Finite Group Databasebeta

← #13156610

Subgroups — #13156610

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
1C2×#88810C211yesmaximal, characteristic
2#11723439C211yesmaximal, characteristic, complemented
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic, complemented
5C3×#63864C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591606—31nomaximal
9#10590862—31nomaximal
10#10590370—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88810—22no—
19#88803—22no—
20#88799—22no—
21#88796—22no—
22C3×#3722—22no—
23C3×#3717—22no—
24C2×#3756—31no—
25C2×#3744—31no—
26#64084—31no—
27C3×#947—31no—
28C2×C3×#215—31no—
29C2×C3×#204—31no—
30C3×#875—31no—
31#63864D311yescharacteristic, complemented
32#63860—31no—
33#63860—31no—
34#63860—61no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3719—61no—
38C2×#3717D311yescharacteristic, complemented
39C2×#3716—31no—
40C2×#3716—31no—
41C2×#3716—61no—
42C2×#3714—31no—
43C2×#3714—31no—
44C2×#3713—31no—
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D412yes—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64#5168—91no—
65#5059—91no—
66#4696—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—61no—
76C2×#634—121no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×D12—61no—
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86C2×C32×C8—31no—
87C32×C42—31no—
88#3756—62no—
89#3752—62no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91#3744—62no—
92C3×#215—62no—
93C3×(C2×D3)⋊C4—62no—
94C3×(C2×D3)⋊C4—62no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×#204—62no—
105#3722—22no—
106(C2×D3)⋊Dic3—64no—
107(C2×D3)⋊Dic3—62no—
108(C2×D3)⋊Dic3—62no—
109#3719—62no—
110#3719—62no—
111#3719—64no—
112#3717—22no—
113#3716—64no—
114#3716—62no—
115#3716—62no—
116#3714—62no—
117#3714—62no—
118#3713—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—42no—
123C2×C3×C3⋊D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3⋊(C3⋊Dic3)—41no—
127C32×Dic3—22no—
128C32×Dic3—41no—
129C2×C3×SD16—91no—
130C3×#124—91no—
131#947—31no—
132C2×#216—91no—
133C2×#216—91no—
134C2×#216—181no—
135C2×#215—31no—
136#936—91no—
137C2×#214—91no—
138C2×#213—91no—
139C2×#213—91no—
140C2×Dic12—91no—
141C2×D24—91no—
142C2×#204—31no—
143C2×#204—91no—
144C2×#204—91no—
145C2×#204—181no—
146#875—31no—
147#875—91no—
148#875—91no—
149#875—181no—
150C3×C4×C8—91no—
151C8×Dic3—31no—
152C4×C3⋊C8—31no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—21no—
155C22×C3×D3—62no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C32×Q8—61no—
162C32×Q8—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—32no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—62no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—121no—
177C2×C3⋊Dic3—121no—
178C2×C3⋊Dic3—122no—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182C4⋊(C3⋊D3)—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—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3D1212yes—
202C2×C3×Dic3#6712yes—
203C3×D12—62no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—121no—
213C3×D12—121no—
214C3×D12—121no—
215C3×Dic6—22no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—21no—
219C3×Dic6—21no—
220C3×Dic6—41no—
221C32×C8—32no—
222C3⋊(C3⋊C8)D1212yes—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8D1212yes—
228C3×C3⋊C8#6712yes—
229#495—271noSylow
230C2×C33#12711yescharacteristic
231C2×C33#13512yes—
232C3×C3⋊D3—81no—
233C3×C3⋊D3—81no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×D4—91no—
238C2×C3×D4—91no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—61no—
244C2×D12—61no—
245C2×Dic6—91no—
246C2×Dic6—91no—
247C2×Dic6—91no—
248C2×Dic6—181no—
249C2×Dic6—181no—
250C2×Dic6—181no—
251C2×Dic6—361no—
252C2×Dic6D3211yescharacteristic
253C3×SD16—182no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—61no—
257C2×C3×C8—91no—
258C3×C22⋊C4—182no—
259C3×C42—31no—
260C3×C42—31no—
261C3×C42—61no—
262#217—182no—
263#216—182no—
264#216—182no—
265#216—184no—
266#215—62no—
267#214—182no—
268#213—182no—
269#213—182no—
270(C2×D3)⋊C4—62no—
271(C2×D3)⋊C4—62no—
272(C2×D3)⋊C4—182no—
273(C2×D3)⋊C4—182no—
274(C2×D3)⋊C4—184no—
275C4×Dic3D3211yescharacteristic
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—61no—
279C2×C3⋊C8D3211yescharacteristic
280Dic12—182no—
281D24—182no—
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×C32C2×D1211yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3⋊D3—42no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3⋊D3—42no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—124no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
304C2×C3×D3—122no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—122no—
