Finite Group Databasebeta

← #13150893

Subgroups — #13150893

Order 1728.

1447 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1447 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×C4⋊#272C211yesmaximal, characteristic, complemented
2C2×#88677C211yesmaximal, characteristic, complemented
3#64138⋊C3C212yesmaximal, complemented
4#11721323C212yesmaximal, complemented
5#11721320C211yesmaximal, characteristic, complemented
6#11721163C212yesmaximal, complemented
7#11721157C214yesmaximal, complemented
8#11721151C212yesmaximal, complemented
9#10595062—32nomaximal
10C23×#272C2211yescharacteristic
11C23×#272C2211yescharacteristic, complemented
12C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic, complemented
13C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×C4⋊#272—24no—
15C2×C4⋊#272C2212yes—
16C2×C4⋊#272C2212yescomplemented
17C2×#638⋊C3C2212yes—
18#88677—24no—
19C2×#6200C2212yescomplemented
20C2×#6200C2214yescomplemented
21C2×#6199C2211yescharacteristic, complemented
22C2×#6199C2212yescomplemented
23#3757⋊C3C2212yescomplemented
24#3752⋊C3C2214yescomplemented
25#3750⋊C3C2214yescomplemented
26#88451C2212yescomplemented
27#88451C2214yescomplemented
28C22×C3×D12—32no—
29C22×C3×D12—61no—
30C2×D3×#67—32no—
31#64138D312yescomplemented
32C3×#929—32no—
33#63977—32no—
34#63976—32no—
35D3×#217—32no—
36#63922—34no—
37#63886—34no—
38#63885—34no—
39#63884—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272—22no—
42C22×#272—42no—
43C22×#272—42no—
44C22×#272D414yescomplemented
45C22×#272C2312yes—
46C22×#272C2311yescharacteristic
47C22×#272C2311yescharacteristic
48C22×(C3⋊D3)⋊C3C2312yes—
49C2×((C3⋊D3)⋊C3)⋊C2—22no—
50C2×((C3⋊D3)⋊C3)⋊C2—42no—
51C2×C4×H(3,𝔽3)—22no—
52C2×C4×H(3,𝔽3)C2312yes—
53C4⋊#272—24no—
54C4⋊#272—24no—
55#638⋊C3—24no—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2314yes—
58#6200—24no—
59#6200—44no—
60#6199—22no—
61#6199—42no—
62#5804—91no—
63#5653—92no—
64C23×C3×D3—32no—
65C23×C3×D3—32no—
66C23×C3×D3—61no—
67C23×C3×D3—61no—
68C22×D32—32no—
69C22×C32×C4—21no—
70C22×C32×C4C2×D312yescomplemented
71C2×#638C2×D312yes—
72C2×C3×#67—32no—
73C2×C3×D12—34no—
74C2×C3×D12—34no—
75C2×C3×D12—68no—
76C2×C3×D12—62no—
77C2×C3×D12—62no—
78C2×C3×D12—68no—
79D3×#67—68no—
80C2×#626—32no—
81C2×#626—34no—
82C2×#625—32no—
83C2×#625—34no—
84C2×D3×Dic3—32no—
85C2×D3×Dic3—34no—
86#3757C2×D312yescomplemented
87#3752C2×D314yescomplemented
88#3750C2×D314yescomplemented
89C3×#217—32no—
90C3×(C2×D3)⋊C4—34no—
91C3×Dic3⋊C4—34no—
92(C2×D3)⋊Dic3—34no—
93(C2×D3)⋊Dic3—34no—
94(C2×D3)⋊Dic3—34no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)C2×D412yes—
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×#272—22no—
99C2×#272—22no—
100C2×#272—22no—
101C2×#272—24no—
102C2×#272—44no—
103C2×#272—44no—
104C2×#272C2×D412yes—
105C2×#272C2×D412yes—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3—24no—
108((C3⋊D3)⋊C3)⋊C2—44no—
109C4×H(3,𝔽3)—22no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3—24no—
112C22×C3×D4—91no—
113C22×#67—91no—
114C2×D3×D4—92no—
115C22×D12—32no—
116C22×D12—61no—
117C3×#123—92no—
118#953—92no—
119C2×#217—91no—
120#939—94no—
121#937—92no—
122#937—92no—
123#935—92no—
124#929—32no—
125#929—92no—
126#895—94no—
127#894—94no—
128#884—94no—
129#882—94no—
130#882—92no—
131D3×C22⋊C4—92no—
132C23×C32—21no—
133C23×C32C22×D312yes—
134C22×C3⋊D3C22×D312yes—
135C22×C3×D3—32no—
136C22×C3×D3—32no—
137C22×C3×D3—38no—
138C22×C3×D3—32no—
139C22×C3×D3—34no—
140C22×C3×D3—64no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—62no—
