Finite Group Databasebeta

← C2×#11723708 #13178731

Subgroups — C2×#11723708 #13178731

Order 1728.

1522 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic
3C2×C3×#3755C212yesmaximal, complemented
4#11723708C218yesmaximal, direct product factor
5C2×#88811C212yesmaximal, complemented
6C2×#88808C211yesmaximal, characteristic, complemented
7C2×#63940—31nomaximal
8C2×D3×#215—32nomaximal
9C22×#1113—21no—
10C2×C33×Q8C2211yescharacteristic, complemented
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3C2214yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634—21no—
17C2×C3×#634C2211yescharacteristic, complemented
18C2×#6295—21no—
19C4⋊#1113—22no—
20C4⋊#1113C2214yes—
21C3×#3755C2218yescomplemented
22C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
23#88811C2218yescomplemented
24#88808C2214yescomplemented
25C2×Q8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×Q8—32no—
27C2×C3×D3×Q8—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30C2×#3755D312yescomplemented
31C2×C3×#215—32no—
32C2×C3×#215—32no—
33C2×C3×#215—62no—
34#63940—38no—
35D3×#215—316no—
36C2×#3717—32no—
37C2×#3716—32no—
38C2×#3715—32no—
39C2×#3710—31no—
40C2×D3×C3⋊C8—32no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3D411yescharacteristic
43C2×#1113—24no—
44C2×#1113—44no—
45C33×Q8—21no—
46C33×Q8C2312yescomplemented
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—22no—
52C3×C4⋊(C3⋊D3)C2314yes—
53C3×C4×C3⋊D3—24no—
54C3×C4×C3⋊D3D412yes—
55C3×C4×C3⋊D3C2312yes—
56C3×#634—21no—
57C3×#634—24no—
58C3×#634C2312yescomplemented
59#6295—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C22×#215—91no—
62C2×D3×SD16—92no—
63C22×D32—62no—
64C22×D32—61no—
65C2×C32×Q8—21no—
66C2×C32×Q8—22no—
67C2×C32×Q8—41no—
68C2×C32×Q8C2×D311yescharacteristic, complemented
69C2×C32×Q8C2×D312yescomplemented
70Q8×C3⋊D3—22no—
71Q8×C3⋊D3C2×D314yescomplemented
72C2×C4⋊(C3⋊D3)C2×D312yes—
73C2×C4×C3⋊D3—21no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×#634—21no—
76C2×#634C2×D311yescharacteristic, complemented
77C3×D3×Q8—38no—
78C3×D3×Q8—64no—
79C3×D3×Q8—64no—
80C3×D3×Q8—64no—
81C2×C3×D12—32no—
82C2×C3×D12—32no—
83C2×C3×D12—62no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—62no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—62no—
88C2×C3×Dic6—32no—
89C2×C3×Dic6—62no—
90C2×C3×Dic6—61no—
91C2×C3×Dic6—62no—
92C2×#626—62no—
93C2×#625—61no—
94#3802—34no—
95#3802—62no—
96D3×D12—38no—
97D3×D12—64no—
98#3755C2×D318yescomplemented
99C2×C3⋊(C3⋊C8)C2×D312yescomplemented
100C3×#215—38no—
101C3×#215—38no—
102C3×#215—68no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—32no—
105C2×C3×C3⋊C8—62no—
106#3717—38no—
107#3716—38no—
108#3715—38no—
109#3710—34no—
110D3×C3⋊C8—38no—
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—24no—
113C2×C3×C3⋊D3—44no—
114C2×C3×C3⋊D3SD1614yescomplemented
115C2×C3×C3⋊D3C2×D412yes—
116#1113—48no—
117C33×C4—22no—
118C33×C4C2×D411yescharacteristic
