Finite Group Databasebeta

← C2×#19137217 #25255216

Subgroups — C2×#19137217 #25255216

Order 11008.

3502 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063283C211yesmaximal, characteristic
2C22×#13063282C211yesmaximal, characteristic
3C22×C4⋊C4×D43C211yesmaximal, characteristic
4#19137217C2116yesmaximal, direct product factor
5C2×#14582798C211yesmaximal, characteristic, complemented
6C2×#14582797C211yesmaximal, characteristic, complemented
7C2×#14582795C211yesmaximal, characteristic, complemented
8C2×#14582773C212yesmaximal
9C2×#14582770C212yesmaximal
10C2×#14582768C212yesmaximal, complemented
11C2×#14582763C212yesmaximal, complemented
12C2×C43×#389C211yesmaximal, characteristic, complemented
13C23×C4×D43C2212yes—
14C23×C4×D43C2211yescharacteristic
15C2×#13063283C4116yescomplemented
16C2×#13063283C2218yes—
17C2×#13063282—28no—
18C2×#13063282C2218yes—
19C22×C4⋊Dic43C2211yescharacteristic
20C22×C4⋊Dic43C2212yes—
21C22×C4⋊Dic43C2211yescharacteristic
22C22×Dic43⋊C4C2211yescharacteristic
23C22×Dic43⋊C4C2212yes—
24C22×Dic43⋊C4C2211yescharacteristic
25C22×Dic43⋊C4C2212yes—
26C22×Dic43⋊C4C2212yes—
27C2×C4⋊C4×D43—28no—
28C2×C4⋊C4×D43C2218yes—
29C22×(C2×D43)⋊C4C2211yescharacteristic
30C22×(C2×D43)⋊C4C2211yescharacteristic
31C22×(C2×D43)⋊C4C2212yes—
32C22×C4×Dic43C2211yescharacteristic
33C22×C4×Dic43C2212yes—
34#14582798C2218yescomplemented
35#14582797C2218yescomplemented
36#14582795C2218yescomplemented
37#14582773C22116yes—
38#14582770C22116yes—
39#14582768C22116yescomplemented
40#14582763C22116yescomplemented
41C2×C43⋊#95C2212yes—
42C2×C43⋊#95C2212yes—
43C2×C43⋊#95C2212yes—
44C22×C4⋊C4×C43C2211yescharacteristic
45C22×C4⋊C4×C43C2212yes—
46C22×C4⋊C4×C43C2211yescharacteristic
47C22×C42×C43C2211yescharacteristic
48C43×#389C2218yescomplemented
49C2×#95×C43C2212yes—
50C24×D43C2311yescharacteristic
51C23×Dic43C2311yescharacteristic
52C23×Dic43C2311yescharacteristic
53C23×Dic43C2312yes—
54C23×Dic43C2312yes—
55C23×Dic43C2311yescharacteristic
56C22×C4×D43—28no—
57C22×C4×D43—44no—
58C22×C4×D43—44no—
59C22×C4×D43C2×C418yes—
60C22×C4×D43D418yes—
61C22×C4×D43Q818yes—
62C22×C4×D43C2318yes—
63C22×C4×D43C2314yes—
64#13063283C2×C4164yescomplemented
65#13063282—232no—
66C2×C4⋊Dic43—24no—
67C2×C4⋊Dic43—44no—
68C2×C4⋊Dic43C2×C418yescomplemented
69C2×C4⋊Dic43C2314yes—
70C2×C4⋊Dic43C2318yes—
71C2×C4⋊Dic43C2314yes—
72C2×Dic43⋊C4—28no—
73C2×Dic43⋊C4—24no—
74C2×Dic43⋊C4—24no—
75C2×Dic43⋊C4—44no—
76C2×Dic43⋊C4—44no—
77C2×Dic43⋊C4C2318yes—
78C2×Dic43⋊C4C2314yes—
79C2×Dic43⋊C4C2314yes—
80C2×Dic43⋊C4C2318yes—
81C2×Dic43⋊C4C2318yes—
82C4⋊C4×D43—232no—
83C2×(C2×D43)⋊C4—24no—
84C2×(C2×D43)⋊C4—24no—
85C2×(C2×D43)⋊C4C2×C4116yes—
86C2×(C2×D43)⋊C4C2314yes—
87C2×(C2×D43)⋊C4C2314yes—
88C2×(C2×D43)⋊C4C2318yes—
89C2×C4×Dic43—24no—
90C2×C4×Dic43C2×C4116yescomplemented
