Finite Group Databasebeta

← #13178676

Subgroups — #13178676

Order 1728.

1280 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64147C212yesmaximal, complemented
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11723718C214yesmaximal, complemented
4#11723717C212yesmaximal, complemented
5#11723715C212yesmaximal, complemented
6C4⋊#6296C211yesmaximal, characteristic, complemented
7C4×#6295C211yesmaximal, characteristic, complemented
8#11723667C212yesmaximal, complemented
9#10595253—31nomaximal
10#10595249—32nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2214yes—
13C22×C3×C3⋊Dic3C2212yes—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×#6296C2212yes—
16C2×#6295C2211yescharacteristic
17C3×#3757C2212yes—
18C3×#3757C2214yes—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22#88815C2211yescharacteristic
23#88815C2212yes—
24#88814C2214yes—
25#88814C2212yes—
26#88813C2214yes—
27#64147D312yescomplemented
28D4×C3⋊Dic3D311yescharacteristic, complemented
29C3×#938—32no—
30C3×#938—32no—
31C3×#938—62no—
32C3×D4×Dic3—32no—
33C3×D4×Dic3—61no—
34#63980—32no—
35#63978—34no—
36#63959—34no—
37#63959—34no—
38#63957—32no—
39#63955—34no—
40#63953—34no—
41#63919—31no—
42#63918—32no—
43C4×#626—32no—
44C4×#625—31no—
45#63849—32no—
46#63841—32no—
47#63841—32no—
48C23×C33C2312yes—
49C22×C3×C3⋊D3C2312yes—
50C33×D4—22no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—44no—
53C2×C3×C3⋊Dic3—24no—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C2×C3×C3⋊Dic3C2314yes—
56C2×C3×C3⋊Dic3C2312yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C3×C4×C3⋊D3—24no—
60#6296—42no—
61#6295—22no—
62#5616—92no—
63#5563—91no—
64C2×C32×D4—21no—
65C2×C32×D4—22no—
66C2×C32×D4—41no—
67C2×C32×D4C2×D312yes—
68C2×C32×D4C2×D311yescharacteristic
69C2×#638C2×D314yes—
70C22×C3⋊Dic3C2×D312yes—
71C2×C4×C3⋊D3C2×D312yes—
72C2×C3×#67—34no—
73C2×C3×#67—34no—
74C2×C3×#67—64no—
75C22×C3×Dic3—34no—
76C22×C3×Dic3—62no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—62no—
80C2×#627—34no—
81C2×#627—32no—
82C2×#626—32no—
83C2×#625—31no—
84#3757C2×D312yes—
85#3757C2×D314yes—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87C4⋊(C3⋊Dic3)C2×D312yes—
88C4×C3⋊Dic3C2×D311yescharacteristic
89C3×#217—34no—
90C3×#217—34no—
91C3×#217—34no—
92C3×#217—62no—
93C3×#217—64no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3—62no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—61no—
101#3724—31no—
102#3724—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—34no—
105#3722—34no—
106#3722—32no—
107(C2×D3)⋊Dic3—34no—
108(C2×D3)⋊Dic3—34no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—32no—
111Dic32—34no—
112Dic32—34no—
113Dic32—34no—
114C22×C33—22no—
115C22×C33—22no—
116C22×C33—22no—
117C22×C33C2411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—22no—
121C33×C4#4112yes—
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3—21no—
125C3×C3⋊Dic3—24no—
126C3×C3⋊Dic3#4112yes—
127C3×#121—92no—
128C3×C4×D4—91no—
129#939—94no—
130#938—32no—
131#938—32no—
132#938—62no—
133#936—94no—
134D4×Dic3—32no—
135D4×Dic3—61no—
136#930—92no—
137C4×#67—92no—
138#926—92no—
139#926—92no—
140#892—92no—
141#892—92no—
142#888—92no—
143#885—94no—
144#885—94no—
145#877—94no—
146C4×D12—91no—
147#869—91no—
148C23×C32—22no—
149C23×C32—24no—
