Finite Group Databasebeta

← C2×#20246070 #26375020

Subgroups — C2×#20246070 #26375020

Order 12032.

3198 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C22×C4×Dic94C211yesmaximal, characteristic, complemented
3#20246070C2116yesmaximal, direct product factor
4C2×#15689029C211yesmaximal, characteristic, complemented
5C2×#15689021C211yesmaximal, characteristic, complemented
6C2×#15689018C212yesmaximal, complemented
7C2×#15689017C212yesmaximal
8C2×#15689011C212yesmaximal
9C2×#15689004C212yesmaximal, complemented
10C2×#15688983C211yesmaximal, characteristic
11C2×C4×C4⋊Dic47C211yesmaximal, characteristic, complemented
12C22×#117×C47C211yesmaximal, characteristic
13C23×Dic94C2211yescharacteristic
14C2×#13094003C4116yescomplemented
15C2×#13094003C2218yes—
16C2×C4×Dic94—28no—
17C2×C4×Dic94C2218yescomplemented
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×C22⋊Dic47C2212yes—
25C22×C22⋊Dic47C2212yes—
26C22×C4×Dic47C2212yes—
27C22×C4×Dic47C2212yes—
28#15689029C2218yescomplemented
29#15689021C2218yescomplemented
30#15689018C22116yescomplemented
31#15689017C22116yes—
32#15689011C22116yes—
33#15689004C22116yescomplemented
34#15688983C2218yes—
35C4×C4⋊Dic47C2218yescomplemented
36C2×C47⋊#95C2212yes—
37C2×C47⋊#95C2212yes—
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C24×C4×C47C2211yescharacteristic
42C2×#117×C47—28no—
43C2×#117×C47C2218yes—
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C22⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2211yescharacteristic
48C22×C42×C47C2211yescharacteristic
49C22×C42×C47C2211yescharacteristic
50C22×Dic94—24no—
51C22×Dic94—24no—
52C22×Dic94C2×C418yes—
53C22×Dic94C2314yes—
54C23×Dic47C2312yes—
55C23×Dic47C2312yes—
56C23×Dic47C2312yes—
57C23×Dic47C2312yes—
58#13094003C2×C4164yescomplemented
59C4×Dic94—232no—
60C2×C4⋊Dic47—24no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47C2×C418yes—
63C2×C4⋊Dic47C2314yes—
64C2×C4⋊Dic47C2314yes—
65C2×C4⋊Dic47C2314yes—
66C2×Dic47⋊C4—28no—
67C2×Dic47⋊C4—44no—
68C2×Dic47⋊C4C2×C4116yes—
69C2×Dic47⋊C4C2318yes—
70C2×Dic47⋊C4C2318yes—
71C2×Dic47⋊C4C2318yes—
72C2×C22⋊Dic47—44no—
73C2×C22⋊Dic47C2×C4116yes—
74C2×C22⋊Dic47C2318yes—
75C2×C22⋊Dic47C2318yes—
76C2×C4×Dic47—28no—
77C2×C4×Dic47—44no—
78C2×C4×Dic47C2318yes—
79C2×C4×Dic47C2318yes—
80C47⋊#95C2318yes—
81C47⋊#95C2318yes—
82C47⋊#95C2318yes—
83C47⋊#95C2318yes—
84C47⋊#95C2318yes—
85C25×C47C2311yescharacteristic
86C23×C4×C47—24no—
87C23×C4×C47—24no—
88C23×C4×C47C2×C418yes—
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2311yescharacteristic
91C23×C4×C47C2311yescharacteristic
92C23×C4×C47C2314yes—
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96#117×C47—232no—
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47—24no—
99C2×C4⋊C4×C47C2314yes—
100C2×C4⋊C4×C47C2314yes—
101C2×C22⋊C4×C47—24no—
102C2×C22⋊C4×C47—24no—
103C2×C22⋊C4×C47C2314yes—
104C2×C22⋊C4×C47C2314yes—
105C2×C42×C47—24no—
106C2×C42×C47D418yes—
107C2×C42×C47C2314yes—
108C2×C42×C47C2314yes—
109C24×C47—22no—
110C24×C47—22no—
111C24×C47—22no—
112C24×C47—22no—
113C24×C47—22no—
114C24×C47—22no—
