Finite Group Databasebeta

← C2×#10594231 #11903163

Subgroups — C2×#10594231 #11903163

Order 1152.

2717 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
1C2×#64090C211yesmaximal, characteristic, complemented
2C2×C3×#67⋊C4C212yesmaximal, complemented
3C2×#63980C211yesmaximal, characteristic, complemented
4C2×#63978C212yesmaximal, complemented
5C22×#3724C211yesmaximal, characteristic, complemented
6C2×#63912C211yesmaximal, characteristic, complemented
7C2×C4×#625C211yesmaximal, characteristic, complemented
8C2×#63874C211yesmaximal, characteristic, complemented
9C2×#63870C211yesmaximal, characteristic, complemented
10C2×#63859C212yesmaximal, complemented
11C2×#63850C212yesmaximal, complemented
12#10594231C2116yesmaximal, direct product factor
13C2×#5540—32nomaximal
14C22×#638C2211yescharacteristic
15C23×C3⋊Dic3C2211yescharacteristic
16C22×C4×C3⋊D3C2211yescharacteristic
17C22×C3×#67C2212yes—
18C23×C3×Dic3C2212yes—
19C22×C3×C4×D3C2212yes—
20C22×#626C2212yes—
21C22×#625C2211yescharacteristic
22C2×C32×C22⋊C4C2211yescharacteristic
23C2×#3752C2211yescharacteristic
24C2×#3750C2211yescharacteristic
25C2×C4×C3⋊Dic3C2211yescharacteristic
26#64090C2218yescomplemented
27C2×C3×(C2×D3)⋊C4C2212yes—
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C3×#67⋊C4C22116yescomplemented
31#63980C2218yescomplemented
32#63978C22116yescomplemented
33C2×#3724C2212yes—
34C2×#3724C2211yescharacteristic
35C2×#3724C2211yescharacteristic
36C2×#3724C2211yescharacteristic
37C2×#3724C2218yescomplemented
38C2×Dic3⋊Dic3C2212yes—
39C2×#3722C2212yes—
40C2×(C2×D3)⋊Dic3C2212yes—
41C2×(C2×D3)⋊Dic3C2212yes—
42#63912C2218yescomplemented
43C4×#625C2218yescomplemented
44#63874C2218yescomplemented
45#63870C2218yescomplemented
46#63859C22116yescomplemented
47#63850C22116yescomplemented
48C2×C3×C4×D4—32no—
49C2×#939—32no—
50C2×#939—32no—
51C2×#930—32no—
52C2×#929—32no—
53C2×C4×#67—32no—
54C22×Dic3⋊C4—32no—
55#5540—332no—
56C2×#894—32no—
57C2×#889—32no—
58C2×#884—32no—
59C2×#67⋊C4—32no—
60C2×#67⋊C4—61no—
61C2×#67⋊C4D312yescomplemented
62C2×#879—32no—
63C2×#876—32no—
64C2×#872—32no—
65C24×C32C2311yescharacteristic
66C23×C3⋊D3C2311yescharacteristic
67C23×C3×D3C2312yes—
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2311yescharacteristic
70C2×#638—24no—
71C2×#638C2314yes—
72C22×C3⋊Dic3—24no—
73C22×C3⋊Dic3C2314yes—
74C22×C3⋊Dic3C2311yescharacteristic
75C22×C3⋊Dic3C2311yescharacteristic
76C22×C3⋊Dic3C2311yescharacteristic
77C2×C4×C3⋊D3—24no—
78C2×C4×C3⋊D3C2314yes—
79C2×C3×#67—28no—
80C2×C3×#67C2318yes—
81C22×C3×Dic3—44no—
82C22×C3×Dic3C2312yes—
83C22×C3×Dic3C2312yes—
84C22×C3×Dic3C2312yes—
85C22×C3×Dic3C2318yes—
