Finite Group Databasebeta

← C2×#20241722 #26182124

Subgroups — C2×#20241722 #26182124

Order 12032.

1326 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689180C212yesmaximal, complemented
2#20241722C218yesmaximal, direct product factor
3C2×#15688849C212yesmaximal, complemented
4C2×#15688847C212yesmaximal
5C22×#117×C47C211yesmaximal, characteristic
6#15689180—24no—
7#15689180C2218yescomplemented
8C2×C4⋊(C47⋊C8)—22no—
9C2×C22⋊(C47⋊C8)C2214yes—
10C2×#13093935—22no—
11#15688849C2218yescomplemented
12#15688847C2218yes—
13C24×C4×C47C411yescharacteristic
14C2×#117×C47—22no—
15C2×#117×C47—22no—
16C2×#117×C47—24no—
17C2×#117×C47C414yes—
18C2×#117×C47C2214yes—
19C22×C4⋊C4×C47C411yescharacteristic
20C22×C4⋊C4×C47C2211yescharacteristic
21C22×C22⋊C4×C47C411yescharacteristic
22C22×C22⋊C4×C47C2211yescharacteristic
23C22×C42×C47C411yescharacteristic
24C22×C42×C47C2211yescharacteristic
25C22×C47⋊C8—24no—
26C4⋊(C47⋊C8)—28no—
27C22⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)C2318yes—
29#13093935—28no—
30C25×C47C2×C411yescharacteristic
31C23×C4×C47—24no—
32C23×C4×C47—22no—
33C23×C4×C47—21no—
34C23×C4×C47—21no—
35C23×C4×C47—24no—
36C23×C4×C47C2×C414yes—
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47C2×C411yescharacteristic
40C23×C4×C47C2×C411yescharacteristic
41C23×C4×C47C2×C411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43#117×C47—28no—
44#117×C47—28no—
45#117×C47—28no—
46#117×C47—28no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47—22no—
51C2×C4⋊C4×C47C2×C414yes—
52C2×C4⋊C4×C47C2×C412yes—
53C2×C4⋊C4×C47C2312yes—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47—22no—
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47C2×C414yes—
59C2×C22⋊C4×C47C2×C412yes—
60C2×C22⋊C4×C47C2312yes—
61C2×C42×C47—22no—
62C2×C42×C47—21no—
63C2×C42×C47—21no—
64C2×C42×C47C2×C414yes—
65C2×C42×C47C2×C412yes—
66C2×C42×C47D412yes—
67C2×C42×C47D412yes—
68C2×C42×C47D414yes—
69C2×C42×C47C2312yes—
70C24×C47—21no—
71C24×C47—21no—
72C24×C47—22no—
73C24×C47—42no—
74C24×C47—42no—
75C24×C47M1614yes—
76C24×C47C22×C411yescharacteristic
77C24×C47C22×C412yes—
78C22×C4×C47—216no—
79C22×C4×C47—28no—
80C22×C4×C47—28no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—28no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47—24no—
88C22×C4×C47—24no—
89C22×C4×C47—28no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—42no—
95C22×C4×C47—42no—
96C22×C4×C47—42no—
97C22×C4×C47—42no—
98C22×C4×C47C22⋊C412yes—
99C22×C4×C47C22⋊C411yescharacteristic
100C22×C4×C47C22⋊C411yescharacteristic
101C22×C4×C47C22⋊C412yes—
102C22×C4×C47C22⋊C412yes—
103C22×C4×C47M1614yes—
104C22×C4×C47C22×C412yes—
105C22×C4×C47C22×C412yes—
106C22×C4×C47C22×C412yes—
107C22×C4×C47C22×C412yes—
108C22×C4×C47C22×C411yescharacteristic
109C22×C4×C47C22×C412yes—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2×D412yes—
112C22×C4×C47C2411yescharacteristic
113C4⋊C4×C47—24no—
114C4⋊C4×C47—28no—
115C4⋊C4×C47—24no—
116C22⋊C4×C47—24no—
117C22⋊C4×C47—28no—
118C22⋊C4×C47—24no—
119C42×C47—24no—
120C42×C47—24no—
