Finite Group Databasebeta

← C2×#10594974 #11901440

Subgroups — C2×#10594974 #11901440

Order 1152.

2497 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3750C211yesmaximal, characteristic, complemented
2C2×C3×#930C211yesmaximal, characteristic
3C2×C3×#924C211yesmaximal, characteristic, complemented
4C2×#67⋊Dic3C212yesmaximal, complemented
5C2×#63962C212yesmaximal, complemented
6#10594974C2116yesmaximal, direct product factor
7C2×#63887C212yesmaximal, complemented
8C2×#63884C212yesmaximal, complemented
9C2×#63858C212yesmaximal, complemented
10C2×#63846C212yesmaximal, complemented
11C2×#5645—31nomaximal
12C2×#5602—31nomaximal
13C23×C32×C4C2211yescharacteristic
14C23×C3⋊Dic3C2211yescharacteristic
15C23×C3⋊Dic3C2211yescharacteristic
16C22×C3×#67C2212yes—
17C22×C3×D12C2211yescharacteristic
18C22×C3×Dic6C2211yescharacteristic
19C22×D3×Dic3C2212yes—
20C2×#3750C2212yes—
21C2×#3750C2212yes—
22C2×#3750C2218yescomplemented
23C2×C3×#217C2212yes—
24C3×#930C2218yes—
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C3×#924C2218yescomplemented
29C2×C3×Dic3⋊C4C2212yes—
30#67⋊Dic3C22116yescomplemented
31C2×Dic3⋊Dic3C2212yes—
32C2×Dic3⋊Dic3C2212yes—
33C2×Dic3⋊Dic3C2212yes—
34#63962C22116yescomplemented
35C2×(C2×D3)⋊Dic3C2212yes—
36C2×(C2×D3)⋊Dic3C2212yes—
37C2×(C2×D3)⋊Dic3C2212yes—
38C2×Dic32C2212yes—
39#63887C22116yescomplemented
40#63884C22116yescomplemented
41#63858C22116yescomplemented
42#63846C22116yescomplemented
43C2×C3×C22⋊Q8—31no—
44C2×C3×#121—31no—
45C2×#946—32no—
46C2×#935—32no—
47C2×#935—32no—
48C2×D4×Dic3—32no—
49C2×#930D311yescharacteristic
50C2×#924D311yescharacteristic, complemented
51C22×Dic3⋊C4—31no—
52C22×Dic3⋊C4—31no—
53C22×Dic3⋊C4—61no—
54#5645—316no—
55#5602—316no—
56C2×#898—32no—
57C2×#894—32no—
58C2×#890—32no—
59C2×#889—32no—
60C2×#886—32no—
61C2×#67⋊C4—32no—
62C2×#879—32no—
63C2×#878—32no—
64C24×C32C2311yescharacteristic
65C23×C3×D3C2312yes—
66C22×C32×C4—42no—
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2314yes—
70C22×C3⋊Dic3—24no—
71C22×C3⋊Dic3—42no—
72C22×C3⋊Dic3C2312yes—
73C22×C3⋊Dic3C2312yes—
74C22×C3⋊Dic3C2314yes—
75C22×C3⋊Dic3C2314yes—
76C2×C3×#67—28no—
77C2×C3×#67C2318yes—
78C22×C3×Dic3C2312yes—
79C22×C3×Dic3C2312yes—
80C22×C3×Dic3C2312yes—
81C2×C3×D12—42no—
82C2×C3×D12C2314yes—
83C2×C3×Dic6—42no—
84C2×C3×Dic6C2314yes—
85C2×D3×Dic3—44no—
86C2×D3×Dic3C2318yes—
87#3750C2318yes—
88#3750C2318yes—
89C3×#217C2318yes—
90C3×(C2×D3)⋊C4C2318yes—
91C3×C4⋊Dic3C2314yes—
92C3×C4⋊Dic3C2314yes—
93C3×Dic3⋊C4C2318yes—
94Dic3⋊Dic3C2318yes—
95Dic3⋊Dic3C2318yes—
96Dic3⋊Dic3C2318yes—
97(C2×D3)⋊Dic3C2318yes—
