Finite Group Databasebeta

← #13161958

Subgroups — #13161958

Order 1728.

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

normal subgroups only

This group has 882 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×#64149C211yesmaximal, characteristic, complemented
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic, complemented
4#11723677C211yesmaximal, characteristic, complemented
5C2×#88811C211yesmaximal, characteristic, complemented
6C2×#88810C211yesmaximal, characteristic, complemented
7#11722071C211yesmaximal, characteristic, complemented
8#10591475—31nomaximal
9#10591472—31nomaximal
10#10591469—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—22no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
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#88811—22no—
23#88810—22no—
24#64149D311yescharacteristic, complemented
25C2×#3754D311yescharacteristic, complemented
26C2×#3753D311yescharacteristic, complemented
27C3×#940—31no—
28C3×#940—31no—
29C3×#940—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×#3717—31no—
40C2×#3716—31no—
41C2×#3715—31no—
42C2×#3714—31no—
43C2×#3714—31no—
44C2×#3713—31no—
45#63555—31no—
46#63554—31no—
47#63554—31no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C33×D4—21no—
52C33×D4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—44no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D412yes—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)—21no—
61C3×#634—21no—
62C3×#634—21no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C3×C3⋊(C3⋊C8)D412yescomplemented
65#5176—91no—
66#5154—91no—
67#5074—91no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—41no—
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×#638—22no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic
79C2×C3×#67—62no—
80C2×C3×#67—62no—
81C2×C3×#67—64no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C2×C3×D12—61no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—61no—
91#3754—22no—
92#3753—22no—
93C4×C3⋊Dic3C2×D311yescharacteristic
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96C3×#214—62no—
97C3×#214—62no—
98C3×#214—64no—
99C3×#213—62no—
100C3×#213—62no—
101C3×#213—64no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—61no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C2×C3×C3⋊C8—61no—
111#3722—62no—
112(C2×D3)⋊Dic3—62no—
113(C2×D3)⋊Dic3—62no—
114#3717—62no—
115#3716—62no—
116#3715—62no—
117#3714—62no—
118#3714—62no—
119#3713—62no—
120C22×C33—41no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—42no—
126C2×C3×C3⋊D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×D8—91no—
133C3×#127—91no—
134#947—91no—
135C2×#215—91no—
136#940—31no—
137#940—31no—
138#940—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×#213—91no—
148C2×#213—91no—
149C2×D24—91no—
150C2×#204—91no—
151#875—91no—
152#815—91no—
153#815—91no—
154#803—91no—
155C23×C32—21no—
156C23×C32—22no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—41no—
160C23×C32—41no—
161C23×C32—42no—
162C22×C3⋊D3—21no—
163C22×C3⋊D3—21no—
164C22×C3×D3—62no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—61no—
168C22×C3×D3—61no—
169C22×C3×D3—121no—
170C32×D4—21no—
171C32×D4—21no—
172C32×D4—22no—
173C32×D4—22no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—42no—
183C32×D4—42no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191#638—44no—
192C2×C3⋊Dic3—21no—
193C2×C3⋊Dic3#6712yes—
194C4⋊(C3⋊D3)—21no—
195C4⋊(C3⋊D3)—21no—
196C4⋊(C3⋊D3)—21no—
197C4⋊(C3⋊D3)—21no—
198#634—21no—
199#634—21no—
200C3×#67—128no—
201C3×#67—124no—
202C3×#67—124no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—121no—
209C3×D12—62no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—61no—
218C3×D12—121no—
219C3×D12—121no—
220C3×D12—121no—