121C42×C47C22⋊C414yes—
122C42×C47C22⋊C414yes—
123C42×C47C2×D414yes—
124C42×C47C2×D414yes—
125C2×C47⋊C8—28no—
126C2×C47⋊C8—48no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47—22no—
130C23×C47—22no—
131C23×C47—24no—
132C23×C47—24no—
133C23×C47—48no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—42no—
137C23×C47—42no—
138C23×C47—48no—
139C23×C47—42no—
140C23×C47—42no—
141C23×C47C2×C22⋊C411yescharacteristic
142C23×C47C2×C22⋊C411yescharacteristic
143C23×C47C2×M1612yes—
144C23×C47C2×M1612yes—
145C23×C47C2×M1614yes—
146C23×C47C23×C411yescharacteristic
147C2×C4×C47—216no—
148C2×C4×C47—216no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—24no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47—24no—
159C2×C4×C47—24no—
160C2×C4×C47—22no—
161C2×C4×C47—22no—
162C2×C4×C47—42no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47—42no—
168C2×C4×C47—42no—
169C2×C4×C47—42no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47C2×C22⋊C412yes—
174C2×C4×C47C2×C22⋊C412yes—
175C2×C4×C47C2×C22⋊C412yes—
176C2×C4×C47C2×C22⋊C414yes—
177C2×C4×C47C2×M1614yes—
178C2×C4×C47C22×D412yes—
179C47⋊C8—48no—
180C2×#1464—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—24no—
186C22×C47—24no—
187C22×C47—24no—
188C22×C47—44no—
189C22×C47—44no—
190C22×C47—48no—
191C22×C47—42no—
192C22×C47—42no—
193C22×C47—48no—
194C22×C47#40712yes—
195C22×C47#40712yes—
196C22×C47#41012yes—
197C22×C47#41312yes—
198C22×C47C22×C22⋊C411yescharacteristic
199C22×C47C22×M1611yescharacteristic
200C22×C47C22×M1611yescharacteristic
201C4×C47—22no—
202C4×C47—22no—
203C4×C47—24no—
204C4×C47—22no—
205C4×C47—22no—
206C4×C47—24no—
207C4×C47—42no—
208C4×C47—42no—
209C22×#117D4711yescharacteristic
210C2×#432—472no—
211#1464—478no—
212C2×#324—472no—
213C2×#323—472no—
214C2×C47—21no—
215C2×C47—21no—
216C2×C47—22no—
217C2×C47—44no—
218C2×C47#146414yescomplemented
219C2×C47C2×#40711yescharacteristic
220C2×C47C2×#41011yescharacteristic
221C2×C47C2×#41311yescharacteristic
222C24×C4Dic4711yescharacteristic
223C2×#117—22no—
224C2×#117—22no—
225C2×#117—24no—
226C2×#117Dic4714yes—
227C2×#117C2×D4714yes—
228C22×C4⋊C4Dic4711yescharacteristic
229C22×C4⋊C4C2×D4711yescharacteristic
230C22×C22⋊C4Dic4711yescharacteristic
231C22×C22⋊C4C2×D4711yescharacteristic
232C22×C42Dic4711yescharacteristic
233C22×C42C2×D4711yescharacteristic
234#432—478no—
235#432—944no—
236C2×C4⋊C8—942no—
237C2×C22⋊C8—474no—
238C2×#97—942no—
239#324—478no—
240#323—478no—
241C47C2×#146411yescharacteristic, complemented, Sylow
242C25C2×Dic4711yescharacteristic
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×Dic4714yes—
249C23×C4C2×Dic4711yescharacteristic
250C23×C4C2×Dic4711yescharacteristic
251C23×C4C2×Dic4711yescharacteristic
252C23×C4C2×Dic4711yescharacteristic
253C23×C4C2×Dic4711yescharacteristic
254C23×C4C22×D4711yescharacteristic
255C22×C8—944no—
256#117—28no—
257#117—28no—
258#117—28no—
259#117—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C2×Dic4714yes—
265C2×C4⋊C4C2×Dic4712yes—