150C23×C32—42no—
151C23×C32C22×D314yes—
152C23×C32C22×D312yes—
153C22×C3⋊D3C22×D312yes—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—62no—
157C32×D4—22no—
158C32×D4—24no—
159C32×D4—24no—
160C32×D4—44no—
161C32×D4—44no—
162C2×C32×C4—21no—
163C2×C32×C4—22no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D312yes—
167#638—44no—
168C2×C3⋊Dic3—24no—
169C2×C3⋊Dic3C22×D312yes—
170C2×C3⋊Dic3C22×D312yes—
171C2×C3⋊Dic3C22×D314yes—
172C2×C3⋊Dic3C22×D311yescharacteristic
173C2×C3⋊Dic3C22×D311yescharacteristic
174C4×C3⋊D3—24no—
175C3×#67—124no—
176C3×#67—124no—
177C3×#67—128no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—68no—
191C2×C3×Dic3—124no—
192C3×C4×D3—64no—
193C3×C4×D3—64no—
194C3×C4×D3—124no—
195#627—124no—
196#627—122no—
197#626—64no—
198#625—62no—
199#555—271noSylow
200C2×C33—42no—
201C2×C33C2×#4111yescharacteristic
202C2×C33C2×#4111yescharacteristic
203C2×C33H(5,𝔽2)11yescharacteristic
204C3×C3⋊D3—42no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—41no—
208C2×C3×D4—94no—
209C2×C3×D4D3212yes—
210C2×C3×D4D3211yescharacteristic
211C22×C3×C4—92no—
212C22×C3×C4—92no—
213C2×#67—34no—
214C2×#67—34no—
215C2×#67—64no—
216C2×#67—92no—
217C22×Dic3—34no—
218C22×Dic3—62no—
219C2×D12—91no—
220C2×C4×D3—32no—
221C2×C4×D3—32no—
222C2×C4×D3—62no—
223C2×Dic6—92no—
224C2×Dic6—94no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—91no—
227C3×C22⋊C4—94no—
228C3×C22⋊C4—92no—
229C3×C42—91no—
230#217—34no—
231#217—34no—
232#217—34no—
233#217—62no—
234#217—64no—
235#217—94no—
236#217—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—94no—
240(C2×D3)⋊C4—94no—
241C4⋊Dic3—32no—
242C4⋊Dic3—32no—
243C4⋊Dic3—32no—
244C4⋊Dic3—61no—
245C4⋊Dic3—62no—
246C4⋊Dic3—92no—
247C4⋊Dic3—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4—94no—
250C4×Dic3—32no—
251C4×Dic3—61no—
252C4×Dic3—94no—
253C4×Dic3—94no—
254C4×Dic3—94no—
255C22×C32—22no—
256C22×C32—24no—
257C22×C32—22no—
258C22×C32—24no—
259C22×C32—24no—
260C22×C32—24no—
261C22×C32—22no—
262C22×C32—21no—
263C22×C32—22no—
264C22×C32—41no—
265C22×C32—44no—
266C22×C32—44no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—44no—
270C22×C32—44no—
271C22×C32—82no—
272C22×C32—82no—
273C22×C32C23×D311yescharacteristic
274C22×C32C23×D312yes—
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3×D3—64no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287C32×C4—24no—
288C32×C4—22no—
289C32×C4—42no—
290C32×C4#23512yes—
291C32×C4#23714yes—
292C3⋊Dic3—22no—
293C3⋊Dic3—22no—
294C3⋊Dic3—24no—
295C3⋊Dic3—21no—
296C3⋊Dic3#23712yes—
297C3×Dic3—34no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—64no—
303C3×Dic3—64no—
304C3×Dic3—64no—
305C3×Dic3—122no—
306C3×Dic3—121no—
307C3×Dic3—122no—
308C3×Dic3—124no—
309C4⋊Q8—271no—
310#124—274no—
311#121—92no—
312#121—272no—
313C4×D4—91no—
314C4×D4—271no—
315#117—272no—
316#117—272no—
317C33#55511yescharacteristic, complemented, Sylow
318C23×C3—22no—
319C23×C3—24no—
320C23×C3—42no—
321C23×C3—92no—
322C23×C3C2×D3214yes—
323C23×C3C2×D3212yes—
324C22×D3—32no—
325C22×D3—32no—
326C22×D3—62no—
327C3×D4—24no—
328C3×D4—24no—
329C3×D4—22no—
330C3×D4—44no—
331C3×D4—44no—
332C3×D4—364no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—94no—
340C2×C3×C4—91no—
341C2×C3×C4—184no—
342C2×C3×C4—184no—
343C2×C3×C4C2×D3212yes—
