Finite Group Databasebeta

← C2×#11722482 #13164712

Subgroups — C2×#11722482 #13164712

Order 1728.

2943 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3C22×#634⋊C3C211yesmaximal, characteristic, complemented
4C2×#88671C214yesmaximal, complemented
5C2×#88652C212yesmaximal, complemented
6C2×#88651C212yesmaximal, complemented
7#11722482C2116yesmaximal, direct product factor
8C2×#88649C212yesmaximal, complemented
9C2×((C3⋊D3)⋊C3)⋊Q8C212yesmaximal, complemented
10C2×#64298—31nomaximal
11C2×#64297—31nomaximal
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6235C2212yes—
14#88740C2218yescomplemented
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2212yes—
17C2×#6229C2211yescharacteristic
18C2×#638⋊C3C2212yes—
19C22×(C3⋊Dic3)⋊C3C2212yes—
20#3828⋊C3C2218yescomplemented
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
22C2×C4×(C3⋊D3)⋊C3C2212yes—
23C2×#634⋊C3C2212yes—
24C2×#634⋊C3C2212yes—
25C2×#634⋊C3C2211yescharacteristic
26C2×#634⋊C3C2218yescomplemented
27C2×#6200C2214yes—
28C2×#6199C2214yes—
29#88671C22132yescomplemented
30C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
31C2×#6196C2214yes—
32#88652C22116yescomplemented
33#88651C22116yescomplemented
34#88649C22116yescomplemented
35((C3⋊D3)⋊C3)⋊Q8C22116yescomplemented
36C2×#3828D311yescharacteristic, complemented
37C22×#634D311yescharacteristic, complemented
38C2×C3×#235—31no—
39C2×C3×#235—31no—
40C2×C3×#235—31no—
41C2×C3×#235—61no—
42C22×C3×Dic6—31no—
43C2×#3805—34no—
44C2×#3804—34no—
45C2×#3799—32no—
46C2×#3797—32no—
47#64298—316no—
48#64297—316no—
49C2×#3796—32no—
50C2×#3795—32no—
51C2×#3795—32no—
52C2×D3×Dic6—32no—
53C2×D3×Dic6—32no—
54C2×D3×Dic6—32no—
55C23×H(3,𝔽3)C2311yescharacteristic
56C22×#272C2312yes—
57C22×(C3⋊D3)⋊C3C2312yes—
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2314yes—
61#6235C2318yes—
62C2×#1084C2312yes—
63C4⋊#272C2314yes—
64C4×#272C2318yes—
65#6229C2314yes—
66#638⋊C3C2318yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2314yes—
69C2×(C3⋊Dic3)⋊C3C2318yes—
70C4⋊((C3⋊D3)⋊C3)C2314yes—
71C4×(C3⋊D3)⋊C3C2318yes—
72#634⋊C3C2318yes—
73#634⋊C3C2318yes—
74#634⋊C3C2314yes—
75#6200C23116yes—
76#6199C23116yes—
77((C3⋊D3)⋊C3)⋊C4C23116yes—
78#6196C23116yes—
79C2×#1010—91no—
80C2×#1010—91no—
81C2×#1007—91no—
82C22×C32×C4—21no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C2×D311yescharacteristic
85C2×#638C2×D312yes—
86C22×C3⋊Dic3C2×D312yes—
87#3828C2×D318yescomplemented
88C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
89C2×C4×C3⋊D3C2×D312yes—
90C2×#634C2×D311yescharacteristic
91C2×#634C2×D312yes—
92C2×#634C2×D312yes—
93C2×#634C2×D318yescomplemented
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—32no—
97C2×C3×#67—62no—
98C22×C3×Dic3—32no—
99C3×#235—38no—
100C3×#235—38no—
101C3×#235—38no—
102C3×#235—68no—
103C2×C3×D12—31no—
104C2×C3×D12—31no—
105C2×C3×D12—31no—
106C2×C3×D12—61no—
107C2×C3×C4×D3—32no—
108C2×C3×C4×D3—32no—
109C2×C3×C4×D3—32no—
110C2×C3×C4×D3—62no—
111C2×C3×Dic6—31no—
112C2×C3×Dic6—31no—
113C2×C3×Dic6—32no—
114C2×C3×Dic6—32no—
115C2×C3×Dic6—38no—
116C2×C3×Dic6—31no—
117C2×C3×Dic6—61no—
118C2×#627—34no—
119C2×#627—34no—
120C2×#626—34no—
