Finite Group Databasebeta

← #13141172

Subgroups — #13141172

Order 1728.

1237 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic
2C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4C2×#88451C212yesmaximal, complemented
5#11720334C212yesmaximal, complemented
6#10591713—31nomaximal
7#10591710—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C23×#272C2211yescharacteristic
10C2×#6235—22no—
11C2×#6235—22no—
12C2×#6235C414yescomplemented
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3—22no—
15C22×(C3⋊Dic3)⋊C3—24no—
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2212yes—
18#3757⋊C3—22no—
19#88451—24no—
20C23×C3⋊Dic3D311yescharacteristic, complemented
21C22×C3×#67—31no—
22C22×C3×#67—31no—
23C22×C3×#67—61no—
24C23×C3×Dic3—31no—
25C2×#3757D311yescharacteristic, complemented
26C2×C3×#217—31no—
27C2×(C2×D3)⋊Dic3—32no—
28C2×(C2×D3)⋊Dic3—32no—
29#63579—32no—
30#63579—32no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)—22no—
33C23×H(3,𝔽3)C2×C412yes—
34C23×H(3,𝔽3)D412yes—
35C23×H(3,𝔽3)D412yes—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×#272—21no—
38C22×#272—21no—
39C22×#272—22no—
40C22×#272—22no—
41C22×#272C2×C412yescomplemented
42#6235—28no—
43C2×#1084—21no—
44C2×#1084—21no—
45C2×#1084C2×C412yescomplemented
46C2×(C3⋊Dic3)⋊C3—28no—
47C2×(C3⋊Dic3)⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3—22no—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3D414yescomplemented
53#5220—91no—
54#5213—91no—
55#4958—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C2×D311yescharacteristic
59C23×C3×D3—31no—
60C23×C3×D3—31no—
61C23×C3×D3—61no—
62C22×C3⋊Dic3—22no—
63C22×C3⋊Dic3—24no—
64C22×C3⋊Dic3C2×D312yes—
65C22×C3⋊Dic3C2×D312yes—
66C2×C3×#67—34no—
67C2×C3×#67—34no—
68C2×C3×#67—64no—
69C2×C3×#67—64no—
70C2×C3×#67—64no—
71C2×C3×#67—62no—
72C2×C3×#67—62no—
73C2×C3×#67—62no—
74C2×C3×#67—62no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—64no—
80C22×C3×Dic3—61no—
81C22×C3×Dic3—62no—
82#3757—22no—
83C3×#217—62no—
84(C2×D3)⋊Dic3—64no—
85(C2×D3)⋊Dic3—64no—
86C22×H(3,𝔽3)—24no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)—24no—
89C22×H(3,𝔽3)—22no—
90C22×H(3,𝔽3)C22⋊C414yescomplemented
91C22×H(3,𝔽3)C22×C411yescharacteristic
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C22×H(3,𝔽3)C2×D411yescharacteristic
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)#4112yes—
96C2×#272—22no—
97C2×#272—24no—
98C2×#272—41no—
99C2×#272—42no—
100C2×#272—41no—
101#1084—22no—
102(C3⋊Dic3)⋊C3—24no—
103(C3⋊Dic3)⋊C3—42no—
104C22×C3×D4—91no—
105C23×C3×C4—91no—
106C22×#67—31no—
107C22×#67—31no—
108C22×#67—61no—
109C23×Dic3—31no—
110C23×Dic3—91no—
111C2×C3×C22⋊C4—91no—
112C2×#217—31no—
113C2×#217—91no—
114C2×#217—92no—
115C2×(C2×D3)⋊C4—92no—
116C2×(C2×D3)⋊C4—92no—
117C3⋊#95—92no—
118C3⋊#95—92no—
119C3⋊#95—92no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—21no—
128C23×C32—42no—
129C23×C32—42no—
130C23×C32C4×D312yes—
131C23×C32C4×D312yes—
132C23×C32D1212yes—
133C23×C32D1212yes—
134C23×C32#6712yes—
135C23×C32#6712yes—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D311yescharacteristic
138C22×C3×D3—32no—
139C22×C3×D3—32no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—122no—
152C22×C3×D3—122no—
153C2×C3⋊Dic3—28no—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3#6714yescomplemented
160C3×#67—616no—
161C3×#67—68no—
162C3×#67—68no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—68no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179#386—271noSylow
180C2×H(3,𝔽3)—24no—
181C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
182C2×H(3,𝔽3)C4×D412yes—
183C2×H(3,𝔽3)#12011yescharacteristic
184C2×H(3,𝔽3)#12112yes—
185C2×H(3,𝔽3)#12311yescharacteristic
186#272—42no—
187C24×C3—31no—
