Finite Group Databasebeta

← #13161959

Subgroups — #13161959

Order 1728.

794 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#64145C211yesmaximal, characteristic, complemented
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic
4#11723677C211yesmaximal, characteristic, complemented
5C2×#88812C211yesmaximal, characteristic, complemented
6C2×#88811C211yesmaximal, characteristic, complemented
7#11722070C211yesmaximal, characteristic, complemented
8#10591474—31nomaximal
9#10591473—31nomaximal
10#10591470—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C3×#3757—22no—
16C3×#3754—22no—
17C3×#3753—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#88814—22no—
22#88812—22no—
23#88811—22no—
24#64145D311yescharacteristic, complemented
25C2×#3754D311yescharacteristic, complemented
26C2×#3753D311yescharacteristic
27C3×#936—31no—
28C3×#936—31no—
29C3×#936—61no—
30C2×C3×#214—31no—
31C2×C3×#214—31no—
32C2×C3×#214—61no—
33C2×C3×#213—31no—
34C2×C3×#213—31no—
35C2×C3×#213—61no—
36#63864—31no—
37#63863—31no—
38#63860—31no—
39C2×#3719—31no—
40C2×#3719—31no—
41C2×#3718—31no—
42C2×#3717—31no—
43C2×#3716—31no—
44C2×#3715—31no—
45#63553—31no—
46Dic3×C3⋊C8—31no—
47Dic3×C3⋊C8—31no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D412yes—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×#634—21no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64#5168—91no—
65#5155—91no—
66#5059—91no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—41no—
71C2×C32×D4C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85C2×C3×Dic6—61no—
86#3757—22no—
87#3754—22no—
88#3753—22no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#217—62no—
93C3×#217—62no—
94C3×#217—64no—
95C3×#214—62no—
96C3×#214—62no—
97C3×#214—64no—
98C3×#213—62no—
99C3×#213—62no—
100C3×#213—64no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—61no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8—61no—
110#3722—62no—
111(C2×D3)⋊Dic3—62no—
112(C2×D3)⋊Dic3—62no—
113#3719—62no—
114#3719—62no—
115#3718—62no—
116#3717—62no—
117#3716—62no—
118#3715—62no—
119C22×C33—41no—
120C22×C33—42no—
121C22×C33C2×D411yescharacteristic
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3—41no—
128C3×C3⋊Dic3—41no—
129C2×C3×SD16—91no—
130C2×C3×D8—91no—
131C3×#124—91no—
132#947—91no—
133C2×#216—91no—
134C2×#216—91no—
135C2×#215—91no—
136#936—31no—
137#936—31no—
138#936—61no—
139#936—91no—
140C2×#214—31no—
141C2×#214—31no—
142C2×#214—61no—
143C2×#214—91no—
144C2×#213—31no—
145C2×#213—31no—
146C2×#213—61no—
147C2×Dic12—91no—
148C2×#204—91no—
149#875—91no—
150C8×Dic3—91no—
151C8×Dic3—91no—
152C4×C3⋊C8—91no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—41no—
158C23×C32—41no—
159C23×C32—42no—
160C22×C3⋊D3—21no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×D4—21no—
165C32×D4—21no—
166C32×D4—22no—
167C32×D4—22no—
168C32×D4—21no—
169C32×D4—21no—
170C32×D4—21no—
171C32×D4—21no—
172C32×D4—41no—
173C32×D4—41no—
174C32×D4—41no—
175C32×D4—41no—
176C32×D4—42no—
177C32×D4—42no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—41no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3#6712yes—
188C4⋊(C3⋊D3)—21no—
189C4⋊(C3⋊D3)—21no—
190#634—21no—
191#634—21no—
192#634—21no—
193#634—21no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—121no—
203C3×D12—62no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—121no—
209C3×Dic6—62no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—121no—
219C3×Dic6—121no—
220C3×Dic6—121no—
221C3⋊(C3⋊C8)#6712yes—
222C3⋊(C3⋊C8)#6712yes—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—32no—
227C3×C3⋊C8—62no—
228C3×C3⋊C8—62no—
229#495—271noSylow
230C2×C33—81no—
231C2×C33#12711yescharacteristic
232C2×C33#13512yes—
233C3×C3⋊D3—81no—
234C2×C3×Q8—91no—
235C2×C3×Q8—91no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4—41no—
240C2×C3×D4—91no—
241C2×C3×D4D3211yescharacteristic
242C2×C3×D4D3211yescharacteristic
243C2×C3×D4D3211yescharacteristic
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×Dic6—31no—
248C2×Dic6—31no—
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C2×Dic6—61no—
253C3×SD16—182no—
254C3×D8—182no—
255C2×C3×C8—91no—
256C2×C3×C8—91no—
257C3×C22⋊C4—182no—
258C3×C42—91no—
259#217—62no—
260#217—62no—
261#217—64no—
262#217—182no—
263#216—182no—
264#216—182no—
265#215—182no—
266#214—62no—
267#214—62no—
268#214—64no—
269#214—182no—
270#213—62no—
271#213—62no—
272#213—64no—
273(C2×D3)⋊C4—182no—
274(C2×D3)⋊C4—182no—
275C4×Dic3—31no—
276C4×Dic3—31no—
277C4×Dic3—61no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284Dic12—182no—
285#204—182no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—42no—
293C22×C32—44no—
294C22×C32—41no—
295C22×C32—41no—
296C22×C32—41no—
297C22×C32—42no—
298C22×C32—44no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—82no—
302C22×C32—82no—