119C33×C4C2411yescharacteristic
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3C2×D412yes—
122C22×C3×Q8—91no—
123C2×D3×Q8—32no—
124C2×D3×Q8—61no—
125C2×D3×D4—92no—
126C22×D12—91no—
127C2×C3×SD16—92no—
128C2×#215—32no—
129C2×#215—32no—
130C2×#215—62no—
131C2×#215—92no—
132C2×#215—98no—
133C2×#214—92no—
134C22×C3⋊C8—91no—
135D3×SD16—916no—
136C2×#204—92no—
137C2×D3×C8—92no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3#6711yescharacteristic
140C22×C3×D3—32no—
141C22×C3×D3—61no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—122no—
145C2×D32—68no—
146C2×D32—64no—
147C2×D32—128no—
148C2×D32—124no—
149C32×Q8—21no—
150C32×Q8—22no—
151C32×Q8—22no—
152C32×Q8—22no—
153C32×Q8—24no—
154C32×Q8—42no—
155C32×Q8—42no—
156C32×Q8—42no—
157C32×Q8C22×D312yescomplemented
158C32×Q8C22×D314yescomplemented
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4—42no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3#6711yescharacteristic
171C4⋊(C3⋊D3)—22no—
172C4⋊(C3⋊D3)C22×D314yes—
173C4×C3⋊D3—24no—
174C4×C3⋊D3#6712yes—
175C4×C3⋊D3C22×D312yes—
176#634—21no—
177#634—24no—
178#634C22×D312yescomplemented
179C2×C3×Dic3—32no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—62no—
183C3×D12—34no—
184C3×D12—34no—
185C3×D12—64no—
186C3×D12—62no—
187C3×D12—62no—
188C3×D12—122no—
189C3×C4×D3—34no—
190C3×C4×D3—34no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×C4×D3—68no—
194C3×C4×D3—68no—
195C3×Dic6—34no—
196C3×Dic6—62no—
197C3×Dic6—68no—
198C3×Dic6—68no—
199C3×Dic6—62no—
200C3×Dic6—62no—
201#626—68no—
202#625—64no—
203C3⋊(C3⋊C8)C22×D314yescomplemented
204C3×C3⋊C8—34no—
205C3×C3⋊C8—34no—
206C3×C3⋊C8—64no—
207C22×SD16—271noSylow
208C2×C33C2×SD1612yescomplemented
209C2×C33C22×D411yescharacteristic
210C3×C3⋊D3—44no—
211C3×C3⋊D3C2×SD1614yescomplemented
212C23×D3—182no—
213C23×D3—181no—
214C2×C3×Q8—21no—
215C2×C3×Q8—22no—
216C2×C3×Q8—41no—
217C2×C3×Q8—94no—
218C2×C3×Q8—91no—
219C2×C3×Q8—181no—
220C2×C3×Q8—182no—
221C2×C3×Q8D3211yescharacteristic, complemented
222C2×C3×Q8D3212yescomplemented
223C2×C3×D4—92no—
224C22×C3×C4—91no—
225C22×C3×C4—181no—
226C2×#67—182no—
227D3×Q8—38no—
228D3×Q8—64no—
229D3×Q8—64no—
230D3×Q8—64no—
231D3×D4—98no—
232D3×D4—184no—
233C2×D12—32no—
234C2×D12—32no—
235C2×D12—62no—
236C2×D12—94no—
237C2×D12—182no—
238C2×D12—181no—
239C2×C4×D3—32no—
240C2×C4×D3—61no—
241C2×C4×D3—62no—
242C2×C4×D3—62no—
243C2×Dic6—32no—
244C2×Dic6—61no—
245C2×Dic6—62no—
246C2×Dic6—62no—
247C3×SD16—98no—
248C2×C3×C8—92no—
249#215—38no—
250#215—38no—
251#215—68no—
252#215—98no—
253#214—98no—
254C2×C3⋊C8—32no—
255C2×C3⋊C8—32no—
256C2×C3⋊C8—62no—
257C2×C3⋊C8—94no—
258#204—98no—
259D3×C8—98no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32D3×D412yes—
264C22×C32C2×#6711yescharacteristic
265C2×C3⋊D3—24no—
266C2×C3⋊D3—44no—
267C2×C3⋊D3#21514yescomplemented
268C2×C3⋊D3C2×#6712yes—
269C2×C3×D3—38no—
270C2×C3×D3—34no—
271C2×C3×D3—64no—
272C2×C3×D3—62no—
273C2×C3×D3—64no—