121C2×#625—34no—
122C2×#624—34no—
123#3805—332no—
124#3804—332no—
125C2×D3×Dic3—34no—
126C2×D3×Dic3—34no—
127C2×D3×Dic3—34no—
128#3799—316no—
129#3797—316no—
130#3796—316no—
131#3795—316no—
132#3795—316no—
133D3×Dic6—316no—
134D3×Dic6—316no—
135D3×Dic6—316no—
136C22×H(3,𝔽3)—22no—
137C22×H(3,𝔽3)C2412yes—
138C22×H(3,𝔽3)C2411yescharacteristic
139C2×#272—24no—
140C2×#272C2414yes—
141C2×(C3⋊D3)⋊C3—24no—
142C2×(C3⋊D3)⋊C3C2414yes—
143C4×H(3,𝔽3)C2412yes—
144C4×H(3,𝔽3)C2412yes—
145#1084C2414yes—
146(C3⋊Dic3)⋊C3C2414yes—
147(C3⋊Dic3)⋊C3C2418yes—
148C2×C3×#41—91no—
149C2×C3×#41—91no—
150C22×C3×Q8—91no—
151#1010—916no—
152#1010—916no—
153#1007—916no—
154C2×#239—92no—
155C2×#239—92no—
156C2×D3×Q8—92no—
157C2×D3×Q8—92no—
158C2×D3×Q8—92no—
159C2×D3×Q8—92no—
160C2×#237—92no—
161C2×#237—92no—
162C2×#237—94no—
163C2×#237—94no—
164C2×#235—31no—
165C2×#235—31no—
166C2×#235—31no—
167C2×#235—61no—
168C2×#235—91no—
169C2×#235—92no—
170C2×#235—92no—
171C2×#235—94no—
172C22×Dic6—31no—
173C22×Dic6—91no—
174C22×Dic6—92no—
175C22×Dic6—92no—
176C23×C32—21no—
177C23×C32C22×D311yescharacteristic
178C23×C32C22×D311yescharacteristic
179C22×C3⋊D3C22×D312yes—
180C22×C3×D3—32no—
181C22×C3×D3—32no—
182C22×C3×D3—32no—
183C22×C3×D3—62no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—24no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D314yes—
193#638C22×D318yes—
194C2×C3⋊Dic3C22×D314yes—
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D318yes—
197C4⋊(C3⋊D3)C22×D314yes—
198C4×C3⋊D3C22×D318yes—
199#634C22×D314yes—
200#634C22×D318yes—
201#634C22×D318yes—
202C3×#67—38no—
203C3×#67—38no—
204C3×#67—38no—
205C3×#67—68no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—62no—
212C3×D12—34no—
213C3×D12—34no—
214C3×D12—34no—
215C3×D12—64no—
216C3×C4×D3—38no—
217C3×C4×D3—38no—
218C3×C4×D3—38no—
219C3×C4×D3—68no—
220C3×Dic6—34no—
221C3×Dic6—38no—
222C3×Dic6—38no—
223C3×Dic6—34no—
224C3×Dic6—34no—
225C3×Dic6—64no—
226#627—316no—
227#627—316no—
228#626—316no—
229#625—316no—
230#624—316no—
231D3×Dic3—316no—
232D3×Dic3—316no—
233D3×Dic3—316no—
234C2×2−1+4—271noSylow
235C2×H(3,𝔽3)—22no—
236C2×H(3,𝔽3)2−1+412yescomplemented
237C2×H(3,𝔽3)C2511yescharacteristic
238#272—24no—
239(C3⋊D3)⋊C3—24no—
240C3×#41—98no—
241C3×#41—98no—
242C2×C3×Q8—91no—
243C2×C3×Q8—92no—
244C2×C3×Q8—92no—
245C2×C3×Q8—98no—
246C2×C3×Q8—91no—
247C2×C3×D4—91no—
248C2×C3×D4—92no—
249C2×C3×D4—92no—
250C2×C3×D4—91no—
251C22×C3×C4—31no—
252C22×C3×C4—31no—
253C22×C3×C4—61no—
254C22×C3×C4—92no—
255C22×C3×C4—92no—
256C22×C3×C4—92no—
257C22×C3×C4D3211yescharacteristic
258C2×#67—32no—
259C2×#67—32no—
260C2×#67—32no—
261C2×#67—62no—
262C2×#67—92no—
263C2×#67—94no—
264C2×#67—94no—
265C22×Dic3—32no—
266C22×Dic3—92no—
267C22×Dic3—94no—
268C22×Dic3—94no—
269#239—916no—
270#239—916no—
271D3×Q8—916no—
272D3×Q8—916no—
273D3×Q8—916no—
274D3×Q8—916no—
275#237—916no—
276#237—916no—
277#237—932no—
278#237—932no—
279#235—38no—
280#235—38no—
281#235—38no—
282#235—68no—
283#235—98no—
284#235—916no—
285#235—916no—
286#235—932no—
287C2×D12—31no—
288C2×D12—31no—
289C2×D12—31no—
290C2×D12—61no—
291C2×D12—91no—
292C2×D12—94no—
293C2×C4×D3—32no—
294C2×C4×D3—32no—
