Finite Group Databasebeta

← #13156242

Subgroups — #13156242

Order 1728.

676 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 90 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#11723661C211yesmaximal, characteristic, complemented
2#11723371C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C3×#63817C211yesmaximal, characteristic, complemented
5#11722027C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63559C211yesmaximal, characteristic, complemented
8#10590997—31nomaximal
9#10590987—31nomaximal
10#10590927—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C32×#208C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88811—21no—
19#88810—21no—
20#88797—21no—
21#88791—21no—
22#88791—21no—
23#88791—21no—
24C3×#3712—21no—
25C3×#3710—21no—
26#64117—31no—
27C2×#3745—31no—
28C3×#907—31no—
29C3×#901—31no—
30#63830—31no—
31#63825—31no—
32#63825—31no—
33#63825—31no—
34#63825—61no—
35C2×#3714—31no—
36C2×#3714—31no—
37C2×#3714—61no—
38#63823—31no—
39#63817D311yescharacteristic, complemented
40#63650—31no—
41C3×#823—31no—
42C3×#823—31no—
43C3×#822—31no—
44C3×#819—31no—
45#63577D311yescharacteristic, complemented
46#63559—31no—
47#63559—31no—
48#63559—61no—
49#63559D311yescharacteristic, complemented
50C22×C3⋊(C3⋊D3)—21no—
51C22×C3×C3⋊D3—21no—
52C2×C33×C4C2311yescharacteristic
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))—21no—
55C3×#638—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D411yescharacteristic
59C3×C4×C3⋊D3—21no—
60C3×#634D411yescharacteristic
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66#5142—91no—
67#4897—91no—
68#4749—91no—
69#3828C2×D311yescharacteristic
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—121no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C32×M16—31no—
85C32×M16—31no—
86C2×C32×C8—31no—
87#3748C2×D311yescharacteristic
88#3745—61no—
89#3745—61no—
90#3745—61no—
91#3744—61no—
92C3×#208—21no—
93C3×#208—21no—
94C3×#208—31no—
95C3×#208—31no—
96C3×#208—61no—
97C3×#208C2×D311yescharacteristic
98C3×#208C2×D311yescharacteristic
99C3×#208C2×D311yescharacteristic
100C3×#208C2×D311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×ΓL(1,𝔽25)—61no—
105C3×ΓL(1,𝔽25)—61no—
106C3×D3×C8—61no—
107C3×D3×C8—61no—
108#3717—61no—
109#3717—61no—
110#3717—62no—
111#3717—61no—
112#3716—61no—
113#3715—61no—
114#3714—61no—
115#3714—61no—
116#3714—62no—
117#3714—62no—
118#3714—62no—
119#3714—61no—
120#3714—61no—
121#3714—61no—
122#3714—61no—
123#3714—61no—
124#3714—61no—
125#3713—61no—
126#3712—21no—
127#3710—21no—
128C22×C33#4111yescharacteristic
129C2×C3⋊(C3⋊D3)—41no—
130C2×C3⋊(C3⋊D3)—81no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—41no—
133C2×C3×C3⋊D3—81no—
134C33×C4C2×D411yescharacteristic
135C33×C4C2×D411yescharacteristic
136C3×C3⋊Dic3—21no—
137C3×#131—91no—
138#949—91no—
139#949—91no—
140#932—91no—
141C2×#213—91no—
142#908—91no—
143#908—91no—
144#908—91no—
145#908—181no—
146#907—31no—
147C2×D24—91no—
148C2×D24—91no—
149C2×D24—181no—
150#901—31no—
151C3×#108—91no—
152C3×#100—91no—
153#823—31no—
154#823—31no—
155#823—91no—
156#823—91no—
157#823—91no—
158#823—181no—
159#822—31no—
160#819—31no—
161C22×C3⋊D3—21no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—121no—
167C22×C3⋊D3—121no—
168C22×C3⋊D3—122no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—121no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179#638—21no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—62no—
185C4⋊(C3⋊D3)—62no—
186C4⋊(C3⋊D3)—61no—
187C4⋊(C3⋊D3)—61no—
188C4⋊(C3⋊D3)—61no—
189C4⋊(C3⋊D3)—61no—
190C4⋊(C3⋊D3)—121no—
191C4⋊(C3⋊D3)—121no—
192C4⋊(C3⋊D3)—121no—
193C4⋊(C3⋊D3)—121no—
194C4⋊(C3⋊D3)—122no—
195C4⋊(C3⋊D3)—122no—
196C4⋊(C3⋊D3)D1211yescharacteristic
197C4×C3⋊D3—21no—
198#634D1211yescharacteristic