221C3×Dic6—62no—
222C3×Dic6—61no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—61no—
226C3×Dic6—121no—
227C3⋊(C3⋊C8)#6712yescomplemented
228C3⋊(C3⋊C8)#6712yescomplemented
229C3×C3⋊C8—32no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—32no—
232C3×C3⋊C8—32no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—62no—
235#496—271noSylow
236C2×C33—81no—
237C2×C33#12711yescharacteristic
238C2×C33AGL(1,ℤ/8)12yes—
239C3×C3⋊D3—81no—
240C3×C3⋊D3—81no—
241C2×C3×Q8—91no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—41no—
246C2×C3×D4—91no—
247C2×C3×D4—91no—
248C2×C3×D4—182no—
249C2×C3×D4D3211yescharacteristic
250C2×C3×D4D3211yescharacteristic
251C2×C3×D4D3211yescharacteristic
252C2×#67—62no—
253C2×#67—62no—
254C2×#67—64no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—31no—
259C2×D12—61no—
260C2×D12—61no—
261C2×Dic6—31no—
262C2×Dic6—31no—
263C2×Dic6—61no—
264C3×SD16—182no—
265C3×D8—182no—
266C2×C3×C8—91no—
267C2×C3×C8—91no—
268C3×C42—91no—
269#217—182no—
270#215—182no—
271#214—62no—
272#214—62no—
273#214—64no—
274#214—182no—
275#213—62no—
276#213—62no—
277#213—64no—
278#213—182no—
279#213—182no—
280(C2×D3)⋊C4—182no—
281(C2×D3)⋊C4—182no—
282C4×Dic3—31no—
283C4×Dic3—31no—
284C4×Dic3—61no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—31no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—61no—
290C2×C3⋊C8—61no—
291D24—182no—
292#204—182no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—41no—
297C22×C32—42no—
298C22×C32—42no—
299C22×C32—42no—
300C22×C32—44no—
301C22×C32—42no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—44no—
306C22×C32—42no—
307C22×C32—42no—
308C22×C32—82no—
309C22×C32—82no—
310C22×C32—84no—
311C22×C32D3×D411yescharacteristic
312C22×C32D3×D411yescharacteristic
313C22×C32C2×#6711yescharacteristic
314C2×C3⋊D3—42no—
315C2×C3⋊D3—41no—
316C2×C3⋊D3—41no—
317C2×C3⋊D3—42no—
318C2×C3×D3—122no—
319C2×C3×D3—122no—
320C2×C3×D3—124no—
321C2×C3×D3—122no—
322C2×C3×D3—122no—
323C2×C3×D3—122no—
324C2×C3×D3—122no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—242no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4D3×D411yescharacteristic
339C32×C4D3×D411yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4C2×#6711yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C3⋊Dic3—22no—
345C3⋊Dic3—41no—
346C3×Dic3—64no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—122no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C2×SD16—91no—
353C2×SD16—271no—
354C2×D8—91no—
355C2×D8—271no—
356#127—91no—
357#124—271no—
358#97—271no—
359C33#49611yescharacteristic, complemented, Sylow
360C23×C3—22no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—42no—
367C23×C3—181no—
368C23×C3—181no—
369C22×D3—62no—
370C22×D3—61no—
371C22×D3—61no—
372C22×D3—61no—
373C22×D3—61no—
374C22×D3—121no—
375C3×Q8—181no—
376C3×Q8—181no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—21no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—21no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—42no—
390C3×D4—42no—
391C3×D4—181no—
392C3×D4—181no—
393C3×D4—181no—
394C3×D4—181no—
395C3×D4—364no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—41no—
400C2×C3×C4—92no—
401C2×C3×C4—181no—
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404C2×C3×C4C2×D3211yescharacteristic
405#67—128no—
406#67—124no—
407#67—124no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—62no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—121no—
414D12—62no—
415D12—61no—
416D12—61no—
417D12—61no—
418D12—61no—
419D12—61no—
420D12—61no—
421D12—61no—
422D12—61no—
423D12—121no—
424D12—121no—
425D12—121no—
426Dic6—62no—
427Dic6—61no—
428Dic6—61no—
429Dic6—61no—
430Dic6—61no—
431Dic6—121no—
432C3×C8—92no—
433C3×C8—92no—
434C3⋊C8—32no—
435C3⋊C8—32no—
436C3⋊C8—32no—