91C2×C4×Dic43C2314yes—
92C2×C4×Dic43C2318yes—
93C43⋊#95C2318yes—
94C43⋊#95C2318yes—
95C43⋊#95C2318yes—
96C23×C4×C43C2311yescharacteristic
97C23×C4×C43C2311yescharacteristic
98C23×C4×C43C2312yes—
99C23×C4×C43C2312yes—
100C23×C4×C43C2311yescharacteristic
101C2×C4⋊C4×C43—24no—
102C2×C4⋊C4×C43—44no—
103C2×C4⋊C4×C43C2×C418yescomplemented
104C2×C4⋊C4×C43C2314yes—
105C2×C4⋊C4×C43C2318yes—
106C2×C4⋊C4×C43C2314yes—
107C2×C42×C43—24no—
108C2×C42×C43C2314yes—
109#95×C43C2318yes—
110C24×C43C2411yescharacteristic
111C23×D43—22no—
112C23×D43—22no—
113C23×D43—22no—
114C23×D43—22no—
115C23×D43C22×C414yes—
116C23×D43C2×D414yes—
117C23×D43C2×Q814yes—
118C23×D43C2412yes—
119C22×C4×C43—22no—
120C22×C4×C43—22no—
121C22×C4×C43—24no—
122C22×C4×C43—24no—
123C22×C4×C43—22no—
124C22×C4×C43—24no—
125C22×C4×C43—22no—
126C22×C4×C43—24no—
127C22×C4×C43—24no—
128C22×C4×C43—22no—
129C22×C4×C43—22no—
130C22×C4×C43C22×C418yes—
131C22×C4×C43C22×C414yes—
132C22×C4×C43C2×D414yes—
133C22×C4×C43C2×Q814yes—
134C22×C4×C43C2412yes—
135C22×C4×C43C2414yes—
136C22×C4×C43C2412yes—
137C22×C4×C43C2414yes—
138C22×C4×C43C2412yes—
139C22×Dic43—22no—
140C22×Dic43—22no—
141C22×Dic43—24no—
142C22×Dic43—24no—
143C22×Dic43—22no—
144C22×Dic43—24no—
145C22×Dic43—22no—
146C22×Dic43—24no—
147C22×Dic43—24no—
148C22×Dic43—22no—
149C22×Dic43—22no—
150C22×Dic43C22×C418yes—
151C22×Dic43C22×C414yes—
152C22×Dic43C2×D414yes—
153C22×Dic43C2×Q814yes—
154C22×Dic43C2412yes—
155C22×Dic43C2414yes—
156C22×Dic43C2412yes—
157C22×Dic43C2414yes—
158C22×Dic43C2412yes—
159C2×C4×D43—216no—
160C2×C4×D43—48no—
161C2×C4×D43—48no—
162C2×C4×D43—432no—
163C2×C4×D43C4⋊C4164yescomplemented
164C2×C4×D43C22×C4116yes—
165C2×C4×D43C2×D4116yes—
166C2×C4×D43C2×Q8116yes—
167C4⋊C4×C43—28no—
168C4⋊C4×C43—48no—
169C4⋊C4×C43C22×C4116yescomplemented
170C42×C43—28no—
171(C2×D43)⋊C4—28no—
172(C2×D43)⋊C4—28no—
173(C2×D43)⋊C4C22×C4132yes—
174C4⋊Dic43—28no—
175C4⋊Dic43—48no—
176C4⋊Dic43C22×C4116yescomplemented
177Dic43⋊C4—216no—
178Dic43⋊C4—28no—
179Dic43⋊C4—28no—
180Dic43⋊C4—48no—
181Dic43⋊C4—48no—
182C4×Dic43—28no—
183C4×Dic43C22×C4132yescomplemented
184C23×C43C23×C412yes—
185C23×C43C22×D412yes—
186C23×C43C22×Q812yes—
187C23×C43H(5,𝔽2)12yes—
188C23×C43H(5,𝔽2)12yes—
189C23×C432−1+412yes—
190C23×C432−1+412yes—
191C23×C43C2511yescharacteristic
192C22×D43—28no—
193C22×D43—28no—
194C22×D43—28no—
195C22×D43—28no—
196C22×D43—28no—
197C22×D43—28no—
198C22×D43—22no—
199C22×D43—22no—
200C22×D43—22no—
201C22×D43—22no—
202C22×D43C2×C4⋊C4116yes—
203C22×D43C23×C414yes—
204C22×D43C22×D414yes—
205C22×D43C22×Q814yes—
206C2×C4×C43—216no—
207C2×C4×C43—24no—
208C2×C4×C43—24no—
209C2×C4×C43—22no—
210C2×C4×C43—22no—
211C2×C4×C43—24no—
212C2×C4×C43—22no—
213C2×C4×C43—24no—
214C2×C4×C43—22no—
215C2×C4×C43—24no—
216C2×C4×C43—22no—
217C2×C4×C43—22no—
218C2×C4×C43—48no—
