Finite Group Databasebeta

← C2×#19148305 #26004539

Subgroups — C2×#19148305 #26004539

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#19148305C2132yesmaximal, direct product factor
2C2×#14583610C214yesmaximal, complemented
3C2×#14583602C214yesmaximal, complemented
4C2×#14583587C214yesmaximal, complemented
5C2×#14583552C214yesmaximal, complemented
6C2×#14583498C214yesmaximal, complemented
7C2×#14583480C211yesmaximal, characteristic, complemented
8C2×#14583412C214yesmaximal, complemented
9C2×#14583411C212yesmaximal, complemented
10C2×#14583379C212yesmaximal, complemented
11C22×C4⋊C4×D43C211yesmaximal, characteristic, complemented
12C2×C43×#544C211yesmaximal, characteristic, complemented
13C22×#10621902C2214yes—
14C22×#10621898C2212yes—
15C23×C4×D43C2211yescharacteristic
16C23×C4×D43C2212yes—
17#14583610C22164yescomplemented
18#14583602C22164yescomplemented
19#14583587C22164yescomplemented
20#14583552C22164yescomplemented
21#14583498C22164yescomplemented
22#14583480C22116yescomplemented
23C2×#13063320C2218yescomplemented
24C2×#13063316C2218yescomplemented
25#14583412C22164yescomplemented
26#14583411C22132yescomplemented
27C2×#13063310C2218yes—
28C2×#13063310C2218yescomplemented
29C2×#13063309C2214yes—
30C2×#13063305C2212yes—
31C2×#13063305C2212yes—
32C2×#13063305C2214yescomplemented
33C2×#13063301C2214yes—
34C2×#13063301C2218yescomplemented
35C2×#13063300C2212yes—
36C2×#13063300C2212yes—
37C2×#13063300C2214yes—
38C2×#13063300C2214yescomplemented
39C2×#13063299C2214yes—
40C2×#13063295C2214yes—
41#14583379C22132yescomplemented
42C2×C4×Dic86C2214yescomplemented
43C2×C4×D172C2214yes—
44C2×C4×D172C2214yes—
45C2×C4×D172C2214yescomplemented
46C2×#13063287C2218yes—
47C2×(C22⋊D43)⋊C4C2218yes—
48C2×#13063283C2214yes—
49C2×#13063283C2214yescomplemented
50C22×C4⋊Dic43C2211yescharacteristic
51C22×Dic43⋊C4C2212yes—
52C2×C4⋊C4×D43C2214yes—
53C2×C4⋊C4×D43C2214yes—
54C2×C4⋊C4×D43C22116yescomplemented
55C2×C4⋊C4×D43C2214yescomplemented
56C43×#544C22116yescomplemented
57C2×C4⋊Q8×C43C2214yescomplemented
58C2×#125×C43C2214yescomplemented
59C2×C22⋊Q8×C43C2214yes—
60C2×#117×C43C2212yes—
61C22×C4⋊C4×C43C2211yescharacteristic
62C24×D43C2311yescharacteristic
63C2×#10621902—416no—
64C2×#10621902C23132yes—
65C2×#10621898—216no—
66C2×#10621898C23116yes—
67C22×Dic86C2312yes—
68C22×D172C2314yes—
69C22×D172C2314yes—
70C22×D172C2312yes—
71C22×D172C2314yes—
72C22×C22⋊D43C2314yes—
73C23×Dic43C2311yescharacteristic
74C23×Dic43C2312yes—
75C22×C4×D43—28no—
76C22×C4×D43—48no—
77C22×C4×D43C2314yes—
78C22×C4×D43C2312yes—
79C22×C4×D43C2312yes—
