Finite Group Databasebeta

← C2×#20241736 #26182222

Subgroups — C2×#20241736 #26182222

Order 12032.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689087C211yesmaximal, characteristic, complemented
2C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3#20241736C218yesmaximal, direct product factor
4C2×#15688849C212yesmaximal, complemented
5C2×#15688847C212yesmaximal, complemented
6C22×#123×C47C211yesmaximal, characteristic
7C22×#10625329—21no—
8C23×C47⋊C8—21no—
9#15689087—22no—
10#15689087C2214yescomplemented
11C2×C22⋊(C47⋊C8)—22no—
12C2×C22⋊(C47⋊C8)—21no—
13C2×C22⋊(C47⋊C8)—21no—
14C2×C22⋊(C47⋊C8)C2212yes—
15C2×C22⋊(C47⋊C8)C2212yes—
16C2×C22⋊(C47⋊C8)C2214yescomplemented
17#15688849C2218yescomplemented
18#15688847C2218yescomplemented
19C23×D4×C47C411yescharacteristic
20C24×C4×C47C2211yescharacteristic
21C2×#123×C47—21no—
22C2×#123×C47—21no—
23C2×#123×C47—21no—
24C2×#123×C47—21no—
25C2×#123×C47—22no—
26C2×#123×C47—22no—
27C2×#123×C47C414yes—
28C2×#123×C47C2214yes—
29C22×C4⋊C4×C47C411yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C411yescharacteristic
32C22×C22⋊C4×C47C411yescharacteristic
33C22×C22⋊C4×C47C2211yescharacteristic
34C2×#10625329—24no—
35C2×#10625329—44no—
36C22×C47⋊C8—22no—
37C22×C47⋊C8—22no—
38C22×C47⋊C8—24no—
39C22×C47⋊C8—44no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—24no—
42C22⋊(C47⋊C8)—24no—
43C22⋊(C47⋊C8)—22no—
44C22⋊(C47⋊C8)C2314yes—
45C22⋊(C47⋊C8)C2314yes—
46C25×C47C2×C411yescharacteristic
47C25×C47C2×C411yescharacteristic
48C22×D4×C47—21no—
49C22×D4×C47—22no—
50C22×D4×C47—21no—
51C22×D4×C47—44no—
52C22×D4×C47C2×C414yes—
53C23×C4×C47—44no—
54C23×C4×C47C2×C411yescharacteristic
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47C2×C411yescharacteristic
58C23×C4×C47C2×C412yes—
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D411yescharacteristic
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47D412yes—
64C23×C4×C47D412yes—
65C23×C4×C47C2311yescharacteristic
66C23×C4×C47C2312yes—
67#123×C47—24no—
68#123×C47—24no—
69#123×C47—24no—
70#123×C47—24no—
71#123×C47—24no—
72#123×C47—24no—
73#123×C47—24no—
74#123×C47—24no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47C2×C414yes—
83C2×C4⋊C4×C47C2×C412yes—
84C2×C4⋊C4×C47C2312yes—
85C2×C22⋊C4×C47—21no—
86C2×C22⋊C4×C47—22no—
87C2×C22⋊C4×C47—21no—
88C2×C22⋊C4×C47—21no—
89C2×C22⋊C4×C47—21no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—21no—
92C2×C22⋊C4×C47—21no—
93C2×C22⋊C4×C47—21no—
94C2×C22⋊C4×C47—21no—
95C2×C22⋊C4×C47C2×C414yes—
96C2×C22⋊C4×C47C2×C414yes—
97C2×C22⋊C4×C47C2×C412yes—
98C2×C22⋊C4×C47C2312yes—
99C24×C47—24no—
100C24×C47—24no—
101C24×C47—21no—
102C24×C47—21no—
103C24×C47—24no—
104C24×C47—42no—
105C24×C47—42no—
106C24×C47—42no—
107C24×C47C22⋊C411yescharacteristic
108C24×C47C22⋊C412yes—
109C24×C47C22⋊C411yescharacteristic
110C24×C47C22×C412yes—
111C24×C47C22×C411yescharacteristic
112C24×C47C22×C412yes—
113C2×D4×C47—24no—
114C2×D4×C47—24no—
115C2×D4×C47—48no—
116C2×D4×C47—44no—
117C2×D4×C47—44no—
118C2×D4×C47—48no—
119C22×C4×C47—24no—
120C22×C4×C47—24no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—24no—
128C22×C4×C47—42no—
129C22×C4×C47—44no—
130C22×C4×C47—44no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—44no—
134C22×C4×C47—42no—
135C22×C4×C47—44no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47C22⋊C411yescharacteristic
139C22×C4×C47C22⋊C411yescharacteristic
140C22×C4×C47C22⋊C412yes—
141C22×C4×C47C22⋊C412yes—
142C22×C4×C47C22⋊C412yes—
143C22×C4×C47C22⋊C412yes—
144C22×C4×C47C22⋊C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C22×C412yes—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C22×C412yes—
