Finite Group Databasebeta

← C2×#11721100 #13150832

Subgroups — C2×#11721100 #13150832

Order 1728.

1329 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C22×#6200C211yesmaximal, characteristic
3C2×#88542C211yesmaximal, characteristic, complemented
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5C2×#88451C211yesmaximal, characteristic, complemented
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88449C211yesmaximal, characteristic, complemented
8#11721100C218yesmaximal, direct product factor
9C2×#63905—31nomaximal
10C2×#63853—31nomaximal
11C23×#272C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3—24no—
19C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
20C2×#6200—24no—
21C2×#6200C2214yes—
22#88542C2214yescomplemented
23#3750⋊C3C2214yescomplemented
24#88451C2214yescomplemented
25#88450C2214yescomplemented
26#88449C2214yescomplemented
27C22×C4×C3⋊D3D311yescharacteristic, complemented
28C22×C3×C4×D3—31no—
29C22×#626—31no—
30C22×#625—31no—
31C2×#3750D311yescharacteristic, complemented
32C2×C3×(C2×D3)⋊C4—31no—
33C2×C3×(C2×D3)⋊C4—31no—
34C2×C3×(C2×D3)⋊C4—61no—
35C2×C3×Dic3⋊C4—31no—
36C2×Dic3⋊Dic3—31no—
37C2×Dic3⋊Dic3—31no—
38C2×#3722—31no—
39C2×(C2×D3)⋊Dic3—31no—
40C2×(C2×D3)⋊Dic3—31no—
41C2×(C2×D3)⋊Dic3—31no—
42#63905—38no—
43#63853—38no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C22×#272—22no—
46C22×#272—22no—
47C22×#272—22no—
48C22×#272C2312yes—
49C22×(C3⋊D3)⋊C3—22no—
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3D414yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)C2312yes—
55C2×#1084—22no—
56C2×#1084C2312yes—
57C2×(C3⋊Dic3)⋊C3—22no—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C4×(C3⋊D3)⋊C3—28no—
64#6200—28no—
65C2×#929—91no—
66C2×#894—91no—
67C2×#883—91no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—31no—
70C23×C3×D3—31no—
71C23×C3×D3—31no—
72C23×C3×D3—61no—
73C22×C32×C4—21no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C2×C4×C3⋊D3—24no—
80C2×C4×C3⋊D3C2×D314yescomplemented
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—61no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3—64no—
89C2×#626—34no—
90C2×#626—64no—
91C2×#625—34no—
92C2×#625—64no—
93#3750C2×D314yescomplemented
94C3×(C2×D3)⋊C4—34no—
95C3×(C2×D3)⋊C4—34no—
96C3×(C2×D3)⋊C4—64no—
97C3×Dic3⋊C4—34no—
98Dic3⋊Dic3—34no—
99Dic3⋊Dic3—34no—
100#3722—34no—
101(C2×D3)⋊Dic3—34no—
102(C2×D3)⋊Dic3—34no—
103(C2×D3)⋊Dic3—34no—
104C22×H(3,𝔽3)C2×D412yes—
105C22×H(3,𝔽3)#4112yes—
106C22×H(3,𝔽3)#4112yes—
107C22×H(3,𝔽3)C2411yescharacteristic
108C2×#272—22no—
109C2×#272—22no—
110C2×#272—22no—
111C2×#272—44no—
112C2×(C3⋊D3)⋊C3—28no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3C2×D414yes—
116C4×H(3,𝔽3)—22no—
117#1084—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121C23×C3×C4—91no—
122C22×#67—91no—
123C22×#67—91no—
124C22×D12—91no—
125C22×C4×D3—31no—
126C22×C4×D3—91no—
127C2×C3×C4⋊C4—91no—
128C2×C3×C22⋊C4—91no—
129C2×#217—91no—
130C2×#217—91no—
131#929—98no—
132C2×(C2×D3)⋊C4—31no—
133C2×(C2×D3)⋊C4—31no—
134C2×(C2×D3)⋊C4—61no—
135C2×(C2×D3)⋊C4—91no—
136C2×(C2×D3)⋊C4—91no—
137C2×(C2×D3)⋊C4—91no—
138C2×(C2×D3)⋊C4—91no—
139C2×C4⋊Dic3—91no—
140C2×Dic3⋊C4—31no—
141C2×Dic3⋊C4—91no—
142C2×Dic3⋊C4—91no—
143C2×Dic3⋊C4—91no—
144#894—98no—
145#883—98no—
146C23×C32—21no—
147C23×C32C22×D311yescharacteristic
148C23×C32C22×D311yescharacteristic
149C22×C3⋊D3—22no—
150C22×C3⋊D3—22no—
151C22×C3⋊D3#6714yes—
152C22×C3⋊D3C22×D312yes—
153C22×C3×D3—34no—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—62no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—64no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3—62no—
164C22×C3×D3—62no—
165C22×C3×D3—62no—
166C22×C3×D3—62no—
167C22×C3×D3—122no—
