Finite Group Databasebeta

← #13147903

Subgroups — #13147903

Order 1728.

1277 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1277 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88654C211yesmaximal, characteristic, complemented
2C2×#88652C211yesmaximal, characteristic, complemented
3#11721720C211yesmaximal, characteristic, complemented
4#64084⋊C3C211yesmaximal, characteristic, complemented
5#64081⋊C3C211yesmaximal, characteristic, complemented
6#11721163C214yesmaximal, complemented
7#11721158C211yesmaximal, characteristic, complemented
8#11721157C214yesmaximal, complemented
9#11721114C211yesmaximal, characteristic, complemented
10#10594010—31nomaximal
11#10594000—31nomaximal
12C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4⋊#272C2211yescharacteristic, complemented
15C2×C4⋊#272C2214yescomplemented
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic, complemented
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×#6200C2214yescomplemented
22C2×#6199C2212yescomplemented
23#88654—24no—
24#88652—24no—
25C42×H(3,𝔽3)C2211yescharacteristic, complemented
26#3752⋊C3C2212yes—
27#3752⋊C3C2214yescomplemented
28#3750⋊C3C2212yescomplemented
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
30#88451C2214yescomplemented
31C2×#3802—31no—
32C2×D3×D12—31no—
33C2×#3796—31no—
34C2×#3795—31no—
35#64084D311yescharacteristic, complemented
36#64081D311yescharacteristic, complemented
37C3×#875—31no—
38C3×#874—31no—
39C3×#874—31no—
40C3×#874—61no—
41C3×#872—31no—
42#63922—34no—
43#63922—34no—
44#63914—31no—
45#63911—31no—
46#63886—34no—
47#63885—34no—
48#63863—31no—
49#63860—31no—
50C22×#272C2312yes—
51C22×#272C2312yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C22×(C3⋊D3)⋊C3C2311yescharacteristic
54C2×((C3⋊D3)⋊C3)⋊C2—24no—
55C2×((C3⋊D3)⋊C3)⋊C2—44no—
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C4⋊#272—22no—
59C4⋊#272—22no—
60C4⋊#272—44no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
64C4⋊((C3⋊D3)⋊C3)—22no—
65C4×(C3⋊D3)⋊C3D414yescomplemented
66#634⋊C3—22no—
67#6200—24no—
68#6200—24no—
69#6200—44no—
70#6199—24no—
71#5715—91no—
72#5674—91no—
73#5525—91no—
74C22×D32—32no—
75C22×D32—32no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic, complemented
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—34no—
85C2×C3×D12—34no—
86C2×C3×D12—61no—
87C2×C3×D12—61no—
88C2×C3×D12—64no—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—31no—
91C2×#626—32no—
92C2×#626—32no—
93C2×#626—34no—
94C2×#625—32no—
95C2×#625—32no—
96C2×#625—34no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99#3802—64no—
100D3×D12—64no—
101#3796—64no—
102#3795—64no—
103C32×C42—21no—
104C32×C42C2×D311yescharacteristic, complemented
105C32×C42C2×D311yescharacteristic, complemented
106#3752C2×D312yes—
107#3752C2×D314yescomplemented
108#3750C2×D312yescomplemented
109C4×C3⋊Dic3C2×D311yescharacteristic, complemented
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×C4×Dic3—31no—
114(C2×D3)⋊Dic3—34no—
115(C2×D3)⋊Dic3—34no—
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×#272—22no—
118C2×#272—22no—
119C2×#272—24no—
120C2×#272—24no—
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—24no—
124C2×(C3⋊D3)⋊C3C2×D412yes—
125((C3⋊D3)⋊C3)⋊C2—48no—
126C4×H(3,𝔽3)—22no—
127C4×H(3,𝔽3)C2×D412yes—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3—22no—
130(C3⋊Dic3)⋊C3C2×D412yes—
131C2×#239—91no—
132C2×#237—91no—
133C2×D3×D4—91no—
134C2×D3×D4—91no—
135C2×#235—91no—
136C22×D12—91no—
137C3×#127—91no—
138C3×#124—91no—
139C3×#117—91no—
140#947—91no—
141#940—91no—
142#940—91no—
143#939—94no—
144#937—94no—
145#936—91no—
146#895—94no—
147#884—94no—
148#882—94no—
149#882—94no—
150#875—31no—
151#875—91no—
152#875—91no—
153#874—31no—
154#874—31no—
155#874—61no—
156#874—91no—
157#872—31no—
158#872—91no—
159C22×C3⋊D3C22×D312yes—
160C22×C3⋊D3C22×D311yescharacteristic