98(C2×D3)⋊Dic3C2318yes—
99(C2×D3)⋊Dic3C2318yes—
100Dic32C2318yes—
101C2×#539—91noSylow
102C22×C3×Q8—31no—
103C22×C3×D4—32no—
104C22×C3×D4—31no—
105C23×C3×C4—21no—
106C23×C3×C4C2×D311yescharacteristic
107C23×C3×C4C2×D311yescharacteristic
108C22×#67C2×D312yes—
109C23×Dic3—31no—
110C23×Dic3—31no—
111C23×Dic3—31no—
112C23×Dic3—31no—
113C23×Dic3—32no—
114C23×Dic3—61no—
115C23×Dic3—61no—
116C22×D12C2×D311yescharacteristic
117C22×C4×D3—32no—
118C22×Dic6C2×D311yescharacteristic
119C3×C22⋊Q8—38no—
120C3×#121—38no—
121C2×C3×C4⋊C4—32no—
122C2×C3×C4⋊C4—31no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C22⋊C4—32no—
125C2×C3×C22⋊C4—32no—
126C2×#217—32no—
127C2×#217—32no—
128C2×#217—32no—
129C2×#217C2×D312yes—
130#946—316no—
131#935—316no—
132#935—316no—
133D4×Dic3—316no—
134#930C2×D318yes—
135C2×(C2×D3)⋊C4—32no—
136C2×(C2×D3)⋊C4—32no—
137C2×(C2×D3)⋊C4—32no—
138C2×(C2×D3)⋊C4C2×D312yes—
139C2×C4⋊Dic3—32no—
140C2×C4⋊Dic3—32no—
141C2×C4⋊Dic3—32no—
142C2×C4⋊Dic3C2×D311yescharacteristic
143C2×C4⋊Dic3C2×D311yescharacteristic
144#924C2×D318yescomplemented
145C2×Dic3⋊C4—38no—
146C2×Dic3⋊C4—32no—
147C2×Dic3⋊C4—32no—
148C2×Dic3⋊C4—32no—
149C2×Dic3⋊C4—32no—
150C2×Dic3⋊C4—38no—
151C2×Dic3⋊C4—32no—
152C2×Dic3⋊C4—32no—
153C2×Dic3⋊C4—32no—
154C2×Dic3⋊C4—62no—
155C2×Dic3⋊C4—62no—
156C2×Dic3⋊C4—68no—
157C2×Dic3⋊C4C2×D312yes—
158C2×C4×Dic3—32no—
159C2×C4×Dic3—32no—
160#898—316no—
161#894—316no—
162#890—316no—
163#889—316no—
164#886—316no—
165#67⋊C4—316no—
166#879—316no—
167#878—316no—
168C23×C32—22no—
169C23×C32—22no—
170C23×C32#4114yes—
171C23×C32C2412yes—
172C23×C32C2411yescharacteristic
173C22×C3×D3—24no—
174C22×C3×D3—24no—
175C22×C3×D3—24no—
176C22×C3×D3C2414yes—
177C2×C32×C4—22no—
178C2×C32×C4—22no—
179C2×C32×C4—44no—
180C2×C32×C4C2412yes—
181C2×C32×C4C2412yes—
182C2×C3⋊Dic3—28no—
183C2×C3⋊Dic3—24no—
184C2×C3⋊Dic3—24no—
185C2×C3⋊Dic3—44no—
186C2×C3⋊Dic3C2414yes—
187C2×C3⋊Dic3C2414yes—
188C3×#67—216no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—24no—
192C2×C3×Dic3C2414yes—
193C2×C3×Dic3C2414yes—
194C2×C3×Dic3C2414yes—
195C3×D12—44no—
196C3×Dic6—44no—
197D3×Dic3—48no—
198#539—916no—
199C2×#126—92no—
200C2×#125—92no—
201C2×#123—92no—
202C2×#123—92no—
203C2×C22⋊Q8—31no—
204C2×C22⋊Q8—92no—
205C2×#121—31no—
206C2×C4×D4—92no—
207C22×C4⋊C4—91no—
208C24×C3—21no—
209C24×C3—32no—
210C24×C3C22×D311yescharacteristic
211C24×C3C22×D311yescharacteristic
212C23×D3C22×D312yes—
213C2×C3×Q8—34no—
214C2×C3×Q8—122no—
215C2×C3×D4—38no—
216C2×C3×D4—34no—
217C2×C3×D4—68no—
218C2×C3×D4—122no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—24no—
222C22×C3×C4—32no—
