Finite Group Databasebeta

← #13181063

Subgroups — #13181063

Order 1728.

938 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89040C211yesmaximal, characteristic, complemented
2C3×#64306C211yesmaximal, characteristic
3#11724514C211yesmaximal, characteristic, complemented
4#11724319C214yesmaximal, complemented
5#11724302C212yesmaximal, complemented
6#11724301C212yesmaximal, complemented
7#11724300C214yesmaximal, complemented
8#10596494—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634—21no—
11C32×#235—21no—
12C2×#6296—21no—
13C2×#6294C2212yes—
14#89040—21no—
15#89040—21no—
16#89040C2212yes—
17#89040C2212yes—
18C3×#3811C2212yes—
19C3×#3805—21no—
20C3×#3802C2211yescharacteristic
21C3×C4×D32C2212yes—
22C3×#3798—21no—
23C3×#3797C2211yescharacteristic
24C3×#3796—21no—
25#88994C2212yes—
26C4⋊(C3⋊#157)C2212yes—
27C4×C3⋊#157C2212yes—
28#88934C2218yes—
29D32⋊Dic3C2218yes—
30C3×D3×#41—61no—
31#64306D311yescharacteristic
32C2×#3797—31no—
33C2×D3×Dic6—61no—
34#64286—31no—
35#64233—34no—
36#64222—32no—
37#64221—32no—
38D32⋊Q8—34no—
39C22×C3×C3⋊D3C2311yescharacteristic
40C2×C3×D32C2312yes—
41C2×C3⋊#157C2314yes—
42C2×C33×C4D411yescharacteristic
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C3×C4×C3⋊D3D411yescharacteristic
47C3×C4×C3⋊D3C2311yescharacteristic
48C3×C4×C3⋊D3C2311yescharacteristic
49C3×#634—22no—
50C3×#634—22no—
51C32×#67—22no—
52C32×D12—21no—
53C32×C4×D3—22no—
54C32×Dic6—21no—
55#6296—22no—
56#6296—22no—
57#6294—22no—
58#6294C2314yes—
59C3×#627—21no—
60C3×#626—22no—
61C3×#625—21no—
62C3×#624C2312yes—
63C3×D3×Dic3—22no—
64#5825—91nomaximal
65C32×#41—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—21no—
68C3×#239—61no—
69C3×D3×Q8—61no—
70C3×#237—61no—
71C3×#237—62no—
72C3×D3×D4—61no—
73C3×D3×D4—62no—
74C3×#235—21no—
75C3×#235—21no—
76C3×#235—41no—
77C3×#235—62no—
78C2×C3×C4×D3—62no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—61no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6—121no—
84#3811C2×D312yes—
85C2×#627—62no—
86C2×#627—61no—
87C2×#624—32no—
88#3805—21no—
89C2×D3×Dic3—62no—
90#3802C2×D311yescharacteristic
91C4×D32C2×D312yes—
92#3798—21no—
93#3797—32no—
94#3797—32no—
95#3797—61no—
96#3797—61no—
97#3797C2×D311yescharacteristic
98#3796—21no—
99D3×Dic6—62no—
100D3×Dic6—62no—
101D3×Dic6—62no—
102D3×Dic6—62no—
103C22⋊#157—32no—
104C4⋊#157—32no—
105C4×#157—32no—
106#3773—38no—
107D32⋊C4—38no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3C2×D411yescharacteristic
111C2×C3×C3⋊D3C2411yescharacteristic
112C2×C32×D3—22no—
113C3×D32—24no—
114C3⋊#157—24no—
115C33×C4C2×D411yescharacteristic
116C33×C4C2×D411yescharacteristic
117C3×C3⋊Dic3—24no—
118C3×C3⋊Dic3C2×D411yescharacteristic
119C3×C3⋊Dic3C2×D411yescharacteristic
120C32×Dic3—22no—
121C2×C3×#41—91no—
122D3×#41—61no—
123C2×D3×Q8—181no—
124C22×Dic6—91no—
125#944—92no—
126#936—92no—
127#935—94no—
128#926—91no—
129#924—94no—
130C22×C3⋊D3C22×D311yescharacteristic
131C22×C3×D3—62no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×D32C22×D312yes—
135C2×#157—34no—
136C32×Q8—61no—
137C32×D4—61no—
138C32×D4—62no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4—62no—
144C2×C32×C4#6711yescharacteristic
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3#6711yescharacteristic
147C4×C3⋊D3#6711yescharacteristic
