Finite Group Databasebeta

← C2×#20244374 #26220919

Subgroups — C2×#20244374 #26220919

Order 12032.

1698 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688983C211yesmaximal, characteristic
2C2×C4×C22⋊Dic47C211yesmaximal, characteristic
3#20244374C218yesmaximal, direct product factor
4C2×#15688890C214yesmaximal
5C22×C22⋊Q8×C47C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C22×C22⋊Dic47C2212yes—
8C22×C22⋊Dic47C2212yes—
9C22×C4×Dic47—21no—
10#15688983—22no—
11#15688983C2214yes—
12C4×C22⋊Dic47—22no—
13C4×C22⋊Dic47C2214yes—
14C2×C47⋊#95—21no—
15C2×C47⋊#95—21no—
16#15688890C22116yes—
17C23×Q8×C47C411yescharacteristic, complemented
18C24×C4×C47C2211yescharacteristic
19C2×C22⋊Q8×C47—21no—
20C2×C22⋊Q8×C47—21no—
21C2×C22⋊Q8×C47—21no—
22C2×C22⋊Q8×C47—21no—
23C2×C22⋊Q8×C47—22no—
24C2×C22⋊Q8×C47—22no—
25C2×C22⋊Q8×C47C414yescomplemented
26C2×C22⋊Q8×C47C2214yes—
27C22×C4⋊C4×C47C411yescharacteristic, complemented
28C22×C4⋊C4×C47C412yescomplemented
29C22×C22⋊C4×C47C2212yes—
30C23×Dic47—22no—
31C23×Dic47—22no—
32C2×C4⋊Dic47—24no—
33C2×C4⋊Dic47—44no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—24no—
36C2×C22⋊Dic47—28no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47Q818yescomplemented
39C2×C22⋊Dic47C2314yes—
40C2×C22⋊Dic47C2314yes—
41C2×C4×Dic47—24no—
42C2×C4×Dic47—44no—
43C47⋊#95—24no—
44C47⋊#95—24no—
45C25×C47C2311yescharacteristic
46C22×Q8×C47—22no—
47C22×Q8×C47—21no—
48C22×Q8×C47—21no—
49C22×Q8×C47—44no—
50C22×Q8×C47C2×C414yescomplemented
51C23×C4×C47—44no—
52C23×C4×C47C2×C411yescharacteristic
53C23×C4×C47C2×C412yes—
54C23×C4×C47C2×C412yes—
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47C2×C412yes—
57C23×C4×C47D411yescharacteristic
58C23×C4×C47D411yescharacteristic
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D411yescharacteristic
61C23×C4×C47D412yes—
62C23×C4×C47D412yes—
63C23×C4×C47C2312yes—
64C22⋊Q8×C47—24no—
65C22⋊Q8×C47—24no—
66C22⋊Q8×C47—24no—
67C22⋊Q8×C47—24no—
68C22⋊Q8×C47—24no—
69C22⋊Q8×C47—24no—
70C22⋊Q8×C47—24no—
71C22⋊Q8×C47—24no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47—22no—
77C2×C4⋊C4×C47—24no—
78C2×C4⋊C4×C47C2×C414yescomplemented
79C2×C4⋊C4×C47C2×C418yescomplemented
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47—22no—
82C2×C22⋊C4×C47—22no—
83C2×C22⋊C4×C47—22no—
84C2×C22⋊C4×C47C2×C414yes—
85C2×C22⋊C4×C47C2314yes—
86C24×C47—42no—
87C24×C47—42no—
88C24×C47C22×C411yescharacteristic
89C24×C47C22×C411yescharacteristic
90C24×C47C2×D411yescharacteristic
91C24×C47C2×D411yescharacteristic
92C24×C47C2×D411yescharacteristic
93C24×C47C2×D411yescharacteristic
94C24×C47C2×Q814yes—
95C24×C47#4114yes—
96C24×C47C2411yescharacteristic
97C2×Q8×C47—24no—
98C2×Q8×C47—24no—
99C2×Q8×C47—44no—
100C2×Q8×C47—48no—
101C2×Q8×C47—48no—
102C2×Q8×C47—44no—
103C22×C4×C47—24no—
104C22×C4×C47—24no—
105C22×C4×C47—24no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—44no—
111C22×C4×C47—44no—
112C22×C4×C47—44no—
113C22×C4×C47—44no—
114C22×C4×C47—42no—
115C22×C4×C47—42no—
116C22×C4×C47—42no—
117C22×C4×C47—44no—
