Finite Group Databasebeta

← C2×#11721068 #13150641

Subgroups — C2×#11721068 #13150641

Order 1728.

1279 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#272C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88437C212yesmaximal, complemented
5#11721068C218yesmaximal, direct product factor
6C2×#88436C212yesmaximal, complemented
7C2×#63830—31nomaximal
8C2×#63823—31nomaximal
9C23×#272—21no—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272—22no—
12C2×C4⋊#272—21no—
13C2×C4⋊#272C2212yes—
14C2×C4⋊#272C2214yescomplemented
15C2×#638⋊C3—21no—
16#3828⋊C3—22no—
17#3828⋊C3C2214yescomplemented
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×#634⋊C3C2211yescharacteristic, complemented
21#3748⋊C3C2214yescomplemented
22C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
23#88437C2218yescomplemented
24#88436C2218yescomplemented
25C2×#3828D311yescharacteristic, complemented
26C2×C3×#235—31no—
27C22×C3×D12—31no—
28C22×C3×D12—31no—
29C22×C3×D12—61no—
30C2×#3748D311yescharacteristic, complemented
31C2×C3×#208—31no—
32#63830—38no—
33C2×#3716—32no—
34C2×#3715—32no—
35C2×#3714—32no—
36#63823—38no—
37C2×#3713—32no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×#272—22no—
40C22×#272—24no—
41C22×#272—44no—
42C22×(C3⋊D3)⋊C3—21no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)D412yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2312yes—
47C4⋊#272—22no—
48C4⋊#272—24no—
49C4⋊#272C2314yes—
50#638⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)C2312yescomplemented
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—21no—
56#634⋊C3C2312yescomplemented
57(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
58C2×#949—91no—
59C2×#932—91no—
60C2×#908—91no—
61C23×C3×D3—62no—
62C23×C3×D3—61no—
63C23×C3×D3—61no—
64C22×C32×C4—21no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C2×#638—21no—
68#3828—22no—
69#3828C2×D314yescomplemented
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
71C2×C4×C3⋊D3—21no—
72C2×#634C2×D311yescharacteristic, complemented
73C2×C3×#67—61no—
74C3×#235—34no—
75C3×#235—62no—
76C2×C3×D12—31no—
77C2×C3×D12—32no—
78C2×C3×D12—34no—
79C2×C3×D12—32no—
80C2×C3×D12—34no—
81C2×C3×D12—62no—
82C2×C3×D12—64no—
83C2×C3×D12—64no—
84C2×C3×D12—62no—
85C2×C3×D12—61no—
86C2×C3×D12—62no—
87C2×C3×D12—61no—
88C2×C3×D12—62no—
89C2×C3×C4×D3—61no—
90C2×C3×Dic6—31no—
91#3748C2×D314yescomplemented
92C2×C3⋊(C3⋊C8)C2×D312yescomplemented
93C3×#208—34no—
94C2×C3×C3⋊C8—32no—
95#3716—38no—
96#3715—38no—
97#3714—38no—
98#3713—38no—
99C22×H(3,𝔽3)—22no—
100C22×H(3,𝔽3)C2×D412yes—
101C22×H(3,𝔽3)C2×D411yescharacteristic
102C2×#272—24no—
103C2×#272—48no—
104C2×#272—44no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—42no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D412yes—
109C4×H(3,𝔽3)C2411yescharacteristic
110(C3⋊Dic3)⋊C3—22no—
111C2×C3×#41—91no—
112C22×C3×D4—91no—
113C2×#235—31no—
114C2×#235—91no—
115C22×D12—31no—
116C22×D12—31no—
117C22×D12—61no—
118C2×C3×M16—91no—
119#949—98no—
120C2×#215—92no—
121C2×#214—92no—
122#932—98no—
123C2×#213—92no—
124C2×#213—92no—
125C2×#208—31no—
126C2×#208—91no—
127#908—98no—
128C2×D24—92no—
129C2×#204—92no—
130C23×C32—21no—
131C23×C32D1211yescharacteristic
132C23×C32#6711yescharacteristic
133C22×C3⋊D3—21no—
134C22×C3×D3—64no—
135C22×C3×D3—68no—
136C22×C3×D3—64no—
137C22×C3×D3—62no—
138C22×C3×D3—61no—
139C22×C3×D3—64no—
140C22×C3×D3—62no—
141C22×C3×D3—128no—
142C22×C3×D3—124no—
143C22×C3×D3—124no—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4—21no—
147C2×C32×C4—22no—
148C2×C32×C4D1211yescharacteristic
149C2×C32×C4D1212yes—
150C2×C32×C4#6711yescharacteristic
151C2×C32×C4#6712yes—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D312yes—
156#638—24no—
