Finite Group Databasebeta

← C2×#11721154 #13150829

Subgroups — C2×#11721154 #13150829

Order 1728.

1623 conjugacy classes of subgroups.

The subgroup lattice has 564 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
3C22×#6199C211yesmaximal, characteristic, complemented
4C2×#88542C211yesmaximal, characteristic, complemented
5C2×#3750⋊C3C211yesmaximal, characteristic, complemented
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88447C211yesmaximal, characteristic, complemented
8#11721154C218yesmaximal, direct product factor
9C2×#63881—31nomaximal
10C2×#63879—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×#6200C418yescomplemented
21C2×#6200C2214yes—
22C2×#6199—24no—
23C2×#6199C2214yescomplemented
24#88542C2214yescomplemented
25#3750⋊C3C2214yescomplemented
26#88450C2214yescomplemented
27#88447C2214yescomplemented
28C22×C4×C3⋊D3D311yescharacteristic, complemented
29C22×C3×C4×D3—31no—
30C22×#626—31no—
31C22×#625—31no—
32C22×D3×Dic3—31no—
33C22×D3×Dic3—31no—
34C2×#3750D311yescharacteristic, complemented
35C2×C3×(C2×D3)⋊C4—31no—
36C2×C3×(C2×D3)⋊C4—31no—
37C2×C3×(C2×D3)⋊C4—61no—
38C2×C3×Dic3⋊C4—31no—
39C2×#3722—31no—
40C2×(C2×D3)⋊Dic3—31no—
41C2×Dic32—31no—
42C2×Dic32—31no—
43#63881—38no—
44#63879—38no—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×#272—22no—
47C22×#272—22no—
48C22×#272C2×C414yescomplemented
49C22×#272C2312yes—
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3C2×C414yescomplemented
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×#1084—22no—
57C2×#1084C2312yes—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
60C2×(C3⋊Dic3)⋊C3D414yescomplemented
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C4×(C3⋊D3)⋊C3—28no—
65#6200C2×C4116yescomplemented
66#6199—28no—
67C2×C4×#67—91no—
68C2×#893—91no—
69C2×#67⋊C4—91no—
70C23×C3⋊D3C2×D311yescharacteristic
71C23×C3×D3—31no—
72C23×C3×D3—31no—
73C23×C3×D3—31no—
74C23×C3×D3—61no—
75C22×C32×C4—21no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C22×C3⋊Dic3C2×D311yescharacteristic
81C2×C4×C3⋊D3—24no—
82C2×C4×C3⋊D3C2×D314yescomplemented
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—31no—
88C22×C3×Dic3—61no—
89C2×C3×C4×D3—34no—
90C2×C3×C4×D3—64no—
91C2×#626—38no—
92C2×#626—34no—
93C2×#625—38no—
94C2×#625—34no—
95C2×D3×Dic3—34no—
96C2×D3×Dic3—34no—
97C2×D3×Dic3—64no—
98C2×D3×Dic3—64no—
99#3750C2×D314yescomplemented
100C3×(C2×D3)⋊C4—34no—
101C3×(C2×D3)⋊C4—34no—
102C3×(C2×D3)⋊C4—64no—
103C3×Dic3⋊C4—34no—
104#3722—34no—
105(C2×D3)⋊Dic3—34no—
106Dic32—34no—
107Dic32—34no—
108C22×H(3,𝔽3)C22×C412yes—
109C22×H(3,𝔽3)C2×D412yes—
110C22×H(3,𝔽3)#4112yes—
111C22×H(3,𝔽3)C2411yescharacteristic
112C2×#272—28no—
113C2×#272—22no—
114C2×#272—22no—
115C2×#272C22×C414yescomplemented
116C2×(C3⋊D3)⋊C3—28no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3C22×C414yescomplemented
120C4×H(3,𝔽3)—22no—
121#1084—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3C22×C414yescomplemented
124(C3⋊Dic3)⋊C3C2×D414yescomplemented
125C23×C3×C4—91no—
126C22×#67—91no—
127C22×#67—91no—
128C23×Dic3—91no—
129C22×D12—91no—
130C22×C4×D3—31no—
131C22×C4×D3—91no—