168C22×C3×D3—122no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4—42no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D312yes—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3C22×D312yes—
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D312yes—
181C4×C3⋊D3—28no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—122no—
194C3×C4×D3—68no—
195#626—68no—
196#625—68no—
197C2×#123—271noSylow
198C2×H(3,𝔽3)#12314yescomplemented
199C2×H(3,𝔽3)C22×D411yescharacteristic
200C2×H(3,𝔽3)C2×#4111yescharacteristic
201C2×H(3,𝔽3)C2×#4111yescharacteristic
202#272—42no—
203(C3⋊D3)⋊C3—24no—
204C24×C3—91no—
205C24×C3—91no—
206C23×D3—31no—
207C23×D3—31no—
208C23×D3—31no—
209C23×D3—61no—
210C23×D3—91no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—61no—
214C22×C3×C4—94no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—184no—
220C22×C3×C4D3211yescharacteristic
221C2×#67—94no—
222C2×#67—94no—
223C2×#67—184no—
224C2×#67—184no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—61no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C2×D12—94no—
235C2×D12—184no—
236C2×C4×D3—34no—
237C2×C4×D3—64no—
238C2×C4×D3—94no—
239C2×C4×D3—184no—
240C3×C4⋊C4—94no—
241C3×C22⋊C4—94no—
242#217—94no—
243#217—94no—
244(C2×D3)⋊C4—34no—
245(C2×D3)⋊C4—34no—
246(C2×D3)⋊C4—64no—
247(C2×D3)⋊C4—94no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251C4⋊Dic3—94no—
252Dic3⋊C4—34no—
253Dic3⋊C4—94no—
254Dic3⋊C4—94no—
255Dic3⋊C4—94no—
256C22×C32—21no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32#23512yes—
261C22×C32#23512yes—
262C22×C32#23512yes—
263C22×C32D3×D412yes—
264C22×C32#23912yes—
265C22×C32C2×#6712yes—
266C22×C32C23×D311yescharacteristic
267C22×C32C23×D311yescharacteristic
268C2×C3⋊D3—28no—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—22no—
271C2×C3⋊D3C2×#6714yes—
272C2×C3×D3—34no—
273C2×C3×D3—68no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—64no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—122no—
284C2×C3×D3—122no—
285C2×C3×D3—128no—
286C2×C3×D3—124no—
287C2×C3×D3—124no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—42no—
291C3⋊Dic3—22no—
292C3⋊Dic3—22no—
293C3⋊Dic3—22no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—122no—
300C22×D4—271no—
301C23×C4—91no—
302#123—278no—
303C2×C4⋊C4—91no—
304C2×C4⋊C4—271no—
305C2×C22⋊C4—91no—
306C2×C22⋊C4—271no—
307C2×C22⋊C4—271no—
308H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
309C23×C3—31no—
310C23×C3—31no—
311C23×C3—61no—
312C23×C3—92no—
313C23×C3—94no—
314C23×C3—92no—
315C23×C3—182no—
316C23×C3—182no—
317C23×C3—182no—
318C23×C3—182no—
319C23×C3—182no—
320C23×C3C2×D3211yescharacteristic
321C22×D3—32no—
322C22×D3—34no—
323C22×D3—32no—
324C22×D3—32no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—62no—
328C22×D3—62no—
329C22×D3—62no—
330C22×D3—62no—
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—64no—
335C22×D3—94no—
336C22×D3—92no—
337C22×D3—122no—
338C22×D3—122no—
339C22×D3—182no—
340C22×D3—182no—
341C2×C3×C4—22no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—122no—
352C2×C3×C4—188no—
353C2×C3×C4—182no—
354C2×C3×C4—182no—
355C2×C3×C4—182no—
356C2×C3×C4—182no—
357C2×C3×C4C2×D3212yes—
358#67—188no—
359#67—188no—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—62no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—122no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—182no—
378D12—188no—
379C4×D3—68no—
380C4×D3—188no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—24no—
385C2×C32#89414yescomplemented
386C2×C32#92914yescomplemented
387C2×C32C2×#23511yescharacteristic
388C2×C32C2×#23511yescharacteristic
389C2×C32C2×#23511yescharacteristic
390C2×C32C2×D3×D411yescharacteristic
