Finite Group Databasebeta

← C2×#11722491 #13164722

Subgroups — C2×#11722491 #13164722

Order 1728.

3351 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C212yesmaximal, complemented
3C2×#88677C212yesmaximal, complemented
4C2×#88672C212yesmaximal, complemented
5#11722491C2116yesmaximal, direct product factor
6C2×#88655C211yesmaximal, characteristic, complemented
7C2×C4×((C3⋊D3)⋊C3)⋊C2C212yesmaximal, complemented
8C2×#88651C211yesmaximal, characteristic, complemented
9C2×#88650C212yesmaximal, complemented
10C2×#88649C212yesmaximal, complemented
11C2×D3×#235—32nomaximal
12C23×#272C2211yescharacteristic
13C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
14C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C22×#1084C2211yescharacteristic, complemented
16C2×C4×#272C2211yescharacteristic
17C2×C4×#272C2211yescharacteristic
18C2×C4×#272C2218yescomplemented
19C2×C4×#272C2211yescharacteristic, complemented
20C2×C4×#272C2211yescharacteristic, complemented
21C2×#638⋊C3C2214yescomplemented
22#3828⋊C3C22116yescomplemented
23C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
24C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
25C2×#634⋊C3C2212yescomplemented
26#88677C22116yescomplemented
27C2×#6200C2214yescomplemented
28C2×#6199C2212yes—
29C2×#6198C2212yescomplemented
30#88672C22116yescomplemented
31C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
32C2×#6196C2212yescomplemented
33#88655C2218yescomplemented
34C4×((C3⋊D3)⋊C3)⋊C2C22116yescomplemented
35#88651C2218yescomplemented
36#88650C22116yescomplemented
37#88649C22116yescomplemented
38C2×#3828D312yescomplemented
39C2×C3×#235—32no—
40C22×C3×C4×D3—32no—
41C22×C3×C4×D3—61no—
42C2×D3×#67—34no—
43C2×#3804—34no—
44D3×#235—332no—
45C2×D3×D12—32no—
46C2×C4×D32—34no—
47C2×#3799—32no—
48C2×#3798—34no—
49C2×#3795—32no—
50C2×D3×Dic6—32no—
51C23×H(3,𝔽3)C2311yescharacteristic
52C22×#272—24no—
53C22×#272C2311yescharacteristic
54C22×#272C2314yes—
55C22×#272C2311yescharacteristic
56C22×(C3⋊D3)⋊C3C2314yes—
57C2×((C3⋊D3)⋊C3)⋊C2—28no—
58C2×((C3⋊D3)⋊C3)⋊C2C2318yes—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2314yescomplemented
62C2×#1084C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64C2×#1084C2314yescomplemented
65C4×#272C2314yes—
66C4×#272C2314yes—
67C4×#272C2314yescomplemented
68C4×#272C2314yescomplemented
69#638⋊C3C23116yescomplemented
70C2×(C3⋊Dic3)⋊C3C2314yes—
71C4⋊((C3⋊D3)⋊C3)C2318yescomplemented
72C4×(C3⋊D3)⋊C3C23116yescomplemented
73#634⋊C3C2318yescomplemented
74#6200C23116yescomplemented
75#6199C2318yes—
76#6198C2318yescomplemented
77((C3⋊D3)⋊C3)⋊C4C23116yescomplemented
78#6196C2318yescomplemented
79C2×D3×#41—92no—
80C22×#235—91no—
81C23×C3×D3—32no—
82C23×C3×D3—61no—
83C22×D32—34no—
84C22×C32×C4—21no—
85C22×C32×C4C2×D312yescomplemented
