Finite Group Databasebeta

← C2×#11723654 #13178453

Subgroups — C2×#11723654 #13178453

Order 1728.

1298 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C212yesmaximal, complemented
2C2×C3×#3746C211yesmaximal, characteristic, complemented
3C2×C3×#3745C211yesmaximal, characteristic, complemented
4C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
5C2×#88811C212yesmaximal, complemented
6#11723654C218yesmaximal, direct product factor
7C2×#63810—31nomaximal
8C2×#63808—31nomaximal
9C2×#63807—31nomaximal
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2212yes—
13C2×#6295—22no—
14C2×#6294—22no—
15#89041—24no—
16#89041C2218yescomplemented
17C2×C33×C8C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#3746C2214yescomplemented
20C3×#3745C2214yescomplemented
21C3×C8×C3⋊D3C2214yescomplemented
22#88811C2218yescomplemented
23C2×#3799—32no—
24C2×#3796—32no—
25C2×#3795—32no—
26C2×#3746D311yescharacteristic, complemented
27C2×#3745D311yescharacteristic, complemented
28C2×C8×C3⋊D3D311yescharacteristic, complemented
29C2×C3×Dic12—31no—
30C2×C3×Dic12—31no—
31C2×C3×Dic12—61no—
32C2×C3×D24—31no—
33C2×C3×D24—31no—
34C2×C3×D24—61no—
35C2×C3×D3×C8—31no—
36C2×C3×D3×C8—31no—
37C2×C3×D3×C8—61no—
38C2×#3717—32no—
39C2×#3716—32no—
40C2×#3715—32no—
41#63810—38no—
42#63808—38no—
43#63807—38no—
44C22×C3×C3⋊D3—22no—
45C22×C3×C3⋊D3D411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—22no—
50C3×C4⋊(C3⋊D3)C2314yes—
51C3×C4×C3⋊D3D412yes—
52C3×C4×C3⋊D3C2312yes—
53C3×#634—22no—
54C3×#634C2314yes—
55#6295—28no—
56#6294—28no—
57C33×C8C2312yes—
58C3×C3⋊(C3⋊C8)C2312yes—
59C2×#919—91no—
60C2×#912—91no—
61C2×#904—91no—
62C2×C4⋊(C3⋊D3)C2×D312yes—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×#634C2×D312yes—
65C2×C3×D12—32no—
66C2×C3×D12—32no—
67C2×C3×D12—62no—
68C2×C3×C4×D3—31no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—62no—
74C2×#626—62no—
75C2×#626—62no—
76C2×#625—62no—
77C2×#624—62no—
78C2×D3×Dic3—62no—
79C2×D3×Dic3—62no—
80#3799—38no—
81#3799—64no—
82#3796—38no—
83#3796—64no—
84#3795—38no—
85#3795—64no—
86C2×C32×C8—21no—
87C2×C32×C8—21no—
88C2×C32×C8—21no—
89C2×C32×C8—41no—
90C2×C32×C8C2×D311yescharacteristic
91C2×C32×C8C2×D311yescharacteristic
92C2×C32×C8C2×D311yescharacteristic
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94#3746C2×D314yescomplemented
95#3745C2×D314yescomplemented
96C8×C3⋊D3C2×D314yescomplemented
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—61no—
100C3×Dic12—34no—
101C3×Dic12—34no—
102C3×Dic12—64no—
103C3×D24—34no—
104C3×D24—34no—
105C3×D24—64no—
106C3×D3×C8—34no—
107C3×D3×C8—34no—
108C3×D3×C8—64no—
109#3717—38no—
110#3716—38no—
111#3715—38no—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—24no—
115C2×C3×C3⋊D3—44no—
116C2×C3×C3⋊D3C2×D412yes—
117C33×C4C2×D411yescharacteristic
118C33×C4C2411yescharacteristic
119C3×C3⋊Dic3—24no—
120C3×C3⋊Dic3C2×D412yes—
121C2×#239—92no—
122C2×#237—92no—
123C2×#235—92no—
124C2×C3×Q16—91no—
125C2×C3×D8—91no—
126C22×C3×C8—91no—
127C2×#215—92no—
128C2×#214—92no—
129#919—98no—
130#912—98no—
131C2×Dic12—31no—
132C2×Dic12—31no—
133C2×Dic12—61no—
134#904—98no—
135C2×D24—31no—
136C2×D24—31no—
137C2×D24—61no—
138C2×#204—92no—
139C2×D3×C8—31no—
140C2×D3×C8—31no—
141C2×D3×C8—61no—
142C22×C3⋊D3—22no—
143C22×C3⋊D3D1211yescharacteristic
144C22×C3×D3—31no—
145C22×C3×D3—31no—
146C22×C3×D3—61no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—122no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3D1211yescharacteristic
159C4⋊(C3⋊D3)—22no—
160C4⋊(C3⋊D3)C22×D314yes—
161C4×C3⋊D3D1212yes—
162C4×C3⋊D3C22×D312yes—
163#634—22no—
164#634C22×D314yes—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—122no—
171C3×D12—34no—