219C2×C4×C43—44no—
220C2×C4×C43—44no—
221C2×C4×C43C2×C4⋊C4116yes—
222C2×C4×C43C23×C418yes—
223C2×C4×C43C23×C414yes—
224C2×C4×C43C22×D414yes—
225C2×C4×C43C22×Q814yes—
226C2×Dic43—216no—
227C2×Dic43—24no—
228C2×Dic43—24no—
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—24no—
232C2×Dic43—22no—
233C2×Dic43—24no—
234C2×Dic43—22no—
235C2×Dic43—24no—
236C2×Dic43—22no—
237C2×Dic43—22no—
238C2×Dic43—48no—
239C2×Dic43—44no—
240C2×Dic43—44no—
241C2×Dic43C2×C4⋊C4116yes—
242C2×Dic43C23×C418yes—
243C2×Dic43C23×C414yes—
244C2×Dic43C22×D414yes—
245C2×Dic43C22×Q814yes—
246C4×D43—432no—
247C4×D43C2×C4⋊C4164yescomplemented
248C2×#2317—431nomaximal, Sylow
249C22×C43C22×C4⋊C414yes—
250C22×C43#52014yes—
251C22×C43#52014yes—
252C22×C43#53514yes—
253C22×C43#53614yes—
254C22×C43#54414yes—
255C22×C43#54414yes—
256C22×C43C24×C411yescharacteristic
257C22×C43C23×D411yescharacteristic
258C22×C43C23×Q811yescharacteristic
259C22×C43C2×H(5,𝔽2)11yescharacteristic
260C22×C43C2×H(5,𝔽2)11yescharacteristic
261C22×C43C2×2−1+411yescharacteristic
262C22×C43C2×2−1+411yescharacteristic
263C2×D43—232no—
264C2×D43—24no—
265C2×D43—24no—
266C2×D43—24no—
267C2×D43—24no—
268C2×D43—24no—
269C2×D43—24no—
270C2×D43C22×C4⋊C418yes—
271C4×C43—28no—
272C4×C43—44no—
273C4×C43—42no—
274C4×C43—42no—
275C4×C43C22×C4⋊C418yes—
276Dic43—28no—
277Dic43—44no—
278Dic43—42no—
279Dic43—42no—
280Dic43C22×C4⋊C418yes—
281C22×#117—431no—
282C22×#117—431no—
283C23×C4⋊C4—431no—
284#2317—4316no—
285C2×#389—431no—
286C2×#389—431no—
287C2×#389—431no—
288C2×#389D4311yescharacteristic, complemented
289C2×#385—432no—
290C2×#385—432no—
291C2×#384—432no—
292C2×#382—432no—
293C2×C43#231718yescomplemented
294C2×C43C23×C4⋊C411yescharacteristic
295C2×C43C2×#52011yescharacteristic
296C2×C43C2×#52011yescharacteristic
297C2×C43C2×#53511yescharacteristic
298C2×C43C2×#53611yescharacteristic
299C2×C43C2×#54411yescharacteristic
300C2×C43C2×#54411yescharacteristic
301D43—28no—
302C24×C4—431no—
303C24×C4—432no—
304C2×#117—438no—
305C2×#117—4316no—
306C2×#117—438no—
307C2×#117—868no—
308C22×C4⋊C4—431no—
309C22×C4⋊C4—432no—
310C22×C4⋊C4—431no—
311C22×C4⋊C4—431no—
312C22×C4⋊C4—431no—
313C22×C4⋊C4—432no—
314C22×C4⋊C4—432no—
315C22×C4⋊C4—432no—
316C22×C4⋊C4—438no—
317C22×C4⋊C4—868no—
318C22×C4⋊C4C2×D4311yescharacteristic
319C22×C4⋊C4C2×D4312yes—
320C22×C4⋊C4C2×D4311yescharacteristic
321C22×C22⋊C4—432no—
322C22×C22⋊C4—431no—
323C22×C22⋊C4—431no—
324C22×C42—432no—
325C22×C42—431no—
326C22×C42C2×D4311yescharacteristic
327#389—438no—
328#389—438no—
329#389—438no—
330#389C2×D4318yescomplemented
331#385—4316no—
332#385—4316no—
333#384—4316no—
334#382—4316no—
335C2×#95—432no—
336C2×#95—432no—
337C2×#95—432no—
338C2×#95C2×D4312yes—
339C43C2×#231711yescharacteristic, complemented, Sylow
340C25—431no—
341C23×C4—431no—
342C23×C4—438no—