86C2×#638C2×D314yescomplemented
87#3828C2×D3116yescomplemented
88C2×C4⋊(C3⋊D3)C2×D312yescomplemented
89C2×C4×C3⋊D3C2×D314yescomplemented
90C2×#634C2×D312yescomplemented
91C2×C3×#67—34no—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—61no—
94C3×#235—316no—
95C2×C3×D12—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—34no—
100C2×C3×C4×D3—32no—
101C2×C3×C4×D3—316no—
102C2×C3×C4×D3—61no—
103C2×C3×C4×D3—61no—
104C2×C3×C4×D3—68no—
105C2×C3×C4×D3—61no—
106C2×C3×C4×D3—61no—
107C2×C3×Dic6—32no—
108D3×#67—332no—
109C2×#627—34no—
110C2×#626—34no—
111C2×#626—34no—
112C2×#625—34no—
113C2×#624—34no—
114#3804—332no—
115C2×D3×Dic3—34no—
116C2×D3×Dic3—34no—
117D3×D12—316no—
118C4×D32—332no—
119#3799—316no—
120#3798—332no—
121#3795—316no—
122D3×Dic6—316no—
123C22×H(3,𝔽3)—22no—
124C22×H(3,𝔽3)C2412yes—
125C22×H(3,𝔽3)C2411yescharacteristic
126C2×#272—22no—
127C2×#272—28no—
128C2×#272#4114yescomplemented
129C2×#272C2412yes—
130C2×#272C2412yes—
131C2×(C3⋊D3)⋊C3—28no—
132C2×(C3⋊D3)⋊C3C2418yes—
133((C3⋊D3)⋊C3)⋊C2—216no—
134C4×H(3,𝔽3)C2412yes—
135C4×H(3,𝔽3)C2412yes—
136#1084C2412yes—
137#1084C2412yes—
138(C3⋊Dic3)⋊C3C2418yes—
139C2×C3×#41—92no—
140C23×C3×C4—91no—
141C22×#67—92no—
142D3×#41—932no—
143C2×#239—92no—
144C2×D3×Q8—92no—
145C2×#237—92no—
146C2×#237—94no—
147C2×D3×D4—92no—
148C2×D3×D4—94no—
149C2×#235—32no—
150C2×#235—92no—
151C2×#235—92no—
152C2×#235—92no—
153C2×#235—94no—
154C2×#235—916no—
155C2×#235—92no—
156C22×D12—91no—
157C22×C4×D3—32no—
158C22×C4×D3—61no—
159C22×C4×D3—92no—
160C22×C4×D3—94no—
161C22×Dic6—91no—
162C23×C32—21no—
163C23×C32C22×D312yes—
164C22×C3⋊D3C22×D314yes—
165C22×C3×D3—32no—
166C22×C3×D3—34no—
167C22×C3×D3—32no—
168C22×C3×D3—38no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—64no—
172C22×C3×D3—68no—
173C22×C3×D3—68no—
174C2×D32—316no—
175C2×D32—616no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—24no—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D312yes—
181C2×C32×C4C22×D318yescomplemented
182#638C22×D3116yescomplemented
183C2×C3⋊Dic3C22×D314yes—
184C4⋊(C3⋊D3)C22×D318yescomplemented
185C4×C3⋊D3C22×D3116yescomplemented
186#634C22×D318yescomplemented
187C3×#67—316no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—64no—
195C3×D12—38no—
196C3×C4×D3—316no—
197C3×C4×D3—38no—
198C3×C4×D3—38no—
199C3×C4×D3—38no—
200C3×C4×D3—38no—
201C3×C4×D3—64no—
202C3×C4×D3—64no—
203C3×C4×D3—64no—
204C3×C4×D3—64no—
205C3×Dic6—38no—
206#627—316no—
207#626—316no—
208#626—316no—
209#625—316no—
210#624—316no—
211D3×Dic3—316no—
212D3×Dic3—316no—
213C22×#41—271noSylow
214C2×H(3,𝔽3)—22no—
215C2×H(3,𝔽3)C2×#4112yescomplemented
216C2×H(3,𝔽3)C2511yescharacteristic
217#272—22no—
218#272C2×#4114yescomplemented
219(C3⋊D3)⋊C3—28no—
220C24×C3—91no—
221C23×D3—32no—
222C23×D3—61no—
