Finite Group Databasebeta

← C22×#88676 #13164818

Subgroups — C22×#88676 #13164818

Order 1728.

5003 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C23×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3C22×#6235C211yesmaximal, characteristic, complemented
4C22×#638⋊C3C211yesmaximal, characteristic, complemented
5C2×#88676C2124yesmaximal, direct product factor
6C22×#6200C211yesmaximal, characteristic, complemented
7C22×#6198C211yesmaximal, characteristic, complemented
8C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
9C22×#3811—31nomaximal
10C22×D3×#67—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic, complemented
12C23×#272C2211yescharacteristic, complemented
13C23×(C3⋊D3)⋊C3—28no—
14C23×(C3⋊D3)⋊C3C2211yescharacteristic
15C23×(C3⋊D3)⋊C3C22112yescomplemented
16C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
17C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
18C22×((C3⋊D3)⋊C3)⋊C2—28no—
19C22×((C3⋊D3)⋊C3)⋊C2C22112yescomplemented
20C2×#6235C22112yescomplemented
21C22×#1084C2211yescharacteristic, complemented
22C2×#638⋊C3C22112yescomplemented
23C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
24#88676C22164yesdirect product factor
25C2×#6200C22112yescomplemented
26C2×#6198C22112yescomplemented
27C2×((C3⋊D3)⋊C3)⋊C4C22112yescomplemented
28C24×C3⋊D3D311yescharacteristic, complemented
29C24×C3×D3—31no—
30C23×D32—31no—
31C23×D32—31no—
32C22×#638D311yescharacteristic, complemented
33C22×C3×#67—31no—
34C22×C3×#67—31no—
35C22×C3×#67—31no—
36C22×C3×#67—61no—
37C2×#3811—324no—
38C2×D3×#67—324no—
39C22×#626—31no—
40C22×#626—31no—
41C22×#626—31no—
42C22×#625—31no—
43C22×#624—31no—
44C22×D3×Dic3—31no—
45C23×H(3,𝔽3)—24no—
46C23×H(3,𝔽3)C2311yescharacteristic
47C23×H(3,𝔽3)C2316yescomplemented
48C22×#272—24no—
49C22×#272C2316yescomplemented
50C22×(C3⋊D3)⋊C3—24no—
51C22×(C3⋊D3)⋊C3—24no—
52C22×(C3⋊D3)⋊C3—248no—
53C22×(C3⋊D3)⋊C3D418yescomplemented
54C22×(C3⋊D3)⋊C3C2316yes—
55C22×(C3⋊D3)⋊C3C23116yescomplemented
56C22×(C3⋊D3)⋊C3C2316yescomplemented
57C22×(C3⋊D3)⋊C3C2316yescomplemented
58C2×((C3⋊D3)⋊C3)⋊C2—248no—
59C2×((C3⋊D3)⋊C3)⋊C2C23116yescomplemented
60#6235C23116yescomplemented
61C2×#1084C2316yescomplemented
62#638⋊C3C23116yescomplemented
63C2×(C3⋊Dic3)⋊C3C2316yescomplemented
64#6200C23116yescomplemented
65#6198C23116yescomplemented
66((C3⋊D3)⋊C3)⋊C4C23116yescomplemented
67C23×#67—91no—
68C22×D3×D4—91no—
69C22×D3×D4—91no—
70C24×C32—21no—
71C24×C32C2×D311yescharacteristic, complemented
72C24×C32C2×D311yescharacteristic, complemented
73C23×C3⋊D3—28no—
74C23×C3⋊D3C2×D311yescharacteristic
75C23×C3⋊D3C2×D3112yescomplemented
76C23×C3⋊D3C2×D311yescharacteristic, complemented
77C23×C3⋊D3C2×D311yescharacteristic, complemented
78C23×C3×D3—31no—
79C23×C3×D3—31no—
80C23×C3×D3—31no—
81C23×C3×D3—31no—
82C23×C3×D3—31no—
83C23×C3×D3—312no—
84C23×C3×D3—68no—
85C23×C3×D3—61no—
86C22×D32—312no—
87C22×D32—312no—
88C22×D32—68no—
