Finite Group Databasebeta

← #13150895

Subgroups — #13150895

Order 1728.

1333 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1333 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
1C2×#88740C211yesmaximal, characteristic, complemented
2C2×#88655C211yesmaximal, characteristic, complemented
3#64139⋊C3C212yesmaximal, complemented
4#11721324C212yesmaximal, complemented
5#11721317C212yesmaximal, complemented
6#11721163C212yesmaximal, complemented
7#11721159C212yesmaximal, complemented
8#11721143C212yesmaximal, complemented
9#11721089C211yesmaximal, characteristic, complemented
10#10595064—32nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
12C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
13C2×#6235C2211yescharacteristic
14C2×#6235C2211yescharacteristic, complemented
15#88740—24no—
16C2×C4⋊#272C2211yescharacteristic
17C2×C4×#272C2211yescharacteristic
18C2×C4×#272C2211yescharacteristic, complemented
19C2×#6229C2211yescharacteristic, complemented
20C2×#638⋊C3C2214yescomplemented
21C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
22C2×#6200C2214yescomplemented
23C2×#6198C2212yes—
24#88655—24no—
25#3752⋊C3C2214yescomplemented
26C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
27#88451C2212yes—
28#88450C2214yescomplemented
29#88447C2212yescomplemented
30C2×C3×#235—32no—
31C2×C3×#235—61no—
32C2×D3×D12—32no—
33#64139D312yescomplemented
34C3×#930—32no—
35#63983—34no—
36#63979—34no—
37#63922—34no—
38#63909—32no—
39#63908—32no—
40#63907—34no—
41#63841—32no—
42C23×H(3,𝔽3)C2311yescharacteristic
43C22×#272C2311yescharacteristic
44C22×#272C2311yescharacteristic
45C22×(C3⋊D3)⋊C3C2314yes—
46C2×((C3⋊D3)⋊C3)⋊C2—24no—
47C2×((C3⋊D3)⋊C3)⋊C2—44no—
48C2×C4×H(3,𝔽3)—22no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51#6235—22no—
52#6235—22no—
53C2×#1084C2311yescharacteristic
54C2×#1084C2311yescharacteristic
55C4⋊#272—22no—
56C4×#272—22no—
57C4×#272D414yescomplemented
58#6229—22no—
59#638⋊C3—44no—
60C2×(C3⋊Dic3)⋊C3C2314yes—
61C4⋊((C3⋊D3)⋊C3)—24no—
62#6200—44no—
63#6198—24no—
64#5830—91no—
65#5607—92no—
66C22×D32—34no—
67C22×C32×C4—21no—
68C22×C32×C4C2×D312yescomplemented
69C2×#638C2×D314yescomplemented
70C2×C4⋊(C3⋊D3)C2×D312yes—
71C2×C3×#67—32no—
72C2×C3×#67—32no—
73C2×C3×#67—34no—
74C2×C3×#67—61no—
75C2×C3×#67—61no—
76C3×#235—68no—
77C3×#235—68no—
78C2×C3×D12—32no—
79C2×C3×D12—32no—
80C2×C3×D12—61no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6—61no—
87C2×#626—34no—
88C2×#626—34no—
89C2×#625—34no—
90D3×D12—68no—
91#3752C2×D314yescomplemented
92C4⋊(C3⋊Dic3)C2×D312yescomplemented
93C3×(C2×D3)⋊C4—34no—
94C3×C4⋊Dic3—32no—
95#3722—34no—
96(C2×D3)⋊Dic3—34no—
97(C2×D3)⋊Dic3—34no—
98Dic32—34no—
99C22×H(3,𝔽3)—21no—
100C22×H(3,𝔽3)—22no—
101C22×H(3,𝔽3)C2411yescharacteristic
102C2×#272—22no—
103C2×#272—21no—
104C2×#272—22no—
105C2×#272C2×D412yes—
106C2×(C3⋊D3)⋊C3—24no—
107C2×(C3⋊D3)⋊C3—24no—
108C2×(C3⋊D3)⋊C3—24no—
109((C3⋊D3)⋊C3)⋊C2—48no—
110C4×H(3,𝔽3)—21no—
111C4×H(3,𝔽3)C2×D412yes—
112#1084—21no—
113#1084C2×D412yes—
114(C3⋊Dic3)⋊C3—24no—
115C2×C3×#41—91no—
116C2×D3×D4—92no—
117C2×#235—32no—
118C2×#235—61no—
119C22×D12—91no—
120C3×#121—92no—
121#947—92no—
122#940—92no—
123#940—94no—
124#938—92no—
125#937—92no—
126#937—92no—
127#937—94no—
128#930—32no—
129#930—92no—
130#930—92no—
131#926—91no—
132#895—92no—
133#892—92no—
134#885—94no—
135#882—94no—
136C23×C32—21no—
137C23×C32C22×D312yes—
138C22×C3⋊D3C22×D314yes—
139C22×C3×D3—32no—
140C22×C3×D3—34no—
