Finite Group Databasebeta

← #13150852

Subgroups — #13150852

Order 1728.

1391 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721323C212yesmaximal, complemented
5#11721307C212yesmaximal, complemented
6#11721159C212yesmaximal, complemented
7C4×#6200C214yesmaximal, complemented
8#11721094C212yesmaximal, complemented
9#10595023—31nomaximal
10#10595022—31nomaximal
11C23×#272C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C2×C4×#272—24no—
15C2×C4×#272C2212yes—
16C2×C4×#272C2212yes—
17C2×#638⋊C3C2212yes—
18#3828⋊C3—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
20C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
21C2×#634⋊C3C2211yescharacteristic, complemented
22C2×#6200C2214yes—
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
25C4×(C3⋊Dic3)⋊C3C2212yescomplemented
26#88451C2214yes—
27#88450C2214yescomplemented
28#88448C2214yescomplemented
29C2×#3828D311yescharacteristic, complemented
30C2×C3×#235—31no—
31C22×C3×C4×D3—31no—
32C22×C3×C4×D3—31no—
33C22×C3×C4×D3—61no—
34#64135D311yescharacteristic, complemented
35C3×#926—31no—
36#63977—32no—
37#63976—32no—
38#63966—32no—
39#63961—32no—
40#63909—32no—
41#63908—32no—
42C4×#626—34no—
43C4×#625—34no—
44#63857—32no—
45#63856—32no—
46C23×H(3,𝔽3)C2311yescharacteristic
47C22×#272—22no—
48C22×#272—24no—
49C22×#272C2312yes—
50C22×(C3⋊D3)⋊C3C2312yes—
51C2×C4×H(3,𝔽3)D414yescomplemented
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×#1084—22no—
55C2×#1084C2312yes—
56C4×#272—24no—
57C4×#272—24no—
58#638⋊C3—24no—
59C2×(C3⋊Dic3)⋊C3C2314yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C4⋊((C3⋊D3)⋊C3)—22no—
62C4×(C3⋊D3)⋊C3—24no—
63#634⋊C3—22no—
64#6200—28no—
65#5828—91no—
66#5791—91no—
67#5524—91no—
68C23×C3×D3—31no—
69C23×C3×D3—31no—
70C23×C3×D3—61no—
71C22×C32×C4—21no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C2×#638C2×D312yes—
75#3828—24no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
77C2×C4×C3⋊D3C2×D312yescomplemented
78C2×#634C2×D311yescharacteristic, complemented
79C2×C3×#67—32no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—61no—
83C3×#235—64no—
84C2×C3×D12—31no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—62no—
91C2×C3×C4×D3—62no—
92C2×C3×C4×D3—68no—
93C2×C3×C4×D3—64no—
94C2×C3×C4×D3—64no—
95C2×C3×Dic6—31no—
96C2×#626—34no—
97C2×#625—34no—
98#3757C2×D312yes—
99C4⋊(C3⋊Dic3)C2×D312yescomplemented
100C4×C3⋊Dic3C2×D312yescomplemented
101C3×#217—32no—
102C3×C4⋊Dic3—32no—
103C3×C4×Dic3—32no—
104#3724—34no—
105Dic3⋊Dic3—34no—
106#3722—34no—
107(C2×D3)⋊Dic3—34no—
108(C2×D3)⋊Dic3—34no—
109(C2×D3)⋊Dic3—34no—
110C22×H(3,𝔽3)—22no—
111C22×H(3,𝔽3)C2×D412yes—
112C22×H(3,𝔽3)C2411yescharacteristic
113C2×#272—28no—
114C2×#272—22no—
115C2×#272—22no—
116C2×#272#4114yescomplemented
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—24no—
119C4×H(3,𝔽3)C2×D412yes—
120C4×H(3,𝔽3)C2×D412yes—
121#1084—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—24no—
124C2×C3×#41—91no—
125C23×C3×C4—91no—
126C2×#235—31no—
127C2×#235—91no—
128C22×C4×D3—31no—
129C22×C4×D3—31no—
130C22×C4×D3—61no—
131C3×#117—91no—
132#953—92no—
133#946—92no—
134#938—92no—
135#937—92no—
136#935—92no—
137#930—92no—
138#929—92no—
139C4×#67—94no—
140C4×#67—94no—
141#926—31no—
142#926—91no—
143#924—92no—
144#897—92no—
145#895—92no—
146#886—92no—
147#882—92no—
148C4×D12—94no—
149C23×C32—21no—
150C23×C32C22×D311yescharacteristic
151C23×C32C22×D311yescharacteristic
152C22×C3⋊D3C22×D312yes—
153C22×C3×D3—32no—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—62no—
157C22×C3×D3—64no—
158C22×C3×D3—64no—
159C22×C3×D3—68no—
160C22×C3×D3—64no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C2×C32×C4—24no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4D1214yescomplemented
167C2×C32×C4#6714yescomplemented
