Finite Group Databasebeta

← C2×#11721121 #13150672

Subgroups — C2×#11721121 #13150672

Order 1728.

1263 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
2C2×C4×#1084C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5C2×#88450C211yesmaximal, characteristic
6C2×#88448C211yesmaximal, characteristic, complemented
7C2×#88447C211yesmaximal, characteristic, complemented
8#11721121C218yesmaximal, direct product factor
9C2×#63877—31nomaximal
10C2×#63836—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
18C2×((C3⋊D3)⋊C3)⋊C4C418yescomplemented
19C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
20C4×#1084C2214yescomplemented
21#3752⋊C3C2214yes—
22#3750⋊C3C2214yescomplemented
23#88450C2214yes—
24#88448C2214yescomplemented
25#88447C2214yescomplemented
26C22×#624—31no—
27C22×D3×Dic3—31no—
28C2×#3752D311yescharacteristic
29C2×#3750D311yescharacteristic, complemented
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×Dic3⋊C4—31no—
32C2×C3×C4×Dic3—31no—
33C2×C3×C4×Dic3—31no—
34C2×C3×C4×Dic3—61no—
35C2×#3724—31no—
36C2×Dic3⋊Dic3—31no—
37C2×#3722—31no—
38C2×(C2×D3)⋊Dic3—31no—
39C2×Dic32—31no—
40C2×Dic32—31no—
41#63877—38no—
42#63836—38no—
43C23×H(3,𝔽3)C2311yescharacteristic
44C22×(C3⋊D3)⋊C3—22no—
45C22×(C3⋊D3)⋊C3—22no—
46C22×(C3⋊D3)⋊C3C2×C414yes—
47C22×(C3⋊D3)⋊C3C2312yes—
48C2×C4×H(3,𝔽3)—22no—
49C2×C4×H(3,𝔽3)C2312yes—
50C2×#1084—22no—
51C2×#1084C2×C414yes—
52C2×#1084C2312yes—
53C2×#1084C2312yes—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3—22no—
56C2×(C3⋊Dic3)⋊C3C2×C414yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60((C3⋊D3)⋊C3)⋊C4C2×C4116yescomplemented
61C2×#892—91no—
62C2×#877—91no—
63C2×#872—91no—
64C23×C3⋊D3C2×D311yescharacteristic
65C23×C3×D3—31no—
66C22×C32×C4—21no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3C2×D311yescharacteristic
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—61no—
80C22×C3×Dic3—61no—
81C2×#624—38no—
82C2×#624—34no—
83C2×D3×Dic3—38no—
84C2×D3×Dic3—34no—
85#3752C2×D314yes—
86#3750C2×D314yescomplemented
87C3×(C2×D3)⋊C4—34no—
88C3×Dic3⋊C4—34no—
89C3×C4×Dic3—34no—
90C3×C4×Dic3—34no—
91C3×C4×Dic3—64no—
92#3724—34no—
93Dic3⋊Dic3—34no—
94#3722—34no—
95(C2×D3)⋊Dic3—34no—
96Dic32—34no—
97Dic32—34no—
98C22×H(3,𝔽3)C22×C412yes—
99C22×H(3,𝔽3)#4112yes—
100C22×H(3,𝔽3)#4112yes—
101C22×H(3,𝔽3)C2411yescharacteristic
102C2×(C3⋊D3)⋊C3—28no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3C22×C414yes—
106C4×H(3,𝔽3)—22no—
107#1084—22no—
108#1084C22×C414yes—
109(C3⋊Dic3)⋊C3—22no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3C22×C414yes—
112C23×Dic3—91no—
113C22×C4×D3—91no—
114C22×C4×D3—91no—
115C2×C3×C4⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C42—91no—
118C2×#217—91no—
119C2×(C2×D3)⋊C4—31no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×C4⋊Dic3—91no—
124C2×Dic3⋊C4—31no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×C4×Dic3—31no—
129C2×C4×Dic3—31no—
130C2×C4×Dic3—61no—
131C2×C4×Dic3—91no—
132C2×C4×Dic3—91no—
133C2×C4×Dic3—91no—
134#892—98no—
135#877—98no—
136#872—98no—
137C23×C32—21no—
138C23×C32C22×D311yescharacteristic
139C23×C32C22×D311yescharacteristic
140C22×C3⋊D3—22no—
141C22×C3⋊D3—22no—
142C22×C3⋊D3C4×D314yes—
143C22×C3⋊D3C22×D312yes—
144C22×C3×D3—34no—
145C22×C3×D3—32no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C2×C32×C4—24no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3C4×D314yes—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3⋊Dic3C22×D312yes—