132C22×C4×D3—91no—
133C22×C4×D3—91no—
134C2×C3×C4⋊C4—91no—
135C2×C3×C22⋊C4—91no—
136C2×#217—91no—
137C4×#67—98no—
138C2×(C2×D3)⋊C4—31no—
139C2×(C2×D3)⋊C4—31no—
140C2×(C2×D3)⋊C4—61no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4—91no—
143C2×Dic3⋊C4—31no—
144C2×Dic3⋊C4—91no—
145C2×C4×Dic3—91no—
146C2×C4×Dic3—91no—
147C2×C4×Dic3—91no—
148#893—98no—
149#67⋊C4—98no—
150C23×C32—21no—
151C23×C32C22×D311yescharacteristic
152C23×C32C22×D311yescharacteristic
153C22×C3⋊D3—22no—
154C22×C3⋊D3—22no—
155C22×C3⋊D3C4×D314yescomplemented
156C22×C3⋊D3C22×D312yes—
157C22×C3×D3—34no—
158C22×C3×D3—34no—
159C22×C3×D3—34no—
160C22×C3×D3—32no—
161C22×C3×D3—32no—
162C22×C3×D3—32no—
163C22×C3×D3—64no—
164C22×C3×D3—62no—
165C22×C3×D3—62no—
166C22×C3×D3—62no—
167C22×C3×D3—62no—
168C22×C3×D3—62no—
169C22×C3×D3—62no—
170C22×C3×D3—62no—
171C22×C3×D3—122no—
172C22×C3×D3—122no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—22no—
176C2×C32×C4—42no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D312yes—
179C2×C3⋊Dic3—22no—
180C2×C3⋊Dic3C4×D314yescomplemented
181C2×C3⋊Dic3#6714yescomplemented
182C2×C3⋊Dic3C22×D312yes—
183C2×C3⋊Dic3C22×D312yes—
184C2×C3⋊Dic3C22×D312yes—
185C4×C3⋊D3—28no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—122no—
198C3×C4×D3—68no—
199#626—316no—
200#625—316no—
201D3×Dic3—68no—
202D3×Dic3—68no—
203C2×C4×D4—271noSylow
204C2×H(3,𝔽3)C4×D414yescomplemented
205C2×H(3,𝔽3)C23×C411yescharacteristic
206C2×H(3,𝔽3)C22×D411yescharacteristic
207C2×H(3,𝔽3)C2×#4111yescharacteristic
208#272—24no—
209(C3⋊D3)⋊C3—24no—
210C24×C3—91no—
211C24×C3—91no—
212C23×D3—31no—
213C23×D3—31no—
214C23×D3—31no—
215C23×D3—61no—
216C23×D3—91no—
217C22×C3×C4—31no—
218C22×C3×C4—31no—
219C22×C3×C4—61no—
220C22×C3×C4—94no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—184no—
226C22×C3×C4D3211yescharacteristic
227C2×#67—94no—
228C2×#67—98no—
229C2×#67—94no—
230C2×#67—98no—
231C22×Dic3—31no—
232C22×Dic3—31no—
233C22×Dic3—31no—
234C22×Dic3—31no—
235C22×Dic3—31no—
236C22×Dic3—61no—
237C22×Dic3—91no—
238C22×Dic3—91no—
239C22×Dic3—91no—
240C22×Dic3—94no—
241C22×Dic3—184no—
242C2×D12—94no—
243C2×D12—98no—
244C2×C4×D3—34no—
245C2×C4×D3—64no—
246C2×C4×D3—94no—
247C2×C4×D3—94no—
248C2×C4×D3—94no—
249C2×C4×D3—184no—
250C2×C4×D3—184no—
251C2×C4×D3—184no—
252C3×C4⋊C4—94no—
253C3×C22⋊C4—94no—
254#217—94no—
255(C2×D3)⋊C4—34no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—64no—
258(C2×D3)⋊C4—94no—
259(C2×D3)⋊C4—94no—
260Dic3⋊C4—34no—
261Dic3⋊C4—94no—
262C4×Dic3—94no—
263C4×Dic3—94no—
264C4×Dic3—94no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32C2×C4×D312yes—
270C22×C32C2×C4×D312yes—
271C22×C32#23512yes—
272C22×C32D3×D412yes—
273C22×C32#23912yes—
274C22×C32C2×#6712yes—
275C22×C32C23×D311yescharacteristic
276C22×C32C23×D311yescharacteristic
277C2×C3⋊D3—28no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3⋊D3C2×C4×D314yescomplemented
281C2×C3×D3—34no—
282C2×C3×D3—34no—
283C2×C3×D3—34no—
284C2×C3×D3—64no—
285C2×C3×D3—68no—
286C2×C3×D3—68no—
287C2×C3×D3—68no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—122no—
295C2×C3×D3—122no—