157C2×C3⋊Dic3—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)C22×D312yescomplemented
160C4×C3⋊D3—24no—
161#634—21no—
162#634C22×D312yescomplemented
163C3×#67—64no—
164C2×C3×Dic3—61no—
165C3×D12—32no—
166C3×D12—34no—
167C3×D12—34no—
168C3×D12—64no—
169C3×D12—64no—
170C3×D12—68no—
171C3×D12—61no—
172C3×D12—62no—
173C3×D12—64no—
174C3×D12—62no—
175C3×D12—64no—
176C3×C4×D3—64no—
177C3×Dic6—32no—
178C3×Dic6—61no—
179C3⋊(C3⋊C8)C22×D314yescomplemented
180C3×C3⋊C8—34no—
181C2×AGL(1,ℤ/8)—271noSylow
182C2×H(3,𝔽3)—22no—
183C2×H(3,𝔽3)AGL(1,ℤ/8)12yescomplemented
184C2×H(3,𝔽3)C22×D411yescharacteristic
185#272—44no—
186(C3⋊D3)⋊C3—42no—
187C24×C3—181no—
188C23×D3—61no—
189C23×D3—61no—
190C23×D3—62no—
191C3×#41—94no—
192C3×#41—182no—
193C2×C3×Q8—91no—
194C2×C3×D4—91no—
195C2×C3×D4—92no—
196C2×C3×D4—94no—
197C2×C3×D4—181no—
198C2×C3×D4—181no—
199C2×C3×D4—182no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—61no—
203C22×C3×C4—181no—
204C22×C3×C4D3211yescharacteristic
205C2×#67—61no—
206C2×#67—181no—
207#235—34no—
208#235—62no—
209#235—94no—
210#235—182no—
211C2×D12—32no—
212C2×D12—34no—
213C2×D12—32no—
214C2×D12—34no—
215C2×D12—31no—
216C2×D12—62no—
217C2×D12—64no—
218C2×D12—61no—
219C2×D12—62no—
220C2×D12—61no—
221C2×D12—62no—
222C2×D12—64no—
223C2×D12—62no—
224C2×D12—91no—
225C2×C4×D3—61no—
226C2×C4×D3—181no—
227C2×Dic6—31no—
228C2×Dic6—91no—
229C3×M16—94no—
230C2×C3×C8—92no—
231#215—98no—
232#214—98no—
233#213—98no—
234#213—98no—
235#208—34no—
236#208—94no—
237C2×C3⋊C8—32no—
238C2×C3⋊C8—92no—
239D24—98no—
240#204—98no—
241C22×C32—24no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—21no—
246C22×C32C2×D1212yes—
247C22×C32C2×D1211yescharacteristic
248C22×C32C2×#6712yes—
249C22×C32C2×#6711yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3—42no—
252C2×C3×D3—68no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—1216no—
257C2×C3×D3—128no—
258C2×C3×D3—128no—
259C2×C3×D3—128no—
260C2×C3×D3—122no—
261C2×C3×D3—124no—
262C2×C3×D3—124no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—22no—
266C32×C4C2×D1211yescharacteristic
267C32×C4C2×D1212yes—
268C32×C4C2×#6711yescharacteristic
269C32×C4C2×#6712yes—
270C32×C4C23×D311yescharacteristic
271C32×C4C23×D311yescharacteristic
272C3⋊Dic3—22no—
273C3×Dic3—62no—
274C2×#41—91no—
275C22×D4—91no—
276AGL(1,ℤ/8)—278no—
277C2×SD16—272no—
278C2×D8—272no—
279C2×M16—91no—
280H(3,𝔽3)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—61no—
284C23×C3—181no—
285C23×C3—184no—
286C23×C3—182no—
287C23×C3—364no—
288C23×C3#62611yescharacteristic
289C22×D3—64no—
290C22×D3—64no—
291C22×D3—62no—
292C22×D3—62no—
293C22×D3—68no—
294C22×D3—64no—
295C22×D3—61no—
296C22×D3—124no—
297C22×D3—124no—
298C22×D3—128no—
299C22×D3—181no—
300C3×Q8—92no—
301C3×Q8—181no—
302C3×D4—92no—
303C3×D4—94no—
304C3×D4—181no—
305C3×D4—184no—
306C3×D4—182no—
307C3×D4—184no—
308C2×C3×C4—31no—
309C2×C3×C4—32no—
310C2×C3×C4—31no—
311C2×C3×C4—32no—
312C2×C3×C4—31no—
313C2×C3×C4—32no—
314C2×C3×C4—31no—
315C2×C3×C4—32no—
316C2×C3×C4—62no—
317C2×C3×C4—61no—
318C2×C3×C4—62no—
319C2×C3×C4—61no—
320C2×C3×C4—184no—
321C2×C3×C4—181no—
322C2×C3×C4#62611yescharacteristic
323C2×C3×C4#62612yes—
324C2×C3×C4C2×D3211yescharacteristic
325C2×C3×C4C2×D3212yes—
326#67—64no—
327#67—184no—
328C2×Dic3—61no—
329C2×Dic3—181no—
330D12—34no—
331D12—34no—
332D12—32no—
333D12—64no—
334D12—62no—
335D12—64no—
336D12—64no—
337D12—62no—
338D12—68no—
339D12—64no—
340D12—61no—
341D12—92no—
342D12—181no—
343C4×D3—64no—
344C4×D3—184no—
345Dic6—32no—
346Dic6—61no—
347Dic6—92no—
348Dic6—181no—
349C3×C8—94no—
350C3⋊C8—34no—
351C3⋊C8—94no—
352C2×C32—24no—
353C2×C32—22no—
354C2×C32—22no—
355C2×C32—21no—
356C2×C32—22no—
357C2×C32#90812yescomplemented
358C2×C32#94912yescomplemented