223C22×C3×C4—32no—
224C22×C3×C4—32no—
225C22×C3×C4—42no—
226C22×C3×C4—42no—
227C22×C3×C4—44no—
228C22×C3×C4C22×D311yescharacteristic
229C22×C3×C4C22×D311yescharacteristic
230C22×C3×C4C22×D311yescharacteristic
231C22×C3×C4C22×D311yescharacteristic
232C22×C3×C4C22×D314yes—
233C22×C3×C4C22×D314yes—
234C2×#67—28no—
235C2×#67C22×D318yes—
236C22×Dic3—34no—
237C22×Dic3—32no—
238C22×Dic3—32no—
239C22×Dic3—32no—
240C22×Dic3—32no—
241C22×Dic3—34no—
242C22×Dic3—34no—
243C22×Dic3—34no—
244C22×Dic3—38no—
245C22×Dic3—64no—
246C22×Dic3—62no—
247C22×Dic3—62no—
248C22×Dic3—64no—
249C22×Dic3—68no—
250C22×Dic3—64no—
251C22×Dic3—64no—
252C22×Dic3—124no—
253C22×Dic3—122no—
254C22×Dic3—122no—
255C22×Dic3—124no—
256C22×Dic3C22×D312yes—
257C22×Dic3C22×D312yes—
258C22×Dic3C22×D312yes—
259C2×D12—42no—
260C2×D12C22×D314yes—
261C2×C4×D3—38no—
262C2×C4×D3—124no—
263C2×Dic6—42no—
264C2×Dic6C22×D314yes—
265C3×C4⋊C4—34no—
266C3×C4⋊C4—38no—
267C3×C4⋊C4—34no—
268C3×C22⋊C4—38no—
269C3×C22⋊C4—38no—
270#217—38no—
271#217—38no—
272#217—38no—
273#217C22×D318yes—
274(C2×D3)⋊C4—38no—
275(C2×D3)⋊C4—38no—
276(C2×D3)⋊C4—38no—
277(C2×D3)⋊C4C22×D318yes—
278C4⋊Dic3—38no—
279C4⋊Dic3—38no—
280C4⋊Dic3—38no—
281C4⋊Dic3C22×D314yes—
282C4⋊Dic3C22×D314yes—
283Dic3⋊C4—38no—
284Dic3⋊C4—38no—
285Dic3⋊C4—38no—
286Dic3⋊C4—38no—
287Dic3⋊C4—38no—
288Dic3⋊C4—38no—
289Dic3⋊C4—38no—
290Dic3⋊C4—68no—
291Dic3⋊C4—68no—
292Dic3⋊C4C22×D318yes—
293C4×Dic3—38no—
294C4×Dic3—38no—
295C22×C32—28no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32C2×#4114yes—
299C22×C32C2×#4112yes—
300C22×C32H(5,𝔽2)12yes—
301C22×C322−1+412yes—
302C22×C32C2511yescharacteristic
303C2×C3×D3—24no—
304C2×C3×D3—24no—
305C2×C3×D3—24no—
306C2×C3×D3—48no—
307C32×C4—22no—
308C32×C4—22no—
309C3⋊Dic3—24no—
310C3⋊Dic3—24no—
311C3×Dic3—24no—
312C3×Dic3—24no—
313C3×Dic3—24no—
314C22×Q8—31no—
315C22×D4—32no—
316C22×D4—31no—
317C23×C4—92no—
318C23×C4—91no—
319C23×C4—91no—
320C23×C4D3211yescharacteristic
321#126—916no—
322#125—916no—
323#123—916no—
324#123—916no—
325C22⋊Q8—38no—
326C22⋊Q8—916no—
327#121—38no—
328C4×D4—916no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—92no—
333C2×C4⋊C4—92no—
334C2×C4⋊C4—92no—
335C2×C4⋊C4—92no—
336C2×C4⋊C4—92no—
337C2×C4⋊C4—98no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—92no—
341C2×C22⋊C4—92no—
342C2×C22⋊C4—92no—
343C2×C42—92no—
344C23×C3—24no—
345C23×C3—24no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—21no—
352C23×C3—24no—
353C23×C3—34no—
354C23×C3—64no—
355C23×C3—64no—
356C23×C3—64no—
357C23×C3#23714yes—
358C23×C3#23914yes—
359C23×C3C23×D312yes—
360C23×C3C23×D312yes—