295C2×C4×D3—32no—
296C2×C4×D3—62no—
297C2×C4×D3—92no—
298C2×C4×D3—94no—
299C2×C4×D3—94no—
300C2×C4×D3—94no—
301C2×C4×D3—94no—
302C2×Dic6—31no—
303C2×Dic6—31no—
304C2×Dic6—32no—
305C2×Dic6—32no—
306C2×Dic6—38no—
307C2×Dic6—31no—
308C2×Dic6—61no—
309C2×Dic6—92no—
310C2×Dic6—92no—
311C2×Dic6—98no—
312C2×Dic6—91no—
313C2×Dic6—916no—
314C2×Dic6—916no—
315C2×Dic6—94no—
316C2×Dic6—94no—
317C2×Dic6—94no—
318C22×C32—22no—
319C22×C32—22no—
320C22×C32—21no—
321C22×C32—22no—
322C22×C32—42no—
323C22×C32C23×D312yes—
324C22×C32C23×D312yes—
325C22×C32C23×D311yescharacteristic
326C22×C32C23×D311yescharacteristic
327C2×C3⋊D3—24no—
328C2×C3⋊D3C23×D314yes—
329C2×C3×D3—34no—
330C2×C3×D3—34no—
331C2×C3×D3—34no—
332C2×C3×D3—64no—
333C2×C3×D3—64no—
334C2×C3×D3—64no—
335C2×C3×D3—64no—
336C2×C3×D3—124no—
337C32×C4—22no—
338C32×C4—22no—
339C32×C4C23×D312yes—
340C32×C4C23×D312yes—
341C32×C4C23×D312yes—
342C32×C4C23×D312yes—
343C3⋊Dic3C23×D318yes—
344C3⋊Dic3C23×D314yes—
345C3×Dic3—34no—
346C3×Dic3—38no—
347C3×Dic3—34no—
348C3×Dic3—34no—
349C3×Dic3—64no—
3502−1+4—2716no—
351C2×#41—91no—
352C2×#41—91no—
353C2×#41—274no—
354C2×#41—272no—
355C2×#41—272no—
356C22×Q8—91no—
357C22×Q8—272no—
358C22×Q8—272no—
359H(3,𝔽3)C2×2−1+411yescharacteristic, complemented, Sylow
360C23×C3—31no—
361C23×C3—31no—
362C23×C3—61no—
363C23×C3—92no—
364C23×C3—92no—
365C23×C3C2×D3211yescharacteristic
366C22×D3—32no—
367C22×D3—32no—
368C22×D3—32no—
369C22×D3—62no—
370C22×D3—92no—
371C3×Q8—94no—
372C3×Q8—98no—
373C3×Q8—98no—
374C3×Q8—94no—
375C3×D4—94no—
376C3×D4—98no—
377C3×D4—98no—
378C3×D4—94no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—34no—
383C2×C3×C4—31no—
384C2×C3×C4—34no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—64no—
388C2×C3×C4—98no—
389C2×C3×C4—92no—
390C2×C3×C4—94no—
391C2×C3×C4—98no—
392C2×C3×C4—92no—
393C2×C3×C4—98no—
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3214yes—
397#67—38no—
398#67—38no—
399#67—38no—
400#67—68no—
401#67—98no—
402#67—916no—
403#67—916no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—34no—
407C2×Dic3—38no—
408C2×Dic3—32no—
409C2×Dic3—62no—
410C2×Dic3—92no—
411C2×Dic3—94no—
412C2×Dic3—98no—
413C2×Dic3—916no—
414C2×Dic3—916no—
415D12—34no—
416D12—34no—
417D12—34no—
418D12—64no—
419D12—94no—
420D12—916no—
421C4×D3—38no—
422C4×D3—38no—
423C4×D3—38no—
424C4×D3—68no—
425C4×D3—98no—
426C4×D3—916no—
427C4×D3—916no—
428C4×D3—916no—
429C4×D3—916no—
430Dic6—34no—
431Dic6—34no—
432Dic6—38no—
433Dic6—38no—
434Dic6—34no—
435Dic6—64no—
436Dic6—98no—
437Dic6—98no—
438Dic6—94no—
439Dic6—916no—
440Dic6—916no—
441Dic6—916no—
442C2×C32—22no—
443C2×C32—22no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—42no—
447C2×C32#100712yescomplemented
448C2×C32#101012yescomplemented
449C2×C32C24×D311yescharacteristic
450C2×C32C24×D311yescharacteristic
451C3⋊D3—24no—
452C3×D3—64no—
453C3×D3—64no—
454C3×D3—64no—
455C3×D3—124no—
456#41—98no—
457#41—98no—
458#41—2732no—
459#41—2716no—
460#41—2716no—
461C2×Q8—92no—
462C2×Q8—92no—
463C2×Q8—98no—
464C2×Q8—91no—
465C2×Q8—91no—
466C2×Q8—274no—
467C2×Q8—2716no—
468C2×Q8—2716no—
469C2×D4—92no—
470C2×D4—92no—
471C2×D4—91no—
472C2×D4—91no—
