Finite Group Databasebeta

← #13141078

Subgroups — #13141078

Order 1728.

754 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 69 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#11722675C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic, complemented
3#11721158C211yesmaximal, characteristic, complemented
4#11721071C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#11720328C211yesmaximal, characteristic, complemented
7#11720325C211yesmaximal, characteristic, complemented
8#10591666—31nomaximal
9#10591665—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88748—21no—
12D4×#272—22no—
13D4×#272—21no—
14#88740—21no—
15#88740C2211yescharacteristic
16C2×C4⋊#272—21no—
17C2×C4⋊#272C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
19C2×#6200—22no—
20#3755⋊C3—21no—
21#3753⋊C3—21no—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
23#3748⋊C3C2211yescharacteristic, complemented
24#3748⋊C3C2211yescharacteristic, complemented
25#88438—21no—
26#88436—21no—
27C3×#1009—31no—
28C3×#1009—31no—
29C3×#1009—61no—
30#64158D311yescharacteristic, complemented
31C3×#949—31no—
32#63914—31no—
33#63911—31no—
34#63825—31no—
35#63823—31no—
36#63664D311yescharacteristic, complemented
37C3×#837—31no—
38#63569—31no—
39#63568—31no—
40#63558—31no—
41#63558—31no—
42C22×#272—22no—
43C22×#272D412yes—
44C22×(C3⋊D3)⋊C3—21no—
45Q8×H(3,𝔽3)D411yescharacteristic, complemented
46D4×H(3,𝔽3)—21no—
47D4×H(3,𝔽3)D411yescharacteristic, complemented
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50#6235—22no—
51#6235—21no—
52C4⋊#272—22no—
53C4⋊#272—21no—
54C4⋊#272—21no—
55C4⋊#272D411yescharacteristic, complemented
56C4×#272—22no—
57C4×#272—21no—
58#6229D411yescharacteristic, complemented
59C2×(C3⋊Dic3)⋊C3—21no—
60C4⋊((C3⋊D3)⋊C3)—21no—
61C4⋊((C3⋊D3)⋊C3)—21no—
62#6200—44no—
63(C3⋊(C3⋊C8))⋊C3—21no—
64(C3⋊(C3⋊C8))⋊C3—21no—
65#5241—91no—
66#5198—91no—
67#4822—91no—
68C32×#41—21no—
69C32×#41C2×D311yescharacteristic
70C32×#41C2×D311yescharacteristic
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
72C3×#239—62no—
73C3×#239—61no—
74C3×#239—61no—
75C3×D3×D4—62no—
76C3×D3×D4—62no—
77C3×D3×D4—64no—
78C3×D3×D4—62no—
79C3×D3×D4—61no—
80C3×D3×D4—61no—
81C3×#235—31no—
82C3×#235—31no—
83C3×#235—61no—
84C3×#235—61no—
85C3×#235—61no—
86C3×#235—62no—
87C2×C3×D12—31no—
88C2×C3×D12—31no—
89C2×C3×D12—31no—
90C2×C3×D12—61no—
91C2×C3×D12—61no—
92C2×C3×D12—61no—
93C2×C3×D12—62no—
94C2×#626—62no—
95C2×#625—62no—
96#3755—21no—
97#3753—21no—
98C4×C3⋊Dic3C2×D311yescharacteristic, complemented
99#3748C2×D311yescharacteristic, complemented
100#3748C2×D311yescharacteristic, complemented
101C3×#215—61no—
102C3×#213—61no—
103C3×C4×Dic3—31no—
104C3×#208—31no—
105C3×#208—31no—
106#3717—61no—
107#3715—61no—
108#3714—61no—
109#3713—61no—
110C22×H(3,𝔽3)—21no—
111C22×H(3,𝔽3)C2×D411yescharacteristic
112C2×#272—22no—
113C2×#272—21no—
114C2×#272—42no—
115C2×#272—82no—
116C2×(C3⋊D3)⋊C3—42no—
117C2×(C3⋊D3)⋊C3—41no—
118C4×H(3,𝔽3)—21no—
119C4×H(3,𝔽3)C2×D411yescharacteristic
120C4×H(3,𝔽3)C2×D411yescharacteristic
121#1084—21no—
122(C3⋊Dic3)⋊C3—22no—
123C3×H(5,𝔽2)—91no—
124#1009—31no—
125#1009—31no—
126#1009—61no—
127C3×AGL(1,ℤ/8)—91no—
128#949—31no—
129#949—91no—
130#949—91no—
131#940—91no—
132#940—91no—
133#932—91no—
134#908—91no—
135#874—91no—
136C3×#104—91no—
137#837—31no—
138#837—91no—
139#837—91no—
140#833—91no—
141#825—91no—
142#823—91no—
143#823—91no—
144#805—91no—
145C22×C3⋊D3—21no—
146C22×C3×D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3—62no—
149C22×C3×D3—64no—
150C22×C3×D3—62no—
151C22×C3×D3—62no—
152C22×C3×D3—61no—
153C32×Q8—21no—
154C32×Q8D1211yescharacteristic, complemented
155C32×Q8#6711yescharacteristic, complemented
156C32×D4—22no—
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—21no—
160C32×D4D1211yescharacteristic, complemented
161C32×D4#6711yescharacteristic, complemented