361C23×C3C23×D311yescharacteristic
362C23×C3C23×D311yescharacteristic
363C22×D3—24no—
364C22×D3—24no—
365C22×D3—24no—
366C22×D3C23×D314yes—
367C3×Q8—124no—
368C3×D4—616no—
369C3×D4—124no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—34no—
377C2×C3×C4—34no—
378C2×C3×C4—34no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—44no—
382C2×C3×C4—44no—
383C2×C3×C4—48no—
384C2×C3×C4—64no—
385C2×C3×C4—64no—
386C2×C3×C4—64no—
387C2×C3×C4C23×D312yes—
388C2×C3×C4C23×D312yes—
389C2×C3×C4C23×D312yes—
390C2×C3×C4C23×D312yes—
391#67—216no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—34no—
396C2×Dic3—34no—
397C2×Dic3—34no—
398C2×Dic3—34no—
399C2×Dic3—64no—
400C2×Dic3—64no—
401C2×Dic3—616no—
402C2×Dic3—68no—
403C2×Dic3—68no—
404C2×Dic3—64no—
405C2×Dic3—64no—
406C2×Dic3—64no—
407C2×Dic3—68no—
408C2×Dic3—64no—
409C2×Dic3—124no—
410C2×Dic3—128no—
411C2×Dic3—124no—
412C2×Dic3—124no—
413C2×Dic3—128no—
414C2×Dic3C23×D314yes—
415C2×Dic3C23×D314yes—
416C2×Dic3C23×D314yes—
417D12—44no—
418C4×D3—128no—
419Dic6—44no—
420C2×C32—24no—
421C2×C32#53914yescomplemented
422C2×C32C22×#4111yescharacteristic
423C2×C32C2×H(5,𝔽2)11yescharacteristic
424C2×C32C2×2−1+411yescharacteristic
425C3×D3—44no—
426C24—32no—
427C24C2×D3211yescharacteristic
428C2×Q8—34no—
429C2×Q8—122no—
430C2×D4—38no—
431C2×D4—34no—
432C2×D4—68no—
433C2×D4—122no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—42no—
438C22×C4—98no—
439C22×C4—94no—
440C22×C4—92no—
441C22×C4—92no—
442C22×C4—94no—
443C22×C4—184no—
444C22×C4—364no—
445C22×C4—362no—
446C22×C4C2×D3211yescharacteristic
447C22×C4C2×D3211yescharacteristic
448C22×C4C2×D3214yes—
449C4⋊C4—34no—
450C4⋊C4—38no—
451C4⋊C4—34no—
452C4⋊C4—98no—
453C4⋊C4—98no—
454C4⋊C4—98no—
455C4⋊C4—98no—
456C4⋊C4—98no—
457C22⋊C4—38no—
458C22⋊C4—38no—
459C22⋊C4—98no—
460C22⋊C4—98no—
461C22⋊C4—98no—
462C42—98no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—216no—
466C22×C3—28no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—28no—
472C22×C3—24no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—64no—
478C22×C3—64no—
479C22×C3—64no—
480C22×C3—128no—
481C22×C3C2×#23714yes—
482C22×C3C2×#23712yes—
483C22×C3#100412yes—
484C22×C3#100412yes—
485C22×C3C2×#23914yes—
486C22×C3C2×#23912yes—
487C22×C3#101012yes—
488C22×C3#101012yes—
489C22×C3C24×D311yescharacteristic
490C22×C3C24×D311yescharacteristic
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—48no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—64no—
502C3×C4—64no—
503C3×C4—64no—
504Dic3—24no—
505Dic3—24no—
506Dic3—24no—
507Dic3—68no—
508Dic3—64no—
509Dic3—64no—
