Finite Group Databasebeta

← C2×#11723660 #13178511

Subgroups — C2×#11723660 #13178511

Order 1728.

1522 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
1C2×C3×#3828C212yesmaximal, complemented
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C2×#88812C211yesmaximal, characteristic, complemented
4C2×#88811C212yesmaximal, complemented
5#11723660C218yesmaximal, direct product factor
6C2×#88810C211yesmaximal, characteristic, complemented
7C2×#63829—32nomaximal
8C2×#63822—31nomaximal
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—22no—
11C3×#3828—24no—
12C3×#3828C2218yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4×C3⋊D3—22no—
15C2×C3×#634C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2214yescomplemented
19#88812C2214yescomplemented
20#88811C2218yescomplemented
21#88810C2214yescomplemented
22C2×#3828D312yescomplemented
23C2×C3×#235—32no—
24C2×C3×#235—32no—
25C2×C3×#235—62no—
26C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
27C22×C3×C3⋊C8—32no—
28C22×C3×C3⋊C8—61no—
29C2×#3719—32no—
30C2×#3718—31no—
31C2×#3717—32no—
32#63829—316no—
33C2×#3716—32no—
34C2×#3715—32no—
35C2×#3714—32no—
36#63822—38no—
37C2×#3713—31no—
38C23×C33D411yescharacteristic
39C22×C3×C3⋊D3—22no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4D412yes—
42C2×C33×C4C2311yescharacteristic
43C2×C33×C4C2312yes—
44C3×#638—28no—
45C2×C3×C3⋊Dic3—22no—
46C3×C4⋊(C3⋊D3)—22no—
47C3×C4⋊(C3⋊D3)C2314yes—
48C3×C4×C3⋊D3—28no—
49C3×#634—22no—
50C3×#634C2314yes—
51C3×C3⋊(C3⋊C8)C2312yes—
52C3×C3⋊(C3⋊C8)C2312yes—
53C2×#950—92no—
54C2×#904—91no—
55C22×C32×C4—21no—
56C22×C32×C4—22no—
57C22×C32×C4—41no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D312yes—
60C2×#638—22no—
61#3828—24no—
62#3828C2×D318yescomplemented
63C2×C4⋊(C3⋊D3)C2×D312yes—
64C2×C4×C3⋊D3—22no—
65C2×#634C2×D312yes—
66C2×C3×#67—64no—
67C2×C3×#67—62no—
68C2×C3×#67—62no—
69C3×#235—38no—
70C3×#235—38no—
71C3×#235—68no—
72C3×#235—68no—
73C3×#235—64no—
74C3×#235—64no—
75C2×C3×D12—32no—
76C2×C3×D12—32no—
77C2×C3×D12—62no—
78C2×C3×C4×D3—64no—
79C2×C3×C4×D3—62no—
80C2×C3×C4×D3—62no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—62no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D314yescomplemented
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—32no—
89C2×C3×C3⋊C8—38no—
90C2×C3×C3⋊C8—61no—
91C2×C3×C3⋊C8—61no—
92C2×C3×C3⋊C8—64no—
93#3719—38no—
94#3718—34no—
95#3717—38no—
96#3716—38no—
97#3715—38no—
98#3714—38no—
99#3713—34no—
100C22×C33—22no—
101C22×C33C2×D412yes—
102C22×C33C2×D411yescharacteristic
103C2×C3×C3⋊D3—24no—
104C2×C3×C3⋊D3—44no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D412yes—
107C33×C4C2411yescharacteristic
108C3×C3⋊Dic3—24no—
109C2×C3×#41—92no—
110C2×#235—32no—
111C2×#235—32no—
112C2×#235—62no—
113C22×C3×C8—91no—
114#950—916no—
115C2×#216—92no—
116C2×#215—92no—
117C2×#214—92no—
118C2×#213—92no—
119C22×C3⋊C8—32no—
120C22×C3⋊C8—61no—
121C2×Dic12—91no—
122#904—98no—
123C2×D24—91no—
124C2×#204—92no—
125C23×C32—21no—
126C23×C32—22no—
127C23×C32—41no—
128C23×C32D1211yescharacteristic
129C23×C32#6712yes—
130C22×C3⋊D3—22no—
131C22×C3×D3—64no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—24no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—24no—
142C2×C32×C4—42no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4—42no—
146C2×C32×C4D1211yescharacteristic
147C2×C32×C4D1212yes—
148C2×C32×C4#6712yes—
149C2×C32×C4#6714yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D312yes—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D314yes—