162C2×C32×C4—22no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)—21no—
171C3×#67—64no—
172C3×#67—62no—
173C3×#67—62no—
174C3×#67—62no—
175C3×#67—61no—
176C3×#67—61no—
177C2×C3×Dic3—61no—
178C3×D12—31no—
179C3×D12—31no—
180C3×D12—61no—
181C3×D12—62no—
182C3×D12—64no—
183C3×D12—62no—
184C3×D12—62no—
185C3×D12—62no—
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—61no—
192C3×C4×D3—64no—
193C3×C4×D3—62no—
194C3×C4×D3—62no—
195C3×C4×D3—62no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×Dic6—31no—
199C3×Dic6—31no—
200C3×Dic6—61no—
201#626—124no—
202#625—124no—
203C3⋊(C3⋊C8)—21no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207#453—271noSylow
208C2×H(3,𝔽3)—21no—
209C2×H(3,𝔽3)—41no—
210C2×H(3,𝔽3)#12011yescharacteristic
211#272—42no—
212#272—41no—
213(C3⋊D3)⋊C3—81no—
214C3×#41—31no—
215C3×#41—31no—
216C3×#41—61no—
217C3×#41—91no—
218C3×#41—181no—
219C3×#41—181no—
220C3×#41D3211yescharacteristic
221C2×C3×D4—91no—
222C2×C3×D4—91no—
223C2×C3×D4—181no—
224C2×C3×D4—182no—
225C2×C3×D4—181no—
226C2×#67—182no—
227C2×#67—182no—
228#239—61no—
229#239—61no—
230#239—62no—
231D3×D4—61no—
232D3×D4—61no—
233D3×D4—62no—
234D3×D4—62no—
235D3×D4—64no—
236D3×D4—62no—
237#235—31no—
238#235—31no—
239#235—61no—
240#235—61no—
241#235—61no—
242#235—62no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×D12—61no—
248C2×D12—61no—
249C2×D12—62no—
250C2×D12—91no—
251C2×D12—182no—
252C3×SD16—181no—
253C3×D8—181no—
254C3×M16—91no—
255C3×M16—91no—
256C3×C42—91no—
257#215—61no—
258#215—181no—
259#215—181no—
260#214—181no—
261#213—61no—
262#213—181no—
263#213—181no—
264#213—181no—
265C4×Dic3—31no—
266C4×Dic3—91no—
267#208—31no—
268#208—31no—
269#208—91no—
270#208—91no—
271D24—181no—
272#204—181no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32D3×D411yescharacteristic
278C22×C32D3×D411yescharacteristic
279C2×C3⋊D3—42no—
280C2×C3⋊D3—41no—
281C2×C3×D3—62no—
282C2×C3×D3—64no—
283C2×C3×D3—61no—
284C2×C3×D3—62no—
285C2×C3×D3—61no—
286C2×C3×D3—62no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—122no—
290C2×C3×D3—121no—
291C2×C3×D3—242no—
292C2×C3×D3—244no—
293C2×C3×D3—242no—
294C2×C3×D3—242no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4C2×D1211yescharacteristic
301C32×C4D3×D411yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4C2×#6711yescharacteristic
304C3⋊Dic3—22no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—61no—
308C3×Dic3—61no—
309H(5,𝔽2)—91no—
310AGL(1,ℤ/8)—91no—
311AGL(1,ℤ/8)—271no—
312#127—271no—
313#104—91no—
314#104—271no—
315#100—271no—
316H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
317C23×C3—92no—
318C23×C3—182no—
319C23×C3—181no—
320C22×D3—32no—
321C22×D3—32no—
322C22×D3—62no—
323C22×D3—62no—
324C22×D3—62no—
325C22×D3—64no—
326C22×D3—61no—
327C22×D3—181no—
328C3×Q8—31no—
329C3×Q8—31no—
330C3×Q8—61no—
331C3×Q8—91no—
332C3×Q8#62611yescharacteristic
333C3×D4—21no—
334C3×D4—31no—
335C3×D4—31no—
336C3×D4—61no—
337C3×D4—61no—
338C3×D4—62no—
339C3×D4—61no—
340C3×D4—91no—
341C3×D4—182no—
342C3×D4—181no—
343C3×D4—181no—
344C3×D4—182no—
345C3×D4—181no—
346C3×D4—181no—
347C3×D4—181no—
348C3×D4#62611yescharacteristic
349C2×C3×C4—21no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—61no—
356C2×C3×C4—182no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4C2×D3211yescharacteristic
360#67—61no—
361#67—61no—
362#67—62no—
363#67—62no—
364#67—64no—
365#67—62no—
366#67—364no—
367#67—364no—
368C2×Dic3—61no—
369C2×Dic3—181no—
370D12—31no—
371D12—31no—
372D12—61no—
373D12—61no—
374D12—61no—
375D12—61no—
376D12—61no—
377D12—62no—
378D12—62no—
379D12—62no—
380D12—64no—
381D12—62no—
382D12—61no—
383D12—61no—
384D12—181no—
385D12—181no—
386D12—364no—
387C4×D3—61no—
388C4×D3—61no—
389C4×D3—62no—
390C4×D3—62no—
391C4×D3—64no—
392C4×D3—62no—
393Dic6—31no—
394Dic6—31no—
395Dic6—61no—
396C3×C8—181no—
397C3×C8—181no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—181no—