141C22×C3×D3—32no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C2×D32—68no—
145C2×D32—124no—
146C2×D32—124no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4C22×D312yes—
152C2×C32×C4C22×D312yes—
153#638—44no—
154C2×C3⋊Dic3C22×D314yes—
155C4⋊(C3⋊D3)—24no—
156C3×#67—64no—
157C3×#67—64no—
158C3×#67—64no—
159C3×#67—64no—
160C3×#67—124no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C3×D12—64no—
167C3×D12—64no—
168C3×D12—64no—
169C3×C4×D3—38no—
170C3×C4×D3—64no—
171C3×C4×D3—64no—
172C3×C4×D3—64no—
173C3×Dic6—64no—
174C3×Dic6—64no—
175#626—68no—
176#626—124no—
177#625—124no—
178#535—271noSylow
179C2×H(3,𝔽3)—41no—
180C2×H(3,𝔽3)C22×D411yescharacteristic
181C2×H(3,𝔽3)H(5,𝔽2)12yes—
182#272—22no—
183#272—41no—
184(C3⋊D3)⋊C3—44no—
185C23×D3—92no—
186C23×D3—94no—
187C3×#41—184no—
188C2×C3×Q8—91no—
189C2×C3×D4—92no—
190C2×C3×D4—91no—
191C2×C3×D4—91no—
192C2×C3×D4—94no—
193C2×C3×D4—91no—
194C22×C3×C4—32no—
195C22×C3×C4—61no—
196C22×C3×C4—91no—
197C22×C3×C4—91no—
198C22×C3×C4D3211yescharacteristic
199C2×#67—32no—
200C2×#67—32no—
201C2×#67—34no—
202C2×#67—61no—
203C2×#67—61no—
204C2×#67—94no—
205C2×#67—94no—
206C2×#67—94no—
207C2×#67—94no—
208D3×D4—188no—
209#235—68no—
210#235—68no—
211C2×D12—32no—
212C2×D12—32no—
213C2×D12—61no—
214C2×D12—92no—
215C2×D12—92no—
216C2×D12—94no—
217C2×D12—184no—
218C2×C4×D3—32no—
219C2×C4×D3—32no—
220C2×C4×D3—61no—
221C2×C4×D3—61no—
222C2×Dic6—32no—
223C2×Dic6—61no—
224C3×C4⋊C4—92no—
225C3×C22⋊C4—94no—
226#217—92no—
227#217—94no—
228(C2×D3)⋊C4—34no—
229(C2×D3)⋊C4—94no—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233C4⋊Dic3—32no—
234C4⋊Dic3—92no—
235C4×Dic3—92no—
236C4×Dic3—94no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—21no—
242C22×C32—42no—
243C22×C32C23×D312yes—
244C2×C3⋊D3—24no—
245C2×C3⋊D3—24no—
246C2×C3⋊D3—24no—
247C2×C3×D3—34no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—62no—
256C2×C3×D3—64no—
257C2×C3×D3—62no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260D32—1216no—
261C32×C4—22no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4D3×D414yes—
265C3⋊Dic3—24no—
266C3×Dic3—34no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C3×Dic3—64no—
271C2×#41—91no—
272C22×D4—271no—
273#127—272no—
274#124—272no—
275#121—92no—
276#121—272no—
277#120—272no—
278#120—272no—
279#117—271no—
280H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
281C23×C3—32no—
282C23×C3—61no—
283C23×C3—94no—
284C23×C3—91no—
285C23×C3—91no—
286C23×C3C2×D3211yescharacteristic
287C22×D3—32no—
288C22×D3—32no—
289C22×D3—34no—
290C22×D3—61no—
291C22×D3—61no—
292C22×D3—94no—
293C22×D3—188no—
294C22×D3—184no—
295C22×D3—364no—
296C22×D3—364no—
297C22×D3—364no—
298C3×Q8—182no—
299C3×D4—182no—
300C3×D4—184no—
301C3×D4—182no—
302C3×D4—182no—
303C3×D4—364no—
304C2×C3×C4—22no—
305C2×C3×C4—32no—
306C2×C3×C4—32no—
307C2×C3×C4—64no—
308C2×C3×C4—64no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—91no—
312C2×C3×C4—94no—
313C2×C3×C4—94no—
314C2×C3×C4—91no—
315C2×C3×C4—182no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318#67—64no—
319#67—64no—
320#67—64no—
321#67—64no—
322#67—124no—
323#67—188no—
324#67—364no—
325#67—364no—
326#67—364no—
327C2×Dic3—32no—
328C2×Dic3—32no—
329C2×Dic3—34no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—94no—
333D12—64no—
334D12—64no—
335D12—64no—
336D12—184no—
337D12—184no—
338D12—364no—
339C4×D3—38no—
340C4×D3—64no—
341C4×D3—64no—
342C4×D3—64no—
343Dic6—64no—
344Dic6—64no—
345C2×C32—22no—
346C2×C32—21no—
347C2×C32—42no—