161C22×C3×D3—31no—
162C22×C3×D3—32no—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—62no—
168C22×C3×D3—62no—
169C2×D32—64no—
170C2×D32—64no—
171C2×D32—124no—
172C2×D32—124no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D312yes—
181C2×C3⋊Dic3C22×D311yescharacteristic
182C4⋊(C3⋊D3)—22no—
183C4×C3⋊D3D1214yescomplemented
184#634—22no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C3×D12—62no—
189C3×D12—62no—
190C3×D12—62no—
191C3×D12—62no—
192C3×D12—62no—
193C3×D12—122no—
194C3×D12—122no—
195C3×D12—124no—
196C3×D12—124no—
197C3×D12—128no—
198C3×C4×D3—34no—
199C3×Dic6—62no—
200#626—64no—
201#626—64no—
202#626—124no—
203#625—64no—
204#625—64no—
205#625—124no—
206D3×Dic3—64no—
207D3×Dic3—64no—
208#535—271noSylow
209C2×H(3,𝔽3)C22×D411yescharacteristic
210C2×H(3,𝔽3)H(5,𝔽2)12yes—
211#272—42no—
212#272—42no—
213(C3⋊D3)⋊C3—22no—
214(C3⋊D3)⋊C3—42no—
215C23×D3—92no—
216C23×D3—92no—
217C23×D3—92no—
218C2×C3×Q8—91no—
219C2×C3×D4—91no—
220C2×C3×D4—91no—
221C2×C3×D4—91no—
222C2×C3×D4—94no—
223C22×C3×C4—91no—
224C2×#67—92no—
225C2×#67—92no—
226C2×#67—94no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—94no—
230C22×Dic3—92no—
231#239—184no—
232#237—184no—
233D3×D4—184no—
234D3×D4—184no—
235#235—184no—
236C2×D12—31no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—34no—
242C2×D12—34no—
243C2×D12—61no—
244C2×D12—61no—
245C2×D12—64no—
246C2×D12—91no—
247C2×D12—92no—
248C2×D12—92no—
249C2×D12—94no—
250C2×D12—184no—
251C2×C4×D3—31no—
252C2×C4×D3—91no—
253C2×C4×D3—92no—
254C2×C4×D3—92no—
255C2×Dic6—31no—
256C2×Dic6—91no—
257C3×C4⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—94no—
260C3×C42—31no—
261C3×C42—31no—
262C3×C42—61no—
263C3×C42—91no—
264C3×C42D3211yescharacteristic
265#217—94no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—34no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—94no—
270(C2×D3)⋊C4—94no—
271(C2×D3)⋊C4—94no—
272Dic3⋊C4—32no—
273Dic3⋊C4—92no—
274C4×Dic3—31no—
275C4×Dic3—91no—
276C22×C32—21no—
277C22×C32C23×D311yescharacteristic
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—24no—
282C2×C3⋊D3C2×D1212yes—
283C2×C3×D3—32no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—64no—
291C2×C3×D3—64no—
292C2×C3×D3—64no—
293C2×C3×D3—64no—
294C2×C3×D3—64no—
295C2×C3×D3—122no—
296C2×C3×D3—122no—
297C2×C3×D3—124no—
298C2×C3×D3—124no—
299D32—128no—
300D32—128no—
301C32×C4—24no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4C2×D1212yes—
306C32×C4D3×D412yes—
307C3⋊Dic3—22no—
308C3⋊Dic3—22no—
309C3⋊Dic3C2×D1212yes—
310C3×Dic3—32no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C2×#41—271no—
314C22×D4—271no—
315#127—91no—
316#127—271no—
317#124—91no—
318#124—271no—
319#121—274no—
320#120—274no—
321#117—91no—
322H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
323C23×C3—91no—
324C23×C3—92no—
325C23×C3—92no—
326C23×C3—92no—
327C22×D3—31no—
328C22×D3—32no—
329C22×D3—32no—
330C22×D3—32no—
331C22×D3—32no—
332C22×D3—32no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—91no—
336C22×D3—92no—
337C22×D3—184no—
338C22×D3—184no—
339C22×D3—184no—
340C22×D3—364no—
341C22×D3—364no—
342C22×D3—364no—
343C3×Q8—182no—
344C3×D4—182no—
345C3×D4—182no—
346C3×D4—182no—
347C3×D4—364no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—94no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4C2×D3212yes—
359C2×C3×C4C2×D3211yescharacteristic
360#67—184no—
361#67—184no—
362#67—184no—
363#67—184no—
364#67—364no—
365#67—364no—
366C2×Dic3—31no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—91no—
370C2×Dic3—92no—
371C2×Dic3—92no—
372C2×Dic3—184no—
373D12—62no—
374D12—62no—
375D12—62no—
376D12—62no—
377D12—62no—
378D12—122no—
379D12—122no—
380D12—124no—
381D12—124no—
382D12—128no—
383D12—182no—
384D12—184no—
385D12—184no—
386D12—364no—
387C4×D3—34no—
388C4×D3—94no—
389C4×D3—184no—
390C4×D3—184no—
391Dic6—62no—