359C2×C32C22×D1211yescharacteristic
360C2×C32C22×#6711yescharacteristic
361C3⋊D3—42no—
362C3×D3—128no—
363C3×D3—122no—
364C3×D3—124no—
365C3×D3—124no—
366C24—181no—
367#41—94no—
368#41—182no—
369C2×Q8—91no—
370C2×D4—91no—
371C2×D4—92no—
372C2×D4—94no—
373C2×D4—181no—
374C2×D4—182no—
375C2×D4—181no—
376C22×C4—181no—
377C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
378SD16—278no—
379D8—278no—
380M16—94no—
381C2×C8—92no—
382C22×C3—22no—
383C22×C3—32no—
384C22×C3—32no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—62no—
388C22×C3—62no—
389C22×C3—64no—
390C22×C3—62no—
391C22×C3—61no—
392C22×C3—182no—
393C22×C3—184no—
394C22×C3—368no—
395C22×C3—362no—
396C22×C3—364no—
397C22×C3C2×#62612yes—
398C22×C3C2×#62611yescharacteristic
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—68no—
402C2×D3—62no—
403C2×D3—124no—
404C2×D3—124no—
405C2×D3—128no—
406C2×D3—128no—
407C2×D3—1216no—
408C2×D3—128no—
409C2×D3—122no—
410C2×D3—182no—
411C2×D3—362no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—31no—
415C3×C4—32no—
416C3×C4—32no—
417C3×C4—31no—
418C3×C4—61no—
419C3×C4—62no—
420C3×C4—61no—
421C3×C4—182no—
422C3×C4C2×#62611yescharacteristic
423C3×C4C2×#62612yes—
424C3×C4C22×D3211yescharacteristic
425Dic3—62no—
426Dic3—182no—
427C32—21no—
428C32C2×#90811yescharacteristic, complemented
429C32C2×#94911yescharacteristic, complemented
430C23—184no—
431C23—181no—
432C23—182no—
433C23—364no—
434C23#620011yescharacteristic
435Q8—92no—
436Q8—181no—
437D4—92no—
438D4—94no—
439D4—184no—
440D4—184no—
441D4—182no—
442D4—181no—
443C2×C4—181no—
444C2×C4—184no—
445C2×C4#620012yes—
446C2×C4#620011yescharacteristic
447C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
448C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C8—94no—
450C2×C3—22no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—32no—
454C2×C3—32no—
455C2×C3—62no—
456C2×C3—62no—
457C2×C3—64no—
458C2×C3—61no—
459C2×C3—62no—
460C2×C3—362no—
461C2×C3—364no—
462C2×C3#6382512yes—
463C2×C3C22×#62611yescharacteristic
464D3—124no—
465D3—124no—
466D3—128no—
467D3—122no—
468D3—362no—
469C22—22no—
470C22—182no—
471C22—184no—
472C22—368no—
473C22—364no—
474C22—362no—
475C22C2×#620012yes—
476C22C2×#620011yescharacteristic
477C4—182no—
478C4C2×#620012yes—
479C4C2×#620011yescharacteristic
480C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
481C3—31no—
482C3—31no—
483C3—61no—
484C3C2×#6382511yescharacteristic
485C2—22no—
486C2—364no—
487C2—362no—
488C2#1172106812yesdirect product factor
489C2C22×#620011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×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×#3828⋊C3C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88437C2semidirect
5. #11721068C2direct (N × H)
6. C2×#88436C2semidirect
14. C2×C4⋊#272C22semidirect
17. #3828⋊C3C22semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. C2×#634⋊C3C22semidirect
21. #3748⋊C3C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #88437C22semidirect
24. #88436C22semidirect
25. C2×#3828D3semidirect
30. C2×#3748D3semidirect
53. C4⋊((C3⋊D3)⋊C3)C23semidirect
56. #634⋊C3C23semidirect
57. (C3⋊(C3⋊C8))⋊C3C23semidirect
69. #3828C2×D3semidirect
70. C2×C4⋊(C3⋊D3)C2×D3semidirect
72. C2×#634C2×D3semidirect
91. #3748C2×D3semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
159. C4⋊(C3⋊D3)C22×D3semidirect
162. #634C22×D3semidirect
179. C3⋊(C3⋊C8)C22×D3semidirect
183. C2×H(3,𝔽3)AGL(1,ℤ/8)semidirect
280. H(3,𝔽3)C2×AGL(1,ℤ/8)semidirect
357. C2×C32#908semidirect
358. C2×C32#949semidirect
428. C32C2×#908semidirect
429. C32C2×#949semidirect
488. C2#11721068direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×#3828⋊C32
C2×#3748⋊C32
C2×#884372
#117210682
C2×#884362
C2×#638303
C2×#638233