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172#638—24no—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D314yes—
175C4⋊(C3⋊D3)—22no—
176C4×C3⋊D3—24no—
177#634—22no—
178C3×#67—64no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C3×D12—62no—
188C3×C4×D3—68no—
189C3×C4×D3—68no—
190C3×C4×D3—64no—
191C3×C4×D3—64no—
192C3×C4×D3—64no—
193C3×C4×D3—64no—
194C3×C4×D3—64no—
195C3×Dic6—62no—
196#626—68no—
197#625—68no—
198#525—271noSylow
199C2×H(3,𝔽3)—22no—
200C2×H(3,𝔽3)C22×D411yescharacteristic
201C2×H(3,𝔽3)C2×#4112yes—
202#272—24no—
203(C3⋊D3)⋊C3—42no—
204C24×C3—91no—
205C23×D3—31no—
206C23×D3—31no—
207C23×D3—61no—
208C3×#41—184no—
209C2×C3×Q8—91no—
210C2×C3×D4—91no—
211C2×C3×D4—92no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—61no—
215C22×C3×C4—92no—
216C22×C3×C4—91no—
217C22×C3×C4—92no—
218C22×C3×C4—92no—
219C22×C3×C4—184no—
220C22×C3×C4D3211yescharacteristic
221C2×#67—32no—
222C2×#67—92no—
223C2×#67—94no—
224C2×#67—94no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—61no—
228#235—64no—
229#235—184no—
230C2×D12—31no—
231C2×D12—91no—
232C2×D12—94no—
233C2×C4×D3—32no—
234C2×C4×D3—32no—
235C2×C4×D3—32no—
236C2×C4×D3—32no—
237C2×C4×D3—32no—
238C2×C4×D3—62no—
239C2×C4×D3—62no—
240C2×C4×D3—64no—
241C2×C4×D3—64no—
242C2×C4×D3—68no—
243C2×C4×D3—92no—
244C2×Dic6—31no—
245C2×Dic6—91no—
246C3×C4⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C42—92no—
249#217—32no—
250#217—92no—
251#217—94no—
252#217—94no—
253(C2×D3)⋊C4—94no—
254(C2×D3)⋊C4—94no—
255(C2×D3)⋊C4—94no—
256(C2×D3)⋊C4—94no—
257C4⋊Dic3—32no—
258C4⋊Dic3—92no—
259C4⋊Dic3—94no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262C4×Dic3—32no—
263C4×Dic3—92no—
264C22×C32—24no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—21no—
268C22×C32—22no—
269C22×C32C2×D1212yes—
270C22×C32C2×#6712yes—
271C22×C32C23×D311yescharacteristic
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—24no—
274C2×C3⋊D3—22no—
275C2×C3×D3—34no—
276C2×C3×D3—34no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—616no—
280C2×C3×D3—68no—
281C2×C3×D3—68no—
282C2×C3×D3—64no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—64no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4C2×D1212yes—
292C32×C4C2×D1212yes—
293C32×C4C2×#6712yes—
294C32×C4C2×#6712yes—
295C3⋊Dic3—24no—
296C3⋊Dic3—22no—
297C3×Dic3—64no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—64no—
301C3×Dic3—62no—
302C2×#41—91no—
303C23×C4—91no—
304#123—272no—
305C22⋊Q8—272no—
306#121—272no—
307#120—272no—
308C4×D4—274no—
309#117—91no—
310H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
311C23×C3—31no—
312C23×C3—31no—
313C23×C3—61no—
314C23×C3—92no—
315C23×C3—92no—
316C23×C3—184no—
317C23×C3—182no—
318C23×C3C2×D3211yescharacteristic
319C22×D3—32no—
320C22×D3—32no—
321C22×D3—32no—
322C22×D3—62no—
323C22×D3—64no—
324C22×D3—64no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—64no—
328C22×D3—68no—
329C22×D3—92no—
330C3×Q8—182no—
331C3×D4—182no—
332C3×D4—184no—
333C2×C3×C4—34no—
334C2×C3×C4—34no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—64no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—92no—
343C2×C3×C4—94no—
344C2×C3×C4—92no—
345C2×C3×C4—184no—
346C2×C3×C4—182no—
347C2×C3×C4—184no—
348C2×C3×C4—184no—
349C2×C3×C4#62614yes—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352#67—64no—
353#67—184no—
354#67—188no—
355#67—188no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—34no—
359C2×Dic3—32no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363C2×Dic3—64no—
364C2×Dic3—94no—
365C2×Dic3—92no—
366D12—62no—
367D12—182no—
368D12—188no—
369C4×D3—64no—
370C4×D3—64no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—68no—
374C4×D3—68no—
375C4×D3—64no—
376C4×D3—184no—
377Dic6—62no—
378Dic6—182no—