89C22×D32—68no—
90C2×#638C2×D3112yescomplemented
91C22×C3⋊Dic3C2×D311yescharacteristic, complemented
92C2×C3×#67—312no—
93C2×C3×#67—312no—
94C2×C3×#67—312no—
95C2×C3×#67—612no—
96C22×C3×Dic3—31no—
97C22×C3×Dic3—31no—
98C22×C3×Dic3—31no—
99C22×C3×Dic3—61no—
100#3811—364no—
101D3×#67—364no—
102C2×#626—312no—
103C2×#626—312no—
104C2×#626—312no—
105C2×#625—312no—
106C2×#624—312no—
107C2×D3×Dic3—312no—
108C22×H(3,𝔽3)—212no—
109C22×H(3,𝔽3)C2×D414yescomplemented
110C22×H(3,𝔽3)C2413yes—
111C22×H(3,𝔽3)C2414yescomplemented
112C2×#272—212no—
113C2×#272C2414yescomplemented
114C2×(C3⋊D3)⋊C3—212no—
115C2×(C3⋊D3)⋊C3—212no—
116C2×(C3⋊D3)⋊C3—232no—
117C2×(C3⋊D3)⋊C3C2×D4124yescomplemented
118C2×(C3⋊D3)⋊C3C2414yes—
119C2×(C3⋊D3)⋊C3C2414yescomplemented
120C2×(C3⋊D3)⋊C3C2414yescomplemented
121((C3⋊D3)⋊C3)⋊C2—232no—
122#1084C2414yescomplemented
123(C3⋊Dic3)⋊C3C2414yescomplemented
124C25×C3—91no—
125C24×D3—31no—
126C24×D3—91no—
127C24×D3—91no—
128C24×D3—91no—
129C24×D3—91no—
130C22×C3×D4—91no—
131C22×C3×D4—91no—
132C22×#67—31no—
133C22×#67—31no—
134C22×#67—31no—
135C22×#67—61no—
136C22×#67—91no—
137C22×#67—91no—
138C22×#67—91no—
139C22×#67—91no—
140C22×#67—924no—
141C22×#67—91no—
142C23×Dic3—91no—
143C2×D3×D4—924no—
144C2×D3×D4—924no—
145C22×D12—91no—
146C22×D12—91no—
147C22×C4×D3—91no—
148C22×C4×D3—91no—
149C23×C32—24no—
150C23×C32—24no—
151C23×C32—21no—
152C23×C32—26no—
153C23×C32—44no—
154C23×C32C22×D311yescharacteristic
155C23×C32C22×D311yescharacteristic
156C23×C32C22×D316yescomplemented
157C23×C32C22×D316yescomplemented
158C22×C3⋊D3—24no—
159C22×C3⋊D3—24no—
160C22×C3⋊D3—248no—
161C22×C3⋊D3#6718yescomplemented
162C22×C3⋊D3C22×D316yes—
163C22×C3⋊D3C22×D3116yescomplemented
164C22×C3⋊D3C22×D316yescomplemented
165C22×C3⋊D3C22×D316yescomplemented
166C22×C3×D3—38no—
167C22×C3×D3—36no—
168C22×C3×D3—36no—
169C22×C3×D3—36no—
170C22×C3×D3—36no—
171C22×C3×D3—36no—
172C22×C3×D3—316no—
173C22×C3×D3—66no—
174C22×C3×D3—648no—
175C22×C3×D3—64no—
176C22×C3×D3—64no—
177C22×C3×D3—64no—
178C22×C3×D3—64no—
179C22×C3×D3—124no—
180C2×D32—316no—
181C2×D32—316no—
182C2×D32—648no—
183C2×D32—648no—
184#638C22×D3116yescomplemented
185C2×C3⋊Dic3C22×D316yescomplemented
186C3×#67—316no—
187C3×#67—316no—
188C3×#67—316no—
189C3×#67—616no—
190C2×C3×Dic3—36no—
191C2×C3×Dic3—36no—
192C2×C3×Dic3—36no—
193C2×C3×Dic3—66no—
194#626—316no—
195#626—316no—
196#626—316no—
197#625—316no—
198#624—316no—
199D3×Dic3—316no—
200C23×D4—271noSylow
201C2×H(3,𝔽3)—24no—
202C2×H(3,𝔽3)C22×D416yescomplemented
203C2×H(3,𝔽3)C2511yescharacteristic
204#272—24no—
205(C3⋊D3)⋊C3—24no—
206(C3⋊D3)⋊C3—24no—
207(C3⋊D3)⋊C3C22×D418yescomplemented
208C24×C3—31no—
209C24×C3—31no—
210C24×C3—61no—
211C24×C3—91no—
212C24×C3—91no—
213C24×C3—912no—
214C24×C3—91no—
215C24×C3—91no—
216C24×C3—188no—
217C24×C3D3211yescharacteristic
218C23×D3—31no—
219C23×D3—31no—
220C23×D3—31no—
221C23×D3—31no—
222C23×D3—31no—
223C23×D3—312no—
224C23×D3—61no—