154#638—28no—
155C2×C3⋊Dic3—22no—
156C4⋊(C3⋊D3)—22no—
157C4⋊(C3⋊D3)C22×D314yes—
158C4×C3⋊D3—28no—
159#634—22no—
160#634C22×D314yes—
161C3×#67—616no—
162C3×#67—68no—
163C3×#67—68no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C3×D12—34no—
168C3×D12—34no—
169C3×D12—64no—
170C3×D12—64no—
171C3×D12—62no—
172C3×D12—62no—
173C3×C4×D3—616no—
174C3×C4×D3—68no—
175C3×C4×D3—68no—
176C3×Dic6—34no—
177C3×Dic6—34no—
178C3×Dic6—64no—
179C3×Dic6—64no—
180C3×Dic6—62no—
181C3×Dic6—62no—
182C3⋊(C3⋊C8)C22×D312yes—
183C3⋊(C3⋊C8)C22×D312yes—
184C3×C3⋊C8—34no—
185C3×C3⋊C8—34no—
186C3×C3⋊C8—62no—
187C3×C3⋊C8—62no—
188C2×#135—271noSylow
189C2×C33—22no—
190C2×C33#13512yescomplemented
191C2×C33C22×D411yescharacteristic
192C3×C3⋊D3—44no—
193C3×#41—98no—
194C3×#41—184no—
195C2×C3×Q8—92no—
196C2×C3×D4—92no—
197C2×C3×D4—182no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—41no—
201C22×C3×C4—182no—
202C22×C3×C4D3211yescharacteristic
203C22×C3×C4D3212yes—
204C2×#67—64no—
205C2×#67—62no—
206C2×#67—62no—
207#235—38no—
208#235—38no—
209#235—68no—
210#235—68no—
211#235—64no—
212#235—64no—
213C2×D12—32no—
214C2×D12—32no—
215C2×D12—62no—
216C2×C4×D3—64no—
217C2×C4×D3—62no—
218C2×C4×D3—62no—
219C2×Dic6—32no—
220C2×Dic6—32no—
221C2×Dic6—62no—
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C2×C3×C8—94no—
225#216—98no—
226#215—98no—
227#214—98no—
228#213—98no—
229C2×C3⋊C8—32no—
230C2×C3⋊C8—32no—
231C2×C3⋊C8—38no—
232C2×C3⋊C8—61no—
233C2×C3⋊C8—61no—
234C2×C3⋊C8—64no—
235Dic12—94no—
236D24—94no—
237#204—98no—
238C22×C32—22no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—22no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—24no—
245C22×C32—41no—
246C22×C32—42no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32C2×D1212yes—
250C22×C32C2×D1211yescharacteristic
251C22×C32C2×#6714yes—
252C22×C32C2×#6712yes—
253C2×C3⋊D3—24no—
254C2×C3⋊D3—44no—
255C2×C3×D3—68no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—128no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C32×C4—21no—
262C32×C4—22no—
263C32×C4—21no—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4—24no—
267C32×C4—41no—
268C32×C4—42no—
269C32×C4—41no—
270C32×C4C2×D1211yescharacteristic
271C32×C4C2×D1212yes—
272C32×C4C2×#6712yes—
273C32×C4C2×#6714yes—
274C32×C4C23×D311yescharacteristic
275C32×C4C23×D312yes—
276C3⋊Dic3—24no—
277C3×Dic3—68no—
278C3×Dic3—64no—
279C3×Dic3—64no—
280C2×#41—92no—
281#135—278no—
282C2×Q16—271no—
283C2×SD16—272no—
284C2×D8—271no—
285C22×C8—91no—
286C33C2×#13511yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—182no—
291C23×C3#62511yescharacteristic
292C23×C3#62612yes—
293C22×D3—64no—
294C22×D3—62no—
295C22×D3—62no—
296C3×Q8—94no—
297C3×Q8—182no—
298C3×D4—94no—
299C3×D4—188no—
300C3×D4—182no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—182no—
314C2×C3×C4—188no—
315C2×C3×C4#62512yes—
316C2×C3×C4#62511yescharacteristic
317C2×C3×C4#62614yes—
318C2×C3×C4#62612yes—
319C2×C3×C4C2×D3211yescharacteristic
320C2×C3×C4C2×D3212yes—
321C2×C3×C4C2×D3212yes—
322C2×C3×C4C2×D3214yes—
323#67—616no—
324#67—68no—
325#67—68no—
326C2×Dic3—64no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329D12—34no—
330D12—34no—
331D12—64no—
332D12—64no—
333D12—62no—
334D12—62no—
335C4×D3—616no—
336C4×D3—68no—
337C4×D3—68no—
338Dic6—34no—
339Dic6—34no—
340Dic6—64no—
341Dic6—64no—
342Dic6—62no—
343Dic6—62no—
344C3×C8—92no—
345C3×C8—92no—
346C3⋊C8—34no—