118C22×C4×C47—44no—
119C22×C4×C47C22⋊C411yescharacteristic
120C22×C4×C47C22⋊C411yescharacteristic
121C22×C4×C47C22⋊C411yescharacteristic
122C22×C4×C47C22⋊C412yes—
123C22×C4×C47C22⋊C412yes—
124C22×C4×C47C22⋊C412yes—
125C22×C4×C47C22⋊C412yes—
126C22×C4×C47C22⋊C412yes—
127C22×C4×C47C22⋊C412yes—
128C22×C4×C47C22⋊C411yescharacteristic
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C414yes—
131C22×C4×C47C22×C414yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×Dic47—28no—
138C22×Dic47—24no—
139C22×Dic47—24no—
140C22×Dic47—44no—
141C22×Dic47—44no—
142C22×Dic47—44no—
143C22×Dic47—44no—
144C22×Dic47—44no—
145C4⋊C4×C47—24no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—28no—
148C4⋊C4×C47—28no—
149C22⋊C4×C47—24no—
150C22⋊C4×C47—24no—
151C22⋊C4×C47—24no—
152C22⋊C4×C47—24no—
153C22⋊Dic47—28no—
154C22⋊Dic47—24no—
155C22⋊Dic47—28no—
156C22⋊Dic47—44no—
157C22⋊Dic47C2×Q818yescomplemented
158C4⋊Dic47—48no—
159C4×Dic47—48no—
160C23×C47—22no—
161C23×C47—22no—
162C23×C47—22no—
163C23×C47—22no—
164C23×C47—22no—
165C23×C47—22no—
166C23×C47—41no—
167C23×C47—41no—
168C23×C47—41no—
169C23×C47—41no—
170C23×C47—41no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—48no—
174C23×C47—48no—
175C23×C47—41no—
176C23×C47C2×C22⋊C411yescharacteristic
177C23×C47C2×C22⋊C411yescharacteristic
178C23×C47C2×C22⋊C411yescharacteristic
179C23×C47C2×C22⋊C411yescharacteristic
180C23×C47C4×Q814yes—
181C23×C47C4×Q814yes—
182C23×C47C22⋊Q814yes—
183C23×C47C22⋊Q814yes—
184C23×C47#12412yes—
185C23×C47#12412yes—
186C23×C47C4⋊Q812yes—
187C23×C47C4⋊Q812yes—
188C23×C47C23×C411yescharacteristic
189C23×C47C22×D411yescharacteristic
190C23×C47C22×D411yescharacteristic
191C23×C47C22×Q812yes—
192C23×C47C2×#4112yes—
193Q8×C47—48no—
194Q8×C47—48no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—44no—
213C2×C4×C47—44no—
214C2×C4×C47—44no—
215C2×C4×C47—44no—
216C2×C4×C47—44no—
217C2×C4×C47—44no—
218C2×C4×C47#9914yes—
219C2×C4×C47C2×C22⋊C412yes—
220C2×C4×C47C2×C22⋊C412yes—
221C2×C4×C47C2×C22⋊C412yes—
222C2×C4×C47C2×C22⋊C412yes—
223C2×Dic47—28no—
224C2×Dic47—44no—
225C2×Dic47—44no—
226C2×Dic47—416no—
227C2×Dic47—44no—
228C2×Dic47—44no—
229C2×Dic47—44no—
230C2×Dic47—88no—
231C2×#1996—471nomaximal, Sylow
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—21no—
235C22×C47—21no—
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—21no—
239C22×C47—21no—
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—21no—
244C22×C47—28no—
245C22×C47—24no—
246C22×C47—28no—
247C22×C47—24no—
248C22×C47—44no—
249C22×C47—44no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47#39112yes—
255C22×C47#39112yes—
256C22×C47#39112yes—
257C22×C47#39112yes—
258C22×C47C2×#9912yes—
259C22×C47#41012yes—
260C22×C47C22×C22⋊C411yescharacteristic
261C22×C47C2×C4×Q812yes—
262C22×C47C2×C22⋊Q812yes—
263C22×C47C2×#12411yescharacteristic
264C22×C47C2×C4⋊Q811yescharacteristic
265C4×C47—22no—
266C4×C47—22no—
267C4×C47—22no—
268C4×C47—44no—
269C4×C47—44no—
270C4×C47C2×#9914yes—
271Dic47—48no—
272Dic47—84no—
