Finite Group Databasebeta

← #13147902

Subgroups — #13147902

Order 1728.

1245 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1245 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×#88648C211yesmaximal, characteristic, complemented
3#11721721C211yesmaximal, characteristic, complemented
4#11721385C211yesmaximal, characteristic, complemented
5#64081⋊C3C211yesmaximal, characteristic, complemented
6#11721162C214yesmaximal, complemented
7#11721158C211yesmaximal, characteristic, complemented
8#11721127C214yesmaximal, complemented
9#11721110C211yesmaximal, characteristic, complemented
10#10594010—31nomaximal
11#10594001—31nomaximal
12C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
13C2×C4⋊#272C2211yescharacteristic
14C2×#6229C2211yescharacteristic, complemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
17C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6200C2214yescomplemented
21C2×#6198C2212yes—
22C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
23#88654—24no—
24#88648—24no—
25C42×H(3,𝔽3)C2211yescharacteristic, complemented
26#88542C2214yescomplemented
27#3752⋊C3C2212yes—
28#3750⋊C3C2212yescomplemented
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
30#88450C2214yescomplemented
31C2×#3802—31no—
32C2×D3×D12—31no—
33C2×#3799—31no—
34C2×#3796—31no—
35#64083D311yescharacteristic, complemented
36#64081D311yescharacteristic, complemented
37C3×#875—31no—
38C3×#875—31no—
39C3×#875—61no—
40C3×#874—31no—
41C3×#872—31no—
42#63923—34no—
43#63922—34no—
44#63914—31no—
45#63911—31no—
46#63910—34no—
47#63885—34no—
48#63864—31no—
49#63863—31no—
50C22×#272C2312yes—
51C22×(C3⋊D3)⋊C3C2312yes—
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
58C2×#1084C2312yes—
59C4⋊#272—22no—
60#6229—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
63C4⋊((C3⋊D3)⋊C3)—22no—
64C4⋊((C3⋊D3)⋊C3)—22no—
65C4⋊((C3⋊D3)⋊C3)—44no—
66C4×(C3⋊D3)⋊C3D414yescomplemented
67#6200—24no—
68#6200—44no—
69#6198—24no—
70((C3⋊D3)⋊C3)⋊C4—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⋊D3)C2×D311yescharacteristic, complemented
78C2×C4⋊(C3⋊D3)C2×D314yescomplemented
79C2×C4×C3⋊D3C2×D311yescharacteristic
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—34no—
85C2×C3×D12—61no—
86C2×C3×C4×D3—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—61no—
90C2×#626—32no—
91C2×#626—32no—
92C2×#626—32no—
93C2×#626—34no—
94C2×#625—32no—
95C2×#625—34no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98#3802—64no—
99D3×D12—64no—
100#3799—64no—
101#3796—64no—
102C32×C42—21no—
103C32×C42C2×D311yescharacteristic, complemented
104C32×C42C2×D311yescharacteristic, complemented
105#3752C2×D312yes—
106#3750C2×D312yescomplemented
107C4×C3⋊Dic3C2×D311yescharacteristic, complemented
108C3×(C2×D3)⋊C4—34no—
109C3×(C2×D3)⋊C4—34no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—64no—
112C3×Dic3⋊C4—32no—
113C3×C4×Dic3—31no—
114#3722—34no—
115(C2×D3)⋊Dic3—34no—
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×#272—22no—
118C2×#272—24no—
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3—22no—
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3—24no—
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#1084—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—31no—
152#875—61no—
153#875—91no—
154#874—31no—
155#874—91no—
156#874—91no—
157#872—31no—
158#872—91no—
159C22×C3⋊D3C22×D312yes—
160C22×C3⋊D3C22×D312yes—
161C22×C3⋊D3C22×D311yescharacteristic
162C22×C3×D3—31no—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—62no—
168C2×D32—64no—
169C2×D32—64no—
170C2×D32—124no—
171C2×D32—124no—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3C22×D312yes—
