Finite Group Databasebeta

← #13156612

Subgroups — #13156612

Order 1728.

762 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#11723677C211yesmaximal, characteristic, complemented
2#11723429C211yesmaximal, characteristic, complemented
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic, complemented
5C2×C3×#3716C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic, complemented
8#10591474—31nomaximal
9#10590864—31nomaximal
10#10590862—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—22no—
19(C2×D3)⋊(C3⋊Dic3)—22no—
20#88800—22no—
21#88796—22no—
22C3×#3716—22no—
23C3×#3714—22no—
24#64145—31no—
25C2×#3746—31no—
26C2×#3744—31no—
27C2×C3×#214—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C2×C3×#204—31no—
31#63864—31no—
32#63863—31no—
33#63860—31no—
34#63860—31no—
35#63860—31no—
36#63860—61no—
37C2×#3719—31no—
38C2×#3719—31no—
39C2×#3719—61no—
40C2×#3716—31no—
41C2×#3716—31no—
42C2×#3716—61no—
43C2×#3716D311yescharacteristic, complemented
44C2×#3714D311yescharacteristic
45C3×C8×Dic3—31no—
46C3×C8×Dic3—31no—
47#63553D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308—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—
60C32×D12—21no—
61C32×D12—21no—
62C32×C3⋊C8D412yes—
63C32×C3⋊C8D412yes—
64#5168—91no—
65#5155—91no—
66#4696—91no—
67C2×C32×D4—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—61no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—121no—
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—21no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×C32×C8—31no—
87C2×C32×C8—31no—
88#3757—62no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90#3746—62no—
91#3744—62no—
92C3×#214—62no—
93C3×#213—62no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—61no—
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C3×D24—62no—
104C3×#204—62no—
105#3722—62no—
106(C2×D3)⋊Dic3—62no—
107(C2×D3)⋊Dic3—62no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—64no—
110(C2×D3)⋊Dic3—62no—
111#3719—64no—
112#3719—62no—
113#3719—62no—
114#3716—22no—
115#3716—64no—
116#3716—62no—
117#3716—62no—
118#3714—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3⋊(C3⋊Dic3)—41no—
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3—41no—
129C2×C3×SD16—91no—
130C2×C3×D8—91no—
131#947—91no—
132C2×#216—91no—
133#936—91no—
134#936—91no—
135#936—91no—
136#936—181no—
137C2×#214—31no—
138C2×#214—91no—
139C2×#213—31no—
140C2×Dic12—91no—
141C2×Dic12—91no—
142C2×Dic12—181no—
143C2×D24—31no—
144C2×#204—31no—
145C2×#204—91no—
146C2×#204—91no—
147C2×#204—181no—
148#875—91no—
149#875—91no—
150C3×C4×C8—91no—
151C8×Dic3—31no—
152C8×Dic3—31no—
153C23×C32—61no—
154C22×C3⋊D3—21no—
155C22×C3×D3—22no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C32×D4—61no—
162C32×D4—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—121no—
176C2×C3⋊Dic3—121no—
177C2×C3⋊Dic3—122no—
178C2×C3⋊Dic3D1212yes—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)—21no—
181#634—21no—
182#634—21no—
183#634—61no—
184#634—61no—
185#634—62no—
186#634—62no—
187#634—61no—
188#634—61no—
189#634—61no—
190#634—61no—
191#634—121no—
192#634—121no—
193#634—121no—
194#634—121no—
195#634—122no—
196#634—122no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—121no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—41no—
208C3×D12—41no—
209C3×D12—62no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—121no—
215C3×Dic6—62no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—121no—
