Finite Group Databasebeta

← #13156251

Subgroups — #13156251

Order 1728.

574 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 88 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×#88798C211yesmaximal, characteristic, complemented
2#11723350C211yesmaximal, characteristic
3C3×#63831C211yesmaximal, characteristic
4C3×#63827C211yesmaximal, characteristic
5#11722074C211yesmaximal, characteristic, complemented
6#11722059C211yesmaximal, characteristic, complemented
7C3×#63560C211yesmaximal, characteristic, complemented
8#10591683—31nomaximal
9#10591644—31nomaximal
10#10590981—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88798—21no—
19#88798—21no—
20#88786—21no—
21D3×C3⋊(C3⋊C8)—21no—
22C3×#3719—21no—
23C3×#3718—21no—
24C3×#3716—21no—
25C3×#3715—21no—
26#64157—31no—
27C2×#3756—31no—
28C3×#951—31no—
29C3×#951—31no—
30C3×#942—31no—
31C3×#909—31no—
32C2×#3718—31no—
33C2×#3718—31no—
34C2×#3718—61no—
35#63831D311yescharacteristic
36#63827D311yescharacteristic
37#63815—31no—
38#63815—31no—
39#63815—61no—
40#63649—31no—
41C3×#836—31no—
42C3×#824—31no—
43#63578—31no—
44#63578—31no—
45#63578—61no—
46#63561—31no—
47#63560—31no—
48#63560—31no—
49#63560D311yescharacteristic, complemented
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C32×#67—21no—
55C2×C32×Dic3—21no—
56C32×D12D411yescharacteristic
57C32×C4×D3—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6D411yescharacteristic
61C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
62C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
63C3×C3⋊(C3⋊C8)—21no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66#5204—91no—
67#5204—91no—
68#4755—91no—
69C32×#41—31no—
70C2×C32×Q8—31no—
71C2×#634C2×D311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C32×M16—31no—
82#3756—61no—
83#3756—61no—
84#3748—61no—
85#3748C2×D311yescharacteristic
86#3748C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—61no—
92C2×C3⋊(C3⋊C8)—61no—
93C2×C3⋊(C3⋊C8)—61no—
94C2×C3⋊(C3⋊C8)—121no—
95C3×#216—61no—
96C3×#216—61no—
97C3×#216—61no—
98C3×#215—61no—
99C3×#214—61no—
100C3×#214—61no—
101C3×#208—21no—
102C3×#208—31no—
103C3×#208—31no—
104C3×#208—31no—
105C3×#208—31no—
106C3×#208—61no—
107C3×#208—61no—
108C3×#208C2×D311yescharacteristic
109C3×#208C2×D311yescharacteristic
110C3×Dic12—61no—
111C3×#204—61no—
112#3719—21no—
113#3718—21no—
114#3718—61no—
115#3718—61no—
116#3718—61no—
117#3718—61no—
118#3718—62no—
119#3718—62no—
120#3716—21no—
121#3715—21no—
122#3711—61no—
123#3711—61no—
124#3711—62no—
125D3×C3⋊C8—61no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—62no—
128C22×C33#4111yescharacteristic
129C2×C32×D3—21no—
130C33×C4C2×D411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—41no—
133C32×Dic3—21no—
134C32×Dic3—41no—
135C3×#137—91no—
136C3×#137—91no—
137#951—31no—
138#951—31no—
139#948—91no—
140#948—91no—
141#948—181no—
142C2×#216—91no—
143C2×#216—91no—
144C2×#216—91no—
145C2×#216—181no—
146#942—31no—
147#909—31no—
148#907—91no—
149C3×#101—91no—
150#836—31no—
151#836—91no—
152#824—31no—
153#824—91no—
154#824—91no—
155#822—91no—
156#822—91no—
157#822—91no—
158#822—181no—
159C32×Q8—31no—
160C32×Q8—61no—
161C32×Q8—61no—
162C32×D4—31no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—21no—
174#634—21no—
175#634—21no—
176C3×#67—22no—
177C3×#67—21no—
