Finite Group Databasebeta

← C2×#10594051 #11900087

Subgroups — C2×#10594051 #11900087

Order 1152.

3445 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3805C211yesmaximal, characteristic
2C2×#64086C211yesmaximal, characteristic, complemented
3C2×C3×#67⋊C4C212yesmaximal, complemented
4C2×Dic3×#67C212yesmaximal, complemented
5C22×#3724C211yesmaximal, characteristic, complemented
6C2×C4×#627C211yesmaximal, characteristic, complemented
7C2×#63902C211yesmaximal, characteristic, complemented
8C2×C4×#625C211yesmaximal, characteristic, complemented
9C2×#63877C212yesmaximal, complemented
10C2×D3×Dic3⋊C4C212yesmaximal, complemented
11C2×#63838C211yesmaximal, characteristic, complemented
12#10594051C2116yesmaximal, direct product factor
13C2×#5546—32nomaximal
14C23×C3⋊Dic3C2211yescharacteristic
15C22×C3×#67C2212yes—
16C23×C3×Dic3C2212yes—
17C22×C3×C4×D3C2212yes—
18C22×#627C2211yescharacteristic
19C22×#625C2211yescharacteristic
20C2×#3805C4116yescomplemented
21C2×#3805C2218yes—
22C22×D3×Dic3C2212yes—
23C22×D3×Dic3C2212yes—
24C2×C32×C22⋊C4C2211yescharacteristic
25C2×#3757C2211yescharacteristic
26C2×#3750C2211yescharacteristic
27C2×#3750C2211yescharacteristic
28C2×C4×C3⋊Dic3C2211yescharacteristic
29C2×C4×C3⋊Dic3C2211yescharacteristic
30#64086C2218yescomplemented
31C2×C3×(C2×D3)⋊C4C2212yes—
32C2×C3×Dic3⋊C4C2212yes—
33C2×C3×C4×Dic3C2212yes—
34C3×#67⋊C4C22116yescomplemented
35Dic3×#67C22116yescomplemented
36C2×#3724C2212yes—
37C2×#3724C2211yescharacteristic
38C2×#3724C2211yescharacteristic
39C2×#3724C2218yescomplemented
40C2×Dic3⋊Dic3C2212yes—
41C2×(C2×D3)⋊Dic3C2212yes—
42C2×Dic32C2212yes—
43C4×#627C2218yescomplemented
44#63902C2218yescomplemented
45C4×#625C2218yescomplemented
46#63877C22116yescomplemented
47D3×Dic3⋊C4C22116yescomplemented
48#63838C2218yescomplemented
49C22×#237—32no—
50C2×C3×C4×D4—32no—
51C2×D4×Dic3—32no—
52C2×C4×#67—32no—
53C2×C4×#67—32no—
54C22×Dic3⋊C4—32no—
55C22×Dic3⋊C4—32no—
56#5546—332no—
57C2×D3×C4⋊C4—32no—
58C2×#887—32no—
59C2×#67⋊C4—32no—
60C2×#67⋊C4D312yescomplemented
61C2×#877—32no—
62C2×#877—32no—
63C2×#877—61no—
64C2×#872—32no—
65C2×C4×Dic6—32no—
66C24×C32C2311yescharacteristic
67C23×C3×D3C2312yes—
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2311yescharacteristic
70C22×C3⋊Dic3C2×C418yescomplemented
71C22×C3⋊Dic3C2311yescharacteristic
72C22×C3⋊Dic3C2311yescharacteristic
73C22×C3⋊Dic3C2311yescharacteristic
74C22×C3⋊Dic3C2311yescharacteristic
75C22×C3⋊Dic3C2314yes—
76C2×C3×#67C2×C4116yescomplemented
77C2×C3×#67C2318yes—
78C22×C3×Dic3—44no—
79C22×C3×Dic3C2312yes—
80C22×C3×Dic3C2312yes—
81C22×C3×Dic3C2318yes—
82C22×C3×Dic3C2312yes—
83C2×C3×C4×D3—44no—
84C2×C3×C4×D3C2318yes—
85C2×#627C2×C418yescomplemented
86C2×#627C2314yes—
87C2×#625C2×C418yescomplemented
88C2×#625C2314yes—