115C24×C47C22×C414yes—
116C24×C47C2412yes—
117C24×C47C2411yescharacteristic
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—232no—
125C22×C4×C47—28no—
126C22×C4×C47—28no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C4116yes—
135C22×C4×C47C22×C414yes—
136C22×C4×C47C2×D414yes—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2×D414yes—
139C22×C4×C47#4114yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2412yes—
143C22×C4×C47C2412yes—
144C22×C4×C47C2412yes—
145C22×C4×C47C2412yes—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47C22×C418yes—
157C22×Dic47C22×C418yes—
158C22×Dic47C2414yes—
159C22×Dic47C2414yes—
160C22×Dic47C2414yes—
161C22×Dic47C2414yes—
162C2×Dic94—232no—
163C2×Dic94—28no—
164C2×Dic94—28no—
165C2×Dic94C22×C4116yes—
166C4⋊C4×C47—28no—
167C4⋊C4×C47—28no—
168C22⋊C4×C47—28no—
169C22⋊C4×C47—28no—
170C42×C47—28no—
171C42×C47C2×D4116yes—
172C22⋊Dic47—48no—
173C22⋊Dic47C22×C4132yes—
174C4⋊Dic47—28no—
175C4⋊Dic47—28no—
176C4⋊Dic47C22×C4116yes—
177Dic47⋊C4—216no—
178Dic47⋊C4—48no—
179Dic47⋊C4C22×C4132yes—
180C4×Dic47—216no—
181C4×Dic47—48no—
182C23×C47—28no—
183C23×C47—28no—
184C23×C47—28no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—44no—
192C23×C47—44no—
193C23×C47—44no—
194C23×C47—44no—
195C23×C47C23×C414yes—
196C23×C47C23×C412yes—
197C23×C47C22×D412yes—
198C23×C47C2×#4112yes—
199C23×C47H(5,𝔽2)12yes—
200C23×C472−1+412yes—
201C23×C472−1+412yes—
202C23×C472−1+412yes—
203C23×C47C2511yescharacteristic
204C2×C4×C47—28no—
205C2×C4×C47—28no—
206C2×C4×C47—28no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—232no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—44no—
221C2×C4×C47—44no—
222C2×C4×C47C4×D4116yes—
223C2×C4×C47C23×C414yes—
224C2×C4×C47C23×C414yes—
225C2×C4×C47C22×D414yes—
226C2×C4×C47C22×D414yes—
227C2×C4×C47C22×D414yes—
228C2×C4×C47C2×#4114yes—
229C2×Dic47—216no—
230C2×Dic47—216no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—48no—
242C2×Dic47—48no—
243C2×Dic47C23×C418yes—
244C2×Dic47C23×C418yes—
245Dic94—232no—
246C2×#2323—471nomaximal, Sylow
247C22×C47—24no—
248C22×C47—24no—
249C22×C47—24no—
250C22×C47—416no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47C2×C4×D414yes—
256C22×C47#51914yes—
257C22×C47#52014yes—
258C22×C47#53614yes—
259C22×C47#53714yes—
260C22×C47#54114yes—
261C22×C47#54214yes—
262C22×C47C24×C411yescharacteristic
263C22×C47C23×D411yescharacteristic
264C22×C47C22×#4111yescharacteristic
265C22×C47C2×H(5,𝔽2)11yescharacteristic
266C22×C47C2×2−1+411yescharacteristic
267C22×C47C2×2−1+411yescharacteristic
268C22×C47C2×2−1+411yescharacteristic
269C4×C47—24no—
270C4×C47—24no—
271C4×C47—24no—
272C4×C47—42no—
273C4×C47—42no—
274C4×C47C2×C4×D418yes—
275Dic47—28no—
276Dic47—28no—
277Dic47—44no—
278Dic47—44no—
279C22×C22⋊Q8—471no—
280C22×C4×Q8—471no—
281C22×#117D4711yescharacteristic
282#2323—4716no—
283C2×#391—471no—
284C2×#389—471no—
285C2×#389—472no—
286C2×#388—472no—
287C2×#388—472no—
288C2×#387—472no—
289C2×#380—471no—
290C2×C4×C4⋊C4—471no—
291C2×C47—44no—
292C2×C47#232318yescomplemented
293C2×C47C22×C4×D411yescharacteristic