343C23×C4—438no—
344C23×C4—438no—
345C23×C4—434no—
346C23×C4—432no—
347C23×C4—431no—
348C23×C4—431no—
349C23×C4—432no—
350C23×C4—438no—
351C23×C4—868no—
352C23×C4—1724no—
353C23×C4—1724no—
354C23×C4C22×D4311yescharacteristic
355C23×C4C22×D4311yescharacteristic
356C23×C4C22×D4312yes—
357C23×C4C22×D4312yes—
358C23×C4C22×D4311yescharacteristic
359#117—4364no—
360#117—8632no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—438no—
364C2×C4⋊C4—434no—
365C2×C4⋊C4—438no—
366C2×C4⋊C4—434no—
367C2×C4⋊C4—434no—
368C2×C4⋊C4—434no—
369C2×C4⋊C4—438no—
370C2×C4⋊C4—438no—
371C2×C4⋊C4—438no—
372C2×C4⋊C4—8632no—
373C2×C4⋊C4—868no—
374C2×C4⋊C4—864no—
375C2×C4⋊C4—864no—
376C2×C4⋊C4—864no—
377C2×C4⋊C4—1724no—
378C2×C4⋊C4—1724no—
379C2×C4⋊C4—1724no—
380C2×C4⋊C4C4×D4318yescomplemented
381C2×C4⋊C4C22×D4314yes—
382C2×C4⋊C4C22×D4318yes—
383C2×C4⋊C4C22×D4314yes—
384C2×C22⋊C4—4316no—
385C2×C22⋊C4—438no—
386C2×C22⋊C4—434no—
387C2×C22⋊C4—434no—
388C2×C22⋊C4—864no—
389C2×C22⋊C4—864no—
390C2×C42—24no—
391C2×C42—4316no—
392C2×C42—438no—
393C2×C42—434no—
394C2×C42—864no—
395C2×C42C22×D4314yes—
396#95—438no—
397#95—438no—
398#95—438no—
399#95C22×D4318yes—
400C24—434no—
401C24—434no—
402C24—434no—
403C24—432no—
404C24—862no—
405C24—862no—
406C24—862no—
407C24—862no—
408C24C23×D4311yescharacteristic
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—434no—
421C22×C4—434no—
422C22×C4—434no—
423C22×C4—432no—
424C22×C4—4364no—
425C22×C4—4316no—
426C22×C4—4316no—
427C22×C4—4316no—
428C22×C4—438no—
429C22×C4—434no—
430C22×C4—432no—
431C22×C4—434no—
432C22×C4—432no—
433C22×C4—8616no—
434C22×C4—864no—
435C22×C4—864no—
436C22×C4—862no—
437C22×C4—862no—
438C22×C4—864no—
439C22×C4—862no—
440C22×C4—864no—
441C22×C4—862no—
442C22×C4—864no—
443C22×C4—862no—
444C22×C4—862no—
445C22×C4—1728no—
446C22×C4—1728no—
447C22×C4—17232no—
448C22×C4C2×C4×D4318yes—
449C22×C4C2×C4×D4314yes—
450C22×C4D4×D4314yes—
451C22×C4Q8×D4314yes—
452C22×C4C23×D4312yes—
453C22×C4C23×D4314yes—
454C22×C4C23×D4312yes—
455C22×C4C23×D4314yes—
456C22×C4C23×D4312yes—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—4316no—
460C4⋊C4—8616no—
461C4⋊C4—868no—
462C4⋊C4—868no—
463C4⋊C4—868no—
464C4⋊C4—1728no—
465C4⋊C4—1728no—
466C4⋊C4—1728no—
467C4⋊C4C2×C4×D43116yescomplemented
468C22⋊C4—4332no—
469C22⋊C4—868no—
470C22⋊C4—868no—
471C42—28no—
472C42—4332no—
473C42—868no—
474C23—4316no—
475C23—434no—
476C23—434no—
477C23—434no—
478C23—868no—
479C23—868no—
480C23—868no—
481C23—868no—
482C23—862no—
483C23—862no—
484C23—862no—
485C23—862no—
486C23—868no—
487C23—868no—
488C23C22×C4×D4312yes—
489C23C2×D4×D4312yes—
490C23C2×Q8×D4312yes—
491C23#1306337012yes—
492C23#1306337112yes—
493C23#1306337212yes—
494C23#1306337312yes—
495C23C24×D4311yescharacteristic
496C2×C4—216no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—4316no—