80C22×C4×D43C2318yes—
81C22×C4×D43C2314yes—
82C22×C4×D43C2314yes—
83C22×C4×D43C23116yes—
84C22×C4×D43C2318yes—
85#13063320C23164yescomplemented
86#13063316C23164yescomplemented
87#13063310C23164yes—
88#13063310C23164yescomplemented
89#13063309C23132yes—
90#13063305C23116yes—
91#13063305C23116yes—
92#13063305C23132yescomplemented
93#13063301C23132yes—
94#13063301C23164yescomplemented
95#13063300C23116yes—
96#13063300C23116yes—
97#13063300C23132yes—
98#13063300C23132yescomplemented
99#13063299C23132yes—
100#13063295C23132yes—
101C4×Dic86C23132yescomplemented
102C4×D172C23132yes—
103C4×D172C23132yes—
104C4×D172C23132yescomplemented
105#13063287C23164yes—
106(C22⋊D43)⋊C4C23164yes—
107#13063283C23132yes—
108#13063283C23132yescomplemented
109C2×C4⋊Dic43C2314yes—
110C2×C4⋊Dic43C2318yes—
111C2×C4⋊Dic43C2314yes—
112C2×C4⋊Dic43C2314yes—
113C2×C4⋊Dic43C2314yes—
114C2×Dic43⋊C4C2318yes—
115C2×Dic43⋊C4C23116yes—
116C2×Dic43⋊C4C2318yes—
117C4⋊C4×D43C23132yes—
118C4⋊C4×D43C23132yes—
119C4⋊C4×D43C23132yescomplemented
120C2×(C2×D43)⋊C4C2314yes—
121C2×(C2×D43)⋊C4C2318yes—
122C2×(C2×D43)⋊C4C2314yes—
123C2×(C2×D43)⋊C4C2318yes—
124C2×(C2×D43)⋊C4C2314yes—
125C2×(C2×D43)⋊C4C2314yes—
126C2×C4×Dic43C2318yes—
127C22×Q8×C43C2314yes—
128C23×C4×C43C2312yes—
129C23×C4×C43C2311yescharacteristic
130C4⋊Q8×C43C23132yescomplemented
131#125×C43C23132yescomplemented
132C22⋊Q8×C43C23132yes—
133#117×C43C23116yes—
134C2×C4⋊C4×C43C2314yes—
135C2×C4⋊C4×C43C2318yes—
136C2×C4⋊C4×C43C2312yes—
137C2×C4⋊C4×C43C2312yes—
138C2×C4⋊C4×C43C2314yes—
139C2×C4⋊C4×C43C2318yes—
140C2×C22⋊C4×C43C2314yes—
141C2×C42×C43C2314yes—
142C24×C43C2411yescharacteristic
143C23×D43—24no—
144C23×D43—24no—
145C23×D43—24no—
146C23×D43C2414yes—
147C23×D43C2414yes—
148C23×D43C2411yescharacteristic
149C23×D43C2411yescharacteristic
150C2×Q8×C43—48no—
151C2×Q8×C43C24116yes—
152C22×C4×C43—44no—
153C22×C4×C43#4118yescomplemented
154C22×C4×C43C2414yes—
155C22×C4×C43C2414yes—
156C22×C4×C43C2414yes—
157C22×C4×C43C2418yes—
158C22×C4×C43C2412yes—
159C22×C4×C43C2414yes—
160C2×C22⋊D43—216no—
161C2×C22⋊D43C24116yes—
162C22×Dic43—24no—
163C22×Dic43—44no—
164C22×Dic43C2414yes—
165C22×Dic43C2414yes—
166C22×Dic43C2412yes—
167C22×Dic43C2414yes—
168C22×Dic43C2418yes—
169#10621902—464no—
170#10621898—264no—
171C2×D172—28no—
172C2×D172—216no—
173C2×D172—216no—
174C2×D172—48no—
175C2×D172C24116yes—
176C2×D172C24116yes—
177C2×D172C2418yes—
178C2×D172C24116yes—
179C2×C4×D43—28no—
180C2×C4×D43—28no—
181C2×C4×D43—216no—
182C2×C4×D43—232no—
183C2×C4×D43—216no—
184C2×C4×D43—432no—
185C2×C4×D43—48no—
186C2×C4×D43—416no—