347C3⋊C8—34no—
348C3⋊C8—62no—
349C3⋊C8—62no—
350C2×C32—22no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—21no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—24no—
357C2×C32—42no—
358C2×C32—41no—
359C2×C32—44no—
360C2×C32—44no—
361C2×C32#90412yescomplemented
362C2×C32#95014yescomplemented
363C2×C32C22×D1211yescharacteristic
364C2×C32C22×#6712yes—
365C3⋊D3—44no—
366C3×D3—128no—
367C3×D3—124no—
368C3×D3—124no—
369#41—98no—
370#41—184no—
371C2×Q8—92no—
372C2×D4—92no—
373C2×D4—182no—
374C22×C4—182no—
375C22×C4#111311yescharacteristic
376Q16—274no—
377SD16—278no—
378D8—274no—
379C2×C8—91no—
380C2×C8—91no—
381C2×C8—94no—
382C22×C3—24no—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—24no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—41no—
393C22×C3—184no—
394C22×C3—364no—
395C22×C3C2×#62512yes—
396C22×C3C2×#62511yescharacteristic
397C22×C3C2×#62614yes—
398C22×C3C2×#62612yes—
399C2×D3—68no—
400C2×D3—64no—
401C2×D3—64no—
402C2×D3—128no—
403C2×D3—124no—
404C2×D3—124no—
405C3×C4—21no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—24no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—41no—
414C3×C4—184no—
415C3×C4C2×#62511yescharacteristic
416C3×C4C2×#62512yes—
417C3×C4C2×#62612yes—
418C3×C4C2×#62614yes—
419C3×C4C22×D3211yescharacteristic
420C3×C4C22×D3212yes—
421Dic3—68no—
422Dic3—64no—
423Dic3—64no—
424C32—21no—
425C32—22no—
426C32—41no—
427C32C2×#90411yescharacteristic, complemented
428C32C2×#95012yescomplemented
429C23—182no—
430C23#629511yescharacteristic
431Q8—94no—
432Q8—182no—
433D4—94no—
434D4—188no—
435D4—182no—
436C2×C4—182no—
437C2×C4—188no—
438C2×C4#629512yes—
439C2×C4#629511yescharacteristic
440C2×C4C2×#111311yescharacteristic
441C2×C4C2×#111312yes—
442C8—92no—
443C8—92no—
444C2×C3—24no—
445C2×C3—22no—
446C2×C3—24no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—24no—
450C2×C3—22no—
451C2×C3—44no—
452C2×C3—44no—
453C2×C3—41no—
454C2×C3—42no—
455C2×C3—364no—
456C2×C3#6382212yescomplemented
457C2×C3#6382914yescomplemented
458C2×C3C22×#62511yescharacteristic
459C2×C3C22×#62612yes—
460D3—128no—
461D3—124no—
462D3—124no—
463C22—22no—
464C22—184no—
465C22—364no—
466C22C2×#629512yes—
467C22C2×#629511yescharacteristic
468C4—184no—
469C4C2×#629512yes—
470C4C2×#629511yescharacteristic
471C4C22×#111311yescharacteristic
472C3—22no—
473C3—21no—
474C3—41no—
475C3C2×#6382211yescharacteristic, complemented
476C3C2×#6382912yescomplemented
477C2—22no—
478C2—364no—
479C2#1172366012yesdirect product factor
480C2C22×#629511yescharacteristic
subgroupG/H
centerC2×C4#6295
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C4#1113
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C2×#88812C2semidirect
4. C2×#88811C2semidirect
5. #11723660C2direct (N × H)
6. C2×#88810C2semidirect
12. C3×#3828C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
19. #88812C22semidirect
20. #88811C22semidirect
21. #88810C22semidirect
22. C2×#3828D3semidirect
26. C22×C3⋊(C3⋊C8)D3semidirect
62. #3828C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
190. C2×C33#135semidirect
286. C33C2×#135semidirect
361. C2×C32#904semidirect
362. C2×C32#950semidirect
427. C32C2×#904semidirect
428. C32C2×#950semidirect
456. C2×C3#63822semidirect
457. C2×C3#63829semidirect
475. C3C2×#63822semidirect
476. C3C2×#63829semidirect
479. C2#11723660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C22×C3×C3⋊(C3⋊C8)2
C2×#888122
C2×#888112
#117236602
C2×#888102
C2×#638293
C2×#638223