Finite Group Databasebeta

← #13161980

Subgroups — #13161980

Order 1728.

652 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 87 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×C3×#3754C211yesmaximal, characteristic, complemented
2C3×#64141C211yesmaximal, characteristic
3#11723663C211yesmaximal, characteristic, complemented
4#11723657C211yesmaximal, characteristic, complemented
5C3×#63660C211yesmaximal, characteristic, complemented
6#11722081C211yesmaximal, characteristic, complemented
7#11722074C211yesmaximal, characteristic, complemented
8#10591507—31nomaximal
9#10591506—31nomaximal
10#10591407—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3754—21no—
15C3×#3754—21no—
16C3×#3754—21no—
17C3×#3753—21no—
18C3×#3748C2211yescharacteristic
19C3×#3748C2211yescharacteristic
20C3×#3748C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88812—21no—
23#88811—21no—
24#88809—21no—
25#88808—21no—
26C2×#3754D311yescharacteristic, complemented
27#64141D311yescharacteristic
28C2×C3×#214—31no—
29C2×C3×#214—31no—
30C2×C3×#214—61no—
31C3×#932—31no—
32C3×#932—31no—
33C3×#932—61no—
34#63833—31no—
35#63831—31no—
36#63827—31no—
37#63817—31no—
38#63815—31no—
39#63814—31no—
40#63660D311yescharacteristic, complemented
41C3×#833—31no—
42C3×#833—31no—
43C3×#833—61no—
44#63578—31no—
45#63577—31no—
46#63577—31no—
47#63561—31no—
48#63560—31no—
49#63560—31no—
50C23×C33—21no—
51C33×D4—21no—
52C33×D4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—21no—
55C2×C3×C3⋊Dic3—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic
57C3×C4×C3⋊D3—21no—
58C3×#634—21no—
59C3×#634—21no—
60C3×#634D411yescharacteristic
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C3×C3⋊(C3⋊C8)D411yescharacteristic
66#5200—91no—
67#5177—91no—
68#4748—91no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76#3828C2×D311yescharacteristic
77C2×#634C2×D311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84#3754—21no—
85#3754—21no—
86#3754—21no—
87#3753—21no—
88#3748C2×D311yescharacteristic
89#3748C2×D311yescharacteristic
90#3748C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#214—61no—
93C3×#214—61no—
94C3×#214—61no—
95C3×#214—61no—
96C3×#214—61no—
97C3×#214—61no—
98C3×#214—62no—
99C3×#214—62no—
100C3×#214—62no—
101C3×#213—61no—
102C3×#213—61no—
103C3×#213—62no—
104C3×#208—31no—
105C3×#208—31no—
106C3×#208—31no—
107C3×#208—31no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—61no—
111C3×#208—61no—
112C3×#208—61no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116#3719—61no—
117#3719—61no—
118#3718—61no—
119#3717—61no—
120#3716—61no—
121#3715—61no—
122#3712—61no—
123#3711—61no—
124#3711—61no—
125#3710—61no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—61no—
128C22×C33—41no—
129C22×C33—81no—
130C22×C33#4111yescharacteristic
131C2×C3×C3⋊D3—21no—
132C33×C4C2×D411yescharacteristic
133C33×C4C2×D411yescharacteristic
134C3×C3⋊Dic3—21no—
135C3×C3⋊Dic3—41no—
136C3×AGL(1,ℤ/8)—91no—
137C2×C3×SD16—91no—
138#951—91no—
139#948—91no—
140#942—91no—
141C2×#214—31no—
142C2×#214—31no—
143C2×#214—61no—
144#932—31no—
145#932—31no—
146#932—61no—
147#909—91no—
148#907—91no—
149#901—91no—
150C3×#100—91no—
151#836—91no—
152#833—31no—
153#833—31no—
154#833—61no—
155#824—91no—
156#824—91no—
157#822—91no—
158#822—91no—
159#819—91no—
160C23×C32—21no—
161C23×C32—22no—
162C23×C32—21no—
163C23×C32—21no—
164C23×C32—41no—
165C23×C32—41no—
166C23×C32—42no—
167C32×D4—21no—