148C4×C3⋊D3#6711yescharacteristic
149C4×C3⋊D3C22×D311yescharacteristic
150C4×C3⋊D3C22×D311yescharacteristic
151#634—22no—
152#634—22no—
153C3×#67—22no—
154C3×#67—22no—
155C3×#67—42no—
156C3×#67—62no—
157C3×#67—62no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—122no—
164C3×D12—21no—
165C3×D12—21no—
166C3×D12—41no—
167C3×D12—62no—
168C3×C4×D3—22no—
169C3×C4×D3—22no—
170C3×C4×D3—42no—
171C3×C4×D3—62no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—62no—
181C3×Dic6—21no—
182C3×Dic6—21no—
183C3×Dic6—41no—
184C3×Dic6—62no—
185C3×Dic6—62no—
186C3×Dic6—62no—
187C3×Dic6—62no—
188C3×Dic6—62no—
189C3×Dic6—122no—
190C3×Dic6—122no—
191#627—21no—
192#627—64no—
193#627—64no—
194#627—62no—
195#627—62no—
196#626—22no—
197#625—21no—
198#624—34no—
199#624—62no—
200#624C22×D312yes—
201D3×Dic3—22no—
202D3×Dic3—64no—
203D3×Dic3—64no—
204#536—271noSylow
205C2×C33—21no—
206C2×C33C22×D411yescharacteristic
207C3×C3⋊D3—21no—
208C3×C3⋊D32−1+412yes—
209C32×D3—42no—
210C3×#41—61no—
211C3×#41—61no—
212C3×#41—121no—
213C3×#41—181no—
214C3×#41—181no—
215C3×#41—181no—
216C2×C3×Q8—91no—
217C2×C3×Q8—181no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—92no—
221C22×C3×C4—91no—
222C22×Dic3—92no—
223#239—61no—
224D3×Q8—61no—
225D3×Q8—182no—
226D3×Q8—182no—
227D3×Q8—182no—
228D3×Q8—182no—
229#237—61no—
230#237—62no—
231D3×D4—61no—
232D3×D4—62no—
233#235—21no—
234#235—62no—
235C2×C4×D3—62no—
236C2×C4×D3—61no—
237C2×C4×D3—61no—
238C2×C4×D3—182no—
239C2×Dic6—61no—
240C2×Dic6—61no—
241C2×Dic6—92no—
242C2×Dic6—92no—
243C2×Dic6—121no—
244C2×Dic6—181no—
245C2×Dic6—181no—
246C2×Dic6—181no—
247C2×Dic6—182no—
248#217—98no—
249#217—92no—
250C4⋊Dic3—92no—
251Dic3⋊C4—98no—
252C4×Dic3—92no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—62no—
258C22×C32C2×#6711yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3C2×#6711yescharacteristic
261C2×C3⋊D3C23×D311yescharacteristic
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—42no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—64no—
270C2×C3×D3—124no—
271C2×C3×D3—122no—
272D32—24no—
273#157—64no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—62no—
283C32×C4C2×#6711yescharacteristic
284C32×C4C2×#6711yescharacteristic
285C3⋊Dic3—24no—
286C3⋊Dic3C2×#6711yescharacteristic
287C3⋊Dic3C2×#6711yescharacteristic
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—42no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—64no—
296C3×Dic3—64no—
297C3×Dic3—124no—
298C2×#41—91no—
299C22×Q8—271no—
300C4⋊Q8—272no—
301#124—272no—
302#123—274no—
303C22⋊Q8—274no—
304#117—271no—
305C33#53611yescharacteristic, complemented, Sylow
306C23×C3—92no—
307C23×C3—91no—
308C22×D3—62no—
309C22×D3—61no—
310C22×D3—61no—
311C3×Q8—61no—
312C3×Q8—61no—
313C3×Q8—121no—
314C3×Q8—182no—
315C3×Q8—182no—
316C3×Q8—181no—
317C3×D4—61no—
318C3×D4—62no—
319C3×D4—62no—
320C3×D4—61no—
321C3×D4—121no—
322C3×D4—122no—
323C3×D4—182no—
324C3×D4—181no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—92no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—122no—
338C2×C3×C4—182no—
339C2×C3×C4—182no—
340C2×C3×C4GO+(4,𝔽2)11yescharacteristic
341#67—22no—
342#67—62no—
343#67—62no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—62no—
349C2×Dic3—94no—
350C2×Dic3—94no—
351C2×Dic3—122no—
352C2×Dic3—182no—
353D12—21no—