86C2×C3×C4×D3—44no—
87C2×C3×C4×D3C2318yes—
88C2×#626—44no—
89C2×#626C2318yes—
90C2×#625—24no—
91C2×#625C2314yes—
92C32×C22⋊C4C2314yes—
93#3752C2314yes—
94#3750C2314yes—
95C4×C3⋊Dic3C2314yes—
96C3×(C2×D3)⋊C4C2318yes—
97C3×Dic3⋊C4C2318yes—
98C3×C4×Dic3C2318yes—
99#3724C2318yes—
100#3724C2314yes—
101#3724C2314yes—
102#3724C2314yes—
103Dic3⋊Dic3C2318yes—
104#3722C2318yes—
105(C2×D3)⋊Dic3C2318yes—
106(C2×D3)⋊Dic3C2318yes—
107C2×#553—91noSylow
108C22×C3×D4—32no—
109C23×C3×C4—32no—
110C23×C3×C4—32no—
111C22×#67—32no—
112C22×#67—32no—
113C22×#67—32no—
114C22×#67—61no—
115C22×#67C2×D312yes—
116C23×Dic3—32no—
117C23×Dic3—61no—
118C23×Dic3C2×D312yes—
119C22×D12—32no—
120C22×C4×D3—32no—
121C22×C4×D3—61no—
122C22×C4×D3C2×D312yes—
123C3×C4×D4—316no—
124C2×C3×C4⋊C4—32no—
125C2×C3×C22⋊C4—21no—
126C2×C3×C22⋊C4—32no—
127C2×C3×C22⋊C4C2×D312yes—
128C2×C3×C42—32no—
129C2×#217—32no—
130C2×#217—32no—
131#939—316no—
132#939—316no—
133#930—316no—
134#929—316no—
135C4×#67—316no—
136C2×(C2×D3)⋊C4—32no—
137C2×(C2×D3)⋊C4—32no—
138C2×(C2×D3)⋊C4—32no—
139C2×(C2×D3)⋊C4—32no—
140C2×(C2×D3)⋊C4—32no—
141C2×(C2×D3)⋊C4—61no—
142C2×(C2×D3)⋊C4C2×D312yes—
143C2×C4⋊Dic3—32no—
144C2×Dic3⋊C4—32no—
145C2×Dic3⋊C4—32no—
146C2×Dic3⋊C4—32no—
147C2×Dic3⋊C4—32no—
148C2×Dic3⋊C4—32no—
149C2×Dic3⋊C4—32no—
150C2×Dic3⋊C4—32no—
151C2×Dic3⋊C4—316no—
152C2×Dic3⋊C4—61no—
153C2×Dic3⋊C4C2×D312yes—
154C2×C4×Dic3—32no—
155C2×C4×Dic3—61no—
156C2×C4×Dic3C2×D312yes—
157#894—316no—
158#889—316no—
159#884—316no—
160#67⋊C4—316no—
161#67⋊C4—68no—
162#67⋊C4C2×D3116yescomplemented
163#879—316no—
164#876—316no—
165#872—316no—
166C23×C32—22no—
167C23×C32—22no—
168C23×C32#4114yes—
169C23×C32C2412yes—
170C23×C32C2411yescharacteristic
171C22×C3⋊D3—22no—
172C22×C3⋊D3—22no—
173C22×C3⋊D3—22no—
174C22×C3⋊D3C2412yes—
175C22×C3×D3—24no—
176C22×C3×D3—24no—
177C22×C3×D3—24no—
178C22×C3×D3C2414yes—
179C2×C32×C4—22no—
180C2×C32×C4—22no—
181C2×C32×C4C2412yes—
182C2×C32×C4C2412yes—
183#638—28no—
184C2×C3⋊Dic3—22no—
185C2×C3⋊Dic3—22no—
186C2×C3⋊Dic3—28no—
187C2×C3⋊Dic3#4114yes—
188C2×C3⋊Dic3C2412yes—
189C2×C3⋊Dic3C2412yes—
190C2×C3⋊Dic3C2412yes—
191C4×C3⋊D3—28no—
192C3×#67—216no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—24no—
196C2×C3×Dic3—48no—
197C2×C3×Dic3C2414yes—
198C2×C3×Dic3C2414yes—
199C2×C3×Dic3C2414yes—
200C3×C4×D3—48no—
201#626—48no—
202#625—28no—
203#553—916no—
204C2×#126—92no—
205C2×#125—91no—
206C2×#123—92no—
207C2×#121—91no—
208C2×#121—92no—
209C2×#121—91no—
210C2×C4×D4—32no—
211C2×C4×D4—91no—
212C2×C4×D4—91no—
213C2×#117—91no—
214C22×C4⋊C4—91no—
215C24×C3—21no—
216C24×C3—32no—
217C24×C3C22×D312yes—
218C23×D3—32no—
219C23×D3—61no—
