Finite Group Databasebeta

← #13141077

Subgroups — #13141077

Order 1728.

656 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 73 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#11722638C211yesmaximal, characteristic, complemented
2#11721870C211yesmaximal, characteristic, complemented
3#11721158C211yesmaximal, characteristic, complemented
4#11721068C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#11720322C211yesmaximal, characteristic, complemented
7#11720320C211yesmaximal, characteristic, complemented
8#10591667—31nomaximal
9#10591665—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic, complemented
12#88722—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—21no—
15#3828⋊C3—21no—
16#3828⋊C3C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)—21no—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×#6200—22no—
20#88549—21no—
21#88545—21no—
22#88538C2211yescharacteristic, complemented
23C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
24#3748⋊C3C2211yescharacteristic, complemented
25#88437—21no—
26#88436—21no—
27#64366D311yescharacteristic, complemented
28C3×#1009—31no—
29C3×#949—31no—
30C3×#949—31no—
31C3×#949—61no—
32#63914—31no—
33#63911—31no—
34#63830—31no—
35#63823—31no—
36#63664D311yescharacteristic, complemented
37C3×#837—31no—
38#63570—31no—
39#63568—31no—
40#63559—31no—
41#63558—31no—
42C22×#272—21no—
43C22×(C3⋊D3)⋊C3—22no—
44C22×(C3⋊D3)⋊C3D412yes—
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
50C4⋊#272—21no—
51C4⋊#272—21no—
52#638⋊C3—22no—
53#638⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55C4⋊((C3⋊D3)⋊C3)—22no—
56C4⋊((C3⋊D3)⋊C3)—21no—
57C4⋊((C3⋊D3)⋊C3)—21no—
58C4⋊((C3⋊D3)⋊C3)D411yescharacteristic, complemented
59C4×(C3⋊D3)⋊C3—22no—
60C4×(C3⋊D3)⋊C3—21no—
61#634⋊C3D411yescharacteristic, complemented
62#6200—44no—
63#6180—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
71#3832—21no—
72D4×C3⋊D3—22no—
73D4×C3⋊D3—21no—
74#3828—21no—
75#3828C2×D311yescharacteristic
76C2×C4⋊(C3⋊D3)—21no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C3×#239—61no—
79C3×D3×D4—62no—
80C3×D3×D4—61no—
81C3×#235—31no—
82C3×#235—61no—
83C2×C3×D12—31no—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C2×C3×D12—61no—
88C2×#626—62no—
89C2×#625—62no—
90C4×C3⋊Dic3C2×D311yescharacteristic, complemented
91#3748C2×D311yescharacteristic, complemented
92C3×#215—62no—
93C3×#215—61no—
94C3×#215—61no—
95C3×#213—62no—
96C3×#213—61no—
97C3×#213—61no—
98C3×C4×Dic3—31no—
99C3×#208—31no—
100C3×#208—31no—
101C3×#208—31no—
102C3×#208—61no—
103#3716—61no—
104#3715—61no—
105#3714—61no—
106#3713—61no—
107C22×H(3,𝔽3)—21no—
108C22×H(3,𝔽3)C2×D411yescharacteristic
109C2×#272—42no—
110C2×#272—41no—
111C2×(C3⋊D3)⋊C3—22no—
112C2×(C3⋊D3)⋊C3—21no—
113C2×(C3⋊D3)⋊C3—42no—
114C2×(C3⋊D3)⋊C3—82no—
115C4×H(3,𝔽3)—21no—
116C4×H(3,𝔽3)C2×D411yescharacteristic
117C4×H(3,𝔽3)C2×D411yescharacteristic
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—21no—
120C3×H(5,𝔽2)—91no—
121#1009—31no—
122#1009—91no—
123C3×AGL(1,ℤ/8)—91no—
124#949—31no—
125#949—31no—
126#949—61no—
127#949—91no—
128#940—91no—
129#940—91no—
130#932—91no—
131#908—91no—
132#874—91no—
133C3×#104—91no—
134#837—31no—
135#837—91no—
136#837—91no—
137#833—91no—
138#825—91no—
139#823—91no—
140#823—91no—
141#805—91no—
142C22×C3⋊D3—22no—
143C22×C3⋊D3#6712yes—
144C22×C3×D3—32no—
145C22×C3×D3—62no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—121no—
149C32×Q8—21no—
150C32×Q8D1211yescharacteristic, complemented
151C32×Q8#6711yescharacteristic, complemented
152C32×D4—21no—
153C32×D4—21no—
154C32×D4—21no—
155C32×D4—41no—
156C32×D4D1211yescharacteristic, complemented
157C32×D4#6711yescharacteristic, complemented
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164#638—22no—
