Finite Group Databasebeta

← C2×#19148302 #26004537

Subgroups — C2×#19148302 #26004537

Order 11008.

7910 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19148302C2132yesmaximal, direct product factor
2C2×#14583607C214yesmaximal, complemented
3C2×#14583596C214yesmaximal, complemented
4C2×#14583583C214yesmaximal, complemented
5C2×#14583549C214yesmaximal, complemented
6C2×#14583530C211yesmaximal, characteristic, complemented
7C2×#14583494C214yesmaximal, complemented
8C2×#14583409C214yesmaximal, complemented
9C2×#14583406C212yesmaximal, complemented
10C2×#14583382C212yesmaximal, complemented
11C22×C4⋊C4×D43C211yesmaximal, characteristic, complemented
12C2×C43×#537C211yesmaximal, characteristic, complemented
13C22×#10621900C2214yes—
14C23×C4×D43C2211yescharacteristic
15C23×C4×D43C2212yes—
16#14583607C22164yescomplemented
17#14583596C22164yescomplemented
18#14583583C22164yescomplemented
19#14583549C22164yescomplemented
20#14583530C22116yescomplemented
21#14583494C22164yescomplemented
22C2×#13063324C2214yescomplemented
23C2×#13063318C2218yescomplemented
24C2×#13063316C2214yescomplemented
25C2×#13063315C2218yescomplemented
26#14583409C22164yescomplemented
27#14583406C22132yescomplemented
28C2×#13063307C2218yes—
29C2×#13063306C2214yes—
30C2×#13063306C2218yescomplemented
31C2×#13063305C2214yes—
32C2×#13063304C2214yes—
33C2×#13063302C2218yescomplemented
34C2×#13063301C2212yes—
35C2×#13063301C2212yes—
36C2×#13063296C2212yes—
37C2×#13063296C2212yes—
38C2×#13063296C2214yes—
39C2×#13063293C2214yes—
40#14583382C22132yescomplemented
41C2×Q8×Dic43C2214yescomplemented
42C2×D4×Dic43C2218yes—
43C2×D4×Dic43C2214yescomplemented
44C2×C4×C22⋊D43C2214yes—
45C2×C4×C22⋊D43C2214yes—
46C2×(C22⋊D43)⋊C4C2218yes—
47C2×#13063284C2212yes—
48C2×#13063284C2214yescomplemented
49C2×#13063283C2214yes—
50C22×C4⋊Dic43C2211yescharacteristic
51C22×C4⋊Dic43C2214yescomplemented
52C22×Dic43⋊C4C2212yes—
53C2×C4⋊C4×D43C2214yes—
54C2×C4⋊C4×D43C2214yes—
55C2×C4⋊C4×D43C22116yescomplemented
56C22×#41×C43C2212yes—
57C43×#537C22116yescomplemented
58C2×C22⋊Q8×C43C2214yescomplemented
59C2×#121×C43C2214yes—
60C2×#121×C43C2214yescomplemented
61C22×C4⋊C4×C43C2211yescharacteristic
62C24×D43C2311yescharacteristic
63C2×#10621900—416no—
64C2×#10621900C23132yes—
65C22×Dic86C2314yes—
66C22×C22⋊D43C2314yes—
67C22×C22⋊D43C2314yes—
68C23×Dic43C2311yescharacteristic
69C23×Dic43C2312yes—
70C23×Dic43C2318yes—
71C22×C4×D43—28no—
72C22×C4×D43—48no—
73C22×C4×D43C2314yes—
74C22×C4×D43C2312yes—
75C22×C4×D43C2312yes—
76C22×C4×D43C2318yes—
77C22×C4×D43C2314yes—
78C22×C4×D43C23116yes—