220C23×D3C22×D312yes—
221C2×C3×D4—38no—
222C2×C3×D4—68no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—32no—
228C22×C3×C4—38no—
229C22×C3×C4—38no—
230C22×C3×C4—124no—
231C22×C3×C4—124no—
232C22×C3×C4C22×D312yes—
233C22×C3×C4C22×D312yes—
234C2×#67—28no—
235C2×#67—38no—
236C2×#67—38no—
237C2×#67—38no—
238C2×#67—64no—
239C2×#67—68no—
240C2×#67—68no—
241C2×#67—68no—
242C2×#67—124no—
243C2×#67C22×D318yes—
244C22×Dic3—32no—
245C22×Dic3—38no—
246C22×Dic3—32no—
247C22×Dic3—32no—
248C22×Dic3—44no—
249C22×Dic3—61no—
250C22×Dic3—61no—
251C22×Dic3—64no—
252C22×Dic3—61no—
253C22×Dic3—68no—
254C22×Dic3—124no—
255C22×Dic3C22×D312yes—
256C22×Dic3C22×D312yes—
257C22×Dic3C22×D312yes—
258C22×Dic3C22×D318yes—
259C2×D12—38no—
260C2×D12—124no—
261C2×C4×D3—38no—
262C2×C4×D3—44no—
263C2×C4×D3—64no—
264C2×C4×D3—68no—
265C2×C4×D3—124no—
266C2×C4×D3C22×D318yes—
267C3×C4⋊C4—38no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—38no—
270C3×C22⋊C4C22×D318yes—
271C3×C42—38no—
272#217—38no—
273#217—38no—
274(C2×D3)⋊C4—38no—
275(C2×D3)⋊C4—38no—
276(C2×D3)⋊C4—38no—
277(C2×D3)⋊C4—38no—
278(C2×D3)⋊C4—38no—
279(C2×D3)⋊C4—64no—
280(C2×D3)⋊C4C22×D318yes—
281C4⋊Dic3—38no—
282Dic3⋊C4—38no—
283Dic3⋊C4—38no—
284Dic3⋊C4—38no—
285Dic3⋊C4—38no—
286Dic3⋊C4—38no—
287Dic3⋊C4—38no—
288Dic3⋊C4—38no—
289Dic3⋊C4—64no—
290Dic3⋊C4C22×D318yes—
291C4×Dic3—38no—
292C4×Dic3—64no—
293C4×Dic3C22×D318yes—
294C22×C32—28no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32C2×#4114yes—
298C22×C32C2×#4112yes—
299C22×C32C2×#4112yes—
300C22×C32H(5,𝔽2)12yes—
301C22×C32C2511yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—44no—
306C2×C3×D3—24no—
307C2×C3×D3—24no—
308C2×C3×D3—24no—
309C2×C3×D3—48no—
310C32×C4—22no—
311C32×C4—22no—
312C3⋊Dic3—22no—
313C3⋊Dic3—22no—
314C3⋊Dic3C2×#4114yes—
315C3×Dic3—24no—
316C3×Dic3—24no—
317C3×Dic3—24no—
318C22×D4—32no—
319C22×D4—92no—
320C22×D4—91no—
321C22×D4—91no—
322C23×C4—32no—
323C23×C4—32no—
324C23×C4—91no—
325C23×C4—91no—
326#126—916no—
327#125—98no—
328#123—916no—
329#121—98no—
330#121—916no—
331#121—98no—
332C4×D4—316no—
333C4×D4—98no—
334C4×D4—98no—
335#117—98no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—91no—
338C2×C4⋊C4—91no—
339C2×C4⋊C4—92no—
340C2×C4⋊C4—98no—
341C2×C4⋊C4—91no—
342C2×C4⋊C4—92no—
343C2×C4⋊C4—91no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—92no—
346C2×C22⋊C4—92no—
347C2×C22⋊C4—92no—
348C2×C22⋊C4—91no—
349C2×C22⋊C4D3211yescharacteristic
350C2×C42—32no—
351C2×C42—91no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—24no—
357C23×C3—34no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—64no—
361C23×C3—64no—
362C23×C3—64no—
363C23×C3#23518yes—
