Finite Group Databasebeta

← C2×#20247086 #26384104

Subgroups — C2×#20247086 #26384104

Order 12032.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C212yesmaximal
2C22×C4⋊C4×D47C211yesmaximal, characteristic
3#20247086C2116yesmaximal, direct product factor
4C2×#15689056C212yesmaximal, complemented
5C2×#15689042C212yesmaximal, complemented
6C2×#15689022C211yesmaximal, characteristic, complemented
7C2×#15689021C211yesmaximal, characteristic, complemented
8C2×#15688994C212yesmaximal
9C2×#15688992C211yesmaximal, characteristic, complemented
10C2×#15688980C211yesmaximal, characteristic
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
12C22×C4⋊Q8×C47C211yesmaximal, characteristic, complemented
13C23×C4×D47C2212yes—
14C23×C4×D47C2211yescharacteristic
15C2×#13094000—216no—
16C2×#13094000C22116yes—
17C22×C4⋊Dic47C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2212yes—
20C22×Dic47⋊C4C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C2×C4⋊C4×D47—28no—
25C2×C4⋊C4×D47C2218yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2211yescharacteristic
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×C4×Dic47C2211yescharacteristic
30#15689056C22116yescomplemented
31#15689042C22116yescomplemented
32#15689022C2218yescomplemented
33#15689021C2218yescomplemented
34#15688994C22116yes—
35#15688992C2218yescomplemented
36#15688980C2218yes—
37C4×(C2×D47)⋊C4C2218yes—
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2211yescharacteristic
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C23×Q8×C47C2212yes—
44C2×C4⋊Q8×C47—28no—
45C2×C4⋊Q8×C47C2218yescomplemented
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C4⋊C4×C47C2212yes—
48C22×C4⋊C4×C47C2211yescharacteristic
49C22×C42×C47C2211yescharacteristic
50C24×D47C2311yescharacteristic
51C23×Dic47C2312yes—
52C23×Dic47C2312yes—
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2311yescharacteristic
56C22×C4×D47—24no—
57C22×C4×D47—24no—
58C22×C4×D47—28no—
59C22×C4×D47—28no—
60C22×C4×D47—44no—
61C22×C4×D47Q818yescomplemented
62C22×C4×D47C2318yes—
63C22×C4×D47C2314yes—
64#13094000—264no—
65C2×C4⋊Dic47—24no—
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47—28no—
68C2×C4⋊Dic47C2314yes—
69C2×C4⋊Dic47C2314yes—
70C2×C4⋊Dic47C2318yes—
71C2×Dic47⋊C4—28no—
72C2×Dic47⋊C4—28no—
73C2×Dic47⋊C4—28no—
74C2×Dic47⋊C4—44no—
75C2×Dic47⋊C4C2318yes—
76C2×Dic47⋊C4C2318yes—
77C2×Dic47⋊C4C2318yes—
78C2×Dic47⋊C4C2318yes—
79C4⋊C4×D47—232no—
80C2×(C2×D47)⋊C4—28no—
81C2×(C2×D47)⋊C4D418yes—
82C2×(C2×D47)⋊C4Q818yescomplemented
83C2×(C2×D47)⋊C4C2318yes—
84C2×(C2×D47)⋊C4C2314yes—
85C2×(C2×D47)⋊C4C2314yes—
86C2×C4×Dic47—24no—
87C2×C4×Dic47C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2314yes—
90C47⋊#95C2318yes—
91C47⋊#95C2318yes—
92C47⋊#95C2318yes—
93C22×Q8×C47—28no—
94C22×Q8×C47—28no—
95C22×Q8×C47—44no—
96C22×Q8×C47C2318yes—
97C23×C4×C47C2312yes—
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C4⋊Q8×C47—232no—
103C2×C4⋊C4×C47—28no—
104C2×C4⋊C4×C47—24no—
105C2×C4⋊C4×C47D418yes—
106C2×C4⋊C4×C47C2314yes—
107C2×C4⋊C4×C47C2318yes—
108C2×C4⋊C4×C47C2314yes—
109C2×C42×C47—24no—
110C2×C42×C47C2314yes—
111C24×C47C2411yescharacteristic
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—22no—
116C23×D47C2×D414yes—
117C23×D47C2×Q814yes—
118C23×D47C2×Q814yes—
119C23×D47C2412yes—