160C2×C3×Dic3—34no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—62no—
178#624—316no—
179D3×Dic3—316no—
180C2×#117—271noSylow
181C2×H(3,𝔽3)#11714yescomplemented
182C2×H(3,𝔽3)C23×C411yescharacteristic
183C2×H(3,𝔽3)C2×#4111yescharacteristic
184C2×H(3,𝔽3)C2×#4111yescharacteristic
185(C3⋊D3)⋊C3—24no—
186C24×C3—91no—
187C23×D3—31no—
188C23×D3—91no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—61no—
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4—91no—
196C22×C3×C4—91no—
197C22×C3×C4D3211yescharacteristic
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—31no—
203C22×Dic3—31no—
204C22×Dic3—31no—
205C22×Dic3—61no—
206C22×Dic3—61no—
207C22×Dic3—91no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—94no—
211C22×Dic3—98no—
212C2×C4×D3—94no—
213C2×C4×D3—98no—
214C2×C4×D3—94no—
215C2×C4×D3—98no—
216C3×C4⋊C4—94no—
217C3×C22⋊C4—94no—
218C3×C42—94no—
219#217—94no—
220(C2×D3)⋊C4—34no—
221(C2×D3)⋊C4—94no—
222(C2×D3)⋊C4—94no—
223(C2×D3)⋊C4—94no—
224C4⋊Dic3—94no—
225Dic3⋊C4—34no—
226Dic3⋊C4—94no—
227Dic3⋊C4—94no—
228Dic3⋊C4—94no—
229C4×Dic3—34no—
230C4×Dic3—34no—
231C4×Dic3—64no—
232C4×Dic3—94no—
233C4×Dic3—94no—
234C4×Dic3—94no—
235C22×C32—21no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32C2×C4×D312yes—
240C22×C32C2×C4×D312yes—
241C22×C32#23712yes—
242C22×C32#23712yes—
243C22×C32#23712yes—
244C22×C32#23912yes—
245C22×C32C23×D311yescharacteristic
246C22×C32C23×D311yescharacteristic
247C2×C3⋊D3—28no—
248C2×C3⋊D3—22no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3C2×C4×D314yes—
251C2×C3×D3—34no—
252C2×C3×D3—68no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C32×C4—24no—
256C32×C4—22no—
257C32×C4—22no—
258C3⋊Dic3—22no—
259C3⋊Dic3—22no—
260C3⋊Dic3C2×C4×D314yes—
261C3×Dic3—34no—
262C3×Dic3—34no—
263C3×Dic3—34no—
264C3×Dic3—64no—
265C3×Dic3—64no—
266C3×Dic3—62no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C23×C4—271no—
271#117—278no—
272C2×C4⋊C4—91no—
273C2×C4⋊C4—271no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—271no—
276C2×C42—91no—
277C2×C42—271no—
278H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
279C23×C3—31no—
280C23×C3—31no—
281C23×C3—61no—
282C23×C3—92no—
283C23×C3—94no—
284C23×C3—182no—
285C23×C3—182no—
286C23×C3C2×D3211yescharacteristic
287C22×D3—32no—
288C22×D3—34no—
289C22×D3—62no—
290C22×D3—62no—
291C22×D3—94no—
292C22×D3—92no—
293C22×D3—182no—
294C22×D3—182no—
295C2×C3×C4—22no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—64no—
301C2×C3×C4—62no—
302C2×C3×C4—92no—
303C2×C3×C4—94no—
304C2×C3×C4—92no—
305C2×C3×C4—94no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—92no—
309C2×C3×C4—182no—
310C2×C3×C4—182no—
311C2×C3×C4—182no—
312C2×C3×C4C2×D3212yes—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—34no—
316C2×Dic3—34no—
317C2×Dic3—32no—
318C2×Dic3—34no—
319C2×Dic3—32no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—64no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—64no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—94no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335C2×Dic3—916no—
336C2×Dic3—182no—
337C2×Dic3—182no—
338C4×D3—916no—
339C4×D3—916no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—24no—
344C2×C32#87714yescomplemented
345C2×C32#89214yescomplemented
346C2×C32C22×C4×D311yescharacteristic