179C2×C3⋊Dic3C22×D311yescharacteristic
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)—22no—
182C4⋊(C3⋊D3)—44no—
183C4×C3⋊D3D1214yescomplemented
184C2×C3×Dic3—31no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—62no—
189C3×D12—62no—
190C3×D12—62no—
191C3×D12—62no—
192C3×D12—62no—
193C3×D12—122no—
194C3×D12—124no—
195C3×C4×D3—34no—
196C3×Dic6—62no—
197C3×Dic6—62no—
198C3×Dic6—122no—
199#626—64no—
200#626—64no—
201#626—64no—
202#626—124no—
203#625—64no—
204#625—124no—
205#624—64no—
206D3×Dic3—64no—
207#535—271noSylow
208C2×H(3,𝔽3)C22×D411yescharacteristic
209C2×H(3,𝔽3)H(5,𝔽2)12yes—
210#272—42no—
211(C3⋊D3)⋊C3—22no—
212(C3⋊D3)⋊C3—42no—
213(C3⋊D3)⋊C3—42no—
214C23×D3—92no—
215C23×D3—92no—
216C23×D3—92no—
217C2×C3×Q8—91no—
218C2×C3×D4—91no—
219C2×C3×D4—91no—
220C2×C3×D4—94no—
221C2×C3×D4—91no—
222C22×C3×C4—91no—
223C2×#67—92no—
224C2×#67—92no—
225C2×#67—92no—
226C2×#67—94no—
227C2×#67—92no—
228C2×#67—94no—
229C22×Dic3—92no—
230#239—184no—
231#237—184no—
232D3×D4—184no—
233D3×D4—184no—
234#235—184no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—34no—
240C2×D12—61no—
241C2×D12—91no—
242C2×D12—91no—
243C2×D12—94no—
244C2×D12—92no—
245C2×D12—92no—
246C2×D12—94no—
247C2×D12—184no—
248C2×C4×D3—31no—
249C2×C4×D3—91no—
250C2×C4×D3—92no—
251C2×C4×D3—92no—
252C2×Dic6—31no—
253C2×Dic6—31no—
254C2×Dic6—61no—
255C3×C4⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—94no—
258C3×C42—31no—
259C3×C42—31no—
260C3×C42—61no—
261C3×C42—91no—
262C3×C42D3211yescharacteristic
263#217—94no—
264(C2×D3)⋊C4—34no—
265(C2×D3)⋊C4—34no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—64no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—94no—
270(C2×D3)⋊C4—94no—
271Dic3⋊C4—32no—
272Dic3⋊C4—92no—
273C4×Dic3—31no—
274C4×Dic3—91no—
275C22×C32—21no—
276C22×C32C23×D311yescharacteristic
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—24no—
282C2×C3⋊D3—24no—
283C2×C3⋊D3C2×D1212yes—
284C2×C3×D3—32no—
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—122no—
295C2×C3×D3—124no—
296D32—128no—
297D32—128no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—42no—
302C32×C4C2×D1212yes—
303C32×C4D3×D412yes—
304C3⋊Dic3—22no—
305C3⋊Dic3C2×D1212yes—
306C3×Dic3—32no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—122no—
311C2×#41—271no—
312C22×D4—271no—
313#127—91no—
314#127—271no—
315#124—91no—
316#124—271no—
317#121—274no—
318#120—274no—
319#117—91no—
320H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
321C23×C3—91no—
322C23×C3—92no—
323C23×C3—92no—
324C23×C3—92no—
325C22×D3—31no—
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—32no—
329C22×D3—32no—
330C22×D3—62no—
331C22×D3—91no—
332C22×D3—92no—
333C22×D3—92no—
334C22×D3—184no—
335C22×D3—184no—
336C22×D3—184no—
337C22×D3—364no—
338C22×D3—364no—
339C22×D3—364no—
340C3×Q8—182no—
341C3×D4—182no—
342C3×D4—182no—
343C3×D4—182no—
344C3×D4—364no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—61no—
350C2×C3×C4—62no—
351C2×C3×C4—94no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4C2×D3212yes—
356C2×C3×C4C2×D3211yescharacteristic
357#67—184no—
358#67—184no—
359#67—184no—
360#67—184no—
361#67—364no—
362#67—364no—
363C2×Dic3—31no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—62no—
368C2×Dic3—91no—
369C2×Dic3—92no—
370C2×Dic3—184no—
371D12—62no—
372D12—62no—
373D12—62no—
374D12—62no—
375D12—122no—
376D12—124no—
377D12—182no—
378D12—182no—
379D12—184no—
380D12—184no—
381D12—364no—
382D12—364no—
383C4×D3—34no—
384C4×D3—94no—
385C4×D3—184no—
386C4×D3—184no—
387Dic6—62no—
388Dic6—62no—
389Dic6—122no—
390C2×C32—21no—
391C2×C32—22no—
392C2×C32C22×D1211yescharacteristic
393C2×C32C2×D3×D411yescharacteristic