79#13063324C23132yescomplemented
80#13063318C23164yescomplemented
81#13063316C23132yescomplemented
82#13063315C23164yescomplemented
83#13063307C23164yes—
84#13063306C23132yes—
85#13063306C23164yescomplemented
86#13063305C23132yes—
87#13063304C23132yes—
88#13063302C23164yescomplemented
89#13063301C23116yes—
90#13063301C23116yes—
91#13063296C23116yes—
92#13063296C23132yes—
93#13063296C23116yes—
94#13063293C23132yes—
95Q8×Dic43C23132yescomplemented
96D4×Dic43C23164yes—
97D4×Dic43C23132yescomplemented
98C4×C22⋊D43C23132yes—
99C4×C22⋊D43C23132yes—
100(C22⋊D43)⋊C4C23164yes—
101#13063284C23116yes—
102#13063284C23132yescomplemented
103#13063283C23132yes—
104C2×C4⋊Dic43C2314yes—
105C2×C4⋊Dic43C2318yes—
106C2×C4⋊Dic43C2312yes—
107C2×C4⋊Dic43C2318yes—
108C2×C4⋊Dic43C2314yes—
109C2×C4⋊Dic43C2312yes—
110C2×C4⋊Dic43C23132yescomplemented
111C2×Dic43⋊C4C2318yes—
112C2×Dic43⋊C4C23116yes—
113C2×Dic43⋊C4C2318yes—
114C2×Dic43⋊C4C2314yes—
115C2×Dic43⋊C4C2314yes—
116C4⋊C4×D43C23132yes—
117C4⋊C4×D43C23132yes—
118C2×C22⋊Dic43C2314yes—
119C2×C22⋊Dic43C2318yes—
120C2×C22⋊Dic43C2314yes—
121C2×C22⋊Dic43C2314yes—
122C2×(C2×D43)⋊C4C2314yes—
123C2×(C2×D43)⋊C4C2314yes—
124C2×C4×Dic43C2318yes—
125C2×C4×Dic43C2314yes—
126C2×#41×C43—216no—
127C2×#41×C43C23116yes—
128C22×Q8×C43C2312yes—
129C22×D4×C43C2314yes—
130C22×D4×C43C2312yes—
131C22×D4×C43C2314yes—
132C23×C4×C43C2314yes—
133C23×C4×C43C2312yes—
134C23×C4×C43C2311yescharacteristic
135C22⋊Q8×C43C23132yescomplemented
136#121×C43C23132yes—
137#121×C43C23132yescomplemented
138C2×C4⋊C4×C43C2314yes—
139C2×C4⋊C4×C43C2318yes—
140C2×C4⋊C4×C43C2314yes—
141C2×C22⋊C4×C43C2318yes—
142C24×C43C2414yes—
143C24×C43C2411yescharacteristic
144C23×D43—24no—
145C23×D43—24no—
146C23×D43—24no—
147C23×D43C2414yes—
148C23×D43C2411yescharacteristic
149C23×D43C2411yescharacteristic
150#41×C43—264no—
151C2×Q8×C43—28no—
152C2×Q8×C43C2418yes—
153C2×D4×C43—28no—
154C2×D4×C43—216no—
155C2×D4×C43—48no—
156C2×D4×C43C24116yes—
157C2×D4×C43C2418yes—
158C2×D4×C43C24116yes—
159C22×C4×C43—216no—
160C22×C4×C43—44no—
161C22×C4×C43#4118yescomplemented
162C22×C4×C43C2414yes—
163C22×C4×C43C2414yes—
164C22×C4×C43C24116yes—
165C22×C4×C43C2418yes—
166C22×C4×C43C2412yes—
167C22×C4×C43C2414yes—
168C2×C22⋊D43—216no—
169C2×C22⋊D43—216no—
170C2×C22⋊D43C24116yes—
171C2×C22⋊D43C24116yes—
172C22×Dic43—24no—
173C22×Dic43—44no—
174C22×Dic43—416no—
175C22×Dic43C2414yes—
176C22×Dic43C2414yes—
177C22×Dic43C2414yes—
178C22×Dic43C2412yes—
179C22×Dic43C2414yes—
180C22×Dic43C2418yes—
181C22×Dic43C24132yes—
182#10621900—464no—
183C2×C4×D43—28no—
184C2×C4×D43—28no—
