C47—24no—
120C4⋊C4×C47—22no—
121C4⋊C4×C47—24no—
122C4⋊C4×C47—24no—
123C4⋊C4×C47C2×D414yes—
124C22⋊C4×C47—24no—
125C22⋊C4×C47—24no—
126C42×C47—28no—
127#10625337—416no—
128Dic47⋊C4—48no—
129C2×C47⋊C8—24no—
130C2×C47⋊C8—44no—
131C23×C47—22no—
132C23×C47—22no—
133C23×C47#12011yescharacteristic
134C23×C47#12011yescharacteristic
135C23×C47C2×Q1612yes—
136C23×C47#13712yes—
137C23×C47C22×D411yescharacteristic
138Q8×C47—44no—
139Q8×C47—82no—
140Q8×C47—84no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—24no—
144C2×C4×C47—24no—
145C2×C4×C47—24no—
146C2×C4×C47—24no—
147C2×C4×C47—24no—
148C2×C4×C47—22no—
149C2×C4×C47—42no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—44no—
153C2×C4×C47—42no—
154C2×C4×C47—44no—
155C2×C4×C47—42no—
156C2×C4×C47—81no—
157C2×C4×C47#12012yes—
158C2×C4×C47C2×Q1614yescomplemented
159C2×C4×C47C22×D412yes—
160C2×Dic47—24no—
161C2×Dic47—24no—
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—44no—
165C2×Dic47—48no—
166C2×Dic47—42no—
167C2×Dic47—42no—
168C2×Dic47—44no—
169Dic94—28no—
170Dic94—44no—
171Dic94—48no—
172C47⋊C8—44no—
173C2×#1605—471nomaximal, Sylow
174C22×C47—24no—
175C22×C47—24no—
176C22×C47—21no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—21no—
180C22×C47#45112yes—
181C22×C47#45112yes—
182C22×C47#45612yes—
183C22×C47B(4,𝔽2)12yes—
184C22×C47C2×#12011yescharacteristic
185C22×C47C22×Q1611yescharacteristic
186C22×C47C2×#13711yescharacteristic
187C4×C47—24no—
188C4×C47—22no—
189C4×C47—22no—
190C4×C47—44no—
191C4×C47—42no—
192C4×C47—44no—
193Dic47—44no—
194Dic47—44no—
195C2×#544D4711yescharacteristic
196C2×#451—472no—
197C2×#395—471no—
198#1605—478no—
199C2×#328—472no—
200C2×#324—471no—
201C2×C47—21no—
202C2×C47—21no—
203C2×C47—22no—
204C2×C47#160514yescomplemented
205C2×C47C2×#45111yescharacteristic
206C2×C47C2×#45611yescharacteristic
207C2×C47C2×B(4,𝔽2)11yescharacteristic
208C23×Q8—471no—
209C22×Q16—942no—
210#544—24no—
211#544—22no—
212#544C2×D4714yes—
213C2×C4⋊Q8—22no—
214C2×#125—22no—
215C2×C22⋊Q8—21no—
216C2×C22⋊Q8C2×D4712yes—
217C2×#117—21no—
218C22×C4⋊C4—941no—
219C22×C4⋊C4C2×D4711yescharacteristic
220#451—478no—
221#451—944no—
222C2×#103—942no—
223C2×C22⋊C8—472no—
224#395—474no—
225#395—942no—
226C2×#95—471no—
227C2×#95—941no—
228#328—478no—
229#324—474no—
230C47C2×#160511yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—472no—
233C22×Q8—941no—
234C22×Q8—942no—
235C22×Q8—942no—
236C22×Q8—1884no—
237C22×Q8C22⋊D4712yes—
238C23×C4—941no—
239C23×C4—941no—
240C23×C4C22⋊D4711yescharacteristic
241C23×C4C22⋊D4711yescharacteristic
242C23×C4C22×D4711yescharacteristic
243C2×Q16—948no—
244C2×Q16—1888no—
245C22×C8—942no—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248#125—28no—
249#125—28no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8C22⋊D4714yes—
255C22⋊Q8C22×D4714yes—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—944no—
266C2×C4⋊C4—1884no—
267C2×C4⋊C4C22⋊D4712yes—
268C2×C4⋊C4C22⋊D4712yes—
269C2×C4⋊C4C22×D4712yes—
270C2×C22⋊C4—22no—
271C2×C42—22no—
272#103—948no—
273C22⋊C8—474no—
274C22⋊C8—942no—
275#95—472no—
276#95—941no—
277#95—944no—
278C24C2×C22⋊D4711yescharacteristic
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—44no—
282C2×Q8—82no—
283C2×Q8—948no—
284C2×Q8—944no—
285C2×Q8—1884no—
286C2×Q8—1884no—
287C2×Q8—1884no—
288C2×Q8—1888no—
289C2×Q8—1882no—
290C2×Q8C2×C22⋊D4714yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—944no—
301C22×C4—944no—
302C22×C4—942no—
303C22×C4—942no—
304C22×C4—1882no—
305C22×C4—1882no—
306C22×C4—1881no—
307C22×C4—1881no—
308C22×C4—1882no—
309C22×C4#1062533714yescomplemented
310C22×C4C2×C22⋊D4712yes—
311C22×C4C2×C22⋊D4712yes—
312C22×C4C2×C22⋊D4712yes—
313C22×C4C2×C22⋊D4711yescharacteristic
314C22×C4C2×C22⋊D4712yes—
315C22×C4C23×D4711yescharacteristic
316Q16—18816no—
317C2×C8—944no—
318C2×C8—1884no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—1888no—
329C4⋊C4C2×C22⋊D4714yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—28no—
333C23—22no—
334C23—22no—
335C23#1309399311yescharacteristic
336C23#1309399311yescharacteristic
337C23C2×#1062533712yes—
338C23#1309405512yes—
339C23C22×C22⋊D4711yescharacteristic
340Q8—44no—
341Q8—82no—
342Q8—84no—
343Q8—948no—
344Q8—1884no—
345Q8—1888no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—944no—
363C2×C4—944no—
364C2×C4—1882no—
365C2×C4—1888no—
366C2×C4—1882no—
367C2×C4—1884no—
368C2×C4—1884no—
369C2×C4—1882no—
370C2×C4—1882no—
371C2×C4#1309399312yes—
372C2×C4C2×#1062533714yescomplemented
373C2×C4C22×C22⋊D4712yes—
374C8—1884no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—24no—
381C22#1568927912yes—
382C22#1568927912yes—
383C22#1568930612yes—
384C22#1568931012yes—
385C22C2×#1309399311yescharacteristic
386C22C22×#1062533711yescharacteristic
387C22C2×#1309405511yescharacteristic
388C4—22no—
389C4—22no—
390C4—24no—
391C4—44no—
392C4—42no—
393C4—44no—
394C4—1884no—
395C4—1884no—
396C2—21no—
397C2—22no—
398C2—21no—
399C2#2024236214yesdirect product factor
400C2C2×#1568927911yescharacteristic
401C2C2×#1568930611yescharacteristic
402C2C2×#1568931011yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#544C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenterC2×#544D47
hypocenterC47C2×#1605
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. #20242362C2direct (N × H)
91. C22×C4×C47Q16semidirect
158. C2×C4×C47C2×Q16semidirect
204. C2×C47#1605semidirect
230. C47C2×#1605semidirect
309. C22×C4#10625337semidirect
372. C2×C4C2×#10625337semidirect
399. C2#20242362direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156892782
C2×#156890432
#202423622
C2×#156888602
C2×#156888492
C2×C47×#5442
C2×#160547