144C22×C3×D3—64no—
145C22×C3×D3—64no—
146C22×C3×D3—124no—
147C22×C3×D3—124no—
148C22×C3×D3—124no—
149C22×C3×D3—124no—
150C2×D32—64no—
151C2×D32—124no—
152C2×C32×C4—24no—
153C2×C32×C4—24no—
154C2×C32×C4—22no—
155C2×C32×C4C22×D314yes—
156#638—24no—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D314yes—
159C3×#67—64no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—34no—
162C3×D12—68no—
163C3×D12—68no—
164C3×D12—68no—
165C3×D12—68no—
166#626—64no—
167#626—124no—
168#625—64no—
169#625—124no—
170D3×Dic3—64no—
171D3×Dic3—124no—
172#534—271noSylow
173C2×H(3,𝔽3)—22no—
174C2×H(3,𝔽3)C22×D411yescharacteristic
175C2×H(3,𝔽3)H(5,𝔽2)12yes—
176#272—22no—
177#272—22no—
178#272—42no—
179(C3⋊D3)⋊C3—42no—
180C24×C3—91no—
181C24×C3—91no—
182C23×D3—32no—
183C23×D3—32no—
184C23×D3—61no—
185C23×D3—61no—
186C23×D3—91no—
187C23×D3—92no—
188C2×C3×D4—92no—
189C2×C3×D4—92no—
190C2×C3×D4—92no—
191C2×C3×D4—184no—
192C22×C3×C4—32no—
193C22×C3×C4—61no—
194C22×C3×C4D3211yescharacteristic
195C2×#67—32no—
196C2×#67—92no—
197C2×#67—92no—
198C2×#67—94no—
199C2×#67—92no—
200C2×#67—94no—
201C2×#67—184no—
202C22×Dic3—91no—
203C22×Dic3—92no—
204D3×D4—188no—
205C2×D12—34no—
206C2×D12—34no—
207C2×D12—62no—
208C2×D12—62no—
209C2×D12—68no—
210C2×D12—68no—
211C2×D12—92no—
212C2×D12—94no—
213C2×C4×D3—92no—
214C2×C4×D3—94no—
215C3×C4⋊C4—94no—
216C3×C22⋊C4—94no—
217C3×C22⋊C4—92no—
218#217—32no—
219#217—92no—
220#217—92no—
221#217—94no—
222(C2×D3)⋊C4—34no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—94no—
227Dic3⋊C4—34no—
228Dic3⋊C4—94no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32D3×D414yes—
234C22×C32C23×D312yes—
235C2×C3⋊D3—22no—
236C2×C3⋊D3—24no—
237C2×C3×D3—34no—
238C2×C3×D3—34no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—64no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244C2×C3×D3—62no—
245C2×C3×D3—64no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—122no—
250C2×C3×D3—122no—
251C2×C3×D3—124no—
252C2×C3×D3—128no—
253C2×C3×D3—128no—
254C2×C3×D3—128no—
255C2×C3×D3—128no—
256D32—128no—
257C32×C4—24no—
258C32×C4—24no—
259C3⋊Dic3—22no—
260C3⋊Dic3—24no—
261C3×Dic3—62no—
262C3×Dic3—64no—
263C22×D4—91no—
264C22×D4—271no—
265#123—92no—
266#123—272no—
267#121—274no—
268#120—272no—
269#120—272no—
270C2×C22⋊C4—271no—
271H(3,𝔽3)#53411yescharacteristic, complemented, Sylow
272C23×C3—32no—
273C23×C3—61no—
274C23×C3—92no—
275C23×C3—91no—
276C23×C3—91no—
277C23×C3—94no—
278C23×C3—92no—
279C23×C3—182no—
280C23×C3—362no—
281C23×C3—362no—
282C23×C3C2×D3211yescharacteristic
283C22×D3—32no—
284C22×D3—32no—
285C22×D3—38no—
286C22×D3—34no—
287C22×D3—32no—
288C22×D3—61no—
289C22×D3—61no—
290C22×D3—64no—
291C22×D3—62no—
292C22×D3—64no—
293C22×D3—64no—
294C22×D3—92no—
295C22×D3—124no—
296C22×D3—124no—
297C22×D3—124no—
298C22×D3—124no—
299C22×D3—184no—
300C22×D3—182no—
301C22×D3—362no—
302C22×D3—364no—
303C3×D4—184no—
304C3×D4—184no—
305C3×D4—184no—
306C2×C3×C4—22no—
307C2×C3×C4—34no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—62no—
311C2×C3×C4—94no—
312C2×C3×C4—92no—
313C2×C3×C4C2×D3212yes—
314#67—64no—
315#67—184no—
316#67—184no—
317#67—184no—
318#67—364no—
319#67—364no—
320C2×Dic3—32no—
321C2×Dic3—34no—
322C2×Dic3—92no—
323C2×Dic3—94no—
324C2×Dic3—182no—
325C2×Dic3—362no—
326D12—68no—
327D12—68no—
328D12—68no—
329D12—68no—
330D12—184no—
331D12—364no—
332C4×D3—184no—
333C4×D3—364no—
334C2×C32—24no—
335C2×C32—24no—
336C2×C32—22no—
337C2×C32—21no—
338C2×C32C2×D3×D412yes—
339C2×C32#100914yes—
340C3⋊D3—42no—
341C3×D3—64no—