294C2×C47C2×#51911yescharacteristic
295C2×C47C2×#52011yescharacteristic
296C2×C47C2×#53611yescharacteristic
297C2×C47C2×#53711yescharacteristic
298C2×C47C2×#54111yescharacteristic
299C2×C47C2×#54211yescharacteristic
300C23×Q8—471no—
301C24×C4C2×D4711yescharacteristic
302C2×C22⋊Q8—4716no—
303C2×C22⋊Q8—478no—
304C2×C4×Q8—478no—
305C2×C4×Q8—948no—
306C2×#117—28no—
307C2×#117C2×D4718yes—
308C22×C4⋊C4—472no—
309C22×C4⋊C4—471no—
310C22×C4⋊C4—472no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C4⋊C4C2×D4711yescharacteristic
316C22×C22⋊C4—472no—
317C22×C22⋊C4—472no—
318C22×C22⋊C4C2×D4711yescharacteristic
319C22×C22⋊C4C2×D4711yescharacteristic
320C22×C42—472no—
321C22×C42—472no—
322C22×C42C2×D4711yescharacteristic
323C22×C42C2×D4711yescharacteristic
324#391—478no—
325#389—4716no—
326#389—478no—
327#388—4716no—
328#388—4716no—
329#387—4716no—
330#380—478no—
331C4×C4⋊C4—478no—
332C2×#95—472no—
333C2×#95—472no—
334C2×#95—472no—
335C2×#95—472no—
336C2×#95—472no—
337C47C2×#232311yescharacteristic, complemented, Sylow
338C25C22×D4711yescharacteristic
339C22×Q8—478no—
340C22×Q8—474no—
341C22×Q8—944no—
342C22×Q8—944no—
343C23×C4—24no—
344C23×C4—24no—
345C23×C4—472no—
346C23×C4—472no—
347C23×C4—472no—
348C23×C4—472no—
349C23×C4C4×D4718yes—
350C23×C4C22×D4711yescharacteristic
351C23×C4C22×D4711yescharacteristic
352C23×C4C22×D4714yes—
353C23×C4C22×D4711yescharacteristic
354C23×C4C22×D4711yescharacteristic
355C23×C4C22×D4711yescharacteristic
356C23×C4C22×D4711yescharacteristic
357C22⋊Q8—4764no—
358C4×Q8—9432no—
359#117—232no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—4716no—
363C2×C4⋊C4—478no—
364C2×C4⋊C4—478no—
365C2×C4⋊C4—474no—
366C2×C4⋊C4—478no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—478no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—948no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—1884no—
374C2×C4⋊C4C22×D4714yes—
375C2×C4⋊C4C22×D4714yes—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—4716no—
379C2×C22⋊C4—478no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—1884no—
382C2×C22⋊C4C22×D4714yes—
383C2×C22⋊C4C22×D4714yes—
384C2×C42—24no—
385C2×C42—478no—
386C2×C42—478no—
387C2×C42—948no—
388C2×C42—1884no—
389C2×C42C22⋊D4718yes—
390C2×C42C22×D4714yes—
391C2×C42C22×D4714yes—
392#95—478no—
393#95—478no—
394#95—478no—
395#95—478no—
396#95—478no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24C2×C4×D4714yes—
404C24C23×D4712yes—
405C24C23×D4711yescharacteristic
406C2×Q8—4716no—
407C2×Q8—9432no—
408C2×Q8—948no—
409C2×Q8—948no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—232no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—478no—
426C22×C4—478no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—474no—
430C22×C4—474no—
431C22×C4—944no—
432C22×C4—944no—
433C22×C4—944no—
434C22×C4—944no—
435C22×C4—944no—
436C22×C4—944no—
437C22×C4—944no—
438C22×C4—944no—
439C22×C4—944no—
440C22×C4—944no—
441C22×C4C2×C4×D4714yes—
442C22×C4C2×C4×D47116yes—
443C22×C4C2×C4×D4714yes—
444C22×C4#1062535014yes—
445C22×C4C2×C22⋊D4714yes—
446C22×C4C2×C22⋊D4714yes—