187C2×C4×D43C24116yes—
188C2×C4×D43C2418yes—
189C2×C4×D43C2418yes—
190C2×C4×D43C24116yes—
191C2×C4×D43C24116yes—
192C2×C4×D43C24132yes—
193C2×Dic86—28no—
194C2×Dic86C2418yes—
195C4⋊C4×C43C24116yes—
196C4⋊C4×C43C2418yes—
197C4⋊C4×C43C2418yes—
198C4⋊C4×C43C24116yes—
199C4⋊C4×C43C24132yes—
200C22⋊C4×C43C24116yes—
201C42×C43C24116yes—
202(C2×D43)⋊C4C24116yes—
203(C2×D43)⋊C4C24132yes—
204(C2×D43)⋊C4C24116yes—
205(C2×D43)⋊C4C24132yes—
206(C2×D43)⋊C4C24116yes—
207(C2×D43)⋊C4C24116yes—
208C4⋊Dic43C24116yes—
209C4⋊Dic43C24116yes—
210C4⋊Dic43C24116yes—
211C4⋊Dic43C24116yes—
212Dic43⋊C4C24132yes—
213Dic43⋊C4C24132yes—
214C4×Dic43C24132yes—
215C23×C43—22no—
216C23×C43—22no—
217C23×C43C2×#4114yes—
218C23×C43C2512yes—
219C23×C43C2511yescharacteristic
220C22×D43—232no—
221C22×D43—28no—
222C22×D43—22no—
223C22×D43—22no—
224C22×D43—22no—
225C22×D43—22no—
226C22×D43—28no—
227C22×D43—28no—
228C22×D43—28no—
229C22×D43—28no—
230C22×D43—28no—
231C22×D43H(5,𝔽2)14yes—
232C22×D432−1+414yes—
233C22×D43C2518yes—
234C22×D43C2512yes—
235C22×D43C2512yes—
236Q8×C43—416no—
237C2×C4×C43—28no—
238C2×C4×C43—28no—
239C2×C4×C43—28no—
240C2×C4×C43—48no—
241C2×C4×C43C2×#4118yes—
242C2×C4×C43C2×#41116yescomplemented
243C2×C4×C43C2518yes—
244C2×C4×C43C2518yes—
245C2×C4×C43C2518yes—
246C2×C4×C43C2514yes—
247C22⋊D43—232no—
248C2×Dic43—28no—
249C2×Dic43—28no—
250C2×Dic43—24no—
251C2×Dic43—28no—
252C2×Dic43—48no—
253C2×Dic43C2518yes—
254C2×Dic43C2518yes—
255C2×Dic43C2514yes—
256D172—232no—
257D172—232no—
258D172—216no—
259D172—416no—
260C4×D43—232no—
261C4×D43—216no—
262C4×D43—216no—
263C4×D43—232no—
264C4×D43—416no—
265C4×D43—432no—
266Dic86—216no—
267C2×#3492—431nomaximal, Sylow
268C22×C43—22no—
269C22×C43—22no—
270C22×C43—28no—
271C22×C43C22×#4114yes—
272C22×C43C22×#4112yes—
273C22×C43C2×H(5,𝔽2)12yes—
274C22×C43C2×2−1+412yes—
275C22×C43C2611yescharacteristic
276C2×D43—232no—
277C2×D43—22no—
278C2×D43—22no—
279C2×D43—22no—
280C2×D43—22no—
281C2×D43—28no—
282C2×D43—28no—
283C2×D43—28no—
284C2×D43—28no—
285C2×D43—28no—
286C2×D43—416no—
287C2×D43C2×H(5,𝔽2)14yes—
288C2×D43C2×2−1+414yes—
289C4×C43—28no—
290C4×C43—28no—
291C4×C43—28no—
292C4×C43C22×#4118yes—
293Dic43—28no—
294Dic43—24no—
295Dic43—28no—
296#3492—4332no—
297C2×#554—434no—
298C2×#553—434no—
299C2×#552—434no—
300C2×#547—434no—
301C2×#544D4311yescharacteristic, complemented
302C2×#539—434no—
303C2×#537—431no—
304C2×#525—434no—
305C2×#525—432no—
306C2×#520—432no—
307C23×C4⋊C4—431no—
308C2×C43—24no—
309C2×C43#349214yescomplemented
310C2×C43C23×#4111yescharacteristic
311C2×C43C22×H(5,𝔽2)11yescharacteristic