347C2×C32C22×C4×D311yescharacteristic
348C2×C32C2×#23711yescharacteristic
349C2×C32C2×#23711yescharacteristic
350C2×C32C2×#23711yescharacteristic
351C2×C32C2×#23911yescharacteristic
352C3⋊D3—24no—
353C3×D3—64no—
354C24—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4—91no—
359C22×C4—91no—
360C22×C4—274no—
361C22×C4—278no—
362C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
363C4⋊C4—94no—
364C4⋊C4—274no—
365C22⋊C4—94no—
366C22⋊C4—274no—
367C42—94no—
368C42—274no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—32no—
372C22×C3—32no—
373C22×C3—32no—
374C22×C3—32no—
375C22×C3—32no—
376C22×C3—32no—
377C22×C3—61no—
378C22×C3—62no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—94no—
382C22×C3—182no—
383C22×C3—188no—
384C22×C3—182no—
385C22×C3#379612yes—
386C22×C3C4×D3212yes—
387C22×C3#380512yes—
388C22×C3C22×D3211yescharacteristic
389C2×D3—34no—
390C2×D3—62no—
391C2×D3—62no—
392C2×D3—68no—
393C2×D3—94no—
394C2×D3—182no—
395C2×D3—188no—
396C2×D3—182no—
397C3×C4—22no—
398C3×C4—62no—
399C3×C4—62no—
400C3×C4—64no—
401C3×C4—94no—
402C3×C4—94no—
403C3×C4—182no—
404C3×C4—182no—
405C3×C4—182no—
406Dic3—34no—
407Dic3—34no—
408Dic3—34no—
409Dic3—64no—
410Dic3—62no—
411Dic3—62no—
412Dic3—64no—
413Dic3—62no—
414Dic3—62no—
415Dic3—94no—
416Dic3—182no—
417Dic3—182no—
418C32—21no—
419C32C2×#87711yescharacteristic, complemented
420C32C2×#89211yescharacteristic, complemented
421C23—92no—
422C23—94no—
423C23—182no—
424C23—182no—
425C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C2×C4—22no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—94no—
430C2×C4—94no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—182no—
435C2×C4—182no—
436C2×C4—182no—
437C2×C4—2716no—
438C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—34no—
442C2×C3—34no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—61no—
448C2×C3—64no—
449C2×C3—61no—
450C2×C3—61no—
451C2×C3—184no—
452C2×C3#6387814yes—
453C2×C3C2×#379611yescharacteristic
454C2×C3C2×C4×D3211yescharacteristic
455C2×C3C2×#380511yescharacteristic
456D3—64no—
457D3—184no—
458C22—94no—
459C22—182no—
460C22—188no—
461C22—182no—
462C22#8864912yes—
463C22C4×((C3⋊D3)⋊C3)⋊C212yes—
464C22#8867212yes—
465C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
466C4—22no—
467C4—94no—
468C4—94no—
469C4—182no—
470C4—182no—
471C4—182no—
472C3—31no—
473C3—31no—
474C3—61no—
475C3C2×#6387811yescharacteristic
476C2—184no—
477C2#1172112114yesdirect product factor
478C2C2×#8864911yescharacteristic
479C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C2C2×#8867211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1084C2semidirect
4. C2×#3750⋊C3C2semidirect
6. C2×#88448C2semidirect
7. C2×#88447C2semidirect
8. #11721121C2direct (N × H)
18. C2×((C3⋊D3)⋊C3)⋊C4C4semidirect
20. C4×#1084C22semidirect
22. #3750⋊C3C22semidirect
24. #88448C22semidirect
25. #88447C22semidirect
29. C2×#3750D3semidirect
60. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
86. #3750C2×D3semidirect
181. C2×H(3,𝔽3)#117semidirect
278. H(3,𝔽3)C2×#117semidirect
344. C2×C32#877semidirect
345. C2×C32#892semidirect
419. C32C2×#877semidirect
420. C32C2×#892semidirect
477. C2#11721121direct (N × H)

Maximal subgroups

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

subgroupindex
C22×((C3⋊D3)⋊C3)⋊C42
C2×C4×#10842
C2×#3752⋊C32
C2×#3750⋊C32
C2×#884502
C2×#884482
C2×#884472
#117211212
C2×#638773
C2×#638363