225C23×D3—68no—
226C23×D3—91no—
227C23×D3—91no—
228C23×D3—912no—
229C23×D3—91no—
230C23×D3—912no—
231C23×D3—912no—
232C23×D3—912no—
233C23×D3—188no—
234C23×D3—188no—
235C23×D3—188no—
236C23×D3—188no—
237C2×C3×D4—912no—
238C2×C3×D4—912no—
239C22×C3×C4—91no—
240C22×C3×C4—91no—
241C2×#67—312no—
242C2×#67—312no—
243C2×#67—312no—
244C2×#67—612no—
245C2×#67—912no—
246C2×#67—912no—
247C2×#67—912no—
248C2×#67—912no—
249C2×#67—964no—
250C2×#67—912no—
251C22×Dic3—31no—
252C22×Dic3—31no—
253C22×Dic3—31no—
254C22×Dic3—61no—
255C22×Dic3—91no—
256C22×Dic3—912no—
257D3×D4—964no—
258D3×D4—964no—
259C2×D12—912no—
260C2×D12—912no—
261C2×C4×D3—912no—
262C2×C4×D3—912no—
263C22×C32—212no—
264C22×C32—212no—
265C22×C32—23no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—412no—
269C22×C32D3×D414yescomplemented
270C22×C32C2×#6714yescomplemented
271C22×C32C23×D313yes—
272C22×C32C23×D313yes—
273C22×C32C23×D314yescomplemented
274C22×C32C23×D314yescomplemented
275C2×C3⋊D3—212no—
276C2×C3⋊D3—212no—
277C2×C3⋊D3—232no—
278C2×C3⋊D3C2×#67124yescomplemented
279C2×C3⋊D3C23×D314yes—
280C2×C3⋊D3C23×D314yescomplemented
281C2×C3⋊D3C23×D314yescomplemented
282C2×C3×D3—324no—
283C2×C3×D3—34no—
284C2×C3×D3—34no—
285C2×C3×D3—34no—
286C2×C3×D3—34no—
287C2×C3×D3—34no—
288C2×C3×D3—64no—
289C2×C3×D3—632no—
290C2×C3×D3—612no—
291C2×C3×D3—612no—
292C2×C3×D3—612no—
293C2×C3×D3—612no—
294C2×C3×D3—1212no—
295D32—632no—
296D32—632no—
297C3⋊Dic3C23×D314yescomplemented
298C3×Dic3—34no—
299C3×Dic3—34no—
300C3×Dic3—34no—
301C3×Dic3—64no—
302C25—91no—
303C25—271no—
304C22×D4—91no—
305C22×D4—91no—
306C22×D4—271no—
307C22×D4—271no—
308C22×D4—2724no—
309C23×C4—271no—
310H(3,𝔽3)C23×D411yescharacteristic, complemented, Sylow
311C23×C3—24no—
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—36no—
315C23×C3—36no—
316C23×C3—64no—
317C23×C3—64no—
318C23×C3—61no—
319C23×C3—66no—
320C23×C3—96no—
321C23×C3—98no—
322C23×C3—96no—
323C23×C3—916no—
324C23×C3—96no—
325C23×C3—96no—
326C23×C3—124no—
327C23×C3—184no—
328C23×C3—184no—
329C23×C3—1848no—
330C23×C3—184no—
331C23×C3C2×D3211yescharacteristic
332C23×C3C2×D3216yes—
333C22×D3—36no—
334C22×D3—38no—
335C22×D3—36no—
336C22×D3—36no—
337C22×D3—36no—
338C22×D3—36no—
339C22×D3—316no—
340C22×D3—66no—
341C22×D3—648no—
342C22×D3—64no—
343C22×D3—64no—
344C22×D3—64no—
345C22×D3—64no—
346C22×D3—98no—
347C22×D3—96no—
348C22×D3—96no—
349C22×D3—916no—
350C22×D3—96no—
351C22×D3—916no—
352C22×D3—916no—
353C22×D3—916no—
354C22×D3—124no—
355C22×D3—1848no—
356C22×D3—184no—
357C22×D3—184no—
358C22×D3—1848no—
359C22×D3—1848no—
360C22×D3—1848no—
361C3×D4—916no—
362C3×D4—916no—
363C2×C3×C4—96no—
364C2×C3×C4—96no—
365#67—316no—
366#67—316no—
367#67—316no—
368#67—616no—
369#67—916no—
370#67—916no—
371#67—916no—
372#67—916no—
373#67—916no—
374C2×Dic3—36no—
375C2×Dic3—36no—
376C2×Dic3—36no—
377C2×Dic3—66no—
378C2×Dic3—96no—
379C2×Dic3—916no—
380D12—916no—
381D12—916no—