364C23×C3C23×D314yes—
365C23×C3C23×D312yes—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3—24no—
369C22×D3—34no—
370C22×D3—62no—
371C22×D3—64no—
372C22×D3—64no—
373C22×D3—64no—
374C22×D3—64no—
375C22×D3—122no—
376C22×D3—122no—
377C22×D3C23×D314yes—
378C3×D4—616no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—34no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—64no—
389C2×C3×C4—64no—
390C2×C3×C4—64no—
391C2×C3×C4—128no—
392C2×C3×C4—128no—
393C2×C3×C4C23×D314yes—
394C2×C3×C4C23×D314yes—
395#67—216no—
396#67—616no—
397#67—616no—
398#67—616no—
399#67—128no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—38no—
404C2×Dic3—34no—
405C2×Dic3—34no—
406C2×Dic3—34no—
407C2×Dic3—48no—
408C2×Dic3—64no—
409C2×Dic3—62no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—616no—
413C2×Dic3—64no—
414C2×Dic3—64no—
415C2×Dic3—64no—
416C2×Dic3—128no—
417C2×Dic3—122no—
418C2×Dic3C23×D314yes—
419C2×Dic3C23×D314yes—
420C2×Dic3C23×D314yes—
421D12—128no—
422C4×D3—48no—
423C4×D3—616no—
424C4×D3—128no—
425C2×C32—24no—
426C2×C32#55314yescomplemented
427C2×C32C22×#4111yescharacteristic
428C2×C32C22×#4111yescharacteristic
429C2×C32C2×H(5,𝔽2)11yescharacteristic
430C3⋊D3—42no—
431C3×D3—44no—
432C24—32no—
433C24—91no—
434C24C2×D3211yescharacteristic
435C2×D4—38no—
436C2×D4—68no—
437C2×D4—98no—
438C2×D4—94no—
439C2×D4—94no—
440C2×D4—184no—
441C2×D4—184no—
442C2×D4—364no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—38no—
447C22×C4—38no—
448C22×C4—91no—
449C22×C4—94no—
450C22×C4—91no—
451C22×C4—91no—
452C22×C4—94no—
453C22×C4—124no—
454C22×C4—124no—
455C22×C4—184no—
456C22×C4—184no—
457C22×C4C2×D3211yescharacteristic
458C22×C4C2×D3211yescharacteristic
459C4⋊C4—38no—
460C4⋊C4—94no—
461C4⋊C4—94no—
462C4⋊C4—98no—
463C4⋊C4—94no—
464C4⋊C4—98no—
465C4⋊C4—94no—
466C22⋊C4—38no—
467C22⋊C4—98no—
468C22⋊C4—98no—
469C22⋊C4—98no—
470C22⋊C4—94no—
471C22⋊C4C2×D3214yes—
472C42—38no—
473C42—94no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—216no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—48no—
485C22×C3—64no—
486C22×C3—64no—
487C22×C3—64no—
488C22×C3—128no—
489C22×C3C2×#23518yes—
490C22×C3C2×#23514yes—
491C22×C3#100414yes—
492C22×C3D3×#4114yes—
493C22×C3C24×D312yes—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—24no—
497C2×D3—48no—
498C2×D3—64no—
499C2×D3—64no—
500C2×D3—64no—
501C2×D3—64no—
502C2×D3—122no—
503C2×D3—122no—
504C2×D3—128no—
505C2×D3—128no—
506C3×C4—24no—
507C3×C4—24no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—64no—
511C3×C4—64no—
512C3×C4—64no—
513Dic3—24no—
514Dic3—24no—
515Dic3—24no—
516Dic3—38no—
517Dic3—64no—
518Dic3—64no—
519Dic3—64no—