178C3×#67—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—41no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C3×D12—21no—
186C3×D12#6711yescharacteristic
187C3×D12#6711yescharacteristic
188C3×C4×D3—22no—
189C3×C4×D3—21no—
190C3×C4×D3—21no—
191C3×Dic6—22no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—41no—
198C3×Dic6—62no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—61no—
203C3×Dic6—121no—
204C3×Dic6#6711yescharacteristic
205C3×Dic6#6711yescharacteristic
206C32×C8—61no—
207C3⋊(C3⋊C8)—21no—
208C3⋊(C3⋊C8)—21no—
209C3⋊(C3⋊C8)—31no—
210C3⋊(C3⋊C8)—31no—
211C3⋊(C3⋊C8)—31no—
212C3⋊(C3⋊C8)—31no—
213C3⋊(C3⋊C8)—31no—
214C3⋊(C3⋊C8)—31no—
215C3⋊(C3⋊C8)—61no—
216C3⋊(C3⋊C8)—61no—
217C3⋊(C3⋊C8)—61no—
218C3⋊(C3⋊C8)—61no—
219C3⋊(C3⋊C8)—61no—
220C3⋊(C3⋊C8)—61no—
221C3⋊(C3⋊C8)—121no—
222C3⋊(C3⋊C8)—121no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—41no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—121no—
232#473—271noSylow
233C2×C33—21no—
234C2×C33#12111yescharacteristic
235C32×D3—41no—
236C3×#41—31no—
237C3×#41—31no—
238C3×#41—61no—
239C2×C3×Q8—31no—
240C2×C3×Q8—31no—
241C2×C3×Q8—61no—
242C2×C3×Q8—91no—
243#235D3211yescharacteristic
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—61no—
247C2×Dic6D3211yescharacteristic
248C3×Q16—181no—
249C3×Q16—181no—
250C3×SD16—181no—
251C3×SD16—181no—
252C3×M16—31no—
253C3×M16—31no—
254C3×M16—61no—
255C3×M16—91no—
256C3×M16—91no—
257#216—61no—
258#216—61no—
259#216—61no—
260#216—181no—
261#216—181no—
262#216—181no—
263#216—181no—
264#216—181no—
265#216—181no—
266#216—182no—
267#216—182no—
268#215—61no—
269#214—61no—
270#214—61no—
271#208—31no—
272#208—31no—
273#208—31no—
274#208—31no—
275#208—61no—
276#208—61no—
277#208—181no—
278#208—181no—
279#208—182no—
280#208D3211yescharacteristic
281C2×C3⋊C8—91no—
282C2×C3⋊C8—91no—
283C2×C3⋊C8—91no—
284C2×C3⋊C8—181no—
285C2×C3⋊C8—181no—
286C2×C3⋊C8—181no—
287C2×C3⋊C8—181no—
288C2×C3⋊C8—181no—
289C2×C3⋊C8—182no—
290C2×C3⋊C8—361no—
291Dic12—61no—
292#204—61no—
293ΓL(1,𝔽25)—181no—
294D3×C8—181no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—61no—
300C22×C32#23511yescharacteristic
301C22×C32#23711yescharacteristic
302C22×C32#23711yescharacteristic
303C2×C3×D3—22no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—61no—
315C32×C4—121no—
316C32×C4D3×D411yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4C2×#6711yescharacteristic
321C32×C4C2×#6711yescharacteristic
322C3⋊Dic3—41no—
323C3×Dic3—22no—
324C3×Dic3—21no—
325C3×Dic3—21no—
326C3×Dic3—42no—
327C3×Dic3—41no—
328C3×Dic3—41no—
329C3×Dic3—122no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332#137—91no—
333#137—91no—
334C2×Q16—271no—
335#131—271no—
336#108—271no—
337#101—91no—
338#101—271no—
339C33#47311yescharacteristic, complemented, Sylow
340C3×Q8—31no—
341C3×Q8—31no—
342C3×Q8—61no—
343C3×Q8—62no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—121no—
349C3×Q8—181no—
350C3×Q8—181no—
351C3×D4—31no—
352C3×D4—31no—
353C3×D4—61no—
354C3×D4—62no—
355C3×D4—61no—
356C3×D4—61no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—62no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—121no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371#67—21no—
372C2×Dic3—21no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—121no—
376D12#62511yescharacteristic
377C4×D3—21no—
378Dic6—21no—
379Dic6—21no—
380Dic6—62no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—121no—