401C3⋊C8—181no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32#88211yescharacteristic
410C2×C32#93711yescharacteristic
411C3⋊D3—81no—
412C3×D3—124no—
413C3×D3—122no—
414C3×D3—122no—
415C3×D3—121no—
416C3×D3—122no—
417C3×D3—121no—
418C3×D3—241no—
419#41—91no—
420#41—181no—
421#41—181no—
422#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C2×D4—91no—
424C2×D4—91no—
425C2×D4—181no—
426C2×D4—182no—
427C2×D4—181no—
428C2×D4—542no—
429SD16—181no—
430SD16—541no—
431D8—181no—
432D8—541no—
433M16—91no—
434M16—91no—
435C42—91no—
436C22×C3—21no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3—62no—
442C22×C3—61no—
443C22×C3—181no—
444C22×C3—182no—
445C22×C3—362no—
446C22×C3—361no—
447C22×C3—362no—
448C22×C3—722no—
449C22×C3#381111yescharacteristic
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—62no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—64no—
456C2×D3—122no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—121no—
460C2×D3—242no—
461C2×D3—242no—
462C2×D3—242no—
463C2×D3—244no—
464C2×D3—361no—
465C2×D3—362no—
466C3×C4—21no—
467C3×C4—31no—
468C3×C4—31no—
469C3×C4—31no—
470C3×C4—31no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—62no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—182no—
477C3×C4—181no—
478C3×C4C2×#62611yescharacteristic
479C3×C4#381111yescharacteristic
480Dic3—62no—
481Dic3—61no—
482Dic3—61no—
483Dic3—62no—
484Dic3—182no—
485C32—21no—
486C32#482211yescharacteristic, complemented
487C32#519811yescharacteristic, complemented
488C23—92no—
489C23—182no—
490C23—181no—
491Q8—91no—
492Q8#620011yescharacteristic, complemented
493D4—21no—
494D4—91no—
495D4—182no—
496D4—182no—
497D4—181no—
498D4—181no—
499D4—181no—
500D4—181no—
501D4—181no—
502D4—1084no—
503D4#620011yescharacteristic, complemented
504C2×C4—21no—
505C2×C4—182no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C8—181no—
510C8—181no—
511C2×C3—21no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—41no—
515C2×C3—61no—
516C2×C3—61no—
517C2×C3—61no—
518C2×C3—121no—
519C2×C3—121no—
520C2×C3—121no—
521C2×C3—122no—
522C2×C3—362no—
523C2×C3—361no—
524C2×C3—721no—
525C2×C3#6398111yescharacteristic
526D3—122no—
527D3—122no—
528D3—121no—
529D3—121no—
530D3—124no—
531D3—122no—
532D3—241no—
533D3—721no—
534C22—21no—
535C22—181no—
536C22—182no—
537C22—362no—
538C22—362no—
539C22—361no—
540C22—722no—
541C22#8867711yescharacteristic
542C4—21no—
543C4—182no—
544C4—181no—
545C4C2×#620011yescharacteristic
546C4#8867711yescharacteristic
547C3—31no—
548C3—31no—
549C3—61no—
550C3#1059166711yescharacteristic
551C2—21no—
552C2—41no—
553C2—362no—
554C2—361no—
555C2—721no—
556C2#1172132311yescharacteristic
subgroupG/H
centerC2#11721323
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63981
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#453
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11722675C2semidirect
2. #64158⋊C3C2semidirect
3. #11721158C2semidirect
4. #11721071C2semidirect
5. #63664⋊C3C2semidirect
6. #11720328C2semidirect
7. #11720325C2semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
22. C4×(C3⋊Dic3)⋊C3C22semidirect
23. #3748⋊C3C22semidirect
24. #3748⋊C3C22semidirect
30. #64158D3semidirect
36. #63664D3semidirect
45. Q8×H(3,𝔽3)D4semidirect
47. D4×H(3,𝔽3)D4semidirect
55. C4⋊#272D4semidirect
58. #6229D4semidirect
71. C2×C4⋊(C3⋊D3)C2×D3semidirect
98. C4×C3⋊Dic3C2×D3semidirect
99. #3748C2×D3semidirect
100. #3748C2×D3semidirect
154. C32×Q8D12semidirect
155. C32×Q8#67semidirect
160. C32×D4D12semidirect
161. C32×D4#67semidirect
316. H(3,𝔽3)#453semidirect
486. C32#4822semidirect
487. C32#5198semidirect
492. Q8#6200semidirect
503. D4#6200semidirect

Maximal subgroups

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

subgroupindex
#117226752
#64158⋊C32
#117211582
#117210712
#63664⋊C32
#117203282
#117203252
#105916663
#105916653