Finite Group Databasebeta

← #13156614

Subgroups — #13156614

Order 1728.

762 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 106 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#11723429C211yesmaximal, characteristic, complemented
2C2×#88799C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3715C211yesmaximal, characteristic, complemented
6C2×C3×#3713C211yesmaximal, characteristic
7#11722070C211yesmaximal, characteristic, complemented
8#10591606—31nomaximal
9#10591474—31nomaximal
10#10591470—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18(C2×D3)⋊(C3⋊Dic3)—22no—
19#88799—22no—
20#88794—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3715—22no—
23C3×#3713—22no—
24C2×#3756—31no—
25#64145—31no—
26C2×#3754—31no—
27C3×#947—31no—
28C2×C3×#215—31no—
29C3×#936—31no—
30C2×C3×#214—31no—
31C2×C3×#213—31no—
32C2×C3×#213—31no—
33#63863D311yescharacteristic, complemented
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×#3715D311yescharacteristic, complemented
44C2×#3713D311yescharacteristic
45#63553—31no—
46Dic3×C3⋊C8—31no—
47Dic3×C3⋊C8—31no—
48C22×C32×D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308—21no—
54C2×C3×C3⋊Dic3D412yes—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64#5168—91no—
65#5155—91no—
66#5059—91no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C32×D4—31no—
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—61no—
76C2×#634—121no—
77C2×C3×D12—21no—
78C2×C3×D12—21no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86#3757—62no—
87#3756—62no—
88#3754—62no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#217—62no—
93C3×#215—62no—
94C3×#214—62no—
95C3×#213—62no—
96C3×#213—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8—61no—
107(C2×D3)⋊Dic3—22no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—62no—
110(C2×D3)⋊Dic3—64no—
111#3719—62no—
112#3719—62no—
113#3719—64no—
114#3716—62no—
115#3716—62no—
116#3716—64no—
117#3715—22no—
118#3713—22no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C2×C32×D3—42no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3⋊(C3⋊Dic3)—41no—
127C3×C3⋊Dic3—22no—
128C32×Dic3—41no—
129C2×C3×SD16—91no—
130C2×C3×D8—91no—
131C3×#124—91no—
132#947—31no—
133C2×#216—91no—
134C2×#216—91no—
135C2×#216—181no—
136C2×#215—31no—
137#936—31no—
138#936—91no—
139#936—91no—
140#936—181no—
141C2×#214—31no—
142C2×#214—91no—
143C2×#214—91no—
144C2×#214—181no—
145C2×#213—31no—
146C2×#213—31no—
147C2×Dic12—91no—
148C2×#204—91no—
149#875—91no—
150C8×Dic3—91no—
151C8×Dic3—91no—
152C4×C3⋊C8—91no—
153C23×C32—61no—
154C23×C32—61no—
155C22×C3×D3—22no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C32×Q8—61no—
162C32×Q8—61no—
163C32×D4—61no—
164C32×D4—61no—
165C32×D4—61no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—62no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—121no—
180C2×C3⋊Dic3—121no—
181C2×C3⋊Dic3—122no—
182C2×C3⋊Dic3#6712yes—
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—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—41no—
202C2×C3×Dic3—62no—
203C3×D12—22no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—41no—
213C3×D12—41no—
214C3×D12—41no—
215C3×Dic6—22no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—21no—
219C3×Dic6—21no—
220C3×Dic6—41no—
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#12711yescharacteristic
231C2×C33#13512yes—
232C32×D3—81no—
233C32×D3—81no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—61no—
243C2×D12D3211yescharacteristic
244C2×D12D3211yescharacteristic
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—
254C3×D8—182no—
255C2×C3×C8—91no—
256C2×C3×C8—91no—
257C3×C22⋊C4—182no—
258C3×C42—91no—
259#217—62no—
260#217—182no—
261#217—182no—
262#217—184no—
263#216—182no—
264#216—182no—
265#216—184no—
266#215—62no—
267#214—62no—
268#214—182no—
269#214—182no—
270#214—184no—
271#213—62no—
272#213—62no—
273(C2×D3)⋊C4—62no—
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—121no—
291C22×C32—121no—
292C22×C32—122no—
293C22×C32—122no—
294C22×C32D3×D411yescharacteristic
295C22×C32D3×D411yescharacteristic
296C22×C32C2×#6711yescharacteristic
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—44no—
300C2×C3×D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—42no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—42no—
308C2×C3×D3—82no—
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—121no—
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C32×C4C2×#6711yescharacteristic
324C3⋊Dic3—22no—
325C3⋊Dic3—122no—
326C3⋊Dic3—122no—
327C3⋊Dic3—122no—
328C3⋊Dic3—124no—