382C4×D3—916no—
383C4×D3—916no—
384C2×C32—24no—
385C2×C32—24no—
386C2×C32—26no—
387C2×C32—21no—
388C2×C32—44no—
389C2×C32C2×D3×D416yescomplemented
390C2×C32C22×#6716yescomplemented
391C2×C32C24×D311yescharacteristic
392C2×C32C24×D311yescharacteristic
393C3⋊D3—24no—
394C3⋊D3—24no—
395C3⋊D3C22×#6718yescomplemented
396C3×D3—38no—
397C3×D3—64no—
398C3×D3—64no—
399C3×D3—64no—
400C3×D3—64no—
401C3×D3—124no—
402C24—91no—
403C24—91no—
404C24—91no—
405C24—91no—
406C24—912no—
407C24—188no—
408C24—2712no—
409C24—548no—
410C24((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
411C2×D4—912no—
412C2×D4—912no—
413C2×D4—2712no—
414C2×D4—2712no—
415C2×D4—2764no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—2712no—
419C22×C3—212no—
420C22×C3—33no—
421C22×C3—33no—
422C22×C3—34no—
423C22×C3—34no—
424C22×C3—34no—
425C22×C3—34no—
426C22×C3—612no—
427C22×C3—612no—
428C22×C3—63no—
429C22×C3—64no—
430C22×C3—64no—
431C22×C3—94no—
432C22×C3—924no—
433C22×C3—94no—
434C22×C3—94no—
435C22×C3—94no—
436C22×C3—1212no—
437C22×C3—1812no—
438C22×C3—1812no—
439C22×C3—1832no—
440C22×C3—1812no—
441C22×C3D3×#6714yes—
442C22×C3C22×D3213yes—
443C22×C3C22×D3214yes—
444C2×D3—324no—
445C2×D3—34no—
446C2×D3—34no—
447C2×D3—34no—
448C2×D3—34no—
449C2×D3—34no—
450C2×D3—64no—
451C2×D3—632no—
452C2×D3—612no—
453C2×D3—612no—
454C2×D3—612no—
455C2×D3—612no—
456C2×D3—94no—
457C2×D3—924no—
458C2×D3—94no—
459C2×D3—94no—
460C2×D3—1212no—
461C2×D3—1832no—
462C2×D3—1812no—
463C2×D3—1812no—
464C2×D3—1832no—
465C2×D3—1832no—
466C2×D3—1832no—
467C3×C4—94no—
468C3×C4—94no—
469Dic3—34no—
470Dic3—34no—
471Dic3—34no—
472Dic3—64no—
473Dic3—94no—
474C32—21no—
475C32C22×D3×D411yescharacteristic, complemented
476C32C23×#6711yescharacteristic, complemented
477C23—24no—
478C23—96no—
479C23—98no—
480C23—96no—
481C23—96no—
482C23—96no—
483C23—916no—
484C23—1848no—
485C23—184no—
486C23—184no—
487C23—184no—
488C23—2716no—
489C23—5448no—
490C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
491C23C2×((C3⋊D3)⋊C3)⋊C216yescomplemented
492D4—916no—
493D4—916no—
494D4—2716no—
495D4—2716no—
496C2×C4—96no—
497C2×C4—96no—
498C2×C4—2716no—
499C2×C3—24no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—36no—
503C2×C3—36no—
504C2×C3—64no—
505C2×C3—64no—
506C2×C3—61no—
507C2×C3—66no—
508C2×C3—98no—
509C2×C3—124no—
510C2×C3—184no—
511C2×C3—184no—
512C2×C3—184no—
513C2×C3C2×D3×#6716yes—
514C2×C3C23×D3211yescharacteristic
515D3—38no—
516D3—64no—
517D3—64no—
518D3—64no—
519D3—64no—
520D3—98no—
521D3—124no—
522D3—184no—
523D3—184no—
524C22—212no—
525C22—94no—
526C22—924no—
527C22—94no—
528C22—94no—
529C22—94no—
530C22—1832no—
531C22—1812no—
532C22—1812no—
533C22—1812no—
534C22—5432no—
535C22#8867614yesdirect product factor
536C22C22×((C3⋊D3)⋊C3)⋊C213yes—
537C22C22×((C3⋊D3)⋊C3)⋊C214yescomplemented
538C4—94no—
539C4—94no—
540C3—31no—
541C3—31no—
542C3—61no—
543C3C22×D3×#6711yescharacteristic
544C2—24no—
545C2—98no—