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D412yes—
153C22×C4×C47C2×D412yes—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2×D412yes—
158C22×C4×C47C2411yescharacteristic
159C4⋊C4×C47—24no—
160C4⋊C4×C47—24no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C22⋊C4×C47—24no—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—24no—
168C22⋊C4×C47—24no—
169C22⋊C4×C47—22no—
170C22⋊C4×C47—22no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173#10625329—48no—
174C2×C47⋊C8—24no—
175C2×C47⋊C8—24no—
176C2×C47⋊C8—44no—
177C2×C47⋊C8—48no—
178C2×C47⋊C8—44no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—22no—
186C23×C47—24no—
187C23×C47—22no—
188C23×C47—24no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—416no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—88no—
204C23×C47#9914yes—
205C23×C47C2×C22⋊C412yes—
206C23×C47C2×C22⋊C411yescharacteristic
207C23×C47C2×C22⋊C412yes—
208C23×C47C2×C22⋊C411yescharacteristic
209C23×C47#13114yes—
210C23×C47C23×C411yescharacteristic
211D4×C47—48no—
212D4×C47—48no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—41no—
229C2×C4×C47—42no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—41no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47C2×C22⋊C412yes—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C411yescharacteristic
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C47⋊C8—44no—
250C47⋊C8—44no—
251C2×#1468—471nomaximal, Sylow
252C22×C47—22no—
253C22×C47—21no—
254C22×C47—21no—
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—22no—
258C22×C47—28no—
259C22×C47—28no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—48no—
264C22×C47—48no—
265C22×C47—44no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—84no—
270C22×C47—84no—
271C22×C47—84no—
272C22×C47#40812yes—
273C22×C47#40812yes—
274C22×C47C2×#9914yes—
275C22×C47C2×#9912yes—
276C22×C47#41312yes—
277C22×C47C22×C22⋊C411yescharacteristic
278C22×C47C2×#13112yes—
279C4×C47—21no—
280C4×C47—21no—
281C4×C47—22no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—22no—
285C4×C47—42no—
286C4×C47—42no—
287C4×C47—42no—
288C22×#123D4711yescharacteristic
289C2×#407—471no—
290C22×C22⋊C8—471no—
291#1468—478no—
292C2×#324—472no—
293C2×#323—472no—
294C2×C47—22no—
295C2×C47—22no—
296C2×C47—24no—
297C2×C47—42no—
298C2×C47—82no—
299C2×C47#146814yescomplemented
300C2×C47C2×#40811yescharacteristic
301C2×C47C22×#9911yescharacteristic
302C2×C47C2×#41311yescharacteristic
303C23×D4Dic4711yescharacteristic
304C24×C4C2×D4711yescharacteristic
305C22×M16—941no—
306C23×C8—941no—
307C2×#123—21no—
308C2×#123—21no—
309C2×#123—22no—
310C2×#123—21no—
311C2×#123—21no—
312C2×#123—22no—
313C2×#123Dic4714yes—
314C2×#123C2×D4714yes—
315C22×C4⋊C4Dic4711yescharacteristic
316C22×C4⋊C4C2×D4711yescharacteristic
317C22×C22⋊C4Dic4711yescharacteristic
318C22×C22⋊C4Dic4711yescharacteristic
319C22×C22⋊C4C2×D4711yescharacteristic
320#407—474no—
321#407—942no—
322C2×C22⋊C8—472no—
323C2×C22⋊C8—474no—
324C2×C22⋊C8—472no—
325C2×C22⋊C8—942no—
326C2×C22⋊C8—941no—
327C2×C22⋊C8—941no—
328#324—478no—
329#323—478no—
330C47C2×#146811yescharacteristic, complemented, Sylow
331C25C2×Dic4711yescharacteristic
332C25C2×Dic4711yescharacteristic
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—44no—
337C22×D4C2×Dic4714yes—
338C23×C4—44no—
339C23×C4C2×Dic4711yescharacteristic
340C23×C4C2×Dic4711yescharacteristic
341C23×C4C2×Dic4711yescharacteristic
342C23×C4C2×Dic4711yescharacteristic
343C23×C4C2×Dic4712yes—
344C23×C4C22⋊D4711yescharacteristic