165#638—21no—
166C2×C3⋊Dic3—21no—
167C4⋊(C3⋊D3)—22no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)#6711yescharacteristic, complemented
171C4×C3⋊D3—22no—
172C4×C3⋊D3—21no—
173#634#6711yescharacteristic, complemented
174C3×#67—62no—
175C3×#67—61no—
176C2×C3×Dic3—61no—
177C3×D12—31no—
178C3×D12—62no—
179C3×D12—62no—
180C3×D12—61no—
181C3×D12—61no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—61no—
186C3×D12—121no—
187C3×C4×D3—62no—
188C3×C4×D3—61no—
189C3×Dic6—31no—
190#626—124no—
191#625—124no—
192C3⋊(C3⋊C8)—21no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—61no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197#453—271noSylow
198C2×H(3,𝔽3)—21no—
199C2×H(3,𝔽3)—41no—
200C2×H(3,𝔽3)#12011yescharacteristic
201#272—81no—
202(C3⋊D3)⋊C3—42no—
203(C3⋊D3)⋊C3—41no—
204C3×#41—31no—
205C3×#41—31no—
206C3×#41—61no—
207C3×#41—91no—
208C3×#41—181no—
209C3×#41—181no—
210C3×#41D3211yescharacteristic
211C2×C3×D4—91no—
212C2×C3×D4—91no—
213C2×C3×D4—181no—
214C2×C3×D4—181no—
215C2×C3×D4—182no—
216C2×#67—182no—
217C2×#67—182no—
218#239—61no—
219#239—181no—
220D3×D4—62no—
221D3×D4—61no—
222D3×D4—181no—
223D3×D4—182no—
224#235—31no—
225#235—61no—
226#235—91no—
227#235—181no—
228C2×D12—31no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×D12—61no—
233C2×D12—91no—
234C2×D12—181no—
235C2×D12—182no—
236C3×SD16—181no—
237C3×D8—181no—
238C3×M16—91no—
239C3×M16—91no—
240C3×C42—91no—
241#215—61no—
242#215—61no—
243#215—62no—
244#215—181no—
245#214—181no—
246#213—61no—
247#213—61no—
248#213—62no—
249#213—181no—
250#213—181no—
251C4×Dic3—31no—
252C4×Dic3—91no—
253#208—31no—
254#208—31no—
255#208—31no—
256#208—61no—
257#208—91no—
258D24—181no—
259#204—181no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—41no—
264C22×C32D3×D411yescharacteristic
265C22×C32C2×#6711yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—21no—
268C2×C3⋊D3—42no—
269C2×C3⋊D3—82no—
270C2×C3×D3—61no—
271C2×C3×D3—62no—
272C2×C3×D3—122no—
273C2×C3×D3—122no—
274C2×C3×D3—122no—
275C2×C3×D3—122no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—242no—
279C2×C3×D3—242no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—41no—
285C32×C4C2×D1211yescharacteristic
286C32×C4D3×D411yescharacteristic
287C32×C4C2×#6711yescharacteristic
288C32×C4C2×#6711yescharacteristic
289C3⋊Dic3—22no—
290C3⋊Dic3—21no—
291C3×Dic3—62no—
292C3×Dic3—61no—
293H(5,𝔽2)—91no—
294AGL(1,ℤ/8)—91no—
295AGL(1,ℤ/8)—271no—
296#127—271no—
297#104—91no—
298#104—271no—
299#100—271no—
300H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
301C23×C3—92no—
302C23×C3—182no—
303C23×C3—181no—
304C22×D3—32no—
305C22×D3—62no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—92no—
309C22×D3—121no—
310C22×D3—182no—
311C3×Q8—31no—
312C3×Q8—31no—
313C3×Q8—61no—
314C3×Q8—91no—
315C3×Q8#62611yescharacteristic
316C3×D4—21no—
317C3×D4—31no—
318C3×D4—31no—
319C3×D4—61no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—91no—
323C3×D4—121no—
324C3×D4—181no—
325C3×D4—181no—
326C3×D4—182no—
327C3×D4—181no—
328C3×D4—182no—
329C3×D4—181no—
330C3×D4—181no—
331C3×D4#62611yescharacteristic
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—121no—
339C2×C3×C4—181no—
340C2×C3×C4—182no—
341C2×C3×C4—181no—
342C2×C3×C4C2×D3211yescharacteristic
343#67—62no—
344#67—61no—
345#67—181no—
346#67—182no—
347#67—364no—
348#67—364no—
349C2×Dic3—61no—
350C2×Dic3—181no—
351D12—31no—
352D12—62no—
353D12—61no—
354D12—61no—
355D12—61no—
356D12—61no—
357D12—62no—
358D12—61no—
359D12—61no—
360D12—91no—
361D12—121no—
362D12—181no—
363D12—181no—
364D12—182no—
365D12—364no—
366C4×D3—62no—
367C4×D3—61no—
368C4×D3—181no—
369C4×D3—182no—
370Dic6—31no—
371Dic6—91no—
372C3×C8—181no—
373C3×C8—181no—
