Finite Group Databasebeta

← #13156611

Subgroups — #13156611

Order 1728.

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

normal subgroups only

This group has 794 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×#88811C211yesmaximal, characteristic, complemented
2#11723441C211yesmaximal, characteristic, complemented
3C2×#88795C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic
5C3×#63864C211yesmaximal, characteristic, complemented
6C2×C3×#3719C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591609—31nomaximal
9#10590865—31nomaximal
10#10590370—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88811—22no—
19#88804—22no—
20#88795—22no—
21#88791—22no—
22C3×#3722—22no—
23C3×#3719—22no—
24C2×#3755—31no—
25C2×#3745—31no—
26#64084—31no—
27C3×#947—31no—
28C2×C3×#216—31no—
29C2×C3×Dic12—31no—
30C3×#875—31no—
31#63864—31no—
32#63864—31no—
33#63864—61no—
34#63864D311yescharacteristic, complemented
35C2×#3719D311yescharacteristic, complemented
36C2×#3717—31no—
37C2×#3717—31no—
38C2×#3717—31no—
39C2×#3717—61no—
40C2×#3716—31no—
41C2×#3715—31no—
42C2×#3714—31no—
43C2×#3714—31no—
44C2×#3714—61no—
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
51C4⋊(C3⋊(C3⋊D3))—21no—
52C4⋊(C3⋊(C3⋊D3))—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D412yes—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64#5191—91no—
65#5059—91no—
66#4696—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—121no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86C2×C32×C8—31no—
87C32×C42—31no—
88#3755—62no—
89#3752—62no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91#3745—62no—
92C3×#216—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×Dic12—62no—
105#3722—22no—
106#3722—64no—
107#3722—62no—
108#3722—62no—
109#3719—22no—
110#3717—62no—
111#3717—62no—
112#3717—62no—
113#3717—64no—
114#3716—62no—
115#3715—62no—
116#3714—64no—
117#3714—62no—
118#3714—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3⋊(C3⋊D3)—41no—
121C2×C3⋊(C3⋊D3)—42no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—22no—
128C32×Dic3—41no—
129C2×C3×Q16—91no—
130C3×#124—91no—
131#947—31no—
132#947—91no—
133C2×#216—31no—
134C2×#215—91no—
135C2×#215—91no—
136C2×#215—91no—
137C2×#215—181no—
138C2×#214—91no—
139C2×#213—91no—
140C2×Dic12—31no—
141C2×D24—91no—
142C2×D24—91no—
143C2×D24—181no—
144C2×#204—91no—
145C2×#204—91no—
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—61no—
155C22×C3⋊D3—62no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—121no—
159C22×C3⋊D3—121no—
160C22×C3⋊D3—122no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×Q8—61no—
165C32×Q8—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—32no—
170C2×C32×C4—41no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—61no—
179C4⋊(C3⋊D3)—61no—
180C4⋊(C3⋊D3)—62no—
181C4⋊(C3⋊D3)—62no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—121no—
187C4⋊(C3⋊D3)—121no—
188C4⋊(C3⋊D3)—121no—
189C4⋊(C3⋊D3)—121no—
190C4⋊(C3⋊D3)—122no—
191C4⋊(C3⋊D3)—122no—
192#634—21no—
193#634—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3D1212yes—
202C2×C3×Dic3#6712yes—
203C3×D12—62no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—121no—
209C3×Dic6—22no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—41no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220C3×Dic6—121no—
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×Q8—91no—
238C2×C3×D4—91no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—91no—
243C2×D12—91no—
244C2×D12—91no—
245C2×D12—181no—
246C2×D12—181no—
247C2×D12—181no—
248C2×D12—361no—
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C2×Dic6D3211yescharacteristic
253C3×Q16—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#216—62no—
263#215—182no—
264#215—182no—
265#215—182no—
266#215—184no—
267#214—182no—
268#213—182no—
269(C2×D3)⋊C4—62no—
270(C2×D3)⋊C4—62no—
271(C2×D3)⋊C4—182no—
272(C2×D3)⋊C4—182no—
273(C2×D3)⋊C4—184no—
274(C2×D3)⋊C4—182no—
275C4×Dic3D3211yescharacteristic
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—61no—
279C2×C3⋊C8D3211yescharacteristic
280Dic12—62no—
281D24—182no—
282D24—182no—
283D24—184no—
284#204—182no—
285#204—182no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32C2×D1211yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3⋊D3—41no—
294C2×C3⋊D3—42no—
295C2×C3⋊D3—122no—
296C2×C3⋊D3—122no—
297C2×C3⋊D3—122no—
298C2×C3⋊D3—124no—
299C2×C3⋊D3—121no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—121no—
302C2×C3⋊D3—121no—
303C2×C3⋊D3—124no—
304C2×C3⋊D3—122no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—242no—
307C2×C3⋊D3—242no—
308C2×C3⋊D3—244no—
309C2×C3×D3—122no—
310C2×C3×D3—121no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—121no—
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×C4—62no—
324C32×C4—121no—
325C32×C4C2×D1211yescharacteristic
326C32×C4C2×D1211yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4D3×D411yescharacteristic
329C32×C4D3×D411yescharacteristic
330C32×C4C2×#6711yescharacteristic