296C2×C3×D3—128no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—42no—
300C3⋊Dic3—22no—
301C3⋊Dic3C2×C4×D314yescomplemented
302C3⋊Dic3C2×#6714yescomplemented
303C3×Dic3—34no—
304C3×Dic3—34no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—122no—
309C22×D4—271no—
310C23×C4—91no—
311C23×C4—271no—
312C4×D4—278no—
313C2×C4⋊C4—91no—
314C2×C22⋊C4—91no—
315C2×C22⋊C4—271no—
316C2×C42—271no—
317H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
318C23×C3—31no—
319C23×C3—31no—
320C23×C3—61no—
321C23×C3—92no—
322C23×C3—94no—
323C23×C3—92no—
324C23×C3—94no—
325C23×C3—182no—
326C23×C3—182no—
327C23×C3—182no—
328C23×C3—182no—
329C23×C3C2×D3211yescharacteristic
330C22×D3—32no—
331C22×D3—32no—
332C22×D3—34no—
333C22×D3—34no—
334C22×D3—32no—
335C22×D3—34no—
336C22×D3—62no—
337C22×D3—64no—
338C22×D3—62no—
339C22×D3—62no—
340C22×D3—62no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—62no—
344C22×D3—94no—
345C22×D3—92no—
346C22×D3—122no—
347C22×D3—122no—
348C22×D3—182no—
349C22×D3—182no—
350C2×C3×C4—22no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—94no—
359C2×C3×C4—94no—
360C2×C3×C4—92no—
361C2×C3×C4—92no—
362C2×C3×C4—122no—
363C2×C3×C4—188no—
364C2×C3×C4—182no—
365C2×C3×C4—182no—
366C2×C3×C4C2×D3212yes—
367#67—916no—
368#67—916no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—34no—
372C2×Dic3—34no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—62no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—94no—
381C2×Dic3—94no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—92no—
385C2×Dic3—122no—
386C2×Dic3—182no—
387C2×Dic3—188no—
388D12—916no—
389C4×D3—68no—
390C4×D3—188no—
391C4×D3—188no—
392C4×D3—188no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—24no—
397C2×C32#89314yescomplemented
398C2×C32C4×#6714yescomplemented
399C2×C32C22×C4×D311yescharacteristic
400C2×C32C22×C4×D311yescharacteristic
401C2×C32C2×#23511yescharacteristic
402C2×C32C2×D3×D411yescharacteristic
403C2×C32C2×#23911yescharacteristic
404C2×C32C22×#6711yescharacteristic
405C3⋊D3—24no—
406C3×D3—64no—
407C3×D3—64no—
408C3×D3—64no—
409C3×D3—124no—
410C24—91no—
411C24—91no—
412C2×D4—274no—
413C2×D4—278no—
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—94no—
419C22×C4—184no—
420C22×C4—274no—
421C22×C4—544no—
422C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C4⋊C4—94no—
424C22⋊C4—94no—
425C22⋊C4—274no—
426C42—274no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—32no—
430C22×C3—32no—
431C22×C3—32no—
432C22×C3—32no—
433C22×C3—32no—
434C22×C3—32no—
435C22×C3—61no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—94no—
440C22×C3—94no—
441C22×C3—182no—
442C22×C3—188no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3—182no—
446C22×C3—188no—
447C22×C3#379912yes—
448C22×C3C4×D3212yes—
449C22×C3D3×#6712yes—
450C22×C3C22×D3211yescharacteristic
451C2×D3—34no—
452C2×D3—34no—
453C2×D3—34no—
454C2×D3—64no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—68no—
458C2×D3—68no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—68no—
464C2×D3—94no—
465C2×D3—122no—
466C2×D3—128no—
467C2×D3—122no—
468C2×D3—182no—
469C2×D3—188no—
470C2×D3—182no—