188C24×C3—31no—
189C24×C3—61no—
190C24×C3—91no—
191C24×C3D3211yescharacteristic
192C23×D3—31no—
193C23×D3—31no—
194C23×D3—61no—
195C2×C3×D4—94no—
196C2×C3×D4—182no—
197C2×C3×D4—182no—
198C22×C3×C4—92no—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C22×C3×C4—184no—
202C22×C3×C4—182no—
203C2×#67—34no—
204C2×#67—34no—
205C2×#67—64no—
206C2×#67—62no—
207C2×#67—62no—
208C2×#67—62no—
209C2×#67—62no—
210C2×#67—64no—
211C2×#67—64no—
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—32no—
215C22×Dic3—32no—
216C22×Dic3—61no—
217C22×Dic3—64no—
218C22×Dic3—62no—
219C22×Dic3—92no—
220C22×Dic3—92no—
221C22×Dic3—184no—
222C22×Dic3—182no—
223C3×C22⋊C4—182no—
224#217—62no—
225#217—182no—
226#217—184no—
227(C2×D3)⋊C4—184no—
228(C2×D3)⋊C4—184no—
229C22×C32—24no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—24no—
233C22×C32—22no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—22no—
237C22×C32—24no—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—44no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32—44no—
247C22×C32(C2×D3)⋊C414yescomplemented
248C22×C32(C2×D3)⋊C414yescomplemented
249C22×C32C2×C4×D311yescharacteristic
250C22×C32C2×C4×D311yescharacteristic
251C22×C32C2×D1211yescharacteristic
252C22×C32C2×D1211yescharacteristic
253C22×C32#23512yes—
254C22×C32D3×D412yes—
255C22×C32#23712yes—
256C22×C32C2×#6711yescharacteristic
257C22×C32C2×#6711yescharacteristic
258C22×C32C2×#6712yes—
259C2×C3×D3—64no—
260C2×C3×D3—64no—
261C2×C3×D3—62no—
262C2×C3×D3—62no—
263C2×C3×D3—64no—
264C2×C3×D3—124no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—124no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C2×C3×D3—122no—
272C2×C3×D3—122no—
273C2×C3×D3—121no—
274C3⋊Dic3—24no—
275C3⋊Dic3—42no—
276C3×Dic3—64no—
277C3×Dic3—64no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—122no—
281C22×D4—91no—
282C23×C4—91no—
283C2×C22⋊C4—91no—
284C2×C22⋊C4—272no—
285#95—272no—
286H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—32no—
290C23×C3—32no—
291C23×C3—32no—
292C23×C3—32no—
293C23×C3—32no—
294C23×C3—32no—
295C23×C3—31no—
296C23×C3—31no—
297C23×C3—62no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3—62no—
301C23×C3—62no—
302C23×C3—62no—
303C23×C3—62no—
304C23×C3—61no—
305C23×C3—92no—
306C23×C3—122no—
307C23×C3—122no—
308C23×C3—182no—
309C23×C3—181no—
310C23×C3—181no—
311C23×C3—182no—
312C23×C3#62412yes—
313C23×C3#62612yes—
314C23×C3#62612yes—
315C23×C3C2×D3211yescharacteristic
316C22×D3—32no—
317C22×D3—32no—
318C22×D3—62no—
319C22×D3—62no—
320C22×D3—62no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—62no—
328C22×D3—62no—
329C22×D3—122no—
330C22×D3—122no—
331C3×D4—188no—
332C2×C3×C4—94no—
333C2×C3×C4—92no—
334C2×C3×C4—182no—
335C2×C3×C4—182no—
336C2×C3×C4—188no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4—182no—
340C2×C3×C4—182no—
341C2×C3×C4—182no—
342#67—68no—
343#67—68no—
344#67—616no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—34no—
348C2×Dic3—62no—
349C2×Dic3—68no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—62no—
355C2×Dic3—62no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—94no—
362C2×Dic3—182no—
363C2×Dic3—182no—
364C2×Dic3—188no—
365C2×Dic3—182no—
366C2×Dic3—182no—
367C2×Dic3—182no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—22no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—22no—
375C2×C32—44no—
376C2×C32#67⋊C412yes—
377C2×C32#88211yescharacteristic
378C2×C32#88611yescharacteristic
379C2×C32C2×(C2×D3)⋊C411yescharacteristic
380C2×C32C2×(C2×D3)⋊C411yescharacteristic
381C2×C32C4×#6712yes—
382C2×C32#92911yescharacteristic
383C2×C32#93012yes—
384C2×C32#93912yes—
385C2×C32#95311yescharacteristic
386C3×D3—124no—
387C3×D3—122no—
388C3×D3—122no—
389C24—91no—
