Finite Group Databasebeta

← C2×#11721111 #13150629

Subgroups — C2×#11721111 #13150629

Order 1728.

953 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1084C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#3750⋊C3C211yesmaximal, characteristic, complemented
4C2×#88450C211yesmaximal, characteristic
5C2×#88450C211yesmaximal, characteristic
6C2×#88449C211yesmaximal, characteristic, complemented
7C2×#88447C211yesmaximal, characteristic, complemented
8#11721111C218yesmaximal, direct product factor
9C2×#63858—31nomaximal
10C2×#63848—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C4⋊#1084C2214yescomplemented
19#3752⋊C3C2214yes—
20#3750⋊C3C2214yescomplemented
21#88450C2214yes—
22#88450C2214yes—
23#88449C2214yescomplemented
24#88447C2214yescomplemented
25C2×#3752D311yescharacteristic
26C2×#3750D311yescharacteristic, complemented
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×C4⋊Dic3—31no—
29C2×C3×C4⋊Dic3—31no—
30C2×C3×C4⋊Dic3—61no—
31C2×C3×Dic3⋊C4—31no—
32C2×Dic3⋊Dic3—31no—
33C2×Dic3⋊Dic3—31no—
34C2×#3722—31no—
35C2×#3722—31no—
36C2×(C2×D3)⋊Dic3—31no—
37C2×(C2×D3)⋊Dic3—31no—
38C2×Dic32—31no—
39C2×Dic32—31no—
40#63858—38no—
41#63848—38no—
42C23×H(3,𝔽3)C2311yescharacteristic
43C22×(C3⋊D3)⋊C3—22no—
44C22×(C3⋊D3)⋊C3—22no—
45C22×(C3⋊D3)⋊C3—22no—
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)—22no—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×#1084—22no—
50C2×#1084—22no—
51C2×#1084C2312yes—
52C2×#1084C2312yes—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3—22no—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×#898—91no—
60C2×#898—91no—
61C2×#879—91no—
62C23×C3⋊D3C2×D311yescharacteristic
63C23×C3×D3—31no—
64C22×C32×C4—21no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3C2×D311yescharacteristic
68C22×C3⋊Dic3C2×D311yescharacteristic
69C22×C3⋊Dic3C2×D311yescharacteristic
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—61no—
78C22×C3×Dic3—61no—
79#3752C2×D314yes—
80#3750C2×D314yescomplemented
81C3×(C2×D3)⋊C4—34no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—34no—
84C3×C4⋊Dic3—64no—
85C3×Dic3⋊C4—34no—
86Dic3⋊Dic3—34no—
87Dic3⋊Dic3—34no—
88#3722—34no—
89#3722—34no—
90(C2×D3)⋊Dic3—34no—
91(C2×D3)⋊Dic3—34no—
92Dic32—34no—
93Dic32—34no—
94C22×H(3,𝔽3)#4112yes—
95C22×H(3,𝔽3)#4112yes—
96C22×H(3,𝔽3)#4112yes—
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—22no—
101C2×(C3⋊D3)⋊C3—44no—
102C4×H(3,𝔽3)—22no—
103#1084—22no—
104#1084—22no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—22no—
108C2×C3×C4⋊C4—91no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C22⋊C4—91no—
111C2×#217—91no—
112C2×#217—91no—
113C2×(C2×D3)⋊C4—31no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×C4⋊Dic3—31no—
120C2×C4⋊Dic3—31no—
121C2×C4⋊Dic3—61no—
122C2×C4⋊Dic3—91no—
123C2×Dic3⋊C4—31no—
124C2×Dic3⋊C4—91no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×C4×Dic3—91no—
128C2×C4×Dic3—91no—
129C2×C4×Dic3—91no—
130#898—98no—
131#898—98no—
132#879—98no—
133C23×C32—21no—
134C23×C32C22×D311yescharacteristic
135C23×C32C22×D311yescharacteristic
136C22×C3⋊D3—22no—
137C22×C3⋊D3—22no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3C22×D312yes—