473C2×D4—274no—
474C22×C4—92no—
475C22×C4—92no—
476C22×C4—92no—
477C22×C4—274no—
478C22×C4—274no—
479C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C22×C3—22no—
481C22×C3—32no—
482C22×C3—32no—
483C22×C3—31no—
484C22×C3—31no—
485C22×C3—62no—
486C22×C3—62no—
487C22×C3—62no—
488C22×C3—61no—
489C22×C3—94no—
490C22×C3—94no—
491C22×C3—122no—
492C22×C3—184no—
493C22×C3—184no—
494C22×C3C22×D3212yes—
495C22×C3C22×D3211yescharacteristic
496C2×D3—34no—
497C2×D3—34no—
498C2×D3—34no—
499C2×D3—64no—
500C2×D3—64no—
501C2×D3—64no—
502C2×D3—64no—
503C2×D3—94no—
504C2×D3—124no—
505C2×D3—184no—
506C3×C4—32no—
507C3×C4—32no—
508C3×C4—32no—
509C3×C4—32no—
510C3×C4—62no—
511C3×C4—62no—
512C3×C4—94no—
513C3×C4—98no—
514C3×C4—94no—
515C3×C4C22×D3212yes—
516C3×C4C22×D3212yes—
517Dic3—34no—
518Dic3—34no—
519Dic3—38no—
520Dic3—34no—
521Dic3—64no—
522Dic3—94no—
523Dic3—98no—
524C32—21no—
525C32C2×#100711yescharacteristic, complemented
526C32C2×#101011yescharacteristic, complemented
527C23—92no—
528C23—92no—
529C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530Q8—98no—
531Q8—98no—
532Q8—94no—
533Q8—94no—
534Q8—2716no—
535D4—98no—
536D4—98no—
537D4—94no—
538D4—94no—
539D4—2716no—
540C2×C4—98no—
541C2×C4—92no—
542C2×C4—92no—
543C2×C4—94no—
544C2×C4—98no—
545C2×C4—98no—
546C2×C4—2716no—
547C2×C4—2716no—
548C2×C4C2×((C3⋊D3)⋊C3)⋊C214yes—
549C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
551C2×C3—22no—
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—32no—
555C2×C3—32no—
556C2×C3—62no—
557C2×C3—62no—
558C2×C3—61no—
559C2×C3—62no—
560C2×C3—122no—
561C2×C3—184no—
562C2×C3—184no—
563C2×C3#6429712yes—
564C2×C3C23×D3211yescharacteristic
565D3—64no—
566D3—64no—
567D3—64no—
568D3—124no—
569D3—184no—
570C22—22no—
571C22—94no—
572C22—94no—
573C22—184no—
574C22—184no—
575C22C22×((C3⋊D3)⋊C3)⋊C212yes—
576C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
577C4—94no—
578C4—94no—
579C4—98no—
580C4C22×((C3⋊D3)⋊C3)⋊C212yes—
581C4C22×((C3⋊D3)⋊C3)⋊C212yes—
582C3—31no—
583C3—31no—
584C3—61no—
585C3C2×#6429711yescharacteristic
586C2—22no—
587C2—184no—
588C2—184no—
589C2#1172248212yesdirect product factor
590C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×2−1+4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C2×#3828⋊C3C2semidirect
3. C22×#634⋊C3C2semidirect
4. C2×#88671C2semidirect
5. C2×#88652C2semidirect
6. C2×#88651C2semidirect
7. #11722482C2direct (N × H)
8. C2×#88649C2semidirect
9. C2×((C3⋊D3)⋊C3)⋊Q8C2semidirect
14. #88740C22semidirect
20. #3828⋊C3C22semidirect
26. C2×#634⋊C3C22semidirect
29. #88671C22semidirect
32. #88652C22semidirect
33. #88651C22semidirect
34. #88649C22semidirect
35. ((C3⋊D3)⋊C3)⋊Q8C22semidirect
36. C2×#3828D3semidirect
37. C22×#634D3semidirect
87. #3828C2×D3semidirect
93. C2×#634C2×D3semidirect
236. C2×H(3,𝔽3)2−1+4semidirect
359. H(3,𝔽3)C2×2−1+4semidirect
447. C2×C32#1007semidirect
448. C2×C32#1010semidirect
525. C32C2×#1007semidirect
526. C32C2×#1010semidirect
589. C2#11722482direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#3828⋊C32
C22×#634⋊C32
C2×#886712
C2×#886522
C2×#886512
#117224822
C2×#886492
C2×((C3⋊D3)⋊C3)⋊Q82
C2×#642983
C2×#642973