223C23×D3—92no—
224C23×D3—94no—
225C3×#41—916no—
226C2×C3×Q8—92no—
227C2×C3×D4—92no—
228C2×C3×D4—94no—
229C22×C3×C4—32no—
230C22×C3×C4—61no—
231C22×C3×C4—94no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—91no—
235C22×C3×C4—91no—
236C22×C3×C4—91no—
237C22×C3×C4—98no—
238C22×C3×C4D3211yescharacteristic
239C2×#67—34no—
240C2×#67—94no—
241C2×#67—94no—
242C2×#67—94no—
243C2×#67—916no—
244C2×#67—94no—
245C22×Dic3—32no—
246C22×Dic3—61no—
247C22×Dic3—92no—
248C22×Dic3—94no—
249#239—916no—
250D3×Q8—916no—
251#237—916no—
252#237—932no—
253D3×D4—916no—
254D3×D4—932no—
255#235—316no—
256#235—916no—
257#235—916no—
258#235—916no—
259#235—932no—
260#235—916no—
261C2×D12—32no—
262C2×D12—92no—
263C2×D12—98no—
264C2×D12—92no—
265C2×D12—94no—
266C2×C4×D3—32no—
267C2×C4×D3—32no—
268C2×C4×D3—32no—
269C2×C4×D3—32no—
270C2×C4×D3—316no—
271C2×C4×D3—34no—
272C2×C4×D3—61no—
273C2×C4×D3—61no—
274C2×C4×D3—61no—
275C2×C4×D3—61no—
276C2×C4×D3—68no—
277C2×C4×D3—94no—
278C2×C4×D3—916no—
279C2×C4×D3—932no—
280C2×C4×D3—94no—
281C2×C4×D3—94no—
282C2×C4×D3—94no—
283C2×Dic6—32no—
284C2×Dic6—92no—
285C2×Dic6—98no—
286C2×Dic6—92no—
287C2×Dic6—94no—
288C22×C32—24no—
289C22×C32—24no—
290C22×C32—22no—
291C22×C32—21no—
292C22×C32C23×D314yes—
293C22×C32C23×D312yes—
294C2×C3⋊D3—28no—
295C2×C3⋊D3C23×D318yes—
296C2×C3×D3—38no—
297C2×C3×D3—34no—
298C2×C3×D3—38no—
299C2×C3×D3—34no—
300C2×C3×D3—64no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—616no—
304C2×C3×D3—616no—
305C2×C3×D3—64no—
306C2×C3×D3—68no—
307C2×C3×D3—64no—
308D32—632no—
309C32×C4—22no—
310C32×C4—22no—
311C32×C4C23×D314yes—
312C32×C4C23×D314yes—
313C3⋊Dic3C23×D318yes—
314C3×Dic3—38no—
315C3×Dic3—34no—
316C3×Dic3—34no—
317C3×Dic3—62no—
318C3×Dic3—62no—
319C2×#41—92no—
320C2×#41—272no—
321C2×#41—272no—
322C2×#41—272no—
323C2×#41—2716no—
324C22×Q8—271no—
325C22×D4—271no—
326C22×D4—272no—
327C23×C4—91no—
328C23×C4—272no—
329H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
330C23×C3—32no—
331C23×C3—61no—
332C23×C3—94no—
333C23×C3—91no—
334C23×C3—94no—
335C23×C3—91no—
336C23×C3—184no—
337C23×C3C2×D3211yescharacteristic
338C22×D3—32no—
339C22×D3—32no—
340C22×D3—38no—
341C22×D3—34no—
342C22×D3—61no—
343C22×D3—64no—
344C22×D3—61no—
345C22×D3—68no—
346C22×D3—68no—
347C22×D3—94no—
348C22×D3—98no—
349C22×D3—916no—
350C22×D3—1816no—
351C22×D3—188no—
352C3×Q8—98no—
353C3×D4—98no—
354C3×D4—916no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—38no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—64no—
361C2×C3×C4—916no—
362C2×C3×C4—94no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—94no—
366C2×C3×C4—94no—
367C2×C3×C4—94no—
368C2×C3×C4—94no—
369C2×C3×C4—94no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3214yes—
373#67—316no—
374#67—916no—
375#67—916no—