329C3⋊Dic3—121no—
330C3⋊Dic3—121no—
331C3⋊Dic3—121no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—122no—
338C2×Q16—271no—
339C2×SD16—91no—
340C2×SD16—271no—
341C2×D8—91no—
342#124—91no—
343#124—271no—
344C4×C8—271no—
345C33#49511yescharacteristic, complemented, Sylow
346C23×C3—62no—
347C23×C3—61no—
348C23×C3—61no—
349C23×C3—61no—
350C23×C3—61no—
351C23×C3—121no—
352C22×D3—21no—
353C22×D3—21no—
354C3×Q8—62no—
355C3×Q8—61no—
356C3×Q8—61no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—121no—
360C3×D4—62no—
361C3×D4—61no—
362C3×D4—61no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—121no—
370C3×D4—121no—
371C3×D4—121no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—92no—
379C2×C3×C4—121no—
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×Dic3—21no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—62no—
387C2×Dic3—182no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×Dic3—361no—
392C2×Dic3—361no—
393C2×Dic3—362no—
394D12—21no—
395D12—21no—
396D12—21no—
397D12—21no—
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—92no—
415C3×C8—92no—
416C3⋊C8—32no—
417C3⋊C8—32no—
418C3⋊C8—32no—
419C3⋊C8—32no—
420C3⋊C8—62no—
421C3⋊C8—62no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—22no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—241no—
431C2×C32—241no—
432C2×C32#91212yes—
433C2×C32#91612yes—
434C2×C32#94011yescharacteristic
435C2×C32#94011yescharacteristic
436C2×C32#94011yescharacteristic
437C2×C32#95012yes—
438C3×D3—82no—
439C3×D3—82no—
440C3×D3—81no—
441C3×D3—81no—
442C3×D3—81no—
443C3×D3—81no—
444C2×Q8—31no—
445C2×Q8—271no—
446C2×D4—31no—
447C2×D4—31no—
448Q16—542no—
449SD16—182no—
450SD16—542no—
451D8—182no—
452C2×C8—91no—
453C2×C8—91no—
454C22⋊C4—182no—
455C22⋊C4—542no—
456C42—91no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—124no—
464C22×C3—122no—
465C22×C3—122no—
466C22×C3—122no—
467C22×C3—122no—
468C22×C3—121no—
469C22×C3—121no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—242no—
473C22×C3D3×#6711yescharacteristic
474C22×C3D3×#6711yescharacteristic
475C22×C3#381111yescharacteristic
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—41no—
479C2×D3—41no—
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—122no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—182no—
492C3×C4D3×#6711yescharacteristic
493C3×C4D3×#6711yescharacteristic
494C3×C4D3×#6711yescharacteristic
495C3×C4D3×#6711yescharacteristic
496C3×C4#381111yescharacteristic
497C3×C4#381111yescharacteristic
498Dic3—41no—
499Dic3—62no—
500Dic3—62no—
501Dic3—122no—
502Dic3—362no—
503Dic3—362no—
504Dic3—362no—
505Dic3—364no—
506Dic3—361no—
507Dic3—361no—
508Dic3—361no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#505911yescharacteristic, complemented
514C32#515511yescharacteristic, complemented
515C32#516811yescharacteristic, complemented
516C23—61no—
517C23—61no—
518Q8—61no—
519Q8—61no—
520Q8—541no—
521Q8—541no—
522D4—61no—
523D4—61no—
524D4—61no—
525D4—61no—
526C2×C4—61no—
527C2×C4—92no—
528C2×C4—541no—
529C2×C4D3311yescharacteristic
530C8—92no—
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—242no—
541C2×C3—242no—
542C2×C3—241no—
543C2×C3—241no—
544C2×C3—241no—
545C2×C3—241no—
546C2×C3#6393312yes—
547C2×C3#6393612yes—
548C2×C3#6394712yes—
549C2×C3#6397911yescharacteristic
550C2×C3#6397911yescharacteristic
551C2×C3#6397911yescharacteristic
552D3—81no—
553D3—81no—
554C22—121no—
555C22—121no—
556C22—122no—
557C22—122no—
558C22#8895211yescharacteristic
559C4—121no—
560C4—182no—
561C4—1081no—
562C4#8895211yescharacteristic
563C4#8895211yescharacteristic
564C3—21no—
565C3—21no—
566C3—21no—
567C3—41no—
568C3#1059147011yescharacteristic, complemented
569C3#1059147411yescharacteristic, complemented
570C3#1059160611yescharacteristic, complemented
571C2—241no—
572C2—241no—
573C2#1172285712yes—
574C2#1172303111yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
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. #11723429C2semidirect
2. C2×#88799C2semidirect
3. C2×#88794C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3715C2semidirect
7. #11722070C2semidirect
33. #63863D3semidirect
43. C2×#3715D3semidirect
345. C33#495semidirect
513. C32#5059semidirect
514. C32#5155semidirect
515. C32#5168semidirect
568. C3#10591470semidirect
569. C3#10591474semidirect
570. C3#10591606semidirect

Maximal subgroups

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

subgroupindex
#117234292
C2×#887992
C2×#887942
C3×#638632
C2×C3×#37152
C2×C3×#37132
#117220702
#105916063
#105914743
#105914703