Finite Group Databasebeta

← #13181070

Subgroups — #13181070

Order 1728.

946 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 119 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#11724618C211yesmaximal, characteristic, complemented
2C2×C3×#3798C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11724326C212yesmaximal, complemented
5#11724323C212yesmaximal, complemented
6#11724309C212yesmaximal, complemented
7#11724305C212yesmaximal, complemented
8#11724303C214yesmaximal, complemented
9#10596499—31nomaximal
10C3×#3828—21no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3—21no—
13#89049—21no—
14#89046C2212yes—
15C4⋊#1113C2211yescharacteristic, complemented
16C4×#1113—21no—
17#89042C2212yescomplemented
18#89041—21no—
19#89040C2211yescharacteristic, complemented
20C2×C3×#627C2211yescharacteristic, complemented
21C2×C3×#626—21no—
22C2×C3×#625C2211yescharacteristic, complemented
23C3×#3798—21no—
24C3×#3798—21no—
25C3×#3798C2212yes—
26C3×#3798C2212yescomplemented
27#88993C2212yescomplemented
28#88989C2212yescomplemented
29#88988C2212yescomplemented
30#88938C2214yescomplemented
31#88937C2214yescomplemented
32#88936C2214yescomplemented
33#88935C2214yescomplemented
34C2×C3×#235—61no—
35#64306—31no—
36D3×#235—61no—
37C2×#3798D311yescharacteristic, complemented
38#64283—31no—
39#64239—32no—
40#64236—32no—
41#64226—32no—
42#64224—32no—
43#64223—34no—
44C22×C3×C3⋊D3D411yescharacteristic
45C22×C32×D3—21no—
46C2×#1113—21no—
47C2×C33×C4D411yescharacteristic
48C3×#638—22no—
49C2×C3×C3⋊Dic3C2311yescharacteristic
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4×C3⋊D3—22no—
52C3×C4×C3⋊D3D411yescharacteristic
53C3×C4×C3⋊D3D411yescharacteristic
54C3×C4×C3⋊D3C2311yescharacteristic
55C3×C4×C3⋊D3C2311yescharacteristic
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C32×C4×D3—22no—
59C32×C4×D3—22no—
60#6296C2312yes—
61#6295—22no—
62#6295C2312yes—
63#6294—21no—
64#6294—22no—
65C3×#627—21no—
66C3×#627C2312yes—
67C3×#626—22no—
68C3×#626—22no—
69C3×#625—21no—
70C3×#625C2312yes—
71C3⋊#622C2314yes—
72#5822—91nomaximal
73C22×C32×C4—61no—
74#3828—21no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×#67—61no—
77C2×C3×#67—61no—
78C3×#235—62no—
79C3×#235—62no—
80C3×#235—62no—
81C3×#235—62no—
82C3×#235—61no—
83C3×#235—61no—
84C3×#235—121no—
85C2×C3×D12—61no—
86C2×C3×C4×D3—21no—
87C2×C3×C4×D3—21no—
88C2×C3×C4×D3—41no—
89C2×C3×C4×D3—61no—
90C2×C3×C4×D3—61no—
91C2×C3×Dic6—61no—
92#3811—61no—
93D3×#67—62no—
94C2×#627C2×D311yescharacteristic, complemented
95C2×#626—21no—
96C2×#625C2×D311yescharacteristic, complemented
97#3805—32no—
98#3804—62no—
99#3802—31no—
100D3×D12—61no—
101C4×D32—62no—
102C4×D32—61no—
103#3799—61no—
104#3798—21no—
105#3798—21no—
106#3798—32no—
107#3798—62no—
108#3798C2×D312yes—
109#3798C2×D312yescomplemented
110#3797—31no—
111#3796—61no—
112#3795—61no—
113D3×Dic6—61no—
114#3792—32no—
115#3788—32no—
116#3787—32no—
117#3776—34no—
118#3775—34no—
119#3775—34no—
120#3774—34no—
121C22×C33C2×D411yescharacteristic
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3C2×D411yescharacteristic
125C2×C3×C3⋊D3C2×D411yescharacteristic
126C2×C32×D3—22no—
127C2×C32×D3—42no—
128#1113—42no—
129C33×C4C2×D411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3C2×D411yescharacteristic
133C3×C3⋊Dic3C2411yescharacteristic
134C32×Dic3—22no—
135C2×C3×#41—91no—
136D3×#41—181no—
137C2×#235—61no—
138C2×#235—91no—
139#951—92no—
140#950—94no—
141#949—92no—
142#942—92no—
143#932—92no—
144C2×#208—91no—
145C23×C32—61no—
146C22×C3⋊D3#6711yescharacteristic
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—41no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C2×D32—62no—
153C2×D32—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C32×C4—62no—
159C2×C32×C4—62no—
160C2×C32×C4—61no—
161C2×C32×C4#6711yescharacteristic
162#638—22no—
163C2×C3⋊Dic3C22×D311yescharacteristic
164C4⋊(C3⋊D3)—21no—
165C4×C3⋊D3—22no—
166C4×C3⋊D3#6711yescharacteristic
167C4×C3⋊D3#6711yescharacteristic