120C2×Q8×C47—216no—
121C2×Q8×C47—216no—
122C2×Q8×C47—48no—
123C2×Q8×C47—432no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—22no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×D418yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2×Q814yes—
137C22×C4×C47C2×Q814yes—
138C22×C4×C47#4114yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2414yes—
142C22×C4×C47C2412yes—
143C22×C4×C47C2412yes—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47C2×D414yes—
157C22×Dic47C2×Q814yes—
158C22×Dic47C2×Q814yes—
159C22×Dic47C2414yes—
160C22×Dic47C2414yes—
161C22×Dic47C2412yes—
162C22×Dic47C2412yes—
163C22×Dic47C2412yes—
164C2×C4×D47—232no—
165C2×C4×D47—28no—
166C2×C4×D47—28no—
167C2×C4×D47—216no—
168C2×C4×D47—216no—
169C2×C4×D47—48no—
170C2×C4×D47—432no—
171C2×C4×D47C2×Q8116yescomplemented
172C4⋊C4×C47—216no—
173C4⋊C4×C47—28no—
174C4⋊C4×C47C2×D4116yes—
175C42×C47—28no—
176(C2×D47)⋊C4—216no—
177(C2×D47)⋊C4C2×D4116yes—
178(C2×D47)⋊C4C2×Q8116yescomplemented
179C4⋊Dic47—28no—
180C4⋊Dic47—28no—
181C4⋊Dic47—216no—
182Dic47⋊C4—216no—
183Dic47⋊C4—216no—
184Dic47⋊C4—216no—
185Dic47⋊C4—48no—
186C4×Dic47—28no—
187C23×C47C22×D412yes—
188C23×C47C22×D412yes—
189C23×C47C22×Q812yes—
190C23×C47C22×Q812yes—
191C23×C47C2×#4112yes—
192C23×C47H(5,𝔽2)12yes—
193C23×C472−1+412yes—
194C23×C47C2511yescharacteristic
195C22×D47—28no—
196C22×D47—28no—
197C22×D47—28no—
198C22×D47—28no—
199C22×D47—28no—
200C22×D47—28no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47C4⋊Q8116yescomplemented
206C22×D47C22×D414yes—
207C22×D47C22×Q814yes—
208C22×D47C22×Q814yes—
209Q8×C47—432no—
210C2×C4×C47—216no—
211C2×C4×C47—28no—
212C2×C4×C47—28no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—48no—
223C2×C4×C47C22⋊Q8116yescomplemented
224C2×C4×C47C22×D414yes—
225C2×C4×C47C22×D418yes—
226C2×C4×C47C22×D414yes—
227C2×C4×C47C22×Q814yes—
228C2×C4×C47C22×Q814yes—
229C2×C4×C47C2×#4114yes—
230C2×Dic47—28no—
231C2×Dic47—28no—
232C2×Dic47—28no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—48no—
246C2×Dic47—48no—
247C2×Dic47C22×D414yes—
248C2×Dic47C22×Q814yes—
249C2×Dic47C22×Q814yes—
250C4×D47—232no—
251C4×D47—432no—
252C2×#2440—471nomaximal, Sylow
253C22×C47C2×C22⋊Q814yes—
254C22×C47C2×C4⋊Q814yes—
255C22×C47#54414yes—
256C22×C47D4214yes—
257C22×C47#54714yes—
258C22×C47D4×Q814yes—
259C22×C47#55414yes—
260C22×C47C23×D411yescharacteristic
261C22×C47C23×D411yescharacteristic
262C22×C47C23×Q811yescharacteristic
263C22×C47C23×Q811yescharacteristic
264C22×C47C22×#4111yescharacteristic
265C22×C47C2×H(5,𝔽2)11yescharacteristic
266C22×C47C2×2−1+411yescharacteristic
267C2×D47—24no—
268C2×D47—24no—
269C2×D47—24no—
270C2×D47—232no—
271C2×D47—24no—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47C2×C4⋊Q818yescomplemented
275C4×C47—28no—
276C4×C47—24no—
277C4×C47—24no—
278C4×C47—44no—
279C4×C47C2×C22⋊Q818yescomplemented
280Dic47—24no—
281Dic47—24no—
282Dic47—24no—
283Dic47—44no—
284Dic47—44no—
285C22×C4⋊Q8D4711yescharacteristic, complemented
286C22×C22⋊Q8—472no—
287C23×C4⋊C4—471no—
288#2440—4716no—
289C2×#398—472no—
290C2×#395—472no—
291C2×#389—471no—
292C2×#389—471no—
293C2×#385—472no—
294C2×#384—471no—
295C2×#380—471no—
296C2×C4×C22⋊C4—471no—
297C2×C47#244018yescomplemented
298C2×C47C22×C22⋊Q811yescharacteristic
299C2×C47C22×C4⋊Q811yescharacteristic