546C2—184no—
547C2—184no—
548C2—184no—
549C2C2×#8867616yesdirect product factor
550C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C23×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×(C3⋊D3)⋊C3C2semidirect
2. C23×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. C22×#6235C2semidirect
4. C22×#638⋊C3C2semidirect
5. C2×#88676C2direct (N × H)
6. C22×#6200C2semidirect
7. C22×#6198C2semidirect
8. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
11. C24×H(3,𝔽3)C22semidirect
12. C23×#272C22semidirect
15. C23×(C3⋊D3)⋊C3C22semidirect
16. C23×(C3⋊D3)⋊C3C22semidirect
17. C23×(C3⋊D3)⋊C3C22semidirect
19. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
20. C2×#6235C22semidirect
21. C22×#1084C22semidirect
22. C2×#638⋊C3C22semidirect
23. C22×(C3⋊Dic3)⋊C3C22semidirect
24. #88676C22direct (N × H)
25. C2×#6200C22semidirect
26. C2×#6198C22semidirect
27. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
28. C24×C3⋊D3D3semidirect
32. C22×#638D3semidirect
47. C23×H(3,𝔽3)C23semidirect
49. C22×#272C23semidirect
53. C22×(C3⋊D3)⋊C3D4semidirect
55. C22×(C3⋊D3)⋊C3C23semidirect
56. C22×(C3⋊D3)⋊C3C23semidirect
57. C22×(C3⋊D3)⋊C3C23semidirect
59. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
60. #6235C23semidirect
61. C2×#1084C23semidirect
62. #638⋊C3C23semidirect
63. C2×(C3⋊Dic3)⋊C3C23semidirect
64. #6200C23semidirect
65. #6198C23semidirect
66. ((C3⋊D3)⋊C3)⋊C4C23semidirect
71. C24×C32C2×D3semidirect
72. C24×C32C2×D3semidirect
75. C23×C3⋊D3C2×D3semidirect
76. C23×C3⋊D3C2×D3semidirect
77. C23×C3⋊D3C2×D3semidirect
90. C2×#638C2×D3semidirect
91. C22×C3⋊Dic3C2×D3semidirect
109. C22×H(3,𝔽3)C2×D4semidirect
111. C22×H(3,𝔽3)C24semidirect
113. C2×#272C24semidirect
117. C2×(C3⋊D3)⋊C3C2×D4semidirect
119. C2×(C3⋊D3)⋊C3C24semidirect
120. C2×(C3⋊D3)⋊C3C24semidirect
122. #1084C24semidirect
123. (C3⋊Dic3)⋊C3C24semidirect
156. C23×C32C22×D3semidirect
157. C23×C32C22×D3semidirect
161. C22×C3⋊D3#67semidirect
163. C22×C3⋊D3C22×D3semidirect
164. C22×C3⋊D3C22×D3semidirect
165. C22×C3⋊D3C22×D3semidirect
184. #638C22×D3semidirect
185. C2×C3⋊Dic3C22×D3semidirect
202. C2×H(3,𝔽3)C22×D4semidirect
207. (C3⋊D3)⋊C3C22×D4semidirect
269. C22×C32D3×D4semidirect
270. C22×C32C2×#67semidirect
273. C22×C32C23×D3semidirect
274. C22×C32C23×D3semidirect
278. C2×C3⋊D3C2×#67semidirect
280. C2×C3⋊D3C23×D3semidirect
281. C2×C3⋊D3C23×D3semidirect
297. C3⋊Dic3C23×D3semidirect
310. H(3,𝔽3)C23×D4semidirect
389. C2×C32C2×D3×D4semidirect
390. C2×C32C22×#67semidirect
395. C3⋊D3C22×#67semidirect
410. C24((C3⋊D3)⋊C3)⋊C2semidirect
475. C32C22×D3×D4semidirect
476. C32C23×#67semidirect
491. C23C2×((C3⋊D3)⋊C3)⋊C2semidirect
535. C22#88676direct (N × H)
537. C22C22×((C3⋊D3)⋊C3)⋊C2semidirect
549. C2C2×#88676direct (N × H)

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C23×((C3⋊D3)⋊C3)⋊C22
C22×#62352
C22×#638⋊C32
C2×#886762
C22×#62002
C22×#61982
C22×((C3⋊D3)⋊C3)⋊C42
C22×#38113
C22×D3×#673