274C2×C3×D3—64no—
275C2×C3×D3—124no—
276C2×C3×D3—124no—
277C2×C3×D3—124no—
278C2×C3×D3—244no—
279D32—1216no—
280D32—128no—
281C32×C4—22no—
282C32×C4—24no—
283C32×C4—24no—
284C32×C4—22no—
285C32×C4—21no—
286C32×C4—22no—
287C32×C4—21no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—44no—
291C32×C4—44no—
292C32×C4D3×D412yes—
293C32×C4C2×#6711yescharacteristic
294C32×C4C23×D312yes—
295C32×C4C23×D311yescharacteristic
296C3⋊Dic3—22no—
297C3⋊Dic3C2×#6712yes—
298C3×Dic3—34no—
299C3×Dic3—62no—
300C3×Dic3—64no—
301C3×Dic3—64no—
302C22×Q8—91no—
303C22×D4—271no—
304C2×SD16—92no—
305C2×SD16—272no—
306C2×SD16—278no—
307C22×C8—271no—
308C33C22×SD1611yescharacteristic, complemented, Sylow
309C23×C3—91no—
310C23×C3—182no—
311C22×D3—32no—
312C22×D3—61no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—122no—
316C22×D3—188no—
317C22×D3—184no—
318C22×D3—368no—
319C22×D3—364no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—21no—
323C3×Q8—22no—
324C3×Q8—24no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—92no—
329C3×Q8—184no—
330C3×Q8—181no—
331C3×Q8C2×D3212yescomplemented
332C3×Q8C2×D3214yescomplemented
333C3×D4—94no—
334C3×D4—182no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4—184no—
347C2×C3×C4—181no—
348C2×C3×C4C2×D3212yes—
349C2×C3×C4C2×D3211yescharacteristic
350#67—188no—
351C2×Dic3—32no—
352C2×Dic3—61no—
353C2×Dic3—62no—
354C2×Dic3—62no—
355D12—34no—
356D12—34no—
357D12—64no—
358D12—62no—
359D12—62no—
360D12—122no—
361D12—184no—
362C4×D3—34no—
363C4×D3—34no—
364C4×D3—62no—
365C4×D3—62no—
366C4×D3—68no—
367C4×D3—68no—
368Dic6—34no—
369Dic6—62no—
370Dic6—68no—
371Dic6—68no—
372Dic6—62no—
373Dic6—62no—
374C3×C8—94no—
375C3⋊C8—34no—
376C3⋊C8—34no—
377C3⋊C8—64no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—24no—
381C2×C32—22no—
382C2×C32—42no—
383C2×C32—41no—
384C2×C32D3×SD1614yescomplemented
385C2×C32C2×#21512yescomplemented
386C2×C32C2×D3×D412yes—
387C2×C32C22×#6711yescharacteristic
388C3⋊D3—44no—
389C3⋊D3C2×#21514yescomplemented
390C3×D3—38no—
391C3×D3—64no—
392C3×D3—124no—
393C3×D3—124no—
394C3×D3—244no—
395C24—541no—
396C2×Q8—94no—
397C2×Q8—91no—
398C2×Q8—181no—
399C2×Q8—182no—
400C2×Q8#111311yescharacteristic, complemented
401C2×D4—92no—
402C2×D4—274no—
403C2×D4—542no—
404C2×D4—541no—
405C22×C4—91no—
406C22×C4—181no—
407SD16—98no—
408SD16—278no—
409C2×C8—92no—
410C2×C8—274no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—41no—
414C22×C3—92no—
415C22×C3—94no—
416C22×C3—184no—
417C22×C3—364no—
418C22×C3D3×#6712yes—
419C22×C3#381111yescharacteristic
420C2×D3—34no—
421C2×D3—38no—
422C2×D3—62no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—124no—
427C2×D3—124no—
428C2×D3—124no—
429C2×D3—244no—
430C2×D3—3616no—
431C2×D3—368no—
432C3×C4—24no—
433C3×C4—24no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—44no—
440C3×C4—44no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—92no—
444C3×C4—182no—
445C3×C4D3×#6712yes—