199C3×#67—61no—
200C3×#67—61no—
201C3×#67—121no—
202C3×D12—31no—
203C3×D12—31no—
204C3×D12—61no—
205C3×D12—62no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—121no—
211C3×C4×D3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—121no—
214C3×Dic6—31no—
215C3×Dic6—31no—
216C3×Dic6—61no—
217C32×C8—31no—
218C32×C8—31no—
219C32×C8—61no—
220C32×C8—61no—
221C3⋊(C3⋊C8)—21no—
222C3×C3⋊C8—22no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—21no—
229C3×C3⋊C8—41no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—121no—
233C3×C3⋊C8D1211yescharacteristic
234C3×C3⋊C8D1211yescharacteristic
235C3×C3⋊C8#6711yescharacteristic
236C3×C3⋊C8#6711yescharacteristic
237#472—271noSylow
238C2×C33—21no—
239C2×C33#12111yescharacteristic
240C3⋊(C3⋊D3)—81no—
241C3×C3⋊D3—41no—
242C3×C3⋊D3—81no—
243C3×#41—91no—
244C2×C3×D4—91no—
245#235—31no—
246#235—31no—
247#235—61no—
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—61no—
251C2×D12—91no—
252C2×D12—91no—
253C2×D12—91no—
254C2×D12—181no—
255C2×D12—181no—
256C2×D12—181no—
257C2×D12—361no—
258C3×M16—31no—
259C3×M16—31no—
260C3×M16—31no—
261C3×M16—31no—
262C3×M16—61no—
263C3×M16—61no—
264C3×M16—91no—
265C3×M16—181no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—61no—
269C2×C3×C8—181no—
270#215—181no—
271#215—181no—
272#214—181no—
273#213—181no—
274#213—181no—
275#213—181no—
276#213—181no—
277#213—181no—
278#208—31no—
279#208—31no—
280#208—61no—
281#208D3211yescharacteristic
282#208D3211yescharacteristic
283C2×C3⋊C8D3211yescharacteristic
284D24—181no—
285D24—181no—
286D24—181no—
287D24—181no—
288D24—181no—
289D24—181no—
290D24—182no—
291D24—182no—
292D24—182no—
293D24—181no—
294#204—181no—
295#204—181no—
296#204—182no—
297#204—181no—
298ΓL(1,𝔽25)—61no—
299ΓL(1,𝔽25)—61no—
300D3×C8—61no—
301D3×C8—61no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—41no—
306C22×C32#23511yescharacteristic
307C22×C32#23511yescharacteristic
308C22×C32#23911yescharacteristic
309C2×C3⋊D3—21no—
310C2×C3⋊D3—41no—
311C2×C3⋊D3—81no—
312C2×C3⋊D3—122no—
313C2×C3⋊D3—122no—
314C2×C3⋊D3—122no—
315C2×C3⋊D3—124no—
316C2×C3⋊D3—121no—
317C2×C3⋊D3—121no—
318C2×C3⋊D3—121no—
319C2×C3⋊D3—242no—
320C2×C3⋊D3—244no—
321C2×C3⋊D3—242no—
322C2×C3⋊D3—241no—
323C2×C3⋊D3—241no—
324C2×C3⋊D3—241no—
325C2×C3⋊D3—242no—
326C2×C3×D3—61no—
327C2×C3×D3—61no—
328C2×C3×D3—121no—
329C2×C3×D3—122no—
330C2×C3×D3—121no—
331C2×C3×D3—121no—
332C2×C3×D3—242no—
333C2×C3×D3—241no—
334C2×C3×D3—241no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4C2×D1211yescharacteristic
344C32×C4C2×D1211yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4D3×D411yescharacteristic
348C32×C4C2×#6711yescharacteristic
349C3⋊Dic3—21no—
350C3×Dic3—61no—
351C3×Dic3—61no—
352C3×Dic3—121no—
353AGL(1,ℤ/8)—271no—
354AGL(1,ℤ/8)—271no—
355C2×D8—271no—
356#131—91no—
357#108—91no—
358#100—91no—
359#100—271no—
360C33#47211yescharacteristic, complemented, Sylow
361C23×C3—181no—
362C22×D3—61no—
363C22×D3—61no—
364C22×D3—121no—
365C22×D3—182no—
366C22×D3—181no—
367C22×D3—181no—
368C22×D3—181no—
369C22×D3—361no—
370C22×D3—361no—
371C22×D3—362no—
372C3×Q8—91no—
373C3×D4—91no—
374C3×D4—181no—
375C3×D4—181no—
376C3×D4—181no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—181no—
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385#67—61no—
386#67—61no—
387#67—121no—
388D12—31no—
389D12—31no—
390D12—61no—
391D12—62no—
392D12—61no—
393D12—61no—
394D12—61no—
395D12—61no—
396D12—121no—
397D12—182no—
398D12—182no—
399D12—181no—
400D12—181no—
401D12—181no—
402D12—181no—
403D12—181no—
404D12—181no—
405D12—361no—
406D12—361no—
407D12—361no—
408D12—361no—
409D12—362no—
410D12—362no—
411C4×D3—61no—
412C4×D3—61no—
413C4×D3—121no—
414Dic6—31no—
415Dic6—31no—
416Dic6—61no—
417C3×C8—31no—
418C3×C8—31no—
419C3×C8—31no—