89#3805C2×C4164yescomplemented
90C2×D3×Dic3—44no—
91C2×D3×Dic3C2×C4116yescomplemented
92C2×D3×Dic3C2318yes—
93C2×D3×Dic3C2318yes—
94C32×C22⋊C4C2314yes—
95#3757C2314yes—
96#3750C2314yes—
97#3750C2314yes—
98C4×C3⋊Dic3C2314yes—
99C4×C3⋊Dic3C2314yes—
100C3×(C2×D3)⋊C4C2318yes—
101C3×Dic3⋊C4C2318yes—
102C3×C4×Dic3C2318yes—
103#3724C2314yes—
104#3724C2318yes—
105#3724C2314yes—
106Dic3⋊Dic3C2318yes—
107(C2×D3)⋊Dic3C2318yes—
108Dic32C2318yes—
109C2×#520—91noSylow
110C22×C3×D4—32no—
111C23×C3×C4—32no—
112C23×C3×C4—32no—
113C22×#67—32no—
114C22×#67C2×D312yes—
115C23×Dic3—32no—
116C23×Dic3—32no—
117C23×Dic3—32no—
118C23×Dic3—61no—
119C23×Dic3C2×D312yes—
120C2×#237—332no—
121C2×#237—316no—
122C22×C4×D3—32no—
123C22×C4×D3—32no—
124C22×C4×D3C2×D312yes—
125C22×Dic6—32no—
126C3×C4×D4—316no—
127C2×C3×C4⋊C4—32no—
128C2×C3×C22⋊C4—21no—
129C2×C3×C22⋊C4—32no—
130C2×C3×C22⋊C4C2×D312yes—
131C2×C3×C42—32no—
132C2×#217—32no—
133C2×#217—32no—
134C2×#217—61no—
135D4×Dic3—316no—
136C4×#67—316no—
137C4×#67—316no—
138C2×(C2×D3)⋊C4—32no—
139C2×(C2×D3)⋊C4C2×D312yes—
140C2×C4⋊Dic3—32no—
141C2×Dic3⋊C4—32no—
142C2×Dic3⋊C4—32no—
143C2×Dic3⋊C4—32no—
144C2×Dic3⋊C4—32no—
145C2×Dic3⋊C4—32no—
146C2×Dic3⋊C4—32no—
147C2×Dic3⋊C4—32no—
148C2×Dic3⋊C4—316no—
149C2×Dic3⋊C4—316no—
150C2×Dic3⋊C4—61no—
151C2×Dic3⋊C4—61no—
152C2×Dic3⋊C4C2×D312yes—
153C2×C4×Dic3—32no—
154C2×C4×Dic3—32no—
155C2×C4×Dic3—32no—
156C2×C4×Dic3—32no—
157C2×C4×Dic3—61no—
158C2×C4×Dic3—61no—
159C2×C4×Dic3C2×D312yes—
160D3×C4⋊C4—316no—
161#887—316no—
162#67⋊C4—316no—
163#67⋊C4C2×D3116yescomplemented
164#877—316no—
165#877—316no—
166#877—68no—
167#872—316no—
168C4×Dic6—316no—
169C23×C32—22no—
170C23×C32—22no—
171C23×C32C22×C414yes—
172C23×C32C2412yes—
173C23×C32C2411yescharacteristic
174C22×C3×D3—24no—
175C22×C3×D3—24no—
176C22×C3×D3C22×C418yes—
177C22×C3×D3C2414yes—
178C2×C32×C4—22no—
179C2×C32×C4—22no—
180C2×C32×C4C2412yes—
181C2×C32×C4C2412yes—
182C2×C3⋊Dic3—22no—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3C22×C414yes—
185C2×C3⋊Dic3C22×C414yes—
186C2×C3⋊Dic3C22×C4116yescomplemented
187C2×C3⋊Dic3C2412yes—
188C2×C3⋊Dic3C2412yes—
189C2×C3⋊Dic3C2412yes—
190C2×C3⋊Dic3C2412yes—
191C3×#67C22×C4132yescomplemented
192C2×C3×Dic3—24no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—48no—
195C2×C3×Dic3C22×C418yes—
196C2×C3×Dic3C2414yes—
197C2×C3×Dic3C2414yes—
198C2×C3×Dic3C2414yes—
199C3×C4×D3—48no—
200#627C22×C4116yescomplemented
201#625C22×C4116yescomplemented
202D3×Dic3—48no—
203D3×Dic3C22×C4132yescomplemented
204C22×#41—91no—
205#520—916no—
206C2×C4×Q8—91no—
207C2×C4×Q8—91no—
208C2×C4×D4—32no—
209C2×C4×D4—91no—
210C2×C4×D4—91no—
211C2×C4×D4—92no—
212C2×#117—92no—
213C2×#117—91no—
214C22×C4⋊C4—92no—
215C22×C4⋊C4—91no—
216C24×C3—21no—
217C24×C3—32no—
218C24×C3C22×D312yes—