266C2×C4⋊C4C22×D4712yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4C2×Dic4714yes—
272C2×C22⋊C4C2×Dic4712yes—
273C2×C22⋊C4C22×D4712yes—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C2×Dic4714yes—
278C2×C42C2×Dic4712yes—
279C2×C42C22⋊D4712yes—
280C2×C42C22⋊D4712yes—
281C2×C42C22⋊D4714yes—
282C2×C42C22×D4712yes—
283C4⋊C8—948no—
284C22⋊C8—478no—
285C22⋊C8—944no—
286#97—948no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—42no—
291C24—42no—
292C24#1062532914yes—
293C24C22×Dic4711yescharacteristic
294C24C22×Dic4712yes—
295C22×C4—216no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—28no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4#1062532914yes—
316C22×C4C22⋊Dic4712yes—
317C22×C4C22⋊Dic4711yescharacteristic
318C22×C4C22⋊Dic4711yescharacteristic
319C22×C4C22⋊Dic4712yes—
320C22×C4C22⋊Dic4712yes—
321C22×C4C22×Dic4712yes—
322C22×C4C22×Dic4712yes—
323C22×C4C22×Dic4712yes—
324C22×C4C22×Dic4712yes—
325C22×C4C22×Dic4711yescharacteristic
326C22×C4C22×Dic4712yes—
327C22×C4C2×C22⋊D4712yes—
328C22×C4C2×C22⋊D4712yes—
329C22×C4C23×D4711yescharacteristic
330C2×C8—948no—
331C2×C8—1888no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—28no—
337C22⋊C4—24no—
338C42—24no—
339C42—24no—
340C42C22⋊Dic4714yes—
341C42C22⋊Dic4714yes—
342C42C2×C22⋊D4714yes—
343C42C2×C22⋊D4714yes—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—48no—
351C23—42no—
352C23—42no—
353C23—48no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23C2×C22⋊Dic4711yescharacteristic
359C23C2×C22⋊Dic4711yescharacteristic
360C23C2×#1062532912yes—
361C23C2×#1062532912yes—
362C23C2×#1062532914yes—
363C23C23×Dic4711yescharacteristic
364C2×C4—216no—
365C2×C4—216no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4C2×C22⋊Dic4712yes—
391C2×C4C2×C22⋊Dic4712yes—
392C2×C4C2×C22⋊Dic4712yes—
393C2×C4C2×C22⋊Dic4714yes—
394C2×C4C2×#1062532914yes—
395C2×C4C22×C22⋊D4712yes—
396C8—1888no—
397C22—24no—
398C22—24no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—48no—
406C22—44no—
407C22—48no—
408C22—42no—
409C22—42no—
410C22#1568908712yes—
411C22#1568908712yes—
412C22#1568909712yes—
413C22#1568910712yes—
414C22C22×C22⋊Dic4711yescharacteristic
415C22C22×#1062532911yescharacteristic
416C22C22×#1062532911yescharacteristic
417C4—22no—
418C4—22no—
419C4—24no—
420C4—22no—
421C4—22no—
422C4—24no—
423C4—42no—
424C4—42no—
425C2—22no—
426C2—21no—
427C2—21no—
428C2—44no—
429C2#2024172214yesdirect product factor
430C2C2×#1568908711yescharacteristic
431C2C2×#1568909711yescharacteristic
432C2C2×#1568910711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#117×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenterC22×#117D47
hypocenterC47C2×#1464
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C2×#15689180C2semidirect
2. #20241722C2direct (N × H)
3. C2×#15688849C2semidirect
7. #15689180C22semidirect
11. #15688849C22semidirect
218. C2×C47#1464semidirect
241. C47C2×#1464semidirect
429. C2#20241722direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156891802
#202417222
C2×#156888492
C2×#156888472
C22×#117×C472
C2×#146447