312C2×C43C22×2−1+411yescharacteristic
313D43—24no—
314D43—24no—
315D43—48no—
316C22×#41—434no—
317C22×#41—432no—
318C24×C4—431no—
319C24×C4—432no—
320#554—4364no—
321#553—4364no—
322#552—4364no—
323#547—4364no—
324#544C2×D43116yescomplemented
325#539—4364no—
326#537—4316no—
327C2×C4⋊Q8C2×D4314yescomplemented
328C2×#126—438no—
329C2×#125—438no—
330C2×#125C2×D4314yescomplemented
331#525—4364no—
332#525—4332no—
333C2×#123—438no—
334C2×#123—438no—
335C2×#123—434no—
336C2×C22⋊Q8—434no—
337C2×C22⋊Q8—438no—
338C2×C22⋊Q8—432no—
339C2×C22⋊Q8—432no—
340C2×C22⋊Q8—434no—
341C2×C22⋊Q8C2×D4314yes—
342C2×#121—434no—
343C2×#121—432no—
344C2×#121—434no—
345C2×#121—432no—
346C2×#121—434no—
347C2×#120—434no—
348#520—4332no—
349C2×C4×Q8—434no—
350C2×C4×D4—438no—
351C2×C4×D4—438no—
352C2×C4×D4—434no—
353C2×C4×D4—434no—
354C2×C4×D4—434no—
355C2×#117—434no—
356C2×#117—434no—
357C2×#117C2×D4312yes—
358C22×C4⋊C4—432no—
359C22×C4⋊C4—431no—
360C22×C4⋊C4—434no—
361C22×C4⋊C4—434no—
362C22×C4⋊C4—4316no—
363C22×C4⋊C4—434no—
364C22×C4⋊C4C2×D4311yescharacteristic
365C43C2×#349211yescharacteristic, complemented, Sylow
366C25—431no—
367C2×#41—4332no—
368C2×#41—4316no—
369C2×#41—8616no—
370C2×#41—17216no—
371C22×Q8—432no—
372C22×Q8C22×D4314yes—
373C22×D4—434no—
374C22×D4—434no—
375C22×D4—432no—
376C22×D4—434no—
377C22×D4—434no—
378C23×C4—431no—
379C23×C4—432no—
380C23×C4—434no—
381C23×C4—432no—
382C23×C4—432no—
383C23×C4—438no—
384C23×C4—434no—
385C23×C4—434no—
386C23×C4—4316no—
387C23×C4—438no—
388C23×C4—868no—
389C23×C4—1728no—
390C23×C4C22×D4312yes—
391C23×C4C22×D4311yescharacteristic
392C4⋊Q8C22×D43132yescomplemented
393#126—4364no—
394#125—4364no—
395#125C22×D43132yescomplemented
396#123—4364no—
397#123—4364no—
398#123—4332no—
399C22⋊Q8—4332no—
400C22⋊Q8—4364no—
401C22⋊Q8—4316no—
402C22⋊Q8—4316no—
403C22⋊Q8—4332no—
404C22⋊Q8C22×D43132yes—
405#121—4332no—
406#121—4316no—
407#121—4332no—
408#121—4316no—
409#121—4332no—
410#120—4332no—
411C4×Q8—4332no—
412C4×D4—4364no—
413C4×D4—4364no—
414C4×D4—4332no—
415C4×D4—4332no—
416C4×D4—4332no—
417#117—4332no—
418#117—4332no—
419#117C22×D43116yes—
420C2×C4⋊C4—434no—
421C2×C4⋊C4—438no—
422C2×C4⋊C4—438no—
423C2×C4⋊C4—4316no—
424C2×C4⋊C4—438no—
425C2×C4⋊C4—434no—
426C2×C4⋊C4—434no—
427C2×C4⋊C4—434no—
428C2×C4⋊C4—4332no—
429C2×C4⋊C4—4332no—
430C2×C4⋊C4—4332no—
431C2×C4⋊C4C22×D4314yes—
432C2×C4⋊C4C22×D4318yes—
433C2×C4⋊C4C22×D4312yes—
434C2×C4⋊C4C22×D4312yes—
435C2×C4⋊C4C22×D4314yes—
436C2×C4⋊C4C22×D4318yes—
437C2×C22⋊C4—434no—
438C2×C22⋊C4—438no—
439C2×C22⋊C4—434no—
440C2×C22⋊C4—438no—
441C2×C22⋊C4—434no—
442C2×C22⋊C4—434no—