510Dic3—64no—
511Dic3—64no—
512Dic3—124no—
513C32C2×#53911yescharacteristic, complemented, Sylow
514C23—22no—
515C23—22no—
516C23—34no—
517C23—64no—
518C23—64no—
519C23—64no—
520C23#379614yes—
521C23C22×D3212yes—
522C23C22×D3211yescharacteristic
523Q8—124no—
524D4—616no—
525D4—124no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—34no—
529C2×C4—34no—
530C2×C4—34no—
531C2×C4—44no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—94no—
536C2×C4—94no—
537C2×C4—188no—
538C2×C4—184no—
539C2×C4—184no—
540C2×C4—368no—
541C2×C4—364no—
542C2×C4C22×D3212yes—
543C2×C4C22×D3212yes—
544C2×C3—28no—
545C2×C3—24no—
546C2×C3—24no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—24no—
551C2×C3—124no—
552C2×C3#560214yescomplemented
553C2×C3#564514yescomplemented
554C2×C3C22×#23711yescharacteristic
555C2×C3C2×#100411yescharacteristic
556C2×C3C2×#100411yescharacteristic
557C2×C3C22×#23911yescharacteristic
558C2×C3C2×#101011yescharacteristic
559C2×C3C2×#101011yescharacteristic
560D3—44no—
561C22—22no—
562C22—22no—
563C22—28no—
564C22—64no—
565C22—64no—
566C22—64no—
567C22—128no—
568C22C2×#379614yes—
569C22C2×#379612yes—
570C22#6429812yes—
571C22#6433012yes—
572C22C23×D3211yescharacteristic
573C4—22no—
574C4—22no—
575C4—64no—
576C4—64no—
577C4—64no—
578C4—184no—
579C4—184no—
580C3—21no—
581C3C2×#560211yescharacteristic, complemented
582C3C2×#564511yescharacteristic, complemented
583C2—24no—
584C2—124no—
585C2#1059497414yesdirect product factor
586C2C22×#379611yescharacteristic
587C2C2×#6429811yescharacteristic
588C2C2×#6433011yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC23×C32×C4C22
socleC23×C32C24
prime radicalC2×C3×#930C2
hypercenterC23×C4D32
hypocenterC32C2×#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×#3750C2semidirect
3. C2×C3×#924C2semidirect
4. C2×#67⋊Dic3C2semidirect
5. C2×#63962C2semidirect
6. #10594974C2direct (N × H)
7. C2×#63887C2semidirect
8. C2×#63884C2semidirect
9. C2×#63858C2semidirect
10. C2×#63846C2semidirect
22. C2×#3750C22semidirect
28. C3×#924C22semidirect
30. #67⋊Dic3C22semidirect
34. #63962C22semidirect
39. #63887C22semidirect
40. #63884C22semidirect
41. #63858C22semidirect
42. #63846C22semidirect
50. C2×#924D3semidirect
144. #924C2×D3semidirect
421. C2×C32#539semidirect
513. C32C2×#539semidirect
552. C2×C3#5602semidirect
553. C2×C3#5645semidirect
581. C3C2×#5602semidirect
582. C3C2×#5645semidirect
585. C2#10594974direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37502
C2×C3×#9302
C2×C3×#9242
C2×#67⋊Dic32
C2×#639622
#105949742
C2×#638872
C2×#638842
C2×#638582
C2×#638462
C2×#56453
C2×#56023