172C3×D12—34no—
173C3×D12—64no—
174C3×D12—62no—
175C3×D12—62no—
176C3×D12—122no—
177C3×C4×D3—32no—
178C3×C4×D3—32no—
179C3×C4×D3—32no—
180C3×C4×D3—32no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×Dic6—34no—
184C3×Dic6—34no—
185C3×Dic6—64no—
186C3×Dic6—62no—
187C3×Dic6—62no—
188C3×Dic6—122no—
189#626—68no—
190#626—68no—
191#625—68no—
192#624—68no—
193D3×Dic3—68no—
194D3×Dic3—68no—
195C32×C8—22no—
196C32×C8—22no—
197C32×C8—22no—
198C32×C8—42no—
199C32×C8C22×D312yes—
200C32×C8C22×D312yes—
201C32×C8C22×D312yes—
202C3⋊(C3⋊C8)C22×D312yes—
203C3×C3⋊C8—32no—
204C3×C3⋊C8—32no—
205C3×C3⋊C8—62no—
206C2×#135—271noSylow
207C2×C33#13512yescomplemented
208C2×C33C22×D411yescharacteristic
209C3×C3⋊D3—22no—
210C3×C3⋊D3—44no—
211C2×C3×Q8—92no—
212C2×C3×D4—92no—
213C22×C3×C4—91no—
214C2×#67—182no—
215C2×#67—182no—
216C22×Dic3—182no—
217#239—98no—
218#239—184no—
219#237—98no—
220#237—184no—
221#235—98no—
222#235—184no—
223C2×D12—32no—
224C2×D12—32no—
225C2×D12—62no—
226C2×D12—182no—
227C2×C4×D3—31no—
228C2×C4×D3—31no—
229C2×C4×D3—61no—
230C2×C4×D3—182no—
231C2×C4×D3—182no—
232C2×Dic6—32no—
233C2×Dic6—32no—
234C2×Dic6—62no—
235C3×Q16—94no—
236C3×D8—94no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—41no—
241C2×C3×C8—91no—
242C2×C3×C8—94no—
243C2×C3×C8D3211yescharacteristic
244C2×C3×C8D3211yescharacteristic
245C2×C3×C8D3211yescharacteristic
246#215—98no—
247#214—98no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—61no—
251Dic12—34no—
252Dic12—34no—
253Dic12—64no—
254D24—34no—
255D24—34no—
256D24—64no—
257#204—98no—
258D3×C8—34no—
259D3×C8—34no—
260D3×C8—64no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32C2×D1211yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C2×C3⋊D3—22no—
269C2×C3⋊D3—24no—
270C2×C3⋊D3—44no—
271C2×C3⋊D3C2×D1212yes—
272C2×C3×D3—32no—
273C2×C3×D3—32no—
274C2×C3×D3—62no—
275C2×C3×D3—64no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—124no—
281C2×C3×D3—124no—
282C2×C3×D3—124no—
283C2×C3×D3—244no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4C2×D1211yescharacteristic
293C32×C4D3×D411yescharacteristic
294C32×C4D3×D411yescharacteristic
295C32×C4C23×D311yescharacteristic
296C32×C4C23×D311yescharacteristic
297C32×C4C23×D311yescharacteristic
298C3⋊Dic3—24no—
299C3⋊Dic3C2×D1212yes—
300C3×Dic3—32no—
301C3×Dic3—32no—
302C3×Dic3—62no—
303C3×Dic3—64no—
304C3×Dic3—64no—
305C3×Dic3—124no—
306C2×#41—272no—
307#135—278no—
308C2×Q16—91no—
309C2×SD16—272no—
310C2×D8—91no—
311C22×C8—91no—
312C33C2×#13511yescharacteristic, complemented, Sylow
313C23×C3—91no—
314C23×C3—182no—
315C22×D3—31no—
316C22×D3—31no—
317C22×D3—61no—
318C22×D3—62no—
319C22×D3—62no—
320C22×D3—122no—
321C3×Q8—94no—
322C3×Q8—182no—
323C3×D4—94no—
324C3×D4—182no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—91no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—182no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336#67—188no—
337#67—188no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—61no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—122no—
344C2×Dic3—188no—
345D12—34no—
346D12—34no—
347D12—64no—
348D12—62no—
349D12—62no—
350D12—122no—
351D12—188no—
352C4×D3—32no—
353C4×D3—32no—
354C4×D3—32no—
355C4×D3—32no—
356C4×D3—62no—
357C4×D3—62no—
358C4×D3—188no—
359C4×D3—188no—
360Dic6—34no—
361Dic6—34no—
362Dic6—64no—
363Dic6—62no—
364Dic6—62no—
365Dic6—122no—
366C3×C8—22no—
367C3×C8—22no—
368C3×C8—22no—
369C3×C8—42no—
370C3×C8—92no—
371C3×C8C2×D3212yes—
372C3×C8C2×D3212yes—
373C3×C8C2×D3212yes—
374C3⋊C8—32no—
375C3⋊C8—32no—
376C3⋊C8—62no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—42no—
384C2×C32—41no—
385C2×C32#90412yescomplemented