219C23×D3C22×D312yes—
220C2×C3×D4—316no—
221C2×C3×D4—38no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—32no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—38no—
228C22×C3×C4—38no—
229C22×C3×C4—124no—
230C22×C3×C4—124no—
231C22×C3×C4C22×D312yes—
232C22×C3×C4C22×D312yes—
233C2×#67—316no—
234C2×#67—38no—
235C2×#67C4×D3116yescomplemented
236C2×#67C22×D318yes—
237C22×Dic3—316no—
238C22×Dic3—38no—
239C22×Dic3—32no—
240C22×Dic3—32no—
241C22×Dic3—316no—
242C22×Dic3—38no—
243C22×Dic3—32no—
244C22×Dic3—32no—
245C22×Dic3—38no—
246C22×Dic3—44no—
247C22×Dic3—64no—
248C22×Dic3—61no—
249C22×Dic3—61no—
250C22×Dic3—68no—
251C22×Dic3—61no—
252C22×Dic3—61no—
253C22×Dic3—124no—
254C22×Dic3C22×D312yes—
255C22×Dic3C22×D312yes—
256C22×Dic3C22×D318yes—
257C22×Dic3C22×D312yes—
258#237—3128no—
259C2×C4×D3—316no—
260C2×C4×D3—38no—
261C2×C4×D3—38no—
262C2×C4×D3—44no—
263C2×C4×D3—124no—
264C2×C4×D3C22×D318yes—
265C2×Dic6—316no—
266C2×Dic6—38no—
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#217—64no—
275(C2×D3)⋊C4—38no—
276(C2×D3)⋊C4C22×D318yes—
277C4⋊Dic3—38no—
278Dic3⋊C4—38no—
279Dic3⋊C4—38no—
280Dic3⋊C4—38no—
281Dic3⋊C4—38no—
282Dic3⋊C4—38no—
283Dic3⋊C4—38no—
284Dic3⋊C4—38no—
285Dic3⋊C4—64no—
286Dic3⋊C4—64no—
287Dic3⋊C4C22×D318yes—
288C4×Dic3—38no—
289C4×Dic3—38no—
290C4×Dic3—38no—
291C4×Dic3—38no—
292C4×Dic3—64no—
293C4×Dic3—64no—
294C4×Dic3C22×D318yes—
295C22×C32—28no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32C23×C414yes—
299C22×C32C23×C412yes—
300C22×C32H(5,𝔽2)12yes—
301C22×C322−1+412yes—
302C22×C32C2511yescharacteristic
303C2×C3×D3—216no—
304C2×C3×D3—24no—
305C2×C3×D3—24no—
306C2×C3×D3C23×C418yes—
307C32×C4—22no—
308C32×C4—22no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3C23×C414yes—
312C3⋊Dic3C23×C414yes—
313C3×Dic3—24no—
314C3×Dic3—24no—
315C3×Dic3C23×C418yes—
316C2×#41—98no—
317C2×#41—916no—
318C22×Q8—91no—
319C22×D4—32no—
320C22×D4—91no—
321C23×C4—32no—
322C23×C4—32no—
323C23×C4—92no—
324C23×C4—92no—
325C23×C4—91no—
326C4×Q8—98no—
327C4×Q8—98no—
328C4×D4—316no—
329C4×D4—98no—
330C4×D4—98no—
331C4×D4—916no—
332#117—916no—
333#117—98no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—916no—
336C2×C4⋊C4—92no—
337C2×C4⋊C4—91no—
338C2×C4⋊C4—91no—
339C2×C4⋊C4—92no—
340C2×C4⋊C4—98no—
341C2×C4⋊C4—91no—
342C2×C4⋊C4—91no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—92no—
345C2×C22⋊C4—91no—
346C2×C22⋊C4D3211yescharacteristic
347C2×C42—32no—
348C2×C42—92no—
349C2×C42—91no—
350C2×C42—91no—
351C23×C3—24no—
352C23×C3—24no—
353C23×C3—22no—
354C23×C3—21no—
355C23×C3—24no—
356C23×C3—38no—
357C23×C3—34no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—64no—
361C23×C3—64no—
362C23×C3C2×C4×D318yes—
363C23×C3C23×D314yes—
364C23×C3C23×D312yes—
365C22×D3—24no—
366C22×D3—24no—