471C3×C4—22no—
472C3×C4—62no—
473C3×C4—62no—
474C3×C4—94no—
475C3×C4—94no—
476C3×C4—122no—
477C3×C4—182no—
478C3×C4—182no—
479Dic3—34no—
480Dic3—34no—
481Dic3—62no—
482Dic3—62no—
483Dic3—62no—
484Dic3—94no—
485Dic3—94no—
486Dic3—122no—
487Dic3—182no—
488C32—21no—
489C32C2×#89311yescharacteristic, complemented
490C32C2×C4×#6711yescharacteristic, complemented
491C23—92no—
492C23—92no—
493C23—94no—
494C23—94no—
495C23—182no—
496C23—182no—
497C23—182no—
498C23—182no—
499C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
500D4—2716no—
501C2×C4—22no—
502C2×C4—92no—
503C2×C4—92no—
504C2×C4—94no—
505C2×C4—94no—
506C2×C4—92no—
507C2×C4—92no—
508C2×C4—188no—
509C2×C4—182no—
510C2×C4—182no—
511C2×C4—548no—
512C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—34no—
516C2×C3—34no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—61no—
522C2×C3—64no—
523C2×C3—61no—
524C2×C3—61no—
525C2×C3—184no—
526C2×C3—184no—
527C2×C3#6390414yes—
528C2×C3C2×#379911yescharacteristic
529C2×C3C2×C4×D3211yescharacteristic
530C2×C3C2×D3×#6711yescharacteristic
531D3—64no—
532D3—64no—
533D3—64no—
534D3—124no—
535D3—184no—
536C22—94no—
537C22—94no—
538C22—182no—
539C22—182no—
540C22—188no—
541C22—188no—
542C22—182no—
543C22—182no—
544C22#8865212yes—
545C22C4×((C3⋊D3)⋊C3)⋊C212yes—
546C22#8867612yes—
547C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
548C4—22no—
549C4—94no—
550C4—94no—
551C4—182no—
552C4—182no—
553C3—31no—
554C3—31no—
555C3—61no—
556C3C2×#6390411yescharacteristic
557C2—184no—
558C2—184no—
559C2#1172115414yesdirect product factor
560C2C2×#8865211yescharacteristic
561C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
562C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#6200C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×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. C22×#6199C2semidirect
4. C2×#88542C2semidirect
5. C2×#3750⋊C3C2semidirect
6. C2×#88450C2semidirect
7. C2×#88447C2semidirect
8. #11721154C2direct (N × H)
19. C2×C4×(C3⋊D3)⋊C3C22semidirect
20. C2×#6200C4semidirect
23. C2×#6199C22semidirect
24. #88542C22semidirect
25. #3750⋊C3C22semidirect
26. #88450C22semidirect
27. #88447C22semidirect
28. C22×C4×C3⋊D3D3semidirect
34. C2×#3750D3semidirect
48. C22×#272C2×C4semidirect
52. C22×(C3⋊D3)⋊C3C2×C4semidirect
59. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
60. C2×(C3⋊Dic3)⋊C3D4semidirect
65. #6200C2×C4semidirect
82. C2×C4×C3⋊D3C2×D3semidirect
99. #3750C2×D3semidirect
115. C2×#272C22×C4semidirect
119. C2×(C3⋊D3)⋊C3C22×C4semidirect
123. (C3⋊Dic3)⋊C3C22×C4semidirect
124. (C3⋊Dic3)⋊C3C2×D4semidirect
155. C22×C3⋊D3C4×D3semidirect
180. C2×C3⋊Dic3C4×D3semidirect
181. C2×C3⋊Dic3#67semidirect
204. C2×H(3,𝔽3)C4×D4semidirect
280. C2×C3⋊D3C2×C4×D3semidirect
301. C3⋊Dic3C2×C4×D3semidirect
302. C3⋊Dic3C2×#67semidirect
317. H(3,𝔽3)C2×C4×D4semidirect
397. C2×C32#893semidirect
398. C2×C32C4×#67semidirect
489. C32C2×#893semidirect
490. C32C2×C4×#67semidirect
559. C2#11721154direct (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
C22×#61992
C2×#885422
C2×#3750⋊C32
C2×#884502
C2×#884472
#117211542
C2×#638813
C2×#638793