221C32×C8—32no—
222C32×C8—32no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—22no—
225C3×C3⋊C8D1212yes—
226C3×C3⋊C8D1212yes—
227C3×C3⋊C8#6712yes—
228C3×C3⋊C8#6712yes—
229#495—271noSylow
230C2×C33#12711yescharacteristic
231C2×C33#13512yes—
232C3×C3⋊D3—81no—
233C32×D3—81no—
234C2×C3×Q8—91no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—61no—
238C2×C3×D4—91no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12D3211yescharacteristic
243C2×Dic6—31no—
244C2×Dic6—31no—
245C2×Dic6—61no—
246C2×Dic6—91no—
247C2×Dic6—91no—
248C2×Dic6—91no—
249C2×Dic6—181no—
250C2×Dic6—181no—
251C2×Dic6—181no—
252C2×Dic6—361no—
253C3×SD16—182no—
254C3×D8—182no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—31no—
258C2×C3×C8—31no—
259C2×C3×C8—61no—
260C2×C3×C8—61no—
261C3×C42—91no—
262#217—182no—
263#217—182no—
264#217—182no—
265#217—184no—
266#216—182no—
267#214—62no—
268#214—182no—
269#213—62no—
270(C2×D3)⋊C4—182no—
271(C2×D3)⋊C4—182no—
272(C2×D3)⋊C4—182no—
273C4×Dic3—31no—
274C4×Dic3—31no—
275C4×Dic3—61no—
276C2×C3⋊C8D3211yescharacteristic
277C2×C3⋊C8D3211yescharacteristic
278Dic12—182no—
279Dic12—182no—
280Dic12—184no—
281D24—62no—
282#204—62no—
283#204—182no—
284#204—182no—
285#204—184no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—121no—
291C22×C32—122no—
292C22×C32C2×D1211yescharacteristic
293C22×C32D3×D411yescharacteristic
294C22×C32D3×D411yescharacteristic
295C2×C3⋊D3—41no—
296C2×C3⋊D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—44no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—122no—
304C2×C3×D3—121no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—121no—
308C2×C3×D3—242no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4C2×D1211yescharacteristic
318C32×C4C2×D1211yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4C2×#6711yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C3⋊Dic3—22no—
324C3⋊Dic3—41no—
325C3⋊Dic3—122no—
326C3⋊Dic3—122no—
327C3⋊Dic3—122no—
328C3⋊Dic3—124no—
329C3⋊Dic3—121no—
330C3⋊Dic3—121no—
331C3⋊Dic3—121no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—122no—
335C3×Dic3—122no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C2×Q16—271no—
339C2×SD16—91no—
340C2×SD16—271no—
341C2×D8—91no—
342#124—271no—
343#124—271no—
344C4×C8—91no—
345C33#49511yescharacteristic, complemented, Sylow
346C23×C3—62no—
347C23×C3—61no—
348C23×C3—61no—
349C23×C3—181no—
350C22×D3—21no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—121no—
354C3×Q8—181no—
355C3×Q8—181no—
356C3×D4—61no—
357C3×D4—61no—
358C3×D4—61no—
359C3×D4—61no—
360C3×D4—121no—
361C3×D4—121no—
362C3×D4—181no—
363C3×D4—181no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—92no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—62no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—121no—
379C2×Dic3—182no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C2×Dic3—361no—
384C2×Dic3—361no—
385C2×Dic3—362no—
386D12—21no—
387D12—21no—
388D12—62no—
389D12—61no—
390D12—61no—
391D12—61no—
392D12—61no—
393D12—121no—
394Dic6—62no—
395Dic6—61no—
396Dic6—61no—
397Dic6—61no—
398Dic6—61no—
399Dic6—121no—
400Dic6—182no—
401Dic6—182no—
402Dic6—181no—
403Dic6—181no—
404Dic6—181no—
405Dic6—181no—
406Dic6—181no—
407Dic6—181no—
408Dic6—361no—
409Dic6—361no—
410Dic6—361no—
411Dic6—361no—
412Dic6—362no—
413Dic6—362no—
414C3×C8—32no—
415C3×C8—32no—
416C3×C8—32no—
417C3×C8—32no—
418C3×C8—62no—
419C3×C8—62no—