386C2×C32#91212yescomplemented
387C2×C32#91912yescomplemented
388C2×C32C22×D1211yescharacteristic
389C2×C32C2×D3×D411yescharacteristic
390C2×C32C2×D3×D411yescharacteristic
391C3⋊D3—22no—
392C3⋊D3—44no—
393C3×D3—64no—
394C3×D3—62no—
395C3×D3—62no—
396C3×D3—124no—
397C3×D3—124no—
398C3×D3—244no—
399#41—278no—
400#41—544no—
401C2×Q8—92no—
402C2×D4—92no—
403C2×D4—542no—
404C22×C4—91no—
405C22×C4—542no—
406Q16—94no—
407SD16—278no—
408D8—94no—
409C2×C8—91no—
410C2×C8—94no—
411C2×C8#111311yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—92no—
417C22×C3—182no—
418C22×C3—184no—
419C22×C3—364no—
420C22×C3D3×D1211yescharacteristic
421C22×C3D3×D1211yescharacteristic
422C22×C3#380211yescharacteristic
423C2×D3—32no—
424C2×D3—32no—
425C2×D3—62no—
426C2×D3—64no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—64no—
430C2×D3—64no—
431C2×D3—124no—
432C2×D3—124no—
433C2×D3—124no—
434C2×D3—244no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—92no—
444C3×C4—184no—
445C3×C4D3×D1211yescharacteristic
446C3×C4D3×D1211yescharacteristic
447C3×C4#380211yescharacteristic
448C3×C4C22×D3211yescharacteristic
449C3×C4C22×D3211yescharacteristic
450C3×C4C22×D3211yescharacteristic
451Dic3—32no—
452Dic3—32no—
453Dic3—62no—
454Dic3—64no—
455Dic3—64no—
456Dic3—124no—
457C32—21no—
458C32—21no—
459C32—21no—
460C32—41no—
461C32C2×#90411yescharacteristic, complemented
462C32C2×#91211yescharacteristic, complemented
463C32C2×#91911yescharacteristic, complemented
464C23—91no—
465C23—182no—
466Q8—94no—
467Q8—182no—
468D4—94no—
469D4—182no—
470D4—548no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—182no—
475C2×C4—548no—
476C2×C4C2×#111311yescharacteristic
477C8—92no—
478C8C2×#111312yes—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—41no—
486C2×C3—42no—
487C2×C3—182no—
488C2×C3—364no—
489C2×C3#6380712yescomplemented
490C2×C3#6380812yescomplemented
491C2×C3#6381012yescomplemented
492C2×C3C2×D3×D1211yescharacteristic
493C2×C3C2×D3×D1211yescharacteristic
494C2×C3C2×#380211yescharacteristic
495D3—64no—
496D3—62no—
497D3—62no—
498D3—124no—
499D3—124no—
500D3—244no—
501C22—92no—
502C22—182no—
503C22—184no—
504C22—364no—
505C22C4⋊#111311yescharacteristic
506C4—92no—
507C4—184no—
508C4C4⋊#111311yescharacteristic
509C4C22×#111311yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3C2×#6380711yescharacteristic, complemented
515C3C2×#6380811yescharacteristic, complemented
516C3C2×#6381011yescharacteristic, complemented
517C2—182no—
518C2—364no—
519C2#1172365412yesdirect product factor
520C2C2×C4⋊#111311yescharacteristic
subgroupG/H
centerC22C4⋊#1113
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8#1113
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. C2×C3×#3746C2semidirect
3. C2×C3×#3745C2semidirect
4. C2×C3×C8×C3⋊D3C2semidirect
5. C2×#88811C2semidirect
6. #11723654C2direct (N × H)
16. #89041C22semidirect
19. C3×#3746C22semidirect
20. C3×#3745C22semidirect
21. C3×C8×C3⋊D3C22semidirect
22. #88811C22semidirect
26. C2×#3746D3semidirect
27. C2×#3745D3semidirect
28. C2×C8×C3⋊D3D3semidirect
94. #3746C2×D3semidirect
95. #3745C2×D3semidirect
96. C8×C3⋊D3C2×D3semidirect
207. C2×C33#135semidirect
312. C33C2×#135semidirect
385. C2×C32#904semidirect
386. C2×C32#912semidirect
387. C2×C32#919semidirect
461. C32C2×#904semidirect
462. C32C2×#912semidirect
463. C32C2×#919semidirect
489. C2×C3#63807semidirect
490. C2×C3#63808semidirect
491. C2×C3#63810semidirect
514. C3C2×#63807semidirect
515. C3C2×#63808semidirect
516. C3C2×#63810semidirect
519. C2#11723654direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890412
C2×C3×#37462
C2×C3×#37452
C2×C3×C8×C3⋊D32
C2×#888112
#117236542
C2×#638103
C2×#638083
C2×#638073