376#67—916no—
377#67—916no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—38no—
381C2×Dic3—34no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—64no—
385C2×Dic3—94no—
386C2×Dic3—98no—
387C2×Dic3—916no—
388D12—38no—
389D12—98no—
390D12—98no—
391D12—916no—
392C4×D3—38no—
393C4×D3—38no—
394C4×D3—38no—
395C4×D3—38no—
396C4×D3—316no—
397C4×D3—64no—
398C4×D3—64no—
399C4×D3—64no—
400C4×D3—64no—
401C4×D3—916no—
402C4×D3—916no—
403C4×D3—916no—
404C4×D3—916no—
405Dic6—38no—
406Dic6—98no—
407Dic6—98no—
408Dic6—916no—
409C2×C32—24no—
410C2×C32—24no—
411C2×C32—22no—
412C2×C32—21no—
413C2×C32D3×#4114yescomplemented
414C2×C32C24×D312yes—
415C3⋊D3—28no—
416C3×D3—38no—
417C3×D3—64no—
418C3×D3—64no—
419C3×D3—68no—
420C3×D3—64no—
421C24—91no—
422C24—272no—
423#41—916no—
424#41—2716no—
425#41—2716no—
426#41—2716no—
427C2×Q8—92no—
428C2×Q8—272no—
429C2×Q8—278no—
430C2×D4—94no—
431C2×D4—92no—
432C2×D4—278no—
433C2×D4—272no—
434C2×D4—274no—
435C2×D4—2716no—
436C22×C4—91no—
437C22×C4—91no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—94no—
442C22×C4—98no—
443C22×C4—2716no—
444C22×C4—272no—
445C22×C4—274no—
446C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
447C22×C3—22no—
448C22×C3—32no—
449C22×C3—34no—
450C22×C3—64no—
451C22×C3—64no—
452C22×C3—62no—
453C22×C3—61no—
454C22×C3—98no—
455C22×C3—94no—
456C22×C3—92no—
457C22×C3—92no—
458C22×C3—188no—
459C22×C3—188no—
460C22×C3—182no—
461C22×C3C22×D3212yes—
462C22×C3C22×D3211yescharacteristic
463C2×D3—34no—
464C2×D3—38no—
465C2×D3—34no—
466C2×D3—38no—
467C2×D3—62no—
468C2×D3—62no—
469C2×D3—64no—
470C2×D3—616no—
471C2×D3—64no—
472C2×D3—616no—
473C2×D3—64no—
474C2×D3—68no—
475C2×D3—98no—
476C2×D3—188no—
477C2×D3—1832no—
478C2×D3—1816no—
479C3×C4—34no—
480C3×C4—34no—
481C3×C4—62no—
482C3×C4—62no—
483C3×C4—98no—
484C3×C4—92no—
485C3×C4—92no—
486C3×C4C22×D3212yes—
487C3×C4C22×D3212yes—
488Dic3—34no—
489Dic3—34no—
490Dic3—38no—
491Dic3—62no—
492Dic3—62no—
493Dic3—98no—
494C32—21no—
495C32C2×D3×#4112yescomplemented
496C23—91no—
497C23—94no—
498C23—91no—
499C23—94no—
500C23—184no—
501C23—278no—
502C23—548no—
503C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504Q8—98no—
505Q8—278no—
506D4—916no—
507D4—98no—
508D4—278no—
509D4—2716no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—94no—
513C2×C4—94no—
514C2×C4—94no—
515C2×C4—94no—
516C2×C4—94no—
517C2×C4—94no—
518C2×C4—916no—
519C2×C4—278no—
520C2×C4—2716no—
521C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C2×C4C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
524C2×C3—22no—
525C2×C3—32no—
526C2×C3—34no—
527C2×C3—64no—
528C2×C3—64no—
529C2×C3—61no—
530C2×C3—62no—
531C2×C3—94no—
532C2×C3—188no—
533C2×C3—182no—
534C2×C3#6430612yes—
535C2×C3C23×D3211yescharacteristic
536D3—38no—
537D3—64no—
538D3—64no—