446C3×C4#381111yescharacteristic
447C3×C4C22×D3212yes—
448C3×C4C22×D3211yescharacteristic
449Dic3—34no—
450Dic3—62no—
451Dic3—64no—
452Dic3—64no—
453C32—22no—
454C32—21no—
455C32—41no—
456C32C2×D3×SD1612yescomplemented
457C32C22×#21511yescharacteristic, complemented
458C23—91no—
459C23—182no—
460C23—544no—
461C23—1084no—
462Q8—21no—
463Q8—92no—
464Q8—184no—
465Q8—181no—
466Q8C2×#111312yescomplemented
467D4—94no—
468D4—182no—
469D4—544no—
470C2×C4—21no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—184no—
475C2×C4—181no—
476C2×C4C2×#111311yescharacteristic
477C8—94no—
478C2×C3—21no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—24no—
482C2×C3—41no—
483C2×C3—42no—
484C2×C3—94no—
485C2×C3—364no—
486C2×C3D3×#21514yescomplemented
487C2×C3#6394012yescomplemented
488C2×C3C2×D3×#6712yes—
489C2×C3C2×#381111yescharacteristic
490D3—38no—
491D3—64no—
492D3—124no—
493D3—124no—
494D3—244no—
495C22—92no—
496C22—94no—
497C22—184no—
498C22—364no—
499C22—1088no—
500C22#8904911yescharacteristic
501C4—22no—
502C4—92no—
503C4—182no—
504C4#8904911yescharacteristic
505C4C22×#111311yescharacteristic
506C3—21no—
507C3—22no—
508C3—41no—
509C3C2×D3×#21512yescomplemented
510C3C2×#6394011yescharacteristic, complemented
511C2—94no—
512C2—364no—
513C2#1172370812yesdirect product factor
514C2C2×#8904911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
hypercenterC2×Q8#1113
hypocenterC33C22×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
3. C2×C3×#3755C2semidirect
4. #11723708C2direct (N × H)
5. C2×#88811C2semidirect
6. C2×#88808C2semidirect
10. C2×C33×Q8C22semidirect
12. C3×Q8×C3⋊D3C22semidirect
17. C2×C3×#634C22semidirect
21. C3×#3755C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. #88811C22semidirect
24. #88808C22semidirect
25. C2×Q8×C3⋊D3D3semidirect
30. C2×#3755D3semidirect
46. C33×Q8C23semidirect
58. C3×#634C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
68. C2×C32×Q8C2×D3semidirect
69. C2×C32×Q8C2×D3semidirect
71. Q8×C3⋊D3C2×D3semidirect
76. C2×#634C2×D3semidirect
98. #3755C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
114. C2×C3×C3⋊D3SD16semidirect
157. C32×Q8C22×D3semidirect
158. C32×Q8C22×D3semidirect
178. #634C22×D3semidirect
203. C3⋊(C3⋊C8)C22×D3semidirect
208. C2×C33C2×SD16semidirect
211. C3×C3⋊D3C2×SD16semidirect
221. C2×C3×Q8D32semidirect
222. C2×C3×Q8D32semidirect
267. C2×C3⋊D3#215semidirect
308. C33C22×SD16semidirect
331. C3×Q8C2×D32semidirect
332. C3×Q8C2×D32semidirect
384. C2×C32D3×SD16semidirect
385. C2×C32C2×#215semidirect
389. C3⋊D3C2×#215semidirect
400. C2×Q8#1113semidirect
456. C32C2×D3×SD16semidirect
457. C32C22×#215semidirect
466. Q8C2×#1113semidirect
486. C2×C3D3×#215semidirect
487. C2×C3#63940semidirect
509. C3C2×D3×#215semidirect
510. C3C2×#63940semidirect
513. C2#11723708direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C4⋊#11132
C2×C3×#37552
#117237082
C2×#888112
C2×#888082
C2×#639403
C2×D3×#2153