168C32×D4—21no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—21no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—41no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—42no—
180C32×D4—42no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—41no—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—21no—
189C2×C3⋊Dic3—21no—
190C4⋊(C3⋊D3)#6711yescharacteristic
191C4×C3⋊D3—21no—
192#634—21no—
193#634—21no—
194#634#6711yescharacteristic
195C3×#67—61no—
196C3×#67—61no—
197C3×#67—121no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—121no—
201C3×D12—31no—
202C3×D12—31no—
203C3×D12—61no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×C4×D3—121no—
207C3×Dic6—31no—
208C3×Dic6—31no—
209C3×Dic6—61no—
210C3×Dic6—62no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—121no—
216C3⋊(C3⋊C8)—21no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)#6711yescharacteristic
220C3⋊(C3⋊C8)#6711yescharacteristic
221C3×C3⋊C8—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—62no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—121no—
235C3×C3⋊C8—121no—
236#471—271noSylow
237C2×C33—21no—
238C2×C33—81no—
239C2×C33#12111yescharacteristic
240C3×C3⋊D3—41no—
241C3×#41—91no—
242C2×C3×Q8—91no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—41no—
247C2×C3×D4D3211yescharacteristic
248C2×C3×D4D3211yescharacteristic
249C2×C3×D4D3211yescharacteristic
250#235—31no—
251#235—31no—
252#235—61no—
253C2×Dic6—31no—
254C2×Dic6—31no—
255C2×Dic6—61no—
256C3×SD16—181no—
257C3×SD16—181no—
258C3×SD16—181no—
259C3×D8—181no—
260C3×M16—91no—
261C3×M16—91no—
262C3×M16—91no—
263C2×C3×C8—91no—
264#216—181no—
265#216—181no—
266#215—181no—
267#214—61no—
268#214—61no—
269#214—61no—
270#214—61no—
271#214—61no—
272#214—61no—
273#214—62no—
274#214—62no—
275#214—62no—
276#214—181no—
277#213—61no—
278#213—61no—
279#213—62no—
280#208—31no—
281#208—31no—
282#208—31no—
283#208—31no—
284#208—31no—
285#208—31no—
286#208—61no—
287#208—61no—
288#208—61no—
289#208—181no—
290C2×C3⋊C8—31no—
291C2×C3⋊C8—31no—
292C2×C3⋊C8—61no—
293C2×C3⋊C8—181no—
294Dic12—181no—
295#204—181no—
296ΓL(1,𝔽25)—181no—
297ΓL(1,𝔽25)—181no—
298D3×C8—181no—
299D3×C8—181no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—44no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—81no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—82no—
315C22×C32—82no—
316C22×C32—82no—
317C22×C32—84no—
318C22×C32#23511yescharacteristic
319C22×C32#23711yescharacteristic
320C22×C32#23711yescharacteristic
321C2×C3⋊D3—21no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—121no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4D3×D411yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C3⋊Dic3—21no—
340C3⋊Dic3—41no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—121no—
344C3×Dic3—122no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347#137—271no—
348AGL(1,ℤ/8)—91no—
349C2×SD16—91no—
350#131—271no—
351#108—271no—
352#101—271no—
353#100—91no—
354C33#47111yescharacteristic, complemented, Sylow
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—42no—
362C3×Q8—91no—
363C3×Q8—181no—
364C3×Q8—181no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—91no—
380C3×D4—181no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—61no—
391#67—61no—
392#67—121no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—121no—
396D12—31no—
397D12—31no—
398D12—61no—
399C4×D3—61no—
400C4×D3—61no—
401C4×D3—121no—
402Dic6—31no—
403Dic6—31no—
404Dic6—61no—
405Dic6—62no—
406Dic6—61no—
407Dic6—61no—
408Dic6—61no—
409Dic6—61no—