300C2×C47C2×#54411yescharacteristic
301C2×C47C2×D4211yescharacteristic
302C2×C47C2×#54711yescharacteristic
303C2×C47C2×D4×Q811yescharacteristic
304C2×C47C2×#55411yescharacteristic
305D47—28no—
306C23×Q8C2×D4712yes—
307C24×C4—471no—
308C24×C4—472no—
309C2×C4⋊Q8—28no—
310C2×C4⋊Q8C2×D4718yescomplemented
311C2×C22⋊Q8—4716no—
312C2×C22⋊Q8—9416no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4—472no—
315C22×C4⋊C4—472no—
316C22×C4⋊C4—472no—
317C22×C4⋊C4—471no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4—472no—
320C22×C4⋊C4—478no—
321C22×C4⋊C4—948no—
322C22×C4⋊C4C2×D4712yes—
323C22×C4⋊C4C2×D4711yescharacteristic
324C22×C4⋊C4C2×D4711yescharacteristic
325C22×C22⋊C4—471no—
326C22×C22⋊C4—471no—
327C22×C22⋊C4—472no—
328C22×C42—471no—
329C22×C42C2×D4711yescharacteristic
330#398—4716no—
331#395—4716no—
332#389—478no—
333#389—478no—
334#385—4716no—
335#384—478no—
336#380—478no—
337C4×C22⋊C4—478no—
338C2×#95—471no—
339C2×#95—472no—
340C2×#95—472no—
341C2×#95—471no—
342C2×#95—472no—
343C47C2×#244011yescharacteristic, complemented, Sylow
344C25—471no—
345C22×Q8—28no—
346C22×Q8—28no—
347C22×Q8—44no—
348C22×Q8C22×D4718yes—
349C23×C4—471no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—478no—
353C23×C4—474no—
354C23×C4—472no—
355C23×C4—472no—
356C23×C4—478no—
357C23×C4—948no—
358C23×C4—944no—
359C23×C4—944no—
360C23×C4—948no—
361C23×C4—1884no—
362C23×C4C22×D4712yes—
363C23×C4C22×D4711yescharacteristic
364C23×C4C22×D4711yescharacteristic
365C23×C4C22×D4712yes—
366C23×C4C22×D4711yescharacteristic
367C4⋊Q8—232no—
368C22⋊Q8—9464no—
369C2×C4⋊C4—28no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—478no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—478no—
374C2×C4⋊C4—478no—
375C2×C4⋊C4—478no—
376C2×C4⋊C4—474no—
377C2×C4⋊C4—478no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—948no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—948no—
383C2×C4⋊C4—944no—
384C2×C4⋊C4—9432no—
385C2×C4⋊C4—1884no—
386C2×C4⋊C4C22⋊D4718yes—
387C2×C4⋊C4C22×D4714yes—
388C2×C4⋊C4C22×D4718yes—
389C2×C4⋊C4C22×D4714yes—
390C2×C22⋊C4—478no—
391C2×C22⋊C4—478no—
392C2×C22⋊C4—474no—
393C2×C22⋊C4—474no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—948no—
396C2×C42—24no—
397C2×C42—474no—
398C2×C42—944no—
399C2×C42C22×D4714yes—
400#95—474no—
401#95—474no—
402#95—478no—
403#95—478no—
404#95—478no—
405C24—474no—
406C24—474no—
407C24—474no—
408C24—472no—
409C24—942no—
410C24—942no—
411C24—942no—
412C24—942no—
413C24C23×D4711yescharacteristic
414C2×Q8—216no—
415C2×Q8—216no—
416C2×Q8—48no—
417C2×Q8—432no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—474no—
430C22×C4—472no—
431C22×C4—472no—
432C22×C4—472no—
433C22×C4—4716no—
434C22×C4—474no—
435C22×C4—474no—
436C22×C4—9416no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—9432no—
444C22×C4—948no—
445C22×C4—948no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—9416no—
453C22×C4—1888no—
454C22×C4—18832no—
455C22×C4D4×D4714yes—
456C22×C4Q8×D4714yes—
457C22×C4Q8×D4714yes—
458C22×C4#1062535414yes—
459C22×C4C2×C22⋊D4718yes—
460C22×C4C2×C22⋊D4714yes—
461C22×C4C23×D4714yes—
462C22×C4C23×D4712yes—
463C22×C4C23×D4714yes—
464C22×C4C23×D4712yes—
465C22×C4C23×D4712yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—9416no—
469C4⋊C4—9416no—
470C4⋊C4—9416no—
471C4⋊C4—9416no—