168C4×C3⋊D3C22×D311yescharacteristic
169C4×C3⋊D3C22×D311yescharacteristic
170#634—21no—
171C3×#67—62no—
172C3×#67—62no—
173C3×#67—62no—
174C3×#67—62no—
175C3×#67—62no—
176C3×#67—62no—
177C3×#67—122no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×D12—61no—
184C3×D12—62no—
185C3×D12—62no—
186C3×D12—61no—
187C3×D12—121no—
188C3×C4×D3—22no—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×C4×D3—22no—
192C3×C4×D3—42no—
193C3×C4×D3—42no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—61no—
200C3×C4×D3—61no—
201C3×C4×D3—61no—
202C3×C4×D3—62no—
203C3×C4×D3—62no—
204C3×C4×D3—122no—
205C3×Dic6—62no—
206C3×Dic6—62no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—121no—
210#627—21no—
211#627—32no—
212#627—62no—
213#627C22×D312yes—
214#626—22no—
215#626—22no—
216#626—62no—
217#626—62no—
218#626—62no—
219#625—21no—
220#625—32no—
221#625—62no—
222#625C22×D312yes—
223#624—62no—
224#624—61no—
225D3×Dic3—62no—
226D3×Dic3—62no—
227D3×Dic3—62no—
228#622—34no—
229#575—271noSylow
230C2×C33—21no—
231C2×C33C22×D411yescharacteristic
232C3×C3⋊D3—21no—
233C3×C3⋊D3—21no—
234C3×C3⋊D3—42no—
235C32×D3—42no—
236C3×#41—92no—
237C3×#41—92no—
238C3×#41—181no—
239C3×#41—181no—
240C3×#41—181no—
241C2×C3×Q8—91no—
242C2×C3×D4—91no—
243C2×C3×D4—181no—
244C22×C3×C4—61no—
245C22×C3×C4—61no—
246C22×C3×C4—91no—
247C22×C3×C4—121no—
248C2×#67—61no—
249C2×#67—61no—
250C2×#67—181no—
251#239—181no—
252D3×Q8—181no—
253#237—182no—
254#237—181no—
255D3×D4—181no—
256D3×D4—182no—
257#235—62no—
258#235—62no—
259#235—62no—
260#235—62no—
261#235—61no—
262#235—61no—
263#235—92no—
264#235—92no—
265#235—121no—
266#235—182no—
267#235—181no—
268C2×D12—61no—
269C2×D12—91no—
270C2×C4×D3—21no—
271C2×C4×D3—61no—
272C2×C4×D3—61no—
273C2×C4×D3—181no—
274C2×C4×D3—182no—
275C2×Dic6—61no—
276C2×Dic6—91no—
277#216—94no—
278#215—94no—
279#214—94no—
280#213—94no—
281#208—92no—
282#208—92no—
283C2×C3⋊C8—92no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—62no—
289C22×C32—122no—
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—22no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3C2×#6711yescharacteristic
294C2×C3⋊D3C2×#6711yescharacteristic
295C2×C3×D3—22no—
296C2×C3×D3—22no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—44no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—122no—
308C2×C3×D3—122no—
309C2×C3×D3—122no—
310D32—124no—
311D32—122no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—62no—
321C32×C4C2×#6711yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C3⋊Dic3—22no—
324C3⋊Dic3C2×#6711yescharacteristic
325C3⋊Dic3C23×D311yescharacteristic
326C3×Dic3—22no—
327C3×Dic3—22no—
328C3×Dic3—42no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C3×Dic3—61no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—122no—
336C2×#41—91no—
337C2×#41—271no—
338#137—272no—
339#137—272no—
340AGL(1,ℤ/8)—272no—
341AGL(1,ℤ/8)—272no—
342#135—274no—
343C2×M16—271no—
344C33#57511yescharacteristic, complemented, Sylow
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—91no—
348C23×C3—121no—
349C22×D3—21no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—181no—
353C22×D3—182no—
354C3×Q8—92no—
355C3×Q8—181no—
356C3×Q8—181no—
357C3×D4—92no—
358C3×D4—181no—
359C3×D4—182no—
360C3×D4—182no—
361C3×D4—181no—
362C3×D4—182no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—122no—
379C2×C3×C4—122no—
380C2×C3×C4—121no—
381C2×C3×C4—182no—
382C2×C3×C4GO+(4,𝔽2)11yescharacteristic
383#67—62no—
384#67—62no—
385#67—62no—
386#67—62no—
387#67—62no—
388#67—62no—
389#67—122no—
390#67—182no—
391#67—182no—
392#67—182no—
393C2×Dic3—21no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—181no—
397C2×Dic3—182no—
398D12—61no—
399D12—62no—
400D12—62no—
401D12—61no—
402D12—92no—
403D12—121no—
404D12—182no—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—61no—
408C4×D3—61no—
409C4×D3—61no—
410C4×D3—61no—
411C4×D3—61no—
412C4×D3—61no—
413C4×D3—61no—
414C4×D3—61no—
415C4×D3—62no—
416C4×D3—62no—
417C4×D3—122no—