331C3⋊Dic3—41no—
332C3×Dic3—22no—
333C3×Dic3—22no—
334C3×Dic3—42no—
335C3×Dic3—42no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—122no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C2×Q16—91no—
342C2×SD16—271no—
343C2×SD16—271no—
344C2×D8—271no—
345#124—91no—
346#124—271no—
347C4×C8—91no—
348C33#49511yescharacteristic, complemented, Sylow
349C23×C3—181no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—121no—
353C22×D3—182no—
354C22×D3—181no—
355C22×D3—181no—
356C22×D3—181no—
357C22×D3—361no—
358C22×D3—361no—
359C22×D3—362no—
360C3×Q8—62no—
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—121no—
366C3×Q8—181no—
367C3×Q8—181no—
368C3×D4—181no—
369C3×D4—181no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—32no—
374C2×C3×C4—32no—
375C2×C3×C4—41no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—121no—
380C2×C3×C4—181no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—62no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3#62612yes—
389D12—62no—
390D12—61no—
391D12—61no—
392D12—61no—
393D12—61no—
394D12—121no—
395D12—182no—
396D12—182no—
397D12—181no—
398D12—181no—
399D12—181no—
400D12—181no—
401D12—181no—
402D12—181no—
403D12—361no—
404D12—361no—
405D12—361no—
406D12—361no—
407D12—362no—
408D12—362no—
409Dic6—21no—
410Dic6—21no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—121no—
416Dic6—121no—
417C3×C8—32no—
418C3×C8—32no—
419C3×C8—62no—
420C3×C8—92no—
421C3⋊C8—32no—
422C3⋊C8—32no—
423C3⋊C8—62no—
424C3⋊C8#62612yes—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32#87411yescharacteristic
434C2×C32#90412yes—
435C2×C32#91912yes—
436C2×C32#94011yescharacteristic
437C2×C32#94011yescharacteristic
438C2×C32#95012yes—
439C3⋊D3—81no—
440C3⋊D3—242no—
441C3⋊D3—242no—
442C3⋊D3—242no—
443C3⋊D3—244no—
444C3⋊D3—241no—
445C3⋊D3—241no—
446C3⋊D3—241no—
447C3×D3—242no—
448C3×D3—241no—
449C3×D3—241no—
450C2×Q8—31no—
451C2×Q8—91no—
452C2×D4—91no—
453C2×D4—271no—
454Q16—182no—
455SD16—542no—
456SD16—542no—
457D8—542no—
458C2×C8—31no—
459C2×C8—91no—
460C22⋊C4—182no—
461C22⋊C4—542no—
462C42—31no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—41no—
467C22×C3—362no—
468C22×C3—361no—
469C22×C3D3×D1211yescharacteristic
470C22×C3C2×#62611yescharacteristic
471C22×C3D3×#6711yescharacteristic
472C2×D3—122no—
473C2×D3—122no—
474C2×D3—121no—
475C2×D3—122no—
476C2×D3—121no—
477C2×D3—242no—
478C2×D3—362no—
479C2×D3—362no—
480C2×D3—362no—
481C2×D3—364no—
482C2×D3—364no—
483C2×D3—362no—
484C2×D3—362no—
485C2×D3—361no—
486C2×D3—361no—
487C2×D3—362no—
488C2×D3—361no—
489C2×D3—722no—
490C2×D3—722no—
491C2×D3—724no—
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—62no—
501C3×C4—62no—
502C3×C4—122no—
503C3×C4—122no—
504C3×C4—121no—
505C3×C4—121no—
506C3×C4—361no—
507C3×C4D3×D1211yescharacteristic
508C3×C4D3×D1211yescharacteristic
509C3×C4D3×D1211yescharacteristic
510C3×C4C2×#62611yescharacteristic
511C3×C4D3×#6711yescharacteristic
512C3×C4#381111yescharacteristic
513Dic3—22no—
514Dic3—41no—
515Dic3—122no—
516Dic3—121no—
517Dic3—121no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#469611yescharacteristic, complemented
523C32#505911yescharacteristic, complemented
524C32#519111yescharacteristic, complemented
525C23—181no—
526C23—541no—
527Q8—61no—
528Q8—61no—
529Q8—181no—
530Q8—181no—
531D4—181no—
532D4—181no—
533D4—541no—
534D4—541no—
535C2×C4—32no—
536C2×C4—61no—
537C2×C4—181no—
538C2×C4D3311yescharacteristic
539C8—32no—
540C8—92no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—41no—
548C2×C3—42no—
549C2×C3—721no—
550C2×C3#6380812yes—
551C2×C3#6382912yes—
552C2×C3#6391111yescharacteristic
553C2×C3#6391111yescharacteristic
554C2×C3#6394912yes—
555C2×C3#6397911yescharacteristic
556D3—242no—
557D3—241no—
558D3—241no—
559D3—722no—
560D3—722no—
561D3—722no—
562D3—724no—
563D3—721no—
564D3—721no—
565D3—721no—
566C22—361no—
567C22—362no—
568C22—1081no—
569C22—1082no—
570C22D3×#62611yescharacteristic
571C4—62no—
572C4—121no—
573C4—361no—
574C4D3×#62611yescharacteristic
575C4#8895511yescharacteristic
576C3—21no—
577C3—21no—
578C3—21no—
579C3—41no—
580C3#1059037011yescharacteristic, complemented
581C3#1059086511yescharacteristic, complemented
582C3#1059160911yescharacteristic, complemented
583C2—721no—
584C2—2161no—
585C2#1172285612yes—
586C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#88791C2
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. C2×#88811C2semidirect
2. #11723441C2semidirect
3. C2×#88795C2semidirect
5. C3×#63864C2semidirect
6. C2×C3×#3719C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
34. #63864D3semidirect
35. C2×#3719D3semidirect
47. Dic3×C3⋊C8D3semidirect
348. C33#495semidirect
522. C32#4696semidirect
523. C32#5059semidirect
524. C32#5191semidirect
580. C3#10590370semidirect
581. C3#10590865semidirect
582. C3#10591609semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117234412
C2×#887952
C2×#887912
C3×#638642
C2×C3×#37192
C3×Dic3×C3⋊C82
#105916093
#105908653
#105903703