379C2×C32—24no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32C22×D1211yescharacteristic
385C2×C32D3×#4112yes—
386C2×C32D3×#4112yes—
387C2×C32C22×#6711yescharacteristic
388C3⋊D3—42no—
389C3×D3—68no—
390C3×D3—64no—
391C3×D3—64no—
392C3×D3—122no—
393C24—91no—
394#41—184no—
395C2×Q8—91no—
396C2×D4—92no—
397C2×D4—91no—
398C2×D4—274no—
399C22×C4—91no—
400C22×C4—92no—
401C22×C4—92no—
402C22×C4—92no—
403C22×C4—184no—
404C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C4⋊C4—92no—
406C4⋊C4—274no—
407C22⋊C4—92no—
408C22⋊C4—274no—
409C22⋊C4—274no—
410C42—92no—
411C22×C3—22no—
412C22×C3—31no—
413C22×C3—31no—
414C22×C3—32no—
415C22×C3—32no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—64no—
419C22×C3—61no—
420C22×C3—62no—
421C22×C3—94no—
422C22×C3—188no—
423C22×C3—184no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3C2×#62612yes—
428C22×C3C22×D3211yescharacteristic
429C2×D3—34no—
430C2×D3—34no—
431C2×D3—64no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—64no—
437C2×D3—68no—
438C2×D3—68no—
439C2×D3—616no—
440C2×D3—64no—
441C2×D3—64no—
442C2×D3—62no—
443C2×D3—182no—
444C2×D3—184no—
445C3×C4—32no—
446C3×C4—32no—
447C3×C4—32no—
448C3×C4—32no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—184no—
452C3×C4—182no—
453C3×C4—182no—
454C3×C4C2×#62612yes—
455C3×C4C2×#62612yes—
456Dic3—62no—
457Dic3—62no—
458Dic3—64no—
459Dic3—64no—
460Dic3—62no—
461Dic3—182no—
462Dic3—184no—
463C32—21no—
464C32#552411yescharacteristic, complemented
465C32#582811yescharacteristic, complemented
466C23—92no—
467C23—92no—
468C23—184no—
469C23—182no—
470C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
471Q8—182no—
472D4—184no—
473D4—182no—
474D4—548no—
475C2×C4—94no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—182no—
479C2×C4—184no—
480C2×C4—184no—
481C2×C4—184no—
482C2×C4#620014yescomplemented
483C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
484C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
485C2×C3—22no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—32no—
489C2×C3—32no—
490C2×C3—62no—
491C2×C3—62no—
492C2×C3—64no—
493C2×C3—61no—
494C2×C3—62no—
495C2×C3—184no—
496C2×C3—362no—
497C2×C3#6430612yes—
498C2×C3C22×#62611yescharacteristic
499D3—64no—
500D3—64no—
501D3—68no—
502D3—122no—
503D3—362no—
504C22—22no—
505C22—94no—
506C22—188no—
507C22—182no—
508C22—182no—
509C22—182no—
510C22—184no—
511C22C2×#620012yes—
512C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
513C4—182no—
514C4—182no—
515C4—184no—
516C4C2×#620012yes—
517C4C2×#620012yes—
518C3—31no—
519C3—31no—
520C3—61no—
521C3#1059503411yescharacteristic
522C2—22no—
523C2—184no—
524C2—362no—
525C2#1172249112yes—
526C2C22×#620011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#525
LCSG ▷ C22×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. #64135⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721307C2semidirect
6. #11721159C2semidirect
7. C4×#6200C2semidirect
8. #11721094C2semidirect
19. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. C2×C4×(C3⋊D3)⋊C3C22semidirect
21. C2×#634⋊C3C22semidirect
24. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
25. C4×(C3⋊Dic3)⋊C3C22semidirect
27. #88450C22semidirect
28. #88448C22semidirect
29. C2×#3828D3semidirect
34. #64135D3semidirect
51. C2×C4×H(3,𝔽3)D4semidirect
76. C2×C4⋊(C3⋊D3)C2×D3semidirect
77. C2×C4×C3⋊D3C2×D3semidirect
78. C2×#634C2×D3semidirect
99. C4⋊(C3⋊Dic3)C2×D3semidirect
100. C4×C3⋊Dic3C2×D3semidirect
116. C2×#272#41semidirect
166. C2×C32×C4D12semidirect
167. C2×C32×C4#67semidirect
310. H(3,𝔽3)#525semidirect
464. C32#5524semidirect
465. C32#5828semidirect
482. C2×C4#6200semidirect

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
#64135⋊C32
#117213232
#117213072
#117211592
C4×#62002
#117210942
#105950233
#105950223