374C3⋊C8—61no—
375C3⋊C8—61no—
376C3⋊C8—62no—
377C3⋊C8—61no—
378C3⋊C8—181no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—81no—
386C2×C32#88211yescharacteristic
387C2×C32#95311yescharacteristic
388C3⋊D3—42no—
389C3⋊D3—41no—
390C3×D3—122no—
391C3×D3—121no—
392C3×D3—242no—
393C3×D3—241no—
394C3×D3—241no—
395#41—91no—
396#41—181no—
397#41—181no—
398#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
399C2×D4—91no—
400C2×D4—91no—
401C2×D4—181no—
402C2×D4—182no—
403C2×D4—181no—
404C2×D4—542no—
405SD16—181no—
406SD16—541no—
407D8—181no—
408D8—541no—
409M16—91no—
410M16—91no—
411C42—91no—
412C22×C3—21no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—121no—
419C22×C3—181no—
420C22×C3—182no—
421C22×C3—362no—
422C22×C3—362no—
423C22×C3—361no—
424C22×C3—722no—
425C22×C3D3×#6711yescharacteristic
426C2×D3—61no—
427C2×D3—62no—
428C2×D3—122no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—122no—
432C2×D3—122no—
433C2×D3—122no—
434C2×D3—181no—
435C2×D3—182no—
436C2×D3—242no—
437C2×D3—242no—
438C2×D3—362no—
439C2×D3—722no—
440C3×C4—21no—
441C3×C4—31no—
442C3×C4—31no—
443C3×C4—31no—
444C3×C4—31no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—121no—
450C3×C4—181no—
451C3×C4—182no—
452C3×C4C2×#62611yescharacteristic
453C3×C4D3×#6711yescharacteristic
454Dic3—62no—
455Dic3—61no—
456Dic3—181no—
457Dic3—182no—
458C32—21no—
459C32#482211yescharacteristic, complemented
460C32#524111yescharacteristic, complemented
461C23—92no—
462C23—182no—
463C23—181no—
464Q8—91no—
465Q8#620011yescharacteristic, complemented
466D4—21no—
467D4—91no—
468D4—182no—
469D4—182no—
470D4—181no—
471D4—181no—
472D4—181no—
473D4—181no—
474D4—181no—
475D4—1084no—
476D4#620011yescharacteristic, complemented
477C2×C4—21no—
478C2×C4—182no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
482C8—181no—
483C8—181no—
484C2×C3—21no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—41no—
488C2×C3—61no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—121no—
492C2×C3—121no—
493C2×C3—121no—
494C2×C3—241no—
495C2×C3—362no—
496C2×C3—361no—
497C2×C3—721no—
498C2×C3#6397711yescharacteristic
499D3—122no—
500D3—121no—
501D3—241no—
502D3—241no—
503D3—242no—
504D3—362no—
505D3—361no—
506C22—21no—
507C22—181no—
508C22—182no—
509C22—362no—
510C22—362no—
511C22—361no—
512C22—722no—
513C22#8867611yescharacteristic
514C4—21no—
515C4—182no—
516C4—181no—
517C4C2×#620011yescharacteristic
518C4#8867611yescharacteristic
519C3—31no—
520C3—31no—
521C3—61no—
522C3#1059166611yescharacteristic
523C2—21no—
524C2—41no—
525C2—362no—
526C2—361no—
527C2—721no—
528C2#1172132111yescharacteristic
subgroupG/H
centerC2#11721321
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63977
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. #11722638C2semidirect
2. #11721870C2semidirect
3. #11721158C2semidirect
4. #11721068C2semidirect
5. #63664⋊C3C2semidirect
6. #11720322C2semidirect
7. #11720320C2semidirect
11. C2×C4⋊#272C22semidirect
22. #88538C22semidirect
23. C4×(C3⋊Dic3)⋊C3C22semidirect
24. #3748⋊C3C22semidirect
27. #64366D3semidirect
36. #63664D3semidirect
45. Q8×H(3,𝔽3)D4semidirect
47. D4×H(3,𝔽3)D4semidirect
58. C4⋊((C3⋊D3)⋊C3)D4semidirect
61. #634⋊C3D4semidirect
90. C4×C3⋊Dic3C2×D3semidirect
91. #3748C2×D3semidirect
150. C32×Q8D12semidirect
151. C32×Q8#67semidirect
156. C32×D4D12semidirect
157. C32×D4#67semidirect
170. C4⋊(C3⋊D3)#67semidirect
173. #634#67semidirect
300. H(3,𝔽3)#453semidirect
459. C32#4822semidirect
460. C32#5241semidirect
465. Q8#6200semidirect
476. D4#6200semidirect

Maximal subgroups

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

subgroupindex
#117226382
#117218702
#117211582
#117210682
#63664⋊C32
#117203222
#117203202
#105916673
#105916653