185C2×C4×D43—232no—
186C2×C4×D43—216no—
187C2×C4×D43—432no—
188C2×C4×D43—48no—
189C2×C4×D43C24116yes—
190C2×C4×D43C2418yes—
191C2×C4×D43C2418yes—
192C2×C4×D43C24116yes—
193C2×Dic86—48no—
194C2×Dic86C24116yes—
195C4⋊C4×C43C24116yes—
196C4⋊C4×C43C24116yes—
197C22⋊C4×C43C24132yes—
198C22⋊Dic43C24116yes—
199C22⋊Dic43C24132yes—
200C22⋊Dic43C24116yes—
201C22⋊Dic43C24116yes—
202(C2×D43)⋊C4C24116yes—
203(C2×D43)⋊C4C24116yes—
204C4⋊Dic43C24116yes—
205C4⋊Dic43C2418yes—
206C4⋊Dic43C24132yes—
207C4⋊Dic43C24116yes—
208C4⋊Dic43C2418yes—
209Dic43⋊C4C24132yes—
210Dic43⋊C4C24132yes—
211Dic43⋊C4C24116yes—
212Dic43⋊C4C24116yes—
213C4×Dic43C24132yes—
214C4×Dic43C24116yes—
215C23×C43—22no—
216C23×C43—22no—
217C23×C43—28no—
218C23×C43—28no—
219C23×C43—28no—
220C23×C43C2×#4114yes—
221C23×C43C2518yes—
222C23×C43C2512yes—
223C23×C43C2511yescharacteristic
224C22×D43—232no—
225C22×D43—28no—
226C22×D43—22no—
227C22×D43—22no—
228C22×D43—22no—
229C22×D43—22no—
230C22×D43—28no—
231C22×D43—28no—
232C22×D43H(5,𝔽2)14yes—
233C22×D432−1+414yes—
234C22×D43C2512yes—
235C22×D43C2512yes—
236Q8×C43—216no—
237D4×C43—232no—
238D4×C43—216no—
239D4×C43—416no—
240C2×C4×C43—28no—
241C2×C4×C43—28no—
242C2×C4×C43—232no—
243C2×C4×C43—48no—
244C2×C4×C43C2×#4118yes—
245C2×C4×C43C2×#41116yescomplemented
246C2×C4×C43C2518yes—
247C2×C4×C43C2518yes—
248C2×C4×C43C2514yes—
249C22⋊D43—232no—
250C22⋊D43—232no—
251C2×Dic43—28no—
252C2×Dic43—24no—
253C2×Dic43—28no—
254C2×Dic43—28no—
255C2×Dic43—28no—
256C2×Dic43—48no—
257C2×Dic43—432no—
258C2×Dic43C2518yes—
259C2×Dic43C2518yes—
260C2×Dic43C2518yes—
261C2×Dic43C2514yes—
262C4×D43—232no—
263C4×D43—216no—
264C4×D43—216no—
265C4×D43—416no—
266Dic86—416no—
267C2×#3492—431nomaximal, Sylow
268C22×C43—22no—
269C22×C43—22no—
270C22×C43—28no—
271C22×C43—28no—
272C22×C43—28no—
273C22×C43—28no—
274C22×C43—416no—
275C22×C43C22×#4114yes—
276C22×C43C22×#4112yes—
277C22×C43C2×H(5,𝔽2)12yes—
278C22×C43C2×2−1+412yes—
279C22×C43C2611yescharacteristic
280C2×D43—232no—
281C2×D43—22no—
282C2×D43—22no—
283C2×D43—22no—
284C2×D43—22no—
285C2×D43—28no—
286C2×D43—28no—
287C2×D43C2×H(5,𝔽2)14yes—
288C2×D43C2×2−1+414yes—
289C4×C43—28no—
290C4×C43—28no—
291C4×C43C22×#4118yes—
292Dic43—24no—
293Dic43—28no—
294Dic43—28no—
295Dic43—28no—
296#3492—4332no—
297C2×#554—434no—
298C2×#553—434no—
299C2×#552—434no—
300C2×#547—434no—
301C2×#544—431no—
302C2×#539—434no—
303C2×#537D4311yescharacteristic, complemented
304C2×#525—432no—
305C2×#525—434no—
306C2×#520—432no—
307C23×C4⋊C4—431no—
308C2×C43—24no—
309C2×C43—48no—
310C2×C43#349214yescomplemented