303C22×C32—84no—
304C22×C32D3×D411yescharacteristic
305C22×C32D3×D411yescharacteristic
306C22×C32C2×#6711yescharacteristic
307C2×C3⋊D3—41no—
308C2×C3⋊D3—42no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—122no—
314C2×C3×D3—242no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C3⋊Dic3—22no—
330C3⋊Dic3—41no—
331C3⋊Dic3—41no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—122no—
335C3×Dic3—122no—
336C3×Dic3—122no—
337C3×Dic3—121no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C2×Q16—271no—
342C2×SD16—91no—
343C2×SD16—271no—
344C2×D8—91no—
345#124—91no—
346#124—271no—
347C4×C8—271no—
348C33#49511yescharacteristic, complemented, Sylow
349C23×C3—22no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—42no—
356C23×C3—181no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—121no—
360C3×Q8—181no—
361C3×Q8—181no—
362C3×Q8—181no—
363C3×Q8—181no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—21no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—181no—
379C3×D4—181no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—92no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—121no—
399D12—62no—
400D12—61no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—121no—
405Dic6—62no—
406Dic6—61no—
407Dic6—61no—
408Dic6—61no—
409Dic6—61no—
410Dic6—61no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—121no—
415Dic6—121no—
416Dic6—121no—
417C3×C8—92no—
418C3×C8—92no—
419C3⋊C8—32no—
420C3⋊C8—32no—
421C3⋊C8—32no—
422C3⋊C8—32no—
423C3⋊C8—62no—
424C3⋊C8—62no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—22no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32—82no—
434C2×C32—82no—
435C2×C32—82no—
436C2×C32—84no—
437C2×C32—81no—
438C2×C32—81no—
439C2×C32—81no—
440C2×C32#91212yes—
441C2×C32#91612yes—
442C2×C32#94011yescharacteristic
443C2×C32#94011yescharacteristic
444C2×C32#94011yescharacteristic
445C2×C32#95012yes—
446C3⋊D3—81no—
447C3×D3—242no—
448C3×D3—241no—
449C3×D3—241no—
450C2×Q8—91no—
451C2×Q8—91no—
452C2×D4—91no—
453C2×D4#111311yescharacteristic
454Q16—542no—
455SD16—182no—
456SD16—542no—
457D8—182no—
458C2×C8—91no—
459C2×C8—91no—
460C22⋊C4—182no—
461C22⋊C4—542no—
462C42—91no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—84no—
481C22×C3—362no—
482C22×C3—361no—
483C22×C3D3×#6711yescharacteristic
484C22×C3D3×#6711yescharacteristic
485C22×C3#381111yescharacteristic
486C2×D3—122no—
487C2×D3—122no—
488C2×D3—122no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—242no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—182no—
501C3×C4—361no—
502C3×C4—361no—
503C3×C4D3×#6711yescharacteristic
504C3×C4D3×#6711yescharacteristic
505C3×C4D3×#6711yescharacteristic
506C3×C4D3×#6711yescharacteristic
507C3×C4#381111yescharacteristic
508C3×C4#381111yescharacteristic
509Dic3—62no—
510Dic3—62no—
511Dic3—122no—
512Dic3—122no—
513Dic3—122no—
514Dic3—121no—
515Dic3—121no—
516Dic3—121no—
517Dic3—121no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#505911yescharacteristic, complemented
523C32#515511yescharacteristic, complemented
524C32#516811yescharacteristic, complemented
525C23—21no—
526C23—181no—
527Q8—181no—
528Q8—181no—
529Q8—181no—
530Q8—181no—
531D4—21no—
532D4—21no—
533D4—181no—
534D4—181no—
535C2×C4—92no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4C2×#111311yescharacteristic
539C8—92no—
540C8—92no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—42no—
548C2×C3—41no—
549C2×C3—82no—
550C2×C3—82no—
551C2×C3—82no—
552C2×C3—84no—
553C2×C3—81no—
554C2×C3—81no—
555C2×C3—81no—
556C2×C3—721no—
557C2×C3#6393312yes—
558C2×C3#6393512yes—
559C2×C3#6393612yes—
560C2×C3#6397911yescharacteristic
561C2×C3#6397911yescharacteristic
562C2×C3#6397911yescharacteristic
563D3—242no—
564D3—241no—
565D3—241no—
566C22—42no—
567C22—41no—
568C22—362no—
569C22—361no—
570C22#8904911yescharacteristic
571C4—182no—
572C4—361no—
573C4—361no—
574C4#8904911yescharacteristic
575C4#8904911yescharacteristic
576C3—21no—
577C3—21no—
578C3—21no—
579C3—41no—
580C3#1059147011yescharacteristic, complemented
581C3#1059147311yescharacteristic, complemented
582C3#1059147411yescharacteristic, complemented
583C2—81no—
584C2—721no—
585C2#1172370712yes—
586C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×D4#1113
hypocenterC33#495
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64145C2semidirect
2. C2×C3×#3754C2semidirect
4. #11723677C2semidirect
5. C2×#88812C2semidirect
6. C2×#88811C2semidirect
7. #11722070C2semidirect
24. #64145D3semidirect
25. C2×#3754D3semidirect
348. C33#495semidirect
522. C32#5059semidirect
523. C32#5155semidirect
524. C32#5168semidirect
580. C3#10591470semidirect
581. C3#10591473semidirect
582. C3#10591474semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C3×#37542
C2×C3×#37532
#117236772
C2×#888122
C2×#888112
#117220702
#105914743
#105914733
#105914703