345C23×C4C22⋊D4711yescharacteristic
346C23×C4C22⋊D4712yes—
347C23×C4C22⋊D4711yescharacteristic
348C23×C4C22⋊D4711yescharacteristic
349C23×C4C22⋊D4712yes—
350C23×C4C22×D4711yescharacteristic
351C23×C4C22×D4712yes—
352C2×M16—944no—
353C2×M16—1884no—
354C22×C8—944no—
355C22×C8—942no—
356C22×C8—942no—
357C22×C8—1884no—
358#123—24no—
359#123—24no—
360#123—24no—
361#123—24no—
362#123—24no—
363#123—24no—
364#123—24no—
365#123—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C2×Dic4714yes—
374C2×C4⋊C4C2×Dic4712yes—
375C2×C4⋊C4C22×D4712yes—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4C2×Dic4714yes—
387C2×C22⋊C4C2×Dic4714yes—
388C2×C22⋊C4C2×Dic4712yes—
389C2×C22⋊C4C22×D4712yes—
390C22⋊C8—474no—
391C22⋊C8—474no—
392C22⋊C8—942no—
393C22⋊C8—944no—
394C22⋊C8—942no—
395C22⋊C8—944no—
396C24—24no—
397C24—24no—
398C24—21no—
399C24—21no—
400C24—24no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22⋊Dic4711yescharacteristic
405C24C22⋊Dic4712yes—
406C24C22⋊Dic4711yescharacteristic
407C24C22×Dic4712yes—
408C24C22×Dic4711yescharacteristic
409C24C22×Dic4712yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—48no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C22⋊Dic4712yes—
436C22×C4C22⋊Dic4712yes—
437C22×C4C22⋊Dic4711yescharacteristic
438C22×C4C22⋊Dic4711yescharacteristic
439C22×C4C22⋊Dic4712yes—
440C22×C4C22⋊Dic4712yes—
441C22×C4C22⋊Dic4712yes—
442C22×C4C22×Dic4712yes—
443C22×C4C22×Dic4712yes—
444C22×C4C22×Dic4712yes—
445C22×C4C22×Dic4712yes—
446C22×C4C22×Dic4711yescharacteristic
447C22×C4C2×C22⋊D4712yes—
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4712yes—
454C22×C4C2×C22⋊D4712yes—
455C22×C4C23×D4711yescharacteristic
456M16—1888no—
457C2×C8—944no—
458C2×C8—944no—
459C2×C8—1888no—
460C2×C8—1884no—
461C2×C8—1884no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—416no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—88no—
501C23#1309393914yes—
502C23C2×C22⋊Dic4712yes—
503C23C2×C22⋊Dic4711yescharacteristic
504C23C2×C22⋊Dic4712yes—
505C23C2×C22⋊Dic4711yescharacteristic
506C23#1309403114yes—
507C23C23×Dic4711yescharacteristic
508D4—48no—
509D4—48no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4C2×C22⋊Dic4711yescharacteristic
541C2×C4C2×C22⋊Dic4711yescharacteristic
542C2×C4C2×C22⋊Dic4712yes—
543C2×C4C2×C22⋊Dic4712yes—
544C2×C4C22×C22⋊D4711yescharacteristic
545C2×C4C22×C22⋊D4711yescharacteristic
546C8—1884no—
547C8—1884no—
548C22—28no—
549C22—28no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22#1568909012yes—
569C22#1568909012yes—
570C22C2×#1309393914yes—
571C22C2×#1309393912yes—
572C22#1568910712yes—
573C22C22×C22⋊Dic4711yescharacteristic
574C22C2×#1309403112yes—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—82no—
589C2#2024173614yesdirect product factor
590C2C2×#1568909011yescharacteristic
591C2C22×#1309393911yescharacteristic
592C2C2×#1568910711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#123×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D4×C47C4
cosocleC22×C4×C47C24
hypercenterC22×#123D47
hypocenterC47C2×#1468
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#15689087C2semidirect
2. C22×C22⋊(C47⋊C8)C2semidirect
3. #20241736C2direct (N × H)
4. C2×#15688849C2semidirect
5. C2×#15688847C2semidirect
10. #15689087C22semidirect
16. C2×C22⋊(C47⋊C8)C22semidirect
17. #15688849C22semidirect
18. #15688847C22semidirect
299. C2×C47#1468semidirect
330. C47C2×#1468semidirect
589. C2#20241736direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156890872
C22×C22⋊(C47⋊C8)2
#202417362
C2×#156888492
C2×#156888472
C22×#123×C472
C2×#146847