443C2×C22⋊C4C22×D4314yes—
444C2×C42—438no—
445C2×C42C22×D4314yes—
446C24—431no—
447C24—431no—
448C24—434no—
449C24—434no—
450C24—864no—
451C24—864no—
452C24—864no—
453C24C23×D4311yescharacteristic
454#41—8664no—
455#41—17264no—
456C2×Q8—48no—
457C2×Q8—438no—
458C2×Q8—868no—
459C2×Q8C23×D43116yes—
460C2×D4—4316no—
461C2×D4—4316no—
462C2×D4—438no—
463C2×D4—4316no—
464C2×D4—4316no—
465C2×D4—8616no—
466C2×D4—8616no—
467C2×D4—8616no—
468C2×D4—868no—
469C2×D4—1728no—
470C22×C4—44no—
471C22×C4—434no—
472C22×C4—434no—
473C22×C4—434no—
474C22×C4—432no—
475C22×C4—438no—
476C22×C4—4316no—
477C22×C4—438no—
478C22×C4—438no—
479C22×C4—4316no—
480C22×C4—4316no—
481C22×C4—4332no—
482C22×C4—8616no—
483C22×C4—864no—
484C22×C4—868no—
485C22×C4—868no—
486C22×C4—8616no—
487C22×C4—8632no—
488C22×C4—1728no—
489C22×C4—17216no—
490C22×C4—17232no—
491C22×C4—1724no—
492C22×C4#1062190218yescomplemented
493C22×C4C23×D4314yes—
494C22×C4C23×D4314yes—
495C22×C4C23×D4314yes—
496C22×C4C23×D4318yes—
497C22×C4C23×D4312yes—
498C22×C4C23×D4314yes—
499C4⋊C4—4316no—
500C4⋊C4—4332no—
501C4⋊C4—4332no—
502C4⋊C4—4316no—
503C4⋊C4—4316no—
504C4⋊C4—4316no—
505C4⋊C4C23×D43116yes—
506C4⋊C4C23×D4318yes—
507C4⋊C4C23×D4318yes—
508C4⋊C4C23×D43116yes—
509C4⋊C4C23×D43132yes—
510C22⋊C4—4316no—
511C22⋊C4—4332no—
512C22⋊C4—4316no—
513C22⋊C4—4332no—
514C22⋊C4—4316no—
515C22⋊C4—4316no—
516C22⋊C4C23×D43116yes—
517C42—4332no—
518C42C23×D43116yes—
519C23—22no—
520C23—22no—
521C23—434no—
522C23—434no—
523C23—432no—
524C23—432no—
525C23—438no—
526C23—868no—
527C23—868no—
528C23—8632no—
529C23—868no—
530C23—862no—
531C23—862no—
532C23—862no—
533C23—862no—
534C23—868no—
535C23—868no—
536C23—868no—
537C23C2×#1062190214yes—
538C23C24×D4312yes—
539C23C24×D4311yescharacteristic
540Q8—416no—
541Q8—8616no—
542D4—8632no—
543D4—8632no—
544D4—8632no—
545D4—8616no—
546D4—17216no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—48no—
551C2×C4—438no—
552C2×C4—438no—
553C2×C4—434no—
554C2×C4—8632no—
555C2×C4—868no—
556C2×C4—864no—
557C2×C4—868no—
558C2×C4—868no—
559C2×C4—8616no—
560C2×C4—8616no—
561C2×C4—8632no—
562C2×C4—17216no—
563C2×C4—17232no—
564C2×C4—1728no—
565C2×C4C2×#1062190218yes—
566C2×C4C2×#10621902116yescomplemented
567C2×C4C24×D4318yes—
568C2×C4C24×D4318yes—
569C2×C4C24×D4318yes—
570C2×C4C24×D4314yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—434no—
575C22—434no—
576C22—8632no—
577C22—868no—
578C22—868no—
579C22—862no—
580C22—862no—
581C22—862no—
582C22—862no—
583C22—868no—
584C22—868no—
585C22—868no—
586C22—17216no—
587C22C22×#1062190214yes—
588C22C22×#1062190212yes—