437C3⋊C8—32no—
438C3⋊C8—62no—
439C3⋊C8—62no—
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—22no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—82no—
449C2×C32—82no—
450C2×C32—82no—
451C2×C32—84no—
452C2×C32—81no—
453C2×C32—81no—
454C2×C32—81no—
455C2×C32#91112yes—
456C2×C32#91412yes—
457C2×C32#93212yes—
458C2×C32#94011yescharacteristic
459C2×C32#94011yescharacteristic
460C2×C32#94011yescharacteristic
461C3⋊D3—81no—
462C3⋊D3—81no—
463C3×D3—242no—
464C3×D3—242no—
465C3×D3—241no—
466C3×D3—241no—
467C3×D3—241no—
468C3×D3—241no—
469C2×Q8—91no—
470C2×D4—91no—
471C2×D4—91no—
472C2×D4—182no—
473C2×D4#111311yescharacteristic
474SD16—182no—
475SD16—542no—
476D8—182no—
477D8—542no—
478C2×C8—91no—
479C2×C8—91no—
480C22⋊C4—542no—
481C42—91no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—44no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—84no—
500C22×C3—362no—
501C22×C3—362no—
502C22×C3—361no—
503C22×C3—361no—
504C22×C3D3×#6711yescharacteristic
505C22×C3D3×#6711yescharacteristic
506C22×C3#381111yescharacteristic
507C2×D3—122no—
508C2×D3—122no—
509C2×D3—124no—
510C2×D3—122no—
511C2×D3—122no—
512C2×D3—122no—
513C2×D3—122no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—121no—
517C2×D3—121no—
518C2×D3—242no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—182no—
528C3×C4—361no—
529C3×C4D3×#6711yescharacteristic
530C3×C4D3×#6711yescharacteristic
531C3×C4D3×#6711yescharacteristic
532C3×C4D3×#6711yescharacteristic
533C3×C4#381111yescharacteristic
534C3×C4#381111yescharacteristic
535Dic3—64no—
536Dic3—62no—
537Dic3—62no—
538Dic3—122no—
539Dic3—121no—
540Dic3—121no—
541C32—21no—
542C32—21no—
543C32—21no—
544C32—41no—
545C32#507411yescharacteristic, complemented
546C32#515411yescharacteristic, complemented
547C32#517611yescharacteristic, complemented
548C23—21no—
549C23—181no—
550C23—181no—
551Q8—181no—
552Q8—181no—
553D4—21no—
554D4—21no—
555D4—181no—
556D4—181no—
557D4—181no—
558D4—181no—
559D4—364no—
560C2×C4—92no—
561C2×C4—181no—
562C2×C4C2×#111311yescharacteristic
563C8—92no—
564C8—92no—
565C2×C3—22no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—42no—
572C2×C3—41no—
573C2×C3—82no—
574C2×C3—82no—
575C2×C3—82no—
576C2×C3—84no—
577C2×C3—81no—
578C2×C3—81no—
579C2×C3—81no—
580C2×C3—721no—
581C2×C3—721no—
582C2×C3#6393212yes—
583C2×C3#6393412yes—
584C2×C3#6393712yes—
585C2×C3#6397911yescharacteristic
586C2×C3#6397911yescharacteristic
587C2×C3#6397911yescharacteristic
588D3—242no—
589D3—242no—
590D3—241no—
591D3—241no—
592D3—241no—
593D3—241no—
594C22—42no—
595C22—41no—
596C22—362no—
597C22—362no—
598C22—361no—
599C22—361no—
600C22#8904911yescharacteristic
601C4—182no—
602C4—361no—
603C4#8904911yescharacteristic
604C4#8904911yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059146911yescharacteristic, complemented
610C3#1059147211yescharacteristic, complemented
611C3#1059147511yescharacteristic, complemented
612C2—81no—
613C2—721no—
614C2—721no—
615C2#1172370612yes—
616C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#496
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×#64149C2semidirect
2. C2×C3×#3754C2semidirect
3. C2×C3×#3753C2semidirect
4. #11723677C2semidirect
5. C2×#88811C2semidirect
6. C2×#88810C2semidirect
7. #11722071C2semidirect
24. #64149D3semidirect
25. C2×#3754D3semidirect
26. C2×#3753D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
227. C3⋊(C3⋊C8)#67semidirect
228. C3⋊(C3⋊C8)#67semidirect
359. C33#496semidirect
545. C32#5074semidirect
546. C32#5154semidirect
547. C32#5176semidirect
609. C3#10591469semidirect
610. C3#10591472semidirect
611. C3#10591475semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C3×#37542
C2×C3×#37532
#117236772
C2×#888112
C2×#888102
#117220712
#105914753
#105914723
#105914693