140C22×C3×D3—32no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D312yes—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3C22×D312yes—
154C2×C3⋊Dic3C22×D312yes—
155C2×C3⋊Dic3C22×D312yes—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—122no—
173C2×C3×Dic3—122no—
174C2×#126—271noSylow
175C2×H(3,𝔽3)#12614yescomplemented
176C2×H(3,𝔽3)C2×#4111yescharacteristic
177C2×H(3,𝔽3)C2×#4111yescharacteristic
178C2×H(3,𝔽3)C2×#4111yescharacteristic
179(C3⋊D3)⋊C3—42no—
180C24×C3—91no—
181C23×D3—31no—
182C23×D3—91no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—61no—
186C22×C3×C4—91no—
187C22×C3×C4—91no—
188C22×C3×C4—91no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4D3211yescharacteristic
192C22×Dic3—31no—
193C22×Dic3—31no—
194C22×Dic3—31no—
195C22×Dic3—31no—
196C22×Dic3—31no—
197C22×Dic3—31no—
198C22×Dic3—31no—
199C22×Dic3—61no—
200C22×Dic3—61no—
201C22×Dic3—91no—
202C22×Dic3—91no—
203C22×Dic3—91no—
204C3×C4⋊C4—94no—
205C3×C4⋊C4—94no—
206C3×C22⋊C4—94no—
207#217—94no—
208#217—94no—
209(C2×D3)⋊C4—34no—
210(C2×D3)⋊C4—94no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—94no—
213(C2×D3)⋊C4—94no—
214(C2×D3)⋊C4—94no—
215C4⋊Dic3—34no—
216C4⋊Dic3—34no—
217C4⋊Dic3—64no—
218C4⋊Dic3—94no—
219Dic3⋊C4—34no—
220Dic3⋊C4—94no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223C4×Dic3—94no—
224C4×Dic3—94no—
225C4×Dic3—94no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32#23512yes—
231C22×C32#23512yes—
232C22×C32#23712yes—
233C22×C32#23712yes—
234C22×C32#23712yes—
235C22×C32#23912yes—
236C22×C32C23×D311yescharacteristic
237C22×C32C23×D311yescharacteristic
238C2×C3⋊D3—22no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—44no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—124no—
246C32×C4—24no—
247C32×C4—22no—
248C32×C4—22no—
249C3⋊Dic3—22no—
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3×Dic3—62no—
253C3×Dic3—62no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C3×Dic3—62no—
258C3×Dic3—62no—
259C3×Dic3—122no—
260C3×Dic3—122no—
261#126—278no—
262C2×C4⋊C4—91no—
263C2×C4⋊C4—91no—
264C2×C4⋊C4—271no—
265C2×C22⋊C4—91no—
266C2×C22⋊C4—271no—
267C2×C22⋊C4—271no—
268C2×C42—271no—
269H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
270C23×C3—31no—
271C23×C3—31no—
272C23×C3—61no—
273C23×C3—92no—
274C23×C3—182no—
275C23×C3—182no—
276C23×C3—182no—
277C23×C3C2×D3211yescharacteristic
278C22×D3—32no—
279C22×D3—62no—
280C22×D3—62no—
281C22×D3—62no—
282C22×D3—92no—
283C22×D3—182no—
284C22×D3—182no—
285C22×D3—182no—
286C2×C3×C4—22no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—64no—
292C2×C3×C4—62no—
293C2×C3×C4—92no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—182no—
299C2×C3×C4—182no—
300C2×C3×C4—182no—
301C2×C3×C4—182no—
302C2×C3×C4—182no—
303C2×C3×C4C2×D3212yes—
304C2×Dic3—32no—
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—32no—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—62no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—122no—
324C2×Dic3—122no—
325C2×Dic3—182no—
326C2×Dic3—182no—
327C2×Dic3—182no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—24no—
332C2×C32#87914yescomplemented
333C2×C32#89814yescomplemented