386Dic6#62511yescharacteristic
387C3×C8—61no—
388C3×C8—61no—
389C3×C8—121no—
390C3×C8—181no—
391C3×C8—181no—
392C3⋊C8—21no—
393C3⋊C8—62no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—91no—
399C3⋊C8—91no—
400C3⋊C8—91no—
401C3⋊C8—91no—
402C3⋊C8—91no—
403C3⋊C8—91no—
404C3⋊C8—121no—
405C3⋊C8—181no—
406C3⋊C8—181no—
407C3⋊C8—181no—
408C3⋊C8—181no—
409C3⋊C8—181no—
410C3⋊C8—181no—
411C3⋊C8—361no—
412C3⋊C8—361no—
413C2×C32—21no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32—121no—
425C2×C32#88411yescharacteristic
426C2×C32#93911yescharacteristic
427C2×C32#93911yescharacteristic
428C3×D3—42no—
429C3×D3—41no—
430C3×D3—41no—
431#41—31no—
432C2×Q8—31no—
433C2×Q8—91no—
434Q16—181no—
435Q16—181no—
436Q16—541no—
437Q16—541no—
438SD16—181no—
439SD16—181no—
440M16—31no—
441M16—91no—
442M16—91no—
443M16—541no—
444C2×C8—271no—
445C2×C8—541no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—62no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3#379511yescharacteristic
454C22×C3#380411yescharacteristic
455C22×C3#380511yescharacteristic
456C2×D3—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—62no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—122no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—361no—
472C3×C4#380211yescharacteristic
473C3×C4C2×#62511yescharacteristic
474C3×C4D3×#6711yescharacteristic
475C3×C4D3×#6711yescharacteristic
476C3×C4#381111yescharacteristic
477C3×C4#381111yescharacteristic
478Dic3—21no—
479Dic3—41no—
480Dic3—122no—
481Dic3—121no—
482Dic3—121no—
483C32—21no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32#475511yescharacteristic, complemented
488C32#520411yescharacteristic, complemented
489C32#520411yescharacteristic, complemented
490Q8—31no—
491Q8—61no—
492Q8—61no—
493Q8—181no—
494Q8—181no—
495D4—31no—
496D4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—181no—
500C2×C4D3311yescharacteristic
501C8—61no—
502C8—181no—
503C8—181no—
504C8—271no—
505C8—271no—
506C2×C3—22no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—41no—
514C2×C3—41no—
515C2×C3—42no—
516C2×C3—41no—
517C2×C3—122no—
518C2×C3—121no—
519C2×C3—121no—
520C2×C3#6388511yescharacteristic
521C2×C3#6397111yescharacteristic
522C2×C3#6398011yescharacteristic
523D3—41no—
524C22—61no—
525C22#8896111yescharacteristic
526C4—61no—
527C4—121no—
528C4—361no—
529C4D3×#62511yescharacteristic
530C4#8895411yescharacteristic
531C3—21no—
532C3—21no—
533C3—21no—
534C3—41no—
535C3#1059098111yescharacteristic, complemented
536C3#1059164411yescharacteristic, complemented
537C3#1059168311yescharacteristic, complemented
538C2—21no—
539C2—121no—
540C2#1172295411yescharacteristic
subgroupG/H
centerC2#11722954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
prime radicalC32×#235C22
hypocenterC33#473
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88798C2semidirect
5. #11722074C2semidirect
6. #11722059C2semidirect
7. C3×#63560C2semidirect
49. #63560D3semidirect
339. C33#473semidirect
487. C32#4755semidirect
488. C32#5204semidirect
489. C32#5204semidirect
535. C3#10590981semidirect
536. C3#10591644semidirect
537. C3#10591683semidirect

Maximal subgroups

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

subgroupindex
C2×#887982
#117233502
C3×#638312
C3×#638272
#117220742
#117220592
C3×#635602
#105916833
#105916443
#105909813