348C2×C32—42no—
349C2×C32C2×D3×D412yes—
350C2×C32#100412yes—
351C2×C32#100912yes—
352C3⋊D3—44no—
353C3×D3—64no—
354C3×D3—122no—
355C3×D3—122no—
356C3×D3—124no—
357C3×D3—122no—
358C24—272no—
359#41—184no—
360C2×Q8—91no—
361C2×D4—91no—
362C2×D4—91no—
363C2×D4—94no—
364C2×D4—91no—
365C2×D4—92no—
366C2×D4—272no—
367C2×D4—274no—
368C2×D4—544no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
372C4⋊C4—92no—
373C22⋊C4—94no—
374C22⋊C4—272no—
375C22⋊C4—274no—
376C42—272no—
377C22×C3—22no—
378C22×C3—21no—
379C22×C3—32no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—62no—
383C22×C3—62no—
384C22×C3—61no—
385C22×C3—92no—
386C22×C3—122no—
387C22×C3—184no—
388C22×C3—184no—
389C22×C3—184no—
390C22×C3—182no—
391C22×C3—182no—
392C22×C3—181no—
393C22×C3C22×D3211yescharacteristic
394C2×D3—34no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—64no—
403C2×D3—64no—
404C2×D3—64no—
405C2×D3—122no—
406C2×D3—122no—
407C2×D3—184no—
408C2×D3—184no—
409C2×D3—184no—
410C2×D3—3616no—
411C2×D3—368no—
412C3×C4—21no—
413C3×C4—34no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—92no—
418C3×C4—184no—
419C3×C4—181no—
420C3×C4#381112yes—
421Dic3—34no—
422Dic3—62no—
423Dic3—62no—
424Dic3—62no—
425Dic3—64no—
426Dic3—184no—
427C32—21no—
428C32#560712yescomplemented
429C23—91no—
430C23—94no—
431C23—91no—
432C23—544no—
433C23—1084no—
434C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
435Q8—182no—
436D4—182no—
437D4—182no—
438D4—182no—
439D4—184no—
440D4—364no—
441D4—544no—
442D4—1084no—
443C2×C4—22no—
444C2×C4—91no—
445C2×C4—94no—
446C2×C4—94no—
447C2×C4—91no—
448C2×C4—182no—
449C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
450C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
451C2×C3—32no—
452C2×C3—32no—
453C2×C3—32no—
454C2×C3—41no—
455C2×C3—61no—
456C2×C3—62no—
457C2×C3—122no—
458C2×C3—122no—
459C2×C3—182no—
460C2×C3—364no—
461C2×C3—361no—
462C2×C3#6430712yes—
463C2×C3C2×#381111yescharacteristic
464D3—64no—
465D3—122no—
466D3—122no—
467D3—122no—
468D3—124no—
469D3—364no—
470C22—21no—
471C22—22no—
472C22—92no—
473C22—182no—
474C22—181no—
475C22—184no—
476C22—184no—
477C22—182no—
478C22—184no—
479C22—1088no—
480C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
481C4—21no—
482C4—92no—
483C4—181no—
484C4—184no—
485C4#8867712yes—
486C3—32no—
487C3—61no—
488C3#1059508211yescharacteristic
489C2—41no—
490C2—182no—
491C2—364no—
492C2—361no—
493C2#1172249212yes—
494C2C2×#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)#535
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. C2×#88740C2semidirect
2. C2×#88655C2semidirect
3. #64139⋊C3C2semidirect
4. #11721324C2semidirect
5. #11721317C2semidirect
6. #11721163C2semidirect
7. #11721159C2semidirect
8. #11721143C2semidirect
9. #11721089C2semidirect
12. C22×C4×H(3,𝔽3)C22semidirect
14. C2×#6235C22semidirect
18. C2×C4×#272C22semidirect
19. C2×#6229C22semidirect
20. C2×#638⋊C3C22semidirect
22. C2×#6200C22semidirect
25. #3752⋊C3C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
28. #88450C22semidirect
29. #88447C22semidirect
33. #64139D3semidirect
57. C4×#272D4semidirect
68. C22×C32×C4C2×D3semidirect
69. C2×#638C2×D3semidirect
91. #3752C2×D3semidirect
92. C4⋊(C3⋊Dic3)C2×D3semidirect
280. H(3,𝔽3)#535semidirect
371. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
428. C32#5607semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#886552
#64139⋊C32
#117213242
#117213172
#117211632
#117211592
#117211432
#117210892
#105950643