354D12—62no—
355C4×D3—22no—
356C4×D3—62no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—61no—
360C4×D3—61no—
361C4×D3—61no—
362C4×D3—61no—
363C4×D3—61no—
364C4×D3—61no—
365C4×D3—62no—
366C4×D3—184no—
367C4×D3—184no—
368Dic6—21no—
369Dic6—62no—
370Dic6—62no—
371Dic6—62no—
372Dic6—62no—
373Dic6—62no—
374Dic6—122no—
375Dic6—122no—
376Dic6—182no—
377Dic6—182no—
378Dic6—184no—
379Dic6—184no—
380C2×C32—22no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—42no—
388C2×C32—81no—
389C2×C32—122no—
390C2×C32C22×#6711yescharacteristic
391C3⋊D3—21no—
392C3⋊D3#101012yes—
393C3×D3—42no—
394C3×D3—42no—
395C3×D3—62no—
396C3×D3—62no—
397C3×D3—62no—
398C3×D3—82no—
399C3×D3—121no—
400#41—61no—
401#41—181no—
402#41—181no—
403#41—181no—
404C2×Q8—91no—
405C2×Q8—181no—
406C2×Q8—272no—
407C2×Q8—272no—
408C2×Q8—541no—
409C2×Q8—541no—
410C2×Q8—541no—
411C2×D4—92no—
412C2×D4—91no—
413C22×C4—92no—
414C22×C4—91no—
415C22×C4—272no—
416C4⋊C4—272no—
417C4⋊C4—278no—
418C22⋊C4—272no—
419C22⋊C4—278no—
420C42—272no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—91no—
428C22×C3—91no—
429C22×C3—122no—
430C22×C3—182no—
431C22×C3—184no—
432C22×C3C2×GO+(4,𝔽2)11yescharacteristic
433C2×D3—22no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—64no—
439C2×D3—124no—
440C2×D3—122no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—91no—
452C3×C4—91no—
453C3×C4—122no—
454C3×C4—184no—
455C3×C4C2×GO+(4,𝔽2)11yescharacteristic
456C3×C4C2×GO+(4,𝔽2)11yescharacteristic
457Dic3—22no—
458Dic3—61no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—64no—
463Dic3—64no—
464Dic3—124no—
465Dic3—184no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32—41no—
470C32#582511yescharacteristic, complemented
471C23—92no—
472C23—91no—
473Q8—61no—
474Q8—181no—
475Q8—182no—
476Q8—182no—
477Q8—542no—
478Q8—542no—
479D4—62no—
480D4—61no—
481D4—182no—
482D4—181no—
483C2×C4—62no—
484C2×C4—92no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—182no—
491C2×C4—182no—
492C2×C4—274no—
493C2×C4—274no—
494C2×C4—542no—
495C2×C4D32⋊D311yescharacteristic
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—41no—
501C2×C3—42no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—81no—
505C2×C3—92no—
506C2×C3—122no—
507C2×C3—122no—
508C2×C3—181no—
509C2×C3—242no—
510C2×C3C22×GO+(4,𝔽2)11yescharacteristic
511D3—42no—
512D3—62no—
513D3—62no—
514D3—62no—
515D3—121no—
516C22—62no—
517C22—91no—
518C22—91no—
519C22—182no—
520C22—184no—
521C22C2×D32⋊D311yescharacteristic
522C4—62no—
523C4—91no—
524C4—91no—
525C4—184no—
526C4—544no—
527C4C2×D32⋊D311yescharacteristic
528C4C2×D32⋊D311yescharacteristic
529C3—21no—
530C3—21no—
531C3—41no—
532C3—41no—
533C3#1059649411yescharacteristic, complemented
534C2—21no—
535C2—92no—
536C2—122no—
537C2—181no—
538C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FittingC2×C33×C4D4
socleC2×C33C22×D4
prime radicalC3×#64306C2
hypercenterC2×C4D32⋊D3
hypocenterC33#536
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. #11724514C2semidirect
4. #11724319C2semidirect
5. #11724302C2semidirect
6. #11724301C2semidirect
7. #11724300C2semidirect
305. C33#536semidirect
470. C32#5825semidirect
533. C3#10596494semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×#643062
#117245142
#117243192
#117243022
#117243012
#117243002
#105964943
#58259