539D3—64no—
540D3—68no—
541D3—188no—
542C22—22no—
543C22—92no—
544C22—94no—
545C22—92no—
546C22—98no—
547C22—188no—
548C22—182no—
549C22—188no—
550C22—5416no—
551C22C22×((C3⋊D3)⋊C3)⋊C212yes—
552C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
553C4—92no—
554C4—92no—
555C4—98no—
556C4C22×((C3⋊D3)⋊C3)⋊C212yes—
557C4C22×((C3⋊D3)⋊C3)⋊C212yes—
558C3—32no—
559C3—61no—
560C3C2×#6430611yescharacteristic
561C2—22no—
562C2—94no—
563C2—182no—
564C2—188no—
565C2#1172249112yesdirect product factor
566C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2×C4C2×((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)C22×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×#272C2semidirect
2. C2×#3828⋊C3C2semidirect
3. C2×#88677C2semidirect
4. C2×#88672C2semidirect
5. #11722491C2direct (N × H)
6. C2×#88655C2semidirect
7. C2×C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
8. C2×#88651C2semidirect
9. C2×#88650C2semidirect
10. C2×#88649C2semidirect
14. C22×C4×H(3,𝔽3)C22semidirect
15. C22×#1084C22semidirect
18. C2×C4×#272C22semidirect
19. C2×C4×#272C22semidirect
20. C2×C4×#272C22semidirect
21. C2×#638⋊C3C22semidirect
22. #3828⋊C3C22semidirect
23. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
24. C2×C4×(C3⋊D3)⋊C3C22semidirect
25. C2×#634⋊C3C22semidirect
26. #88677C22semidirect
27. C2×#6200C22semidirect
29. C2×#6198C22semidirect
30. #88672C22semidirect
31. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
32. C2×#6196C22semidirect
33. #88655C22semidirect
34. C4×((C3⋊D3)⋊C3)⋊C2C22semidirect
35. #88651C22semidirect
36. #88650C22semidirect
37. #88649C22semidirect
38. C2×#3828D3semidirect
61. C2×C4×H(3,𝔽3)C23semidirect
64. C2×#1084C23semidirect
67. C4×#272C23semidirect
68. C4×#272C23semidirect
69. #638⋊C3C23semidirect
71. C4⋊((C3⋊D3)⋊C3)C23semidirect
72. C4×(C3⋊D3)⋊C3C23semidirect
73. #634⋊C3C23semidirect
74. #6200C23semidirect
76. #6198C23semidirect
77. ((C3⋊D3)⋊C3)⋊C4C23semidirect
78. #6196C23semidirect
85. C22×C32×C4C2×D3semidirect
86. C2×#638C2×D3semidirect
87. #3828C2×D3semidirect
88. C2×C4⋊(C3⋊D3)C2×D3semidirect
89. C2×C4×C3⋊D3C2×D3semidirect
90. C2×#634C2×D3semidirect
128. C2×#272#41semidirect
181. C2×C32×C4C22×D3semidirect
182. #638C22×D3semidirect
184. C4⋊(C3⋊D3)C22×D3semidirect
185. C4×C3⋊D3C22×D3semidirect
186. #634C22×D3semidirect
215. C2×H(3,𝔽3)C2×#41semidirect
218. #272C2×#41semidirect
329. H(3,𝔽3)C22×#41semidirect
413. C2×C32D3×#41semidirect
446. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
495. C32C2×D3×#41semidirect
523. C2×C4C2×((C3⋊D3)⋊C3)⋊C2semidirect
565. C2#11722491direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#3828⋊C32
C2×#886772
C2×#886722
#117224912
C2×#886552
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#886512
C2×#886502
C2×#886492
C2×D3×#2353