367C22×D3C2×C4×D318yes—
368C22×D3C23×D314yes—
369C3×D4—332no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—38no—
375C2×C3×C4—34no—
376C2×C3×C4—34no—
377C2×C3×C4—34no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—64no—
381C2×C3×C4—64no—
382C2×C3×C4—128no—
383C2×C3×C4—128no—
384C2×C3×C4C23×D314yes—
385C2×C3×C4C23×D314yes—
386#67—332no—
387#67C2×C4×D3132yescomplemented
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—38no—
391C2×Dic3—38no—
392C2×Dic3—332no—
393C2×Dic3—34no—
394C2×Dic3—34no—
395C2×Dic3—332no—
396C2×Dic3—34no—
397C2×Dic3—34no—
398C2×Dic3—48no—
399C2×Dic3—64no—
400C2×Dic3—64no—
401C2×Dic3—616no—
402C2×Dic3—62no—
403C2×Dic3—62no—
404C2×Dic3—62no—
405C2×Dic3—62no—
406C2×Dic3—64no—
407C2×Dic3—64no—
408C2×Dic3—122no—
409C2×Dic3—122no—
410C2×Dic3—128no—
411C2×Dic3C2×C4×D318yes—
412C2×Dic3C23×D314yes—
413C2×Dic3C23×D314yes—
414C2×Dic3C23×D314yes—
415C4×D3—332no—
416C4×D3—48no—
417C4×D3—128no—
418Dic6—332no—
419C2×C32—24no—
420C2×C32#52014yescomplemented
421C2×C32C24×C411yescharacteristic
422C2×C32C2×H(5,𝔽2)11yescharacteristic
423C2×C32C2×2−1+411yescharacteristic
424C3×D3—28no—
425C24—32no—
426C24C2×D3211yescharacteristic
427#41—964no—
428C2×Q8—94no—
429C2×Q8—98no—
430C2×D4—38no—
431C2×D4—316no—
432C2×D4—94no—
433C2×D4—98no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—38no—
438C22×C4—38no—
439C22×C4—98no—
440C22×C4—916no—
441C22×C4—98no—
442C22×C4—94no—
443C22×C4—91no—
444C22×C4—91no—
445C22×C4—98no—
446C22×C4—91no—
447C22×C4—91no—
448C22×C4—124no—
449C22×C4—124no—
450C22×C4—364no—
451C22×C4C2×D3211yescharacteristic
452C22×C4C2×D3211yescharacteristic
453C4⋊C4—38no—
454C4⋊C4—98no—
455C4⋊C4—94no—
456C4⋊C4—94no—
457C4⋊C4—98no—
458C4⋊C4—94no—
459C4⋊C4—94no—
460C22⋊C4—38no—
461C22⋊C4—98no—
462C22⋊C4—94no—
463C22⋊C4C2×D3214yes—
464C42—38no—
465C42—98no—
466C42—94no—
467C42—94no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—216no—
471C22×C3—22no—
472C22×C3—24no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—38no—
477C22×C3—48no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—64no—
481C22×C3—64no—
482C22×C3—616no—
483C22×C3C22×C4×D318yes—
484C22×C3C22×C4×D314yes—
485C22×C3#100414yes—
486C22×C3#101014yes—
487C22×C3C24×D312yes—
488C2×D3—24no—
489C2×D3—24no—
490C2×D3—216no—
491C2×D3C22×C4×D318yes—
492C3×C4—24no—
493C3×C4—24no—
494C3×C4—38no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—64no—
498C3×C4—64no—
499Dic3—24no—
500Dic3—24no—
501Dic3—38no—
502Dic3—38no—
503Dic3—64no—
504Dic3—64no—
505Dic3—64no—
506Dic3—64no—
507Dic3—122no—
508Dic3—122no—
509Dic3C22×C4×D318yes—
510C32C2×#52011yescharacteristic, complemented, Sylow
511C23—22no—
512C23—22no—
513C23—38no—
514C23—34no—
515C23—64no—
516C23—64no—
517C23C4×D3214yes—
518C23C22×D3212yes—
519C23C22×D3211yescharacteristic