420C3×C8—31no—
421C3×C8—61no—
422C3×C8—61no—
423C3×C8—62no—
424C3×C8—61no—
425C3×C8—61no—
426C3×C8—61no—
427C3×C8—61no—
428C3×C8—121no—
429C3×C8—181no—
430C3⋊C8—21no—
431C3⋊C8—21no—
432C3⋊C8—61no—
433C3⋊C8—61no—
434C3⋊C8—121no—
435C3⋊C8#62611yescharacteristic
436C3⋊C8#62611yescharacteristic
437C2×C32—21no—
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32#88411yescharacteristic
449C2×C32#89511yescharacteristic
450C2×C32#93011yescharacteristic
451C3⋊D3—41no—
452C3⋊D3—81no—
453C3⋊D3—242no—
454C3⋊D3—242no—
455C3⋊D3—242no—
456C3⋊D3—244no—
457C3⋊D3—241no—
458C3⋊D3—241no—
459C3⋊D3—241no—
460C3×D3—121no—
461C3×D3—121no—
462C3×D3—241no—
463C3×D3—242no—
464C3×D3—241no—
465C3×D3—241no—
466#41—91no—
467C2×D4—91no—
468C2×D4—271no—
469SD16—541no—
470SD16—541no—
471D8—541no—
472D8—541no—
473D8—541no—
474D8—541no—
475M16—31no—
476M16—31no—
477M16—91no—
478M16—181no—
479C2×C8—31no—
480C2×C8—181no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—41no—
485C22×C3—181no—
486C22×C3—361no—
487C22×C3—721no—
488C22×C3#379811yescharacteristic
489C22×C3#379911yescharacteristic
490C22×C3#379911yescharacteristic
491C2×D3—61no—
492C2×D3—61no—
493C2×D3—121no—
494C2×D3—122no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—242no—
498C2×D3—241no—
499C2×D3—241no—
500C2×D3—362no—
501C2×D3—362no—
502C2×D3—362no—
503C2×D3—364no—
504C2×D3—361no—
505C2×D3—361no—
506C2×D3—361no—
507C2×D3—722no—
508C2×D3—722no—
509C2×D3—724no—
510C2×D3—721no—
511C2×D3—721no—
512C2×D3—721no—
513C2×D3—722no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—181no—
523C3×C4D3×D1211yescharacteristic
524C3×C4D3×D1211yescharacteristic
525C3×C4D3×D1211yescharacteristic
526C3×C4C2×#62611yescharacteristic
527C3×C4D3×#6711yescharacteristic
528C3×C4#381111yescharacteristic
529Dic3—61no—
530Dic3—61no—
531Dic3—121no—
532C32—21no—
533C32—21no—
534C32—21no—
535C32—41no—
536C32#474911yescharacteristic, complemented
537C32#489711yescharacteristic, complemented
538C32#514211yescharacteristic, complemented
539C23—181no—
540C23—541no—
541Q8—91no—
542D4—91no—
543D4—181no—
544D4—181no—
545D4—181no—
546D4—541no—
547D4—541no—
548C2×C4—181no—
549C2×C4D3311yescharacteristic
550C8—31no—
551C8—31no—
552C8—61no—
553C8—61no—
554C8—181no—
555C2×C3—22no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—42no—
565C2×C3—41no—
566C2×C3—361no—
567C2×C3—721no—
568C2×C3#6390611yescharacteristic
569C2×C3#6390911yescharacteristic
570C2×C3#6391011yescharacteristic
571D3—121no—
572D3—121no—
573D3—241no—
574D3—242no—
575D3—241no—
576D3—241no—
577D3—722no—
578D3—722no—
579D3—722no—
580D3—724no—
581D3—721no—
582D3—721no—
583D3—721no—
584C22—181no—
585C22—361no—
586C22—721no—
587C22—1081no—
588C22—2161no—
589C22#8896511yescharacteristic
590C4—181no—
591C4D3×#62611yescharacteristic
592C4#8895511yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059092711yescharacteristic, complemented
598C3#1059098711yescharacteristic, complemented
599C3#1059099711yescharacteristic, complemented
600C2—21no—
601C2—361no—
602C2—721no—
603C2—2161no—
604C2#1172296111yescharacteristic
subgroupG/H
centerC2#11722961
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723661C2semidirect
2. #11723371C2semidirect
3. C2×#88791C2semidirect
4. C3×#63817C2semidirect
5. #11722027C2semidirect
6. C3×#63577C2semidirect
7. C3×#63559C2semidirect
39. #63817D3semidirect
45. #63577D3semidirect
49. #63559D3semidirect
360. C33#472semidirect
536. C32#4749semidirect
537. C32#4897semidirect
538. C32#5142semidirect
597. C3#10590927semidirect
598. C3#10590987semidirect
599. C3#10590997semidirect

Maximal subgroups

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

subgroupindex
#117236612
#117233712
C2×#887912
C3×#638172
#117220272
C3×#635772
C3×#635592
#105909973
#105909873
#105909273