589C22H(5,𝔽2)×D4312yes—
590C222−1+4×D4312yes—
591C22C25×D4311yescharacteristic
592C4—28no—
593C4—28no—
594C4—28no—
595C4—864no—
596C4—868no—
597C4—868no—
598C4C22×#1062190218yes—
599C2—24no—
600C2—864no—
601C2—864no—
602C2—1728no—
603C2#1914830514yesdirect product factor
604C2C23×#1062190211yescharacteristic
605C2C2×H(5,𝔽2)×D4311yescharacteristic
606C2C2×2−1+4×D4311yescharacteristic
subgroupG/H
centerC23C24×D43
commutator, cosocleC22×C43C26
FrattiniC22C25×D43
FittingC2×C43×#544C2
socleC23×C43C25
hypercenterC2×#544D43
hypocenterC43C2×#3492
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. #19148305C2direct (N × H)
2. C2×#14583610C2semidirect
3. C2×#14583602C2semidirect
4. C2×#14583587C2semidirect
5. C2×#14583552C2semidirect
6. C2×#14583498C2semidirect
7. C2×#14583480C2semidirect
8. C2×#14583412C2semidirect
9. C2×#14583411C2semidirect
10. C2×#14583379C2semidirect
11. C22×C4⋊C4×D43C2semidirect
12. C2×C43×#544C2semidirect
17. #14583610C22semidirect
18. #14583602C22semidirect
19. #14583587C22semidirect
20. #14583552C22semidirect
21. #14583498C22semidirect
22. #14583480C22semidirect
23. C2×#13063320C22semidirect
24. C2×#13063316C22semidirect
25. #14583412C22semidirect
26. #14583411C22semidirect
28. C2×#13063310C22semidirect
32. C2×#13063305C22semidirect
34. C2×#13063301C22semidirect
38. C2×#13063300C22semidirect
41. #14583379C22semidirect
42. C2×C4×Dic86C22semidirect
45. C2×C4×D172C22semidirect
49. C2×#13063283C22semidirect
54. C2×C4⋊C4×D43C22semidirect
55. C2×C4⋊C4×D43C22semidirect
56. C43×#544C22semidirect
57. C2×C4⋊Q8×C43C22semidirect
58. C2×#125×C43C22semidirect
85. #13063320C23semidirect
86. #13063316C23semidirect
88. #13063310C23semidirect
92. #13063305C23semidirect
94. #13063301C23semidirect
98. #13063300C23semidirect
101. C4×Dic86C23semidirect
104. C4×D172C23semidirect
108. #13063283C23semidirect
119. C4⋊C4×D43C23semidirect
130. C4⋊Q8×C43C23semidirect
131. #125×C43C23semidirect
153. C22×C4×C43#41semidirect
242. C2×C4×C43C2×#41semidirect
301. C2×#544D43semidirect
309. C2×C43#3492semidirect
324. #544C2×D43semidirect
327. C2×C4⋊Q8C2×D43semidirect
330. C2×#125C2×D43semidirect
365. C43C2×#3492semidirect
392. C4⋊Q8C22×D43semidirect
395. #125C22×D43semidirect
492. C22×C4#10621902semidirect
566. C2×C4C2×#10621902semidirect
603. C2#19148305direct (N × H)

Maximal subgroups

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

subgroupindex
#191483052
C2×#145836102
C2×#145836022
C2×#145835872
C2×#145835522
C2×#145834982
C2×#145834802
C2×#145834122
C2×#145834112
C2×#145833792
C22×C4⋊C4×D432
C2×C43×#5442
C2×#349243