390C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C2×D4—94no—
392C2×D4—182no—
393C2×D4—182no—
394C22×C4—91no—
395C22×C4—92no—
396C22×C4—92no—
397C22×C4—184no—
398C22×C4—182no—
399C22⋊C4—182no—
400C22⋊C4—544no—
401C22×C3—24no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—32no—
406C22×C3—32no—
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—31no—
410C22×C3—31no—
411C22×C3—31no—
412C22×C3—31no—
413C22×C3—32no—
414C22×C3—32no—
415C22×C3—34no—
416C22×C3—34no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—64no—
420C22×C3—64no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—64no—
424C22×C3—64no—
425C22×C3—62no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—62no—
430C22×C3—64no—
431C22×C3—124no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—124no—
435C22×C3—184no—
436C22×C3—182no—
437C22×C3—361no—
438C22×C3—361no—
439C22×C3—362no—
440C22×C3#372214yes—
441C22×C3#380412yes—
442C22×C3C2×#62411yescharacteristic
443C22×C3C2×#62611yescharacteristic
444C22×C3C2×#62611yescharacteristic
445C22×C3D3×#6712yes—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—64no—
451C2×D3—124no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—122no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—122no—
460C2×D3—124no—
461C3×C4—184no—
462C3×C4—182no—
463C3×C4—362no—
464Dic3—62no—
465Dic3—62no—
466Dic3—64no—
467Dic3—64no—
468Dic3—122no—
469Dic3—184no—
470Dic3—362no—
471C32—21no—
472C32#495811yescharacteristic, complemented
473C32#521311yescharacteristic, complemented
474C23—22no—
475C23—22no—
476C23—92no—
477C23—182no—
478C23—181no—
479C23—181no—
480C23—182no—
481C23#619912yes—
482C23#620012yes—
483C23#620012yes—
484C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
485D4—188no—
486C2×C4—92no—
487C2×C4—94no—
488C2×C4—188no—
489C2×C4—182no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—182no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4—182no—
496C2×C3—24no—
497C2×C3—32no—
498C2×C3—32no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—32no—
506C2×C3—32no—
507C2×C3—64no—
508C2×C3—64no—
509C2×C3—62no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—61no—
513C2×C3—62no—
514C2×C3—124no—
515C2×C3—362no—
516C2×C3#6395212yes—
517C2×C3C2×#372211yescharacteristic
518C2×C3#6396611yescharacteristic
519C2×C3#6397711yescharacteristic
520C2×C3#6397812yes—
521D3—122no—
522D3—122no—
523D3—124no—
524C22—24no—
525C22—22no—
526C22—22no—
527C22—24no—
528C22—184no—
529C22—182no—
530C22—361no—
531C22—361no—
532C22—362no—
533C22#8845114yescomplemented
534C22#8867112yes—
535C22C2×#619911yescharacteristic
536C22C2×#620011yescharacteristic
537C22C2×#620011yescharacteristic
538C22#8867612yes—
539C4—182no—
540C4—184no—
541C4—362no—
542C3—31no—
543C3—31no—
544C3—61no—
545C3#1059171111yescharacteristic
546C2—24no—
547C2—362no—
548C2#1172129312yes—
549C2#1172130111yescharacteristic
550C2C2×#8845111yescharacteristic
551C2#1172132111yescharacteristic
552C2#1172132212yes—
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC24×H(3,𝔽3)C22
prime radicalC22×#6235C2
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×(C3⋊Dic3)⋊C3C2semidirect
3. C2×#3757⋊C3C2semidirect
4. C2×#88451C2semidirect
5. #11720334C2semidirect
12. C2×#6235C4semidirect
20. C23×C3⋊Dic3D3semidirect
25. C2×#3757D3semidirect
41. C22×#272C2×C4semidirect
45. C2×#1084C2×C4semidirect
52. C2×(C3⋊Dic3)⋊C3D4semidirect
90. C22×H(3,𝔽3)C22⋊C4semidirect
159. C2×C3⋊Dic3#67semidirect
247. C22×C32(C2×D3)⋊C4semidirect
248. C22×C32(C2×D3)⋊C4semidirect
286. H(3,𝔽3)#386semidirect
472. C32#4958semidirect
473. C32#5213semidirect
533. C22#88451semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C23×(C3⋊Dic3)⋊C32
C2×#3757⋊C32
C2×#884512
#117203342
#105917133
#105917103