420C3⋊C8#62612yes—
421C3⋊C8#62612yes—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—241no—
431C2×C32#87411yescharacteristic
432C2×C32#90412yes—
433C2×C32#91212yes—
434C2×C32#91612yes—
435C2×C32#94011yescharacteristic
436C2×C32#94011yescharacteristic
437C3⋊D3—81no—
438C3×D3—82no—
439C3×D3—81no—
440C3×D3—81no—
441C3×D3—242no—
442C3×D3—241no—
443C3×D3—241no—
444C2×Q8—91no—
445C2×Q8—271no—
446C2×D4—31no—
447C2×D4—91no—
448Q16—542no—
449SD16—182no—
450SD16—542no—
451D8—182no—
452C2×C8—31no—
453C2×C8—31no—
454C22⋊C4—542no—
455C22⋊C4—542no—
456C42—91no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—122no—
462C22×C3—124no—
463C22×C3—122no—
464C22×C3—122no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—362no—
468C22×C3—361no—
469C22×C3D3×D1211yescharacteristic
470C22×C3D3×D1211yescharacteristic
471C22×C3#381111yescharacteristic
472C2×D3—42no—
473C2×D3—41no—
474C2×D3—122no—
475C2×D3—122no—
476C2×D3—121no—
477C2×D3—122no—
478C2×D3—121no—
479C2×D3—242no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—182no—
489C3×C4—361no—
490C3×C4D3×D1211yescharacteristic
491C3×C4D3×D1211yescharacteristic
492C3×C4C2×#62611yescharacteristic
493C3×C4C2×#62611yescharacteristic
494C3×C4D3×#6711yescharacteristic
495C3×C4D3×#6711yescharacteristic
496Dic3—62no—
497Dic3—62no—
498Dic3—122no—
499Dic3—122no—
500Dic3—121no—
501Dic3—121no—
502Dic3—362no—
503Dic3—362no—
504Dic3—362no—
505Dic3—364no—
506Dic3—361no—
507Dic3—361no—
508Dic3—361no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#469611yescharacteristic, complemented
514C32#515511yescharacteristic, complemented
515C32#516811yescharacteristic, complemented
516C23—61no—
517C23—181no—
518Q8—181no—
519Q8—181no—
520Q8—541no—
521Q8—541no—
522D4—61no—
523D4—61no—
524D4—181no—
525D4—181no—
526C2×C4—92no—
527C2×C4—181no—
528C2×C4—541no—
529C2×C4D3311yescharacteristic
530C8—32no—
531C8—32no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—22no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—41no—
539C2×C3—42no—
540C2×C3—242no—
541C2×C3—241no—
542C2×C3—241no—
543C2×C3—721no—
544C2×C3#6380612yes—
545C2×C3#6380712yes—
546C2×C3#6391111yescharacteristic
547C2×C3#6391111yescharacteristic
548C2×C3#6393612yes—
549C2×C3#6397911yescharacteristic
550D3—81no—
551D3—242no—
552D3—241no—
553D3—241no—
554C22—121no—
555C22—122no—
556C22—361no—
557C22—362no—
558C22#8895511yescharacteristic
559C4—182no—
560C4—361no—
561C4—1081no—
562C4D3×#62611yescharacteristic
563C4D3×#62611yescharacteristic
564C3—21no—
565C3—21no—
566C3—21no—
567C3—41no—
568C3#1059086211yescharacteristic, complemented
569C3#1059086411yescharacteristic, complemented
570C3#1059147411yescharacteristic, complemented
571C2—241no—
572C2—721no—
573C2#1172286412yes—
574C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypocenterC33#495
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723677C2semidirect
2. #11723429C2semidirect
3. C2×#88800C2semidirect
4. C2×#88796C2semidirect
5. C2×C3×#3716C2semidirect
7. C3×#63553C2semidirect
43. C2×#3716D3semidirect
47. #63553D3semidirect
345. C33#495semidirect
513. C32#4696semidirect
514. C32#5155semidirect
515. C32#5168semidirect
568. C3#10590862semidirect
569. C3#10590864semidirect
570. C3#10591474semidirect

Maximal subgroups

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

subgroupindex
#117236772
#117234292
C2×#888002
C2×#887962
C2×C3×#37162
C2×C3×#37142
C3×#635532
#105914743
#105908643
#105908623