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×C4—62no—
318C32×C4—121no—
319C32×C4C2×D1211yescharacteristic
320C32×C4C2×D1211yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4D3×D411yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4C2×#6711yescharacteristic
325C3⋊Dic3—122no—
326C3⋊Dic3—122no—
327C3⋊Dic3—122no—
328C3⋊Dic3—124no—
329C3⋊Dic3—121no—
330C3⋊Dic3—121no—
331C3⋊Dic3—121no—
332C3×Dic3—22no—
333C3×Dic3—22no—
334C3×Dic3—42no—
335C3×Dic3—42no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C2×Q16—271no—
339C2×SD16—91no—
340C2×SD16—271no—
341C2×D8—271no—
342#124—91no—
343#124—271no—
344C4×C8—91no—
345C33#49511yescharacteristic, complemented, Sylow
346C23×C3—181no—
347C23×C3—181no—
348C22×D3—62no—
349C22×D3—61no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—121no—
354C3×Q8—62no—
355C3×Q8—61no—
356C3×Q8—61no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—121no—
360C3×D4—181no—
361C3×D4—181no—
362C3×D4—181no—
363C3×D4—181no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—121no—
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—182no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—361no—
383C2×Dic3—361no—
384C2×Dic3—362no—
385C2×Dic3#62612yes—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—61no—
390D12—61no—
391D12—61no—
392D12—61no—
393D12—61no—
394D12—61no—
395D12—121no—
396D12—121no—
397D12—121no—
398Dic6—21no—
399Dic6—21no—
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—62no—
417C3×C8—92no—
418C3⋊C8—32no—
419C3⋊C8—32no—
420C3⋊C8—62no—
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#87411yescharacteristic
431C2×C32#90412yes—
432C2×C32#91612yes—
433C2×C32#94011yescharacteristic
434C2×C32#94011yescharacteristic
435C2×C32#95012yes—
436C3⋊D3—81no—
437C3⋊D3—81no—
438C3×D3—242no—
439C3×D3—242no—
440C3×D3—241no—
441C3×D3—241no—
442C3×D3—241no—
443C3×D3—241no—
444C2×Q8—31no—
445C2×Q8—271no—
446C2×D4—91no—
447C2×D4—91no—
448Q16—542no—
449SD16—182no—
450SD16—542no—
451D8—542no—
452C2×C8—31no—
453C2×C8—91no—
454C22⋊C4—182no—
455C22⋊C4—542no—
456C42—31no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—362no—
462C22×C3—362no—
463C22×C3—361no—
464C22×C3—361no—
465C22×C3D3×D1211yescharacteristic
466C22×C3C2×#62611yescharacteristic
467C22×C3D3×#6711yescharacteristic
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—124no—
471C2×D3—122no—
472C2×D3—122no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—122no—
476C2×D3—122no—
477C2×D3—121no—
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—62no—
489C3×C4—62no—
490C3×C4—122no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4D3×D1211yescharacteristic
495C3×C4D3×D1211yescharacteristic
496C3×C4D3×D1211yescharacteristic
497C3×C4C2×#62611yescharacteristic
498C3×C4D3×#6711yescharacteristic
499C3×C4#381111yescharacteristic
500Dic3—22no—
501Dic3—41no—
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#505911yescharacteristic, complemented
515C32#516811yescharacteristic, complemented
516C23—181no—
517C23—181no—
518Q8—61no—
519Q8—61no—
520Q8—541no—
521Q8—541no—
522D4—181no—
523D4—181no—
524D4—181no—
525D4—181no—
526C2×C4—32no—
527C2×C4—61no—
528C2×C4—541no—
529C2×C4D3311yescharacteristic
530C8—32no—
531C8—92no—
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—721no—
541C2×C3—721no—
542C2×C3#6380612yes—
543C2×C3#6382912yes—
544C2×C3#6391111yescharacteristic
545C2×C3#6391111yescharacteristic
546C2×C3#6394712yes—
547C2×C3#6397911yescharacteristic
548D3—242no—
549D3—242no—
550D3—241no—
551D3—241no—
552D3—241no—
553D3—241no—
554C22—361no—
555C22—361no—
556C22—362no—
557C22—362no—
558C22D3×#62611yescharacteristic
559C4—62no—
560C4—121no—
561C4—1081no—
562C4D3×#62611yescharacteristic
563C4#8895511yescharacteristic
564C3—21no—
565C3—21no—
566C3—21no—
567C3—41no—
568C3#1059037011yescharacteristic, complemented
569C3#1059086211yescharacteristic, complemented
570C3#1059160611yescharacteristic, complemented
571C2—721no—
572C2—721no—
573C2#1172285512yes—
574C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#88810C2
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
2. #11723439C2semidirect
3. C2×#88799C2semidirect
4. C2×#88796C2semidirect
5. C3×#63864C2semidirect
6. C2×C3×#3717C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
31. #63864D3semidirect
38. C2×#3717D3semidirect
47. Dic3×C3⋊C8D3semidirect
345. C33#495semidirect
513. C32#4696semidirect
514. C32#5059semidirect
515. C32#5168semidirect
568. C3#10590370semidirect
569. C3#10590862semidirect
570. C3#10591606semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117234392
C2×#887992
C2×#887962
C3×#638642
C2×C3×#37172
C3×Dic3×C3⋊C82
#105916063
#105908623
#105903703