344C2×C3×C4C2×D3211yescharacteristic
345#67—124no—
346#67—124no—
347#67—128no—
348#67—184no—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—34no—
353C2×Dic3—32no—
354C2×Dic3—34no—
355C2×Dic3—34no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—62no—
359C2×Dic3—64no—
360C2×Dic3—62no—
361C2×Dic3—68no—
362C2×Dic3—124no—
363D12—182no—
364C4×D3—64no—
365C4×D3—64no—
366C4×D3—124no—
367Dic6—362no—
368Dic6—364no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—22no—
373C2×C32—22no—
374C2×C32—21no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—41no—
378C2×C32—42no—
379C2×C32—44no—
380C2×C32—44no—
381C2×C32—84no—
382C2×C32—84no—
383C2×C32C2×#23511yescharacteristic
384C2×C32C2×#23711yescharacteristic
385C2×C32C2×#23712yes—
386C2×C32#100412yes—
387C2×C32D3×#4112yes—
388C2×C32#100911yescharacteristic
389C3⋊D3—42no—
390C3×D3—124no—
391C3×D3—122no—
392C3×D3—122no—
393C2×Q8—272no—
394C2×D4—94no—
395C2×D4—271no—
396C2×D4#111311yescharacteristic
397C22×C4—92no—
398C22×C4—92no—
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—271no—
402C4⋊C4—272no—
403C22⋊C4—94no—
404C22⋊C4—92no—
405C22⋊C4—274no—
406C22⋊C4—272no—
407C42—91no—
408C42—274no—
409C22×C3—24no—
410C22×C3—24no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—22no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—41no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—182no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3C22×D3212yes—
431C22×C3C22×D3211yescharacteristic
432C2×D3—64no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—122no—
440C2×D3—122no—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—92no—
445C3×C4—182no—
446C3×C4—181no—
447C3×C4—184no—
448C3×C4—182no—
449C3×C4#380414yes—
450C3×C4#380512yes—
451Dic3—34no—
452Dic3—62no—
453Dic3—64no—
454Dic3—64no—
455Dic3—62no—
456Dic3—64no—
457Dic3—62no—
458Dic3—62no—
459Dic3—122no—
460Dic3—122no—
461Dic3—124no—
462Dic3—121no—
463C32—22no—
464C32—21no—
465C32—41no—
466C32#556311yescharacteristic, complemented
467C32#561612yescomplemented
468C23—92no—
469C23C2×#111312yes—
470Q8—1082no—
471D4—22no—
472D4—364no—
473D4—542no—
474C2×C4—91no—
475C2×C4—91no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—94no—
479C2×C4—184no—
480C2×C4—184no—
481C2×C4C2×#111311yescharacteristic
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—22no—
487C2×C3—21no—
488C2×C3—44no—
489C2×C3—44no—
490C2×C3—42no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—84no—
495C2×C3—84no—
496C2×C3—362no—
497C2×C3D3×#23712yes—
498C2×C3#6431211yescharacteristic
499C2×C3#6431311yescharacteristic
500C2×C3#6431412yes—
501C2×C3C2×#380412yes—
502C2×C3C2×#380511yescharacteristic
503D3—124no—
504D3—122no—
505D3—122no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—182no—
510C22—182no—
511C22—182no—
512C22C22×#111311yescharacteristic
513C4—92no—
514C4—182no—
515C4—184no—
516C4—181no—
517C4—182no—
518C4#8904612yes—
519C3—22no—
520C3—21no—
521C3—41no—
522C3#1059524912yescomplemented
523C3#1059525311yescharacteristic, complemented
524C2—42no—
525C2—362no—
526C2#1172462211yescharacteristic
527C2#1172462311yescharacteristic
528C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#555
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64147C2semidirect
2. C3×D4×C3⋊Dic