520Q8—916no—
521D4—332no—
522D4—916no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—38no—
526C2×C4—34no—
527C2×C4—34no—
528C2×C4—34no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—932no—
532C2×C4—94no—
533C2×C4—94no—
534C2×C4—916no—
535C2×C4—92no—
536C2×C4—92no—
537C2×C4—92no—
538C2×C4—92no—
539C2×C4—128no—
540C2×C4—128no—
541C2×C4—182no—
542C2×C4—182no—
543C2×C4—368no—
544C2×C4C22×D3212yes—
545C2×C4C22×D3212yes—
546C2×C3—28no—
547C2×C3—24no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—44no—
552C2×C3—68no—
553C2×C3#554618yescomplemented
554C2×C3C23×C4×D312yes—
555C2×C3C2×#100412yes—
556C2×C3C2×#101012yes—
557D3—28no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—38no—
562C22—616no—
563C22—64no—
564C22—64no—
565C22C2×C4×D3214yes—
566C22C2×C4×D3212yes—
567C22#6430912yes—
568C22#6431312yes—
569C22C23×D3211yescharacteristic
570C4—22no—
571C4—22no—
572C4—38no—
573C4—64no—
574C4—64no—
575C4—94no—
576C4—94no—
577C4—182no—
578C4—182no—
579C3—21no—
580C3C2×#554612yescomplemented
581C2—24no—
582C2—68no—
583C2#1059405114yesdirect product factor
584C2C22×C4×D3211yescharacteristic
585C2C2×#6430911yescharacteristic
586C2C2×#6431311yescharacteristic
subgroupG/H
centerC23C22×D32
commutatorC2×C32C24×C4
FrattiniC22C23×D32
FittingC2×C32×C22⋊C4C22
socleC23×C32C24
prime radicalC22×#3805C2
cosocleC22×C32C25
hypercenterC2×C22⋊C4D32
hypocenterC32C2×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#64086C2semidirect
3. C2×C3×#67⋊C4C2semidirect
4. C2×Dic3×#67C2semidirect
5. C22×#3724C2semidirect
6. C2×C4×#627C2semidirect
7. C2×#63902C2semidirect
8. C2×C4×#625C2semidirect
9. C2×#63877C2semidirect
10. C2×D3×Dic3⋊C4C2semidirect
11. C2×#63838C2semidirect
12. #10594051C2direct (N × H)
20. C2×#3805C4semidirect
30. #64086C22semidirect
34. C3×#67⋊C4C22semidirect
35. Dic3×#67C22semidirect
39. C2×#3724C22semidirect
43. C4×#627C22semidirect
44. #63902C22semidirect
45. C4×#625C22semidirect
46. #63877C22semidirect
47. D3×Dic3⋊C4C22semidirect
48. #63838C22semidirect
60. C2×#67⋊C4D3semidirect
70. C22×C3⋊Dic3C2×C4semidirect
76. C2×C3×#67C2×C4semidirect
85. C2×#627C2×C4semidirect
87. C2×#625C2×C4semidirect
89. #3805C2×C4semidirect
91. C2×D3×Dic3C2×C4semidirect
163. #67⋊C4C2×D3semidirect
186. C2×C3⋊Dic3C22×C4semidirect
191. C3×#67C22×C4semidirect
200. #627C22×C4semidirect
201. #625C22×C4semidirect
203. D3×Dic3C22×C4semidirect
235. C2×#67C4×D3semidirect
387. #67C2×C4×D3semidirect
420. C2×C32#520semidirect
510. C32C2×#520semidirect
553. C2×C3#5546semidirect
580. C3C2×#5546semidirect
583. C2#10594051direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#38052
C2×#640862
C2×C3×#67⋊C42
C2×Dic3×#672
C22×#37242
C2×C4×#6272
C2×#639022
C2×C4×#6252
C2×#638772
C2×D3×Dic3⋊C42
C2×#638382
#105940512
C2×#55463