392Dic6—182no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32C22×D1211yescharacteristic
396C2×C32C2×D3×D411yescharacteristic
397C2×C32#100912yes—
398C2×C32#100912yes—
399C3⋊D3—22no—
400C3⋊D3—42no—
401C3×D3—62no—
402C3×D3—122no—
403C3×D3—122no—
404C3×D3—122no—
405C3×D3—122no—
406C3×D3—122no—
407C3×D3—242no—
408C3×D3—242no—
409C24—272no—
410#41—544no—
411C2×Q8—91no—
412C2×D4—91no—
413C2×D4—91no—
414C2×D4—91no—
415C2×D4—94no—
416C2×D4—272no—
417C2×D4—272no—
418C2×D4—274no—
419C2×D4—544no—
420C22×C4—91no—
421C22×C4—272no—
422C4⋊C4—92no—
423C22⋊C4—92no—
424C22⋊C4—94no—
425C22⋊C4—274no—
426C42—91no—
427C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—61no—
431C22×C3—92no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—184no—
435C22×C3—182no—
436C22×C3—182no—
437C22×C3—184no—
438C22×C3—184no—
439C22×C3C22×D3211yescharacteristic
440C2×D3—32no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—64no—
447C2×D3—64no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—62no—
451C2×D3—64no—
452C2×D3—92no—
453C2×D3—122no—
454C2×D3—122no—
455C2×D3—124no—
456C2×D3—124no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—184no—
460C2×D3—368no—
461C2×D3—368no—
462C2×D3—368no—
463C3×C4—22no—
464C3×C4—32no—
465C3×C4—32no—
466C3×C4—62no—
467C3×C4—62no—
468C3×C4—64no—
469C3×C4—62no—
470C3×C4—92no—
471C3×C4—182no—
472C3×C4—182no—
473C3×C4D3×D1212yes—
474Dic3—32no—
475Dic3—62no—
476Dic3—62no—
477Dic3—92no—
478Dic3—182no—
479Dic3—182no—
480C32—21no—
481C32#552511yescharacteristic, complemented
482C32#567411yescharacteristic, complemented
483C23—91no—
484C23—92no—
485C23—92no—
486C23—92no—
487C23—544no—
488C23—1084no—
489Q8—182no—
490D4—182no—
491D4—182no—
492D4—182no—
493D4—364no—
494D4—544no—
495D4—544no—
496D4—1084no—
497C2×C4—91no—
498C2×C4—94no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—544no—
502C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
503C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—32no—
507C2×C3—32no—
508C2×C3—61no—
509C2×C3—62no—
510C2×C3—182no—
511C2×C3—362no—
512C2×C3—362no—
513C2×C3—362no—
514C2×C3C2×D3×D1211yescharacteristic
515C2×C3#6430812yes—
516D3—62no—
517D3—122no—
518D3—122no—
519D3—122no—
520D3—122no—
521D3—122no—
522D3—182no—
523D3—242no—
524D3—242no—
525D3—362no—
526C22—92no—
527C22—182no—
528C22—182no—
529C22—182no—
530C22—182no—
531C22—184no—
532C22—184no—
533C22—184no—
534C22—1088no—
535C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
536C4—22no—
537C4—92no—
538C4—182no—
539C4—182no—
540C4#8865412yes—
541C3—31no—
542C3—31no—
543C3—61no—
544C3#1059400111yescharacteristic
545C2—182no—
546C2—362no—
547C2—362no—
548C2—362no—
549C2C2×#8865411yescharacteristic
550C2#1172249312yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#535
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88654C2semidirect
2. C2×#88652C2semidirect
3. #11721720C2semidirect
4. #64084⋊C3C2semidirect
5. #64081⋊C3C2semidirect
6. #11721163C2semidirect
7. #11721158C2semidirect
8. #11721157C2semidirect
9. #11721114C2semidirect
14. C2×C4⋊#272C22semidirect
15. C2×C4⋊#272C22semidirect
18. C2×#634⋊C3C22semidirect
21. C2×#6200C22semidirect
22. C2×#6199C22semidirect
25. C42×H(3,𝔽3)C22semidirect
27. #3752⋊C3C22semidirect
28. #3750⋊C3C22semidirect
29. C4×(C3⋊Dic3)⋊C3C22semidirect
30. #88451C22semidirect
35. #64084D3semidirect
36. #64081D3semidirect
65. C4×(C3⋊D3)⋊C3D4semidirect
78. C2×#634C2×D3semidirect
104. C32×C42C2×D3semidirect
105. C32×C42C2×D3semidirect
107. #3752C2×D3semidirect
108. #3750C2×D3semidirect
109. C4×C3⋊Dic3C2×D3semidirect
183. C4×C3⋊D3D12semidirect
322. H(3,𝔽3)#535semidirect
427. C42((C3⋊D3)⋊C3)⋊C2semidirect
481. C32#5525semidirect
482. C32#5674semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#886522
#117217202
#64084⋊C32
#64081⋊C32
#117211632
#117211582
#117211572
#117211142
#105940103
#105940003