394C2×C32#100412yes—
395C2×C32#100912yes—
396C3⋊D3—22no—
397C3⋊D3—42no—
398C3⋊D3—42no—
399C3×D3—62no—
400C3×D3—122no—
401C3×D3—122no—
402C3×D3—122no—
403C3×D3—122no—
404C3×D3—242no—
405C24—272no—
406#41—544no—
407C2×Q8—91no—
408C2×D4—91no—
409C2×D4—91no—
410C2×D4—91no—
411C2×D4—94no—
412C2×D4—272no—
413C2×D4—272no—
414C2×D4—274no—
415C2×D4—544no—
416C22×C4—91no—
417C22×C4—272no—
418C4⋊C4—92no—
419C22⋊C4—92no—
420C22⋊C4—94no—
421C22⋊C4—274no—
422C42—91no—
423C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—61no—
427C22×C3—92no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3—182no—
431C22×C3—184no—
432C22×C3—184no—
433C22×C3—182no—
434C22×C3—184no—
435C22×C3C22×D3211yescharacteristic
436C2×D3—32no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—64no—
441C2×D3—64no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—64no—
445C2×D3—64no—
446C2×D3—92no—
447C2×D3—122no—
448C2×D3—124no—
449C2×D3—182no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—184no—
453C2×D3—184no—
454C2×D3—368no—
455C2×D3—368no—
456C2×D3—368no—
457C3×C4—22no—
458C3×C4—32no—
459C3×C4—32no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—92no—
464C3×C4—122no—
465C3×C4—182no—
466C3×C4—182no—
467C3×C4D3×D1212yes—
468Dic3—32no—
469Dic3—62no—
470Dic3—62no—
471Dic3—62no—
472Dic3—92no—
473Dic3—122no—
474Dic3—182no—
475C32—21no—
476C32#552511yescharacteristic, complemented
477C32#571511yescharacteristic, complemented
478C23—91no—
479C23—92no—
480C23—92no—
481C23—92no—
482C23—544no—
483C23—1084no—
484Q8—182no—
485D4—182no—
486D4—182no—
487D4—182no—
488D4—364no—
489D4—544no—
490D4—544no—
491D4—1084no—
492C2×C4—91no—
493C2×C4—94no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—544no—
497C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
498C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—32no—
502C2×C3—32no—
503C2×C3—61no—
504C2×C3—62no—
505C2×C3—182no—
506C2×C3—362no—
507C2×C3—362no—
508C2×C3—362no—
509C2×C3C2×D3×D1211yescharacteristic
510C2×C3#6430712yes—
511D3—62no—
512D3—122no—
513D3—122no—
514D3—122no—
515D3—122no—
516D3—182no—
517D3—242no—
518D3—362no—
519D3—362no—
520C22—92no—
521C22—182no—
522C22—182no—
523C22—182no—
524C22—182no—
525C22—184no—
526C22—184no—
527C22—184no—
528C22—1088no—
529C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C4—22no—
531C4—92no—
532C4—182no—
533C4—182no—
534C4#8865412yes—
535C3—31no—
536C3—31no—
537C3—61no—
538C3#1059400011yescharacteristic
539C2—182no—
540C2—362no—
541C2—362no—
542C2—362no—
543C2C2×#8865411yescharacteristic
544C2#1172249212yes—
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×#88648C2semidirect
3. #11721721C2semidirect
4. #11721385C2semidirect
5. #64081⋊C3C2semidirect
6. #11721162C2semidirect
7. #11721158C2semidirect
8. #11721127C2semidirect
9. #11721110C2semidirect
14. C2×#6229C22semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. C2×#6200C22semidirect
22. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
25. C42×H(3,𝔽3)C22semidirect
26. #88542C22semidirect
28. #3750⋊C3C22semidirect
29. C4×(C3⋊Dic3)⋊C3C22semidirect
30. #88450C22semidirect
35. #64083D3semidirect
36. #64081D3semidirect
66. C4×(C3⋊D3)⋊C3D4semidirect
77. C2×C4⋊(C3⋊D3)C2×D3semidirect
78. C2×C4⋊(C3⋊D3)C2×D3semidirect
103. C32×C42C2×D3semidirect
104. C32×C42C2×D3semidirect
106. #3750C2×D3semidirect
107. C4×C3⋊Dic3C2×D3semidirect
183. C4×C3⋊D3D12semidirect
320. H(3,𝔽3)#535semidirect
423. C42((C3⋊D3)⋊C3)⋊C2semidirect
476. C32#5525semidirect
477. C32#5715semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#886482
#117217212
#117213852
#64081⋊C32
#117211622
#117211582
#117211272
#117211102
#105940103
#105940013