311C2×C43C23×#4111yescharacteristic
312C2×C43C22×H(5,𝔽2)11yescharacteristic
313C2×C43C22×2−1+411yescharacteristic
314D43—24no—
315D43—24no—
316C22×#41—434no—
317C22×#41C2×D4312yes—
318C24×C4—431no—
319C24×C4—432no—
320#554—4364no—
321#553—4364no—
322#552—4364no—
323#547—4364no—
324#544—4316no—
325#539—4364no—
326#537C2×D43116yescomplemented
327C2×C4⋊Q8—434no—
328C2×#126—438no—
329C2×#125—438no—
330C2×#125—434no—
331#525—4332no—
332#525—4364no—
333C2×#123—434no—
334C2×#123—438no—
335C2×#123—438no—
336C2×C22⋊Q8—432no—
337C2×C22⋊Q8—432no—
338C2×C22⋊Q8—434no—
339C2×C22⋊Q8—434no—
340C2×C22⋊Q8—438no—
341C2×C22⋊Q8C2×D4314yescomplemented
342C2×#121—432no—
343C2×#121—434no—
344C2×#121—432no—
345C2×#121C2×D4314yes—
346C2×#121C2×D4314yescomplemented
347C2×#120—434no—
348#520—4332no—
349C2×C4×Q8—434no—
350C2×C4×D4—438no—
351C2×C4×D4—434no—
352C2×C4×D4—434no—
353C2×C4×D4—434no—
354C2×C4×D4—438no—
355C2×#117—432no—
356C2×#117—434no—
357C2×#117—434no—
358C22×C4⋊C4—432no—
359C22×C4⋊C4—431no—
360C22×C4⋊C4—434no—
361C22×C4⋊C4—434no—
362C22×C4⋊C4—434no—
363C22×C4⋊C4—4316no—
364C22×C4⋊C4C2×D4311yescharacteristic
365C43C2×#349211yescharacteristic, complemented, Sylow
366C25—431no—
367C2×#41—216no—
368C2×#41—4332no—
369C2×#41—17216no—
370C2×#41C22×D43116yes—
371C22×Q8—434no—
372C22×Q8C22×D4312yes—
373C22×D4—434no—
374C22×D4—434no—
375C22×D4C22×D4314yes—
376C22×D4C22×D4312yes—
377C22×D4C22×D4314yes—
378C23×C4—431no—
379C23×C4—432no—
380C23×C4—438no—
381C23×C4—434no—
382C23×C4—432no—
383C23×C4—432no—
384C23×C4—438no—
385C23×C4—434no—
386C23×C4—4316no—
387C23×C4—868no—
388C23×C4—1728no—
389C23×C4C22×D4314yes—
390C23×C4C22×D4312yes—
391C23×C4C22×D4311yescharacteristic
392C4⋊Q8—4332no—
393#126—4364no—
394#125—4364no—
395#125—4332no—
396#123—4332no—
397#123—4364no—
398#123—4364no—
399C22⋊Q8—4316no—
400C22⋊Q8—4316no—
401C22⋊Q8—4332no—
402C22⋊Q8—4332no—
403C22⋊Q8—4364no—
404C22⋊Q8C22×D43132yescomplemented
405#121—4316no—
406#121—4332no—
407#121—4316no—
408#121C22×D43132yes—
409#121C22×D43132yescomplemented
410#120—4332no—
411C4×Q8—4332no—
412C4×D4—4364no—
413C4×D4—4332no—
414C4×D4—4332no—
415C4×D4—4332no—
416C4×D4—4364no—
417#117—4316no—
418#117—4332no—
419#117—4332no—
420C2×C4⋊C4—434no—
421C2×C4⋊C4—438no—
422C2×C4⋊C4—438no—
423C2×C4⋊C4—4316no—
424C2×C4⋊C4—432no—
425C2×C4⋊C4—438no—
426C2×C4⋊C4—4332no—
427C2×C4⋊C4—432no—
428C2×C4⋊C4—434no—
429C2×C4⋊C4—434no—
430C2×C4⋊C4—438no—
431C2×C4⋊C4—434no—
432C2×C4⋊C4—4332no—
433C2×C4⋊C4—4332no—
434C2×C4⋊C4C22×D4314yes—
435C2×C4⋊C4C22×D4318yes—
436C2×C4⋊C4C22×D4314yes—
437C2×C22⋊C4—434no—
438C2×C22⋊C4—438no—
439C2×C22⋊C4—434no—