342C3×D3—64no—
343C3×D3—122no—
344C3×D3—122no—
345C3×D3—124no—
346C3×D3—122no—
347C3×D3—124no—
348C24—91no—
349C24—91no—
350C24—271no—
351C2×D4—92no—
352C2×D4—92no—
353C2×D4—92no—
354C2×D4—184no—
355C2×D4—272no—
356C2×D4—274no—
357C2×D4—544no—
358C22×C4—271no—
359C22×C4—272no—
360C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
361C4⋊C4—94no—
362C22⋊C4—92no—
363C22⋊C4—94no—
364C22⋊C4—272no—
365C22⋊C4—274no—
366C22×C3—22no—
367C22×C3—32no—
368C22×C3—34no—
369C22×C3—64no—
370C22×C3—64no—
371C22×C3—61no—
372C22×C3—62no—
373C22×C3—92no—
374C22×C3—92no—
375C22×C3—182no—
376C22×C3—184no—
377C22×C3—182no—
378C22×C3—182no—
379C22×C3—184no—
380C22×C3—182no—
381C22×C3—364no—
382C22×C3—364no—
383C22×C3#381112yes—
384C22×C3C22×D3211yescharacteristic
385C2×D3—34no—
386C2×D3—34no—
387C2×D3—62no—
388C2×D3—62no—
389C2×D3—64no—
390C2×D3—64no—
391C2×D3—64no—
392C2×D3—64no—
393C2×D3—64no—
394C2×D3—64no—
395C2×D3—64no—
396C2×D3—62no—
397C2×D3—122no—
398C2×D3—122no—
399C2×D3—124no—
400C2×D3—128no—
401C2×D3—128no—
402C2×D3—128no—
403C2×D3—128no—
404C2×D3—182no—
405C2×D3—184no—
406C2×D3—368no—
407C2×D3—364no—
408C3×C4—22no—
409C3×C4—64no—
410C3×C4—64no—
411C3×C4—182no—
412C3×C4—184no—
413Dic3—64no—
414Dic3—62no—
415Dic3—182no—
416Dic3—184no—
417C32—21no—
418C32#565312yescomplemented
419C23—91no—
420C23—91no—
421C23—94no—
422C23—92no—
423C23—92no—
424C23—182no—
425C23—362no—
426C23—362no—
427C23—542no—
428C23—1082no—
429C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
430D4—184no—
431D4—184no—
432D4—184no—
433D4—544no—
434D4—1084no—
435C2×C4—22no—
436C2×C4—94no—
437C2×C4—92no—
438C2×C4—542no—
439C2×C4—1082no—
440C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
441C2×C3—22no—
442C2×C3—32no—
443C2×C3—34no—
444C2×C3—64no—
445C2×C3—64no—
446C2×C3—61no—
447C2×C3—62no—
448C2×C3—182no—
449C2×C3—182no—
450C2×C3—362no—
451C2×C3—362no—
452C2×C3#6430812yes—
453C2×C3C2×#381111yescharacteristic
454D3—64no—
455D3—64no—
456D3—122no—
457D3—122no—
458D3—124no—
459D3—124no—
460D3—122no—
461D3—362no—
462C22—22no—
463C22—92no—
464C22—92no—
465C22—182no—
466C22—182no—
467C22—182no—
468C22—182no—
469C22—184no—
470C22—184no—
471C22—364no—
472C22—364no—
473C22—1084no—
474C22#8867712yes—
475C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476C4—22no—
477C4—182no—
478C4—184no—
479C3—32no—
480C3—61no—
481C3#1059508011yescharacteristic
482C2—22no—
483C2—182no—
484C2—182no—
485C2—362no—
486C2—362no—
487C2#1172249312yes—
488C2C2×#8867711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#534
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#272C2semidirect
2. C2×#88677C2semidirect
3. #64138⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721320C2semidirect
6. #11721163C2semidirect
7. #11721157C2semidirect
8. #11721151C2semidirect
11. C23×#272C22semidirect
12. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
16. C2×C4⋊#272C22semidirect
19. C2×#6200C22semidirect
20. C2×#6200C22semidirect
21. C2×#6199C22semidirect
22. C2×#6199C22semidirect
23. #3757⋊C3C22semidirect
24. #3752⋊C3C22semidirect
25. #3750⋊C3C22semidirect
26. #88451C22semidirect
27. #88451C22semidirect
31. #64138D3semidirect
44. C22×#272D4semidirect
70. C22×C32×C4C2×D3semidirect
86. #3757C2×D3semidirect
87. #3752C2×D3semidirect
88. #3750C2×D3semidirect
271. H(3,𝔽3)#534semidirect
360. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
418. C32#5653semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×#886772
#64138⋊C32
#117213232
#117213202
#117211632
#117211572
#117211512
#105950623