334C2×C32C2×#23511yescharacteristic
335C2×C32C2×#23511yescharacteristic
336C2×C32C2×#23711yescharacteristic
337C2×C32C2×#23711yescharacteristic
338C2×C32C2×#23711yescharacteristic
339C2×C32C2×#23911yescharacteristic
340C3⋊D3—42no—
341C3×D3—122no—
342C24—91no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4—91no—
346C22×C4—91no—
347C22×C4—91no—
348C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C4⋊C4—94no—
350C4⋊C4—94no—
351C4⋊C4—274no—
352C22⋊C4—94no—
353C22⋊C4—274no—
354C22⋊C4—274no—
355C42—274no—
356C22×C3—31no—
357C22×C3—31no—
358C22×C3—32no—
359C22×C3—32no—
360C22×C3—32no—
361C22×C3—32no—
362C22×C3—32no—
363C22×C3—32no—
364C22×C3—61no—
365C22×C3—62no—
366C22×C3—62no—
367C22×C3—62no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3—182no—
371C22×C3—364no—
372C22×C3#379512yes—
373C22×C3#379912yes—
374C22×C3#380512yes—
375C22×C3C22×D3211yescharacteristic
376C2×D3—62no—
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—124no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—182no—
383C2×D3—364no—
384C3×C4—22no—
385C3×C4—62no—
386C3×C4—62no—
387C3×C4—64no—
388C3×C4—182no—
389C3×C4—182no—
390C3×C4—182no—
391C3×C4—182no—
392C3×C4—182no—
393Dic3—62no—
394Dic3—62no—
395Dic3—62no—
396Dic3—62no—
397Dic3—62no—
398Dic3—62no—
399Dic3—62no—
400Dic3—122no—
401Dic3—122no—
402Dic3—182no—
403Dic3—182no—
404Dic3—182no—
405C32—21no—
406C32C2×#87911yescharacteristic, complemented
407C32C2×#89811yescharacteristic, complemented
408C23—92no—
409C23—182no—
410C23—182no—
411C23—182no—
412C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C2×C4—22no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—92no—
418C2×C4—92no—
419C2×C4—182no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—34no—
428C2×C3—34no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—61no—
434C2×C3—64no—
435C2×C3—61no—
436C2×C3—61no—
437C2×C3—362no—
438C2×C3#6386114yes—
439C2×C3C2×#379511yescharacteristic
440C2×C3C2×#379911yescharacteristic
441C2×C3C2×#380511yescharacteristic
442D3—122no—
443D3—362no—
444C22—182no—
445C22—182no—
446C22—182no—
447C22—364no—
448C22#8864812yes—
449C22#8865212yes—
450C22#8867212yes—
451C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
452C4—22no—
453C4—182no—
454C4—182no—
455C4—182no—
456C4—182no—
457C4—182no—
458C3—31no—
459C3—31no—
460C3—61no—
461C3C2×#6386111yescharacteristic
462C2—362no—
463C2#1172111114yesdirect product factor
464C2C2×#8864811yescharacteristic
465C2C2×#8865211yescharacteristic
466C2C2×#8867211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1084C2semidirect
3. C2×#3750⋊C3C2semidirect
6. C2×#88449C2semidirect
7. C2×#88447C2semidirect
8. #11721111C2direct (N × H)
18. C4⋊#1084C22semidirect
20. #3750⋊C3C22semidirect
23. #88449C22semidirect
24. #88447C22semidirect
26. C2×#3750D3semidirect
80. #3750C2×D3semidirect
175. C2×H(3,𝔽3)#126semidirect
269. H(3,𝔽3)C2×#126semidirect
332. C2×C32#879semidirect
333. C2×C32#898semidirect
406. C32C2×#879semidirect
407. C32C2×#898semidirect
463. C2#11721111direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#3752⋊C32
C2×#3750⋊C32
C2×#884502
C2×#884492
C2×#884472
#117211112
C2×#638583
C2×#638483