440C2×C22⋊C4—434no—
441C2×C22⋊C4—434no—
442C2×C22⋊C4—434no—
443C2×C22⋊C4C22×D4318yes—
444C2×C42—438no—
445C2×C42—434no—
446C24—431no—
447C24—431no—
448C24—434no—
449C24—864no—
450C24—864no—
451C24—864no—
452C24C23×D4314yes—
453C24C23×D4311yescharacteristic
454#41—264no—
455#41—17264no—
456C2×Q8—28no—
457C2×Q8—4316no—
458C2×Q8—1728no—
459C2×Q8C23×D4318yes—
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—48no—
463C2×D4—4316no—
464C2×D4—4316no—
465C2×D4—8616no—
466C2×D4—8616no—
467C2×D4C23×D43116yes—
468C2×D4C23×D4318yes—
469C2×D4C23×D43116yes—
470C22×C4—216no—
471C22×C4—44no—
472C22×C4—434no—
473C22×C4—434no—
474C22×C4—434no—
475C22×C4—434no—
476C22×C4—432no—
477C22×C4—438no—
478C22×C4—4332no—
479C22×C4—4316no—
480C22×C4—438no—
481C22×C4—438no—
482C22×C4—4316no—
483C22×C4—8616no—
484C22×C4—864no—
485C22×C4—868no—
486C22×C4—868no—
487C22×C4—8632no—
488C22×C4—1728no—
489C22×C4—17232no—
490C22×C4—1724no—
491C22×C4—17216no—
492C22×C4#1062190018yescomplemented
493C22×C4C23×D43116yes—
494C22×C4C23×D4314yes—
495C22×C4C23×D4314yes—
496C22×C4C23×D4318yes—
497C22×C4C23×D4312yes—
498C22×C4C23×D4314yes—
499C4⋊C4—4316no—
500C4⋊C4—4332no—
501C4⋊C4—438no—
502C4⋊C4—4332no—
503C4⋊C4—438no—
504C4⋊C4—4316no—
505C4⋊C4—4316no—
506C4⋊C4—4332no—
507C4⋊C4—4316no—
508C4⋊C4C23×D43116yes—
509C4⋊C4C23×D43116yes—
510C22⋊C4—4316no—
511C22⋊C4—4332no—
512C22⋊C4—4316no—
513C22⋊C4—4316no—
514C22⋊C4—4316no—
515C22⋊C4—4316no—
516C22⋊C4C23×D43132yes—
517C42—4332no—
518C42—4316no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—434no—
525C23—434no—
526C23—432no—
527C23—432no—
528C23—868no—
529C23—868no—
530C23—8632no—
531C23—868no—
532C23—862no—
533C23—862no—
534C23—862no—
535C23—862no—
536C23C2×#1062190014yes—
537C23C24×D4318yes—
538C23C24×D4312yes—
539C23C24×D4311yescharacteristic
540Q8—216no—
541Q8—17216no—
542D4—232no—
543D4—216no—
544D4—416no—
545D4—8632no—
546D4—8632no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—48no—
551C2×C4—438no—
552C2×C4—438no—
553C2×C4—438no—
554C2×C4—434no—
555C2×C4—8632no—
556C2×C4—868no—
557C2×C4—864no—
558C2×C4—868no—
559C2×C4—868no—
560C2×C4—868no—
561C2×C4—8616no—
562C2×C4—8616no—
563C2×C4—17216no—
564C2×C4—1728no—
565C2×C4—17232no—
566C2×C4C2×#1062190018yes—
567C2×C4C2×#10621900116yescomplemented
568C2×C4C24×D4318yes—
569C2×C4C24×D4318yes—
570C2×C4C24×D4314yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—416no—
578C22—434no—
579C22—434no—
580C22—8632no—
581C22—868no—
582C22—868no—
583C22—862no—
584C22—862no—
585C22—862no—
586C22—862no—
587C22C22×#1062190014yes—
588C22C22×#1062190012yes—