520Dic3—64no—
521Dic3—122no—
522C32C2×#55311yescharacteristic, complemented, Sylow
523C23—22no—
524C23—22no—
525C23—34no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—92no—
530C23—182no—
531C23—182no—
532C23—182no—
533C23#379814yes—
534C23C22×D3212yes—
535C23C22×D3211yescharacteristic
536D4—616no—
537D4—188no—
538D4—188no—
539D4—368no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—34no—
543C2×C4—34no—
544C2×C4—34no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—94no—
549C2×C4—92no—
550C2×C4—92no—
551C2×C4—92no—
552C2×C4—128no—
553C2×C4—128no—
554C2×C4—188no—
555C2×C4—188no—
556C2×C4—182no—
557C2×C4—182no—
558C2×C4C22×D3212yes—
559C2×C4C22×D3212yes—
560C2×C3—28no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—24no—
565C2×C3—44no—
566C2×C3—124no—
567C2×C3#554018yescomplemented
568C2×C3C22×#23512yes—
569C2×C3C2×#100412yes—
570C2×C3C2×D3×#4112yes—
571D3—44no—
572D3—124no—
573D3—124no—
574C22—22no—
575C22—22no—
576C22—28no—
577C22—64no—
578C22—64no—
579C22—64no—
580C22—128no—
581C22—182no—
582C22—182no—
583C22—182no—
584C22—364no—
585C22C2×#379814yes—
586C22C2×#379812yes—
587C22#6431212yes—
588C22#6431312yes—
589C22C23×D3211yescharacteristic
590C4—22no—
591C4—22no—
592C4—64no—
593C4—64no—
594C4—64no—
595C4—94no—
596C4—182no—
597C4—182no—
598C3—21no—
599C3C2×#554012yescomplemented
600C2—24no—
601C2—124no—
602C2—362no—
603C2#1059423114yesdirect product factor
604C2C22×#379811yescharacteristic
605C2C2×#6431211yescharacteristic
606C2C2×#6431311yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC2×C32×C22⋊C4C22
socleC23×C32C24
hypercenterC2×C22⋊C4D32
hypocenterC32C2×#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64090C2semidirect
2. C2×C3×#67⋊C4C2semidirect
3. C2×#63980C2semidirect
4. C2×#63978C2semidirect
5. C22×#3724C2semidirect
6. C2×#63912C2semidirect
7. C2×C4×#625C2semidirect
8. C2×#63874C2semidirect
9. C2×#63870C2semidirect
10. C2×#63859C2semidirect
11. C2×#63850C2semidirect
12. #10594231C2direct (N × H)
26. #64090C22semidirect
30. C3×#67⋊C4C22semidirect
31. #63980C22semidirect
32. #63978C22semidirect
37. C2×#3724C22semidirect
42. #63912C22semidirect
43. C4×#625C22semidirect
44. #63874C22semidirect
45. #63870C22semidirect
46. #63859C22semidirect
47. #63850C22semidirect
61. C2×#67⋊C4D3semidirect
162. #67⋊C4C2×D3semidirect
426. C2×C32#553semidirect
522. C32C2×#553semidirect
567. C2×C3#5540semidirect
599. C3C2×#5540semidirect
603. C2#10594231direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#640902
C2×C3×#67⋊C42
C2×#639802
C2×#639782
C22×#37242
C2×#639122
C2×C4×#6252
C2×#638742
C2×#638702
C2×#638592
C2×#638502
#105942312
C2×#55403