391C2×C32C2×#23911yescharacteristic
392C2×C32C22×#6711yescharacteristic
393C3⋊D3—24no—
394C3×D3—64no—
395C3×D3—124no—
396C3×D3—122no—
397C3×D3—122no—
398C24—91no—
399C24—91no—
400C2×D4—274no—
401C2×D4—544no—
402C22×C4—91no—
403C22×C4—91no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—94no—
407C22×C4—184no—
408C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
409C4⋊C4—94no—
410C4⋊C4—274no—
411C22⋊C4—94no—
412C22⋊C4—274no—
413C22⋊C4—274no—
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—32no—
417C22×C3—32no—
418C22×C3—32no—
419C22×C3—32no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—61no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—62no—
426C22×C3—94no—
427C22×C3—188no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3—182no—
431C22×C3—182no—
432C22×C3—182no—
433C22×C3—364no—
434C22×C3#379812yes—
435C22×C3#379912yes—
436C22×C3D3×#6712yes—
437C22×C3C22×D3211yescharacteristic
438C2×D3—34no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—68no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—64no—
449C2×D3—94no—
450C2×D3—122no—
451C2×D3—122no—
452C2×D3—124no—
453C2×D3—124no—
454C2×D3—128no—
455C2×D3—188no—
456C2×D3—182no—
457C2×D3—182no—
458C3×C4—22no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—122no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4—182no—
465C3×C4—182no—
466Dic3—62no—
467Dic3—62no—
468Dic3—62no—
469Dic3—62no—
470Dic3—62no—
471Dic3—122no—
472Dic3—182no—
473Dic3—182no—
474Dic3—182no—
475C32—21no—
476C32C2×#89411yescharacteristic, complemented
477C32C2×#92911yescharacteristic, complemented
478C23—92no—
479C23—94no—
480C23—92no—
481C23—182no—
482C23—182no—
483C23—182no—
484C23—182no—
485C23—182no—
486C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487D4—548no—
488C2×C4—22no—
489C2×C4—92no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—188no—
494C2×C4—182no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—34no—
502C2×C3—34no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—61no—
508C2×C3—64no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—184no—
512C2×C3—362no—
513C2×C3#6385014yes—
514C2×C3C2×#379811yescharacteristic
515C2×C3C2×#379911yescharacteristic
516C2×C3C2×D3×#6711yescharacteristic
517D3—64no—
518D3—122no—
519D3—122no—
520D3—124no—
521D3—184no—
522C22—94no—
523C22—188no—
524C22—182no—
525C22—182no—
526C22—182no—
527C22—182no—
528C22—182no—
529C22—364no—
530C22#8865012yes—
531C22#8865212yes—
532C22#8867612yes—
533C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C4—22no—
535C4—182no—
536C4—182no—
537C4—182no—
538C4—182no—
539C3—31no—
540C3—31no—
541C3—61no—
542C3C2×#6385011yescharacteristic
543C2—184no—
544C2—362no—
545C2#1172110014yesdirect product factor
546C2C2×#8865011yescharacteristic
547C2C2×#8865211yescharacteristic
548C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#6200C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. C2×#88542C2semidirect
4. C2×#3750⋊C3C2semidirect
5. C2×#88451C2semidirect
6. C2×#88450C2semidirect
7. C2×#88449C2semidirect
8. #11721100C2direct (N × H)
19. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. #88542C22semidirect
23. #3750⋊C3C22semidirect
24. #88451C22semidirect
25. #88450C22semidirect
26. #88449C22semidirect
27. C22×C4×C3⋊D3D3semidirect
31. C2×#3750D3semidirect
80. C2×C4×C3⋊D3C2×D3semidirect
93. #3750C2×D3semidirect
198. C2×H(3,𝔽3)#123semidirect
308. H(3,𝔽3)C2×#123semidirect
385. C2×C32#894semidirect
386. C2×C32#929semidirect
476. C32C2×#894semidirect
477. C32C2×#929semidirect
545. C2#11721100direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C22×#62002
C2×#885422
C2×#3750⋊C32
C2×#884512
C2×#884502
C2×#884492
#117211002
C2×#639053
C2×#638533