589C22H(5,𝔽2)×D4312yes—
590C222−1+4×D4312yes—
591C22C25×D4311yescharacteristic
592C4—28no—
593C4—28no—
594C4—864no—
595C4—868no—
596C4—868no—
597C4—868no—
598C4C22×#1062190018yes—
599C2—24no—
600C2—48no—
601C2—864no—
602C2—864no—
603C2#1914830214yesdirect product factor
604C2C23×#1062190011yescharacteristic
605C2C2×H(5,𝔽2)×D4311yescharacteristic
606C2C2×2−1+4×D4311yescharacteristic
subgroupG/H
centerC23C24×D43
commutator, cosocleC22×C43C26
FrattiniC22C25×D43
FittingC2×C43×#537C2
socleC23×C43C25
hypercenterC2×#537D43
hypocenterC43C2×#3492
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #19148302C2direct (N × H)
2. C2×#14583607C2semidirect
3. C2×#14583596C2semidirect
4. C2×#14583583C2semidirect
5. C2×#14583549C2semidirect
6. C2×#14583530C2semidirect
7. C2×#14583494C2semidirect
8. C2×#14583409C2semidirect
9. C2×#14583406C2semidirect
10. C2×#14583382C2semidirect
11. C22×C4⋊C4×D43C2semidirect
12. C2×C43×#537C2semidirect
16. #14583607C22semidirect
17. #14583596C22semidirect
18. #14583583C22semidirect
19. #14583549C22semidirect
20. #14583530C22semidirect
21. #14583494C22semidirect
22. C2×#13063324C22semidirect
23. C2×#13063318C22semidirect
24. C2×#13063316C22semidirect
25. C2×#13063315C22semidirect
26. #14583409C22semidirect
27. #14583406C22semidirect
30. C2×#13063306C22semidirect
33. C2×#13063302C22semidirect
40. #14583382C22semidirect
41. C2×Q8×Dic43C22semidirect
43. C2×D4×Dic43C22semidirect
48. C2×#13063284C22semidirect
51. C22×C4⋊Dic43C22semidirect
55. C2×C4⋊C4×D43C22semidirect
57. C43×#537C22semidirect
58. C2×C22⋊Q8×C43C22semidirect
60. C2×#121×C43C22semidirect
79. #13063324C23semidirect
80. #13063318C23semidirect
81. #13063316C23semidirect
82. #13063315C23semidirect
85. #13063306C23semidirect
88. #13063302C23semidirect
95. Q8×Dic43C23semidirect
97. D4×Dic43C23semidirect
102. #13063284C23semidirect
110. C2×C4⋊Dic43C23semidirect
135. C22⋊Q8×C43C23semidirect
137. #121×C43C23semidirect
161. C22×C4×C43#41semidirect
245. C2×C4×C43C2×#41semidirect
303. C2×#537D43semidirect
310. C2×C43#3492semidirect
326. #537C2×D43semidirect
341. C2×C22⋊Q8C2×D43semidirect
346. C2×#121C2×D43semidirect
365. C43C2×#3492semidirect
404. C22⋊Q8C22×D43semidirect
409. #121C22×D43semidirect
492. C22×C4#10621900semidirect
567. C2×C4C2×#10621900semidirect
603. C2#19148302direct (N × H)

Maximal subgroups

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

subgroupindex
#191483022
C2×#145836072
C2×#145835962
C2×#145835832
C2×#145835492
C2×#145835302
C2×#145834942
C2×#145834092
C2×#145834062
C2×#145833822
C22×C4⋊C4×D432
C2×C43×#5372
C2×#349243