410Dic6—121no—
411C3×C8—91no—
412C3×C8—91no—
413C3×C8—181no—
414C3×C8—181no—
415C3×C8—181no—
416C3⋊C8—31no—
417C3⋊C8—31no—
418C3⋊C8—31no—
419C3⋊C8—31no—
420C3⋊C8—61no—
421C3⋊C8—61no—
422C3⋊C8—62no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—61no—
426C3⋊C8—61no—
427C3⋊C8—61no—
428C3⋊C8—61no—
429C3⋊C8—121no—
430C3⋊C8—121no—
431C2×C32—21no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—41no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—42no—
442C2×C32—82no—
443C2×C32—82no—
444C2×C32—82no—
445C2×C32—84no—
446C2×C32—81no—
447C2×C32—81no—
448C2×C32—81no—
449C2×C32#88411yescharacteristic
450C2×C32#93811yescharacteristic
451C2×C32#93911yescharacteristic
452C3⋊D3—41no—
453C3×D3—121no—
454C3×D3—121no—
455C3×D3—241no—
456#41—91no—
457C2×Q8—91no—
458C2×D4#111311yescharacteristic
459Q16—541no—
460SD16—181no—
461SD16—181no—
462SD16—181no—
463SD16—541no—
464D8—181no—
465M16—91no—
466M16—91no—
467M16—91no—
468M16—541no—
469C2×C8—91no—
470C2×C8—541no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—84no—
489C22×C3—181no—
490C22×C3#380411yescharacteristic
491C22×C3#380411yescharacteristic
492C22×C3#380511yescharacteristic
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—121no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—181no—
505C3×C4—361no—
506C3×C4D3×#6711yescharacteristic
507C3×C4D3×#6711yescharacteristic
508C3×C4D3×#6711yescharacteristic
509C3×C4D3×#6711yescharacteristic
510C3×C4#381111yescharacteristic
511C3×C4#381111yescharacteristic
512Dic3—61no—
513Dic3—61no—
514Dic3—121no—
515Dic3—122no—
516Dic3—121no—
517Dic3—121no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#474811yescharacteristic, complemented
523C32#517711yescharacteristic, complemented
524C32#520011yescharacteristic, complemented
525C23—21no—
526Q8—91no—
527Q8—181no—
528Q8—181no—
529D4—21no—
530D4—21no—
531D4—91no—
532D4—181no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4C2×#111311yescharacteristic
536C8—91no—
537C8—91no—
538C8—181no—
539C8—181no—
540C8—181no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—42no—
551C2×C3—41no—
552C2×C3—82no—
553C2×C3—82no—
554C2×C3—82no—
555C2×C3—84no—
556C2×C3—81no—
557C2×C3—81no—
558C2×C3—81no—
559C2×C3—361no—
560C2×C3#6395811yescharacteristic
561C2×C3#6397011yescharacteristic
562C2×C3#6397111yescharacteristic
563D3—121no—
564D3—121no—
565D3—241no—
566C22—41no—
567C22—81no—
568C22—181no—
569C22#8904611yescharacteristic
570C4—181no—
571C4—361no—
572C4#8904911yescharacteristic
573C4#8904911yescharacteristic
574C3—21no—
575C3—21no—
576C3—21no—
577C3—41no—
578C3#1059140711yescharacteristic, complemented
579C3#1059150611yescharacteristic, complemented
580C3#1059150711yescharacteristic, complemented
581C2—21no—
582C2—81no—
583C2—361no—
584C2#1172372511yescharacteristic
subgroupG/H
centerC2#11723725
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC2×C33#121
prime radicalC3×#64141C2
hypercenterC2×D4#1113
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3754C2semidirect
3. #11723663C2semidirect
4. #11723657C2semidirect
5. C3×#63660C2semidirect
6. #11722081C2semidirect
7. #11722074C2semidirect
26. C2×#3754D3semidirect
40. #63660D3semidirect
354. C33#471semidirect
522. C32#4748semidirect
523. C32#5177semidirect
524. C32#5200semidirect
578. C3#10591407semidirect
579. C3#10591506semidirect
580. C3#10591507semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C3×#641412
#117236632
#117236572
C3×#636602
#117220812
#117220742
#105915073
#105915063
#105914073