273C22×C22⋊Q8D4711yescharacteristic
274C2×#380—471no—
275C2×C4×C22⋊C4—471no—
276#1996—478no—
277C2×#342—474no—
278C2×C47—22no—
279C2×C47—21no—
280C2×C47—22no—
281C2×C47—21no—
282C2×C47—22no—
283C2×C47—22no—
284C2×C47—22no—
285C2×C47#199614yescomplemented
286C2×C47C2×#39111yescharacteristic
287C2×C47C22×#9911yescharacteristic
288C2×C47C2×#41011yescharacteristic
289C23×Q8Dic4711yescharacteristic, complemented
290C24×C4C2×D4711yescharacteristic
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8—21no—
293C2×C22⋊Q8—21no—
294C2×C22⋊Q8—21no—
295C2×C22⋊Q8—22no—
296C2×C22⋊Q8—22no—
297C2×C22⋊Q8Dic4714yescomplemented
298C2×C22⋊Q8C2×D4714yes—
299C22×C4⋊C4—941no—
300C22×C4⋊C4Dic4711yescharacteristic, complemented
301C22×C4⋊C4Dic4712yescomplemented
302C22×C22⋊C4—472no—
303C22×C22⋊C4—472no—
304C22×C22⋊C4C2×D4712yes—
305C22×C42—941no—
306#380—474no—
307#380—942no—
308C4×C22⋊C4—474no—
309C4×C22⋊C4—942no—
310C2×#95—941no—
311C2×#95—941no—
312#342—4716no—
313C47C2×#199611yescharacteristic, complemented, Sylow
314C25C22×D4711yescharacteristic
315C22×Q8—22no—
316C22×Q8—21no—
317C22×Q8—21no—
318C22×Q8—44no—
319C22×Q8C2×Dic4714yescomplemented
320C23×C4—44no—
321C23×C4—942no—
322C23×C4—942no—
323C23×C4C2×Dic4711yescharacteristic
324C23×C4C2×Dic4712yes—
325C23×C4C2×Dic4712yes—
326C23×C4C2×Dic4711yescharacteristic
327C23×C4C2×Dic4712yes—
328C23×C4C22⋊D4711yescharacteristic
329C23×C4C22⋊D4711yescharacteristic
330C23×C4C22⋊D4711yescharacteristic
331C23×C4C22⋊D4711yescharacteristic
332C23×C4C22⋊D4712yes—
333C23×C4C22⋊D4712yes—
334C23×C4C22×D4712yes—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340C22⋊Q8—24no—
341C22⋊Q8—24no—
342C22⋊Q8—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—944no—
350C2×C4⋊C4—1884no—
351C2×C4⋊C4C2×Dic4714yescomplemented
352C2×C4⋊C4C2×Dic4718yescomplemented
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—478no—
358C2×C22⋊C4—474no—
359C2×C22⋊C4—474no—
360C2×C22⋊C4—944no—
361C2×C22⋊C4—942no—
362C2×C22⋊C4—948no—
363C2×C22⋊C4—942no—
364C2×C22⋊C4C2×Dic4714yes—
365C2×C22⋊C4C22×D4714yes—
366C2×C42—944no—
367C2×C42—1884no—
368#95—944no—
369#95—944no—
370C24—42no—
371C24—42no—
372C24Q8×D4714yes—
373C24#1062535414yes—
374C24C22×Dic4711yescharacteristic
375C24C22×Dic4711yescharacteristic
376C24C2×C22⋊D4711yescharacteristic
377C24C2×C22⋊D4711yescharacteristic
378C24C2×C22⋊D4711yescharacteristic
379C24C2×C22⋊D4711yescharacteristic
380C24C23×D4711yescharacteristic
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—44no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—948no—
404C22×C4—944no—
405C22×C4—944no—
406C22×C4—1884no—
407C22×C4—1884no—
408C22×C4—1884no—
409C22×C4—1884no—
410C22×C4—1884no—
411C22×C4C22⋊Dic4712yes—
412C22×C4C22⋊Dic4712yes—
413C22×C4C22⋊Dic4712yes—
414C22×C4C22⋊Dic4711yescharacteristic
415C22×C4C22⋊Dic4711yescharacteristic
416C22×C4C22⋊Dic4711yescharacteristic
417C22×C4C22⋊Dic4712yes—
418C22×C4C22⋊Dic4712yes—
419C22×C4C22⋊Dic4712yes—
420C22×C4C22⋊Dic4711yescharacteristic
421C22×C4C22×Dic4712yes—
422C22×C4C22×Dic4714yes—