447C22×C4C2×C22⋊D4714yes—
448C22×C4C23×D4712yes—
449C22×C4C23×D4712yes—
450C22×C4C23×D4712yes—
451C22×C4C23×D4712yes—
452C22×C4C23×D4712yes—
453C22×C4C23×D4712yes—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—4732no—
457C4⋊C4—4716no—
458C4⋊C4—9416no—
459C4⋊C4—948no—
460C4⋊C4—948no—
461C4⋊C4—1888no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—4732no—
465C22⋊C4—1888no—
466C42—28no—
467C42—9416no—
468C42—1888no—
469C42C2×C22⋊D47116yes—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C22×C4×D4714yes—
484C23C22×C4×D4712yes—
485C23C22×C22⋊D4712yes—
486C23C2×#1062535012yes—
487C23#1309407012yes—
488C23#1309407112yes—
489C23#1309407112yes—
490C23#1309407112yes—
491C23C24×D4711yescharacteristic
492Q8—9432no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—232no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—478no—
512C2×C4—478no—
513C2×C4—9416no—
514C2×C4—9416no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—1888no—
526C2×C4—1888no—
527C2×C4C4×C22⋊D47116yes—
528C2×C4C22×C4×D4714yes—
529C2×C4C22×C4×D4714yes—
530C2×C4C22×C22⋊D4714yes—
531C2×C4C22×C22⋊D4714yes—
532C2×C4C22×C22⋊D4714yes—
533C2×C4C2×#1062535014yes—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—416no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22C2×C4×C22⋊D4714yes—
543C22#1568960014yes—
544C22#1568960314yes—
545C22#1568969414yes—
546C22#1568970114yes—
547C22#1568973514yes—
548C22#1568974414yes—
549C22C23×C4×D4711yescharacteristic
550C22C23×C22⋊D4711yescharacteristic
551C22C22×#1062535011yescharacteristic
552C22C2×#1309407011yescharacteristic
553C22C2×#1309407111yescharacteristic
554C22C2×#1309407111yescharacteristic
555C22C2×#1309407111yescharacteristic
556C4—24no—
557C4—24no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—948no—
562C4—948no—
563C4—1884no—
564C4—1884no—
565C4C2×C4×C22⋊D4718yes—
566C2—44no—
567C2#2024607018yesdirect product factor
568C2C22×C4×C22⋊D4711yescharacteristic
569C2C2×#1568960011yescharacteristic
570C2C2×#1568960311yescharacteristic
571C2C2×#1568969411yescharacteristic
572C2C2×#1568970111yescharacteristic
573C2C2×#1568973511yescharacteristic
574C2C2×#1568974411yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC22×#117×C47C2
socleC24×C47C24
prime radicalC25×C47C23
cosocleC23×C47C25
hypercenterC22×#117D47
hypocenterC47C2×#2323
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic94C2semidirect
3. #20246070C2direct (N × H)
4. C2×#15689029C2semidirect
5. C2×#15689021C2semidirect
6. C2×#15689018C2semidirect
9. C2×#15689004C2semidirect
11. C2×C4×C4⋊Dic47C2semidirect
14. C2×#13094003C4semidirect
17. C2×C4×Dic94C22semidirect
28. #15689029C22semidirect
29. #15689021C22semidirect
30. #15689018C22semidirect
33. #15689004C22semidirect
35. C4×C4⋊Dic47C22semidirect
58. #13094003C2×C4semidirect
292. C2×C47#2323semidirect
337. C47C2×#2323semidirect
567. C2#20246070direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940032
C22×C4×Dic942
#202460702
C2×#156890292
C2×#156890212
C2×#156890182
C2×#156890172
C2×#156890112
C2×#156890042
C2×#156889832
C2×C4×C4⋊Dic472
C22×#117×C472
C2×#232347