418C4×D3—182no—
419C4×D3—182no—
420C4×D3—182no—
421Dic6—62no—
422Dic6—62no—
423Dic6—61no—
424Dic6—61no—
425Dic6—92no—
426Dic6—121no—
427Dic6—182no—
428C3⋊C8—94no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—42no—
437C2×C32—81no—
438C2×C32—122no—
439C2×C32C22×#6711yescharacteristic
440C3⋊D3—21no—
441C3⋊D3—21no—
442C3⋊D3—42no—
443C3×D3—42no—
444C3×D3—44no—
445C3×D3—42no—
446C3×D3—62no—
447C3×D3—121no—
448C3×D3—121no—
449C3×D3—121no—
450C3×D3—122no—
451C3×D3—122no—
452C3×D3—242no—
453#41—92no—
454#41—92no—
455#41—181no—
456#41—181no—
457#41—181no—
458#41—272no—
459#41—272no—
460#41—541no—
461C2×Q8—91no—
462C2×Q8—271no—
463C2×D4—91no—
464C2×D4—181no—
465C2×D4—271no—
466C2×D4—541no—
467C22×C4—61no—
468C22×C4—91no—
469C22×C4—541no—
470Q16—274no—
471SD16—274no—
472SD16—274no—
473D8—274no—
474M16—272no—
475M16—272no—
476C2×C8—272no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—91no—
484C22×C3—91no—
485C22×C3—122no—
486C22×C3—122no—
487C22×C3—122no—
488C22×C3—124no—
489C22×C3—182no—
490C22×C3—361no—
491C22×C3C2×GO+(4,𝔽2)11yescharacteristic
492C2×D3—22no—
493C2×D3—42no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—61no—
497C2×D3—61no—
498C2×D3—62no—
499C2×D3—62no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—122no—
503C2×D3—364no—
504C2×D3—362no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—62no—
514C3×C4—62no—
515C3×C4—91no—
516C3×C4—91no—
517C3×C4—122no—
518C3×C4—182no—
519C3×C4C2×GO+(4,𝔽2)11yescharacteristic
520C3×C4C2×GO+(4,𝔽2)11yescharacteristic
521Dic3—22no—
522Dic3—61no—
523Dic3—61no—
524Dic3—61no—
525Dic3—61no—
526Dic3—62no—
527Dic3—62no—
528Dic3—122no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32—41no—
533C32#582211yescharacteristic, complemented
534C23—61no—
535C23—91no—
536C23—541no—
537Q8—92no—
538Q8—181no—
539Q8—181no—
540Q8—272no—
541D4—92no—
542D4—182no—
543D4—182no—
544D4—181no—
545D4—182no—
546D4—181no—
547D4—272no—
548D4—542no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—91no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—91no—
556C2×C4—91no—
557C2×C4—182no—
558C2×C4—542no—
559C2×C4—541no—
560C2×C4D32⋊D311yescharacteristic
561C8—274no—
562C2×C3—22no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—41no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—81no—
571C2×C3—122no—
572C2×C3—124no—
573C2×C3—122no—
574C2×C3—181no—
575C2×C3—181no—
576C2×C3—362no—
577C2×C3C22×GO+(4,𝔽2)11yescharacteristic
578D3—42no—
579D3—62no—
580D3—121no—
581D3—121no—
582D3—121no—
583D3—122no—
584D3—122no—
585D3—242no—
586C22—62no—
587C22—91no—
588C22—91no—
589C22—122no—
590C22—182no—
591C22—361no—
592C22—1082no—
593C22C2×D32⋊D311yescharacteristic
594C4—62no—
595C4—91no—
596C4—91no—
597C4—182no—
598C4C2×D32⋊D311yescharacteristic
599C4C2×D32⋊D311yescharacteristic
600C3—21no—
601C3—21no—
602C3—41no—
603C3—41no—
604C3#1059649911yescharacteristic, complemented
605C2—21no—
606C2—122no—
607C2—181no—
608C2—181no—
609C2—362no—
610C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC4C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#575
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11724618C2semidirect
2. C2×C3×#3798C2semidirect
3. #11724511C2semidirect
4. #11724326C2semidirect
5. #11724323C2semidirect
6. #11724309C2semidirect
7. #11724305C2semidirect
8. #11724303C2semidirect
15. C4⋊#1113C22semidirect
17. #89042C22semidirect
19. #89040C22semidirect
20. C2×C3×#627C22semidirect
22. C2×C3×#625C22semidirect
26. C3×#3798C22semidirect
27. #88993C22semidirect
28. #88989C22semidirect
29. #88988C22semidirect
30. #88938C22semidirect
31. #88937C22semidirect
32. #88936C22semidirect
33. #88935C22semidirect
37. C2×#3798D3semidirect
94. C2×#627C2×D3semidirect
96. C2×#625C2×D3semidirect
109. #3798C2×D3semidirect
344. C33#575semidirect
533. C32#5822semidirect
604. C3#10596499semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×#37982
#117245112
#117243262
#117243232
#117243092
#117243052
#117243032
#105964993
#58229