423C22×C4C22×Dic4714yes—
424C22×C4C22×Dic4712yes—
425C22×C4C2×C22⋊D4712yes—
426C22×C4C2×C22⋊D4712yes—
427C22×C4C2×C22⋊D4712yes—
428C22×C4C2×C22⋊D4712yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—1888no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—478no—
439C22⋊C4—948no—
440C22⋊C4—944no—
441C22⋊C4—948no—
442C22⋊C4—1884no—
443C42—1888no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—48no—
455C23—48no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23C2×C22⋊Dic4711yescharacteristic
461C23C2×C22⋊Dic4711yescharacteristic
462C23C2×C22⋊Dic4711yescharacteristic
463C23C2×C22⋊Dic4711yescharacteristic
464C23Q8×Dic4714yes—
465C23Q8×Dic4714yes—
466C23#1309400014yes—
467C23#1309400014yes—
468C23#1309401312yes—
469C23#1309401312yes—
470C23#1309402212yes—
471C23#1309402212yes—
472C23C23×Dic4711yescharacteristic
473C23C22×C22⋊D4711yescharacteristic
474C23C22×C22⋊D4711yescharacteristic
475C23C2×Q8×D4712yes—
476C23C2×#1062535412yes—
477Q8—48no—
478Q8—48no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—948no—
503C2×C4—3768no—
504C2×C4—1884no—
505C2×C4—1884no—
506C2×C4—1884no—
507C2×C4—1884no—
508C2×C4—18816no—
509C2×C4—1884no—
510C2×C4#1309393914yes—
511C2×C4C2×C22⋊Dic4712yes—
512C2×C4C2×C22⋊Dic4712yes—
513C2×C4C2×C22⋊Dic4712yes—
514C2×C4C2×C22⋊Dic4712yes—
515C22—24no—
516C22—28no—
517C22—28no—
518C22—24no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—44no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22#1568903012yes—
538C22#1568903012yes—
539C22#1568903012yes—
540C22#1568903012yes—
541C22C2×#1309393912yes—
542C22#1568909712yes—
543C22C22×C22⋊Dic4711yescharacteristic
544C22C2×Q8×Dic4712yes—
545C22C2×#1309400012yes—
546C22C2×#1309401311yescharacteristic
547C22C2×#1309402211yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—44no—
553C4—3764no—
554C4—1888no—
555C4C2×#1309393914yes—
556C2—21no—
557C2—22no—
558C2—22no—
559C2—22no—
560C2—22no—
561C2—22no—
562C2—21no—
563C2#2024437414yesdirect product factor
564C2C2×#1568903011yescharacteristic
565C2C22×#1309393911yescharacteristic
566C2C2×#1568909711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC22×C22⋊Q8×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C23
cosocleC24×C47C24
hypercenterC22×C22⋊Q8D47
hypocenterC47C2×#1996
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20244374C2direct (N × H)
17. C23×Q8×C47C4semidirect
25. C2×C22⋊Q8×C47C4semidirect
27. C22×C4⋊C4×C47C4semidirect
28. C22×C4⋊C4×C47C4semidirect
38. C2×C22⋊Dic47Q8semidirect
50. C22×Q8×C47C2×C4semidirect
78. C2×C4⋊C4×C47C2×C4semidirect
79. C2×C4⋊C4×C47C2×C4semidirect
157. C22⋊Dic47C2×Q8semidirect
285. C2×C47#1996semidirect
289. C23×Q8Dic47semidirect
297. C2×C22⋊Q8Dic47semidirect
300. C22×C4⋊C4Dic47semidirect
301. C22×C4⋊C4Dic47semidirect
313. C47C2×#1996semidirect
319. C22×Q8C2×Dic47semidirect
351. C2×C4⋊C4C2×Dic47semidirect
352. C2×C4⋊C4C2×Dic47semidirect
563. C2#20244374direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156889832
C2×C4×C22⋊Dic472
#202443742
C2×#156888902
C22×C22⋊Q8×C472
C2×#199647