512C2×C4—434no—
513C2×C4—434no—
514C2×C4—434no—
515C2×C4—4364no—
516C2×C4—438no—
517C2×C4—8616no—
518C2×C4—864no—
519C2×C4—864no—
520C2×C4—862no—
521C2×C4—862no—
522C2×C4—864no—
523C2×C4—862no—
524C2×C4—864no—
525C2×C4—862no—
526C2×C4—864no—
527C2×C4—862no—
528C2×C4—862no—
529C2×C4—17232no—
530C2×C4—1728no—
531C2×C4—1724no—
532C2×C4—1724no—
533C2×C4C4⋊C4×D43116yes—
534C2×C4C22×C4×D4318yes—
535C2×C4C22×C4×D4314yes—
536C2×C4C2×D4×D4314yes—
537C2×C4C2×Q8×D4314yes—
538C22—438no—
539C22—864no—
540C22—864no—
541C22—864no—
542C22—864no—
543C22—8632no—
544C22—864no—
545C22—864no—
546C22C2×C4⋊C4×D4314yes—
547C22#1458337814yes—
548C22#1458338114yes—
549C22#1458346514yes—
550C22#1458347214yes—
551C22#1458352614yes—
552C22#1458352914yes—
553C22C23×C4×D4311yescharacteristic
554C22C22×D4×D4311yescharacteristic
555C22C22×Q8×D4311yescharacteristic
556C22C2×#1306337011yescharacteristic
557C22C2×#1306337111yescharacteristic
558C22C2×#1306337211yescharacteristic
559C22C2×#1306337311yescharacteristic
560C4—28no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—438no—
565C4—868no—
566C4—1724no—
567C4—1722no—
568C4—1722no—
569C4C2×C4⋊C4×D4318yes—
570C2—868no—
571C2#1913721718yesdirect product factor
572C2C22×C4⋊C4×D4311yescharacteristic
573C2C2×#1458337811yescharacteristic
574C2C2×#1458338111yescharacteristic
575C2C2×#1458346511yescharacteristic
576C2C2×#1458347211yescharacteristic
577C2C2×#1458352611yescharacteristic
578C2C2×#1458352911yescharacteristic
subgroupG/H
centerC24C23×D43
commutatorC22×C43C24×C4
FrattiniC23C24×D43
FittingC2×C43×#389C2
socleC24×C43C24
prime radicalC24×D43C23
cosocleC23×C43C25
hypercenterC2×#389D43
hypocenterC43C2×#2317
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. #19137217C2direct (N × H)
5. C2×#14582798C2semidirect
6. C2×#14582797C2semidirect
7. C2×#14582795C2semidirect
10. C2×#14582768C2semidirect
11. C2×#14582763C2semidirect
12. C2×C43×#389C2semidirect
15. C2×#13063283C4semidirect
34. #14582798C22semidirect
35. #14582797C22semidirect
36. #14582795C22semidirect
39. #14582768C22semidirect
40. #14582763C22semidirect
48. C43×#389C22semidirect
64. #13063283C2×C4semidirect
68. C2×C4⋊Dic43C2×C4semidirect
90. C2×C4×Dic43C2×C4semidirect
103. C2×C4⋊C4×C43C2×C4semidirect
163. C2×C4×D43C4⋊C4semidirect
169. C4⋊C4×C43C22×C4semidirect
176. C4⋊Dic43C22×C4semidirect
183. C4×Dic43C22×C4semidirect
247. C4×D43C2×C4⋊C4semidirect
288. C2×#389D43semidirect
293. C2×C43#2317semidirect
330. #389C2×D43semidirect
339. C43C2×#2317semidirect
380. C2×C4⋊C4C4×D43semidirect
467. C4⋊C4C2×C4×D43semidirect
571. C2#19137217direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130632832
C22×#130632822
C22×C4⋊C4×D432
#191372172
C2×#145827982
C2×#145827972
C2×#145827952
C2×#145827732
C2×#145827702
C2×#145827682
C2×#145827632
C2×C43×#3892
C2×#231743