472C4⋊C4—948no—
473C4⋊C4—948no—
474C4⋊C4—1888no—
475C4⋊C4C2×C22⋊D47116yes—
476C22⋊C4—4716no—
477C22⋊C4—4716no—
478C22⋊C4—9416no—
479C42—28no—
480C42—948no—
481C23—4716no—
482C23—474no—
483C23—474no—
484C23—474no—
485C23—948no—
486C23—948no—
487C23—948no—
488C23—948no—
489C23—948no—
490C23—948no—
491C23—942no—
492C23—942no—
493C23—942no—
494C23—942no—
495C23C2×D4×D4712yes—
496C23C22×C22⋊D4712yes—
497C23C2×Q8×D4712yes—
498C23C2×Q8×D4712yes—
499C23C2×#1062535412yes—
500C23#1309406912yes—
501C23#1309407212yes—
502C23C24×D4711yescharacteristic
503Q8—432no—
504C2×C4—216no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—474no—
518C2×C4—474no—
519C2×C4—474no—
520C2×C4—948no—
521C2×C4—948no—
522C2×C4—948no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—9432no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—18832no—
537C2×C4—1888no—
538C2×C4—1888no—
539C2×C4#13094000116yescomplemented
540C2×C4C2×D4×D4714yes—
541C2×C4C22×C22⋊D4718yes—
542C2×C4C22×C22⋊D4714yes—
543C2×C4C2×Q8×D4714yes—
544C2×C4C2×Q8×D4714yes—
545C2×C4C2×#1062535414yes—
546C22—478no—
547C22—944no—
548C22—9432no—
549C22—944no—
550C22—944no—
551C22—944no—
552C22—944no—
553C22—944no—
554C22C2×#1309400014yes—
555C22C4⋊Q8×D4714yes—
556C22#1568975014yes—
557C22D4×C22⋊D4714yes—
558C22#1568977614yes—
559C22Q8×C22⋊D4714yes—
560C22#1568983414yes—
561C22C22×D4×D4711yescharacteristic
562C22C23×C22⋊D4711yescharacteristic
563C22C22×Q8×D4711yescharacteristic
564C22C22×Q8×D4711yescharacteristic
565C22C22×#1062535411yescharacteristic
566C22C2×#1309406911yescharacteristic
567C22C2×#1309407211yescharacteristic
568C4—28no—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—944no—
573C4—944no—
574C4—944no—
575C4—1884no—
576C4—1884no—
577C4C2×#1309400018yescomplemented
578C2—948no—
579C2#2024708618yesdirect product factor
580C2C22×#1309400011yescharacteristic
581C2C2×C4⋊Q8×D4711yescharacteristic
582C2C2×#1568975011yescharacteristic
583C2C2×D4×C22⋊D4711yescharacteristic
584C2C2×#1568977611yescharacteristic
585C2C2×Q8×C22⋊D4711yescharacteristic
586C2C2×#1568983411yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×C4⋊Q8×C47C2
socleC24×C47C24
prime radicalC24×D47C23
hypercenterC22×C4⋊Q8D47
hypocenterC47C2×#2440
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20247086C2direct (N × H)
4. C2×#15689056C2semidirect
5. C2×#15689042C2semidirect
6. C2×#15689022C2semidirect
7. C2×#15689021C2semidirect
9. C2×#15688992C2semidirect
12. C22×C4⋊Q8×C47C2semidirect
30. #15689056C22semidirect
31. #15689042C22semidirect
32. #15689022C22semidirect
33. #15689021C22semidirect
35. #15688992C22semidirect
45. C2×C4⋊Q8×C47C22semidirect
61. C22×C4×D47Q8semidirect
82. C2×(C2×D47)⋊C4Q8semidirect
171. C2×C4×D47C2×Q8semidirect
178. (C2×D47)⋊C4C2×Q8semidirect
205. C22×D47C4⋊Q8semidirect
223. C2×C4×C47C22⋊Q8semidirect
274. C2×D47C2×C4⋊Q8semidirect
279. C4×C47C2×C22⋊Q8semidirect
285. C22×C4⋊Q8D47semidirect
297. C2×C47#2440semidirect
310. C2×C4⋊Q8C2×D47semidirect
343. C47C2×#2440semidirect
539. C2×C4#13094000semidirect
577. C4C2×#13094000semidirect
579. C2#20247086direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940002
C22×C4⋊C4×D472
#202470862
C2×#156890562
C2×#156890422
C2×#156890222
C2×#156890212
C2×#156889942
C2×#156889922
C2×#156889802
C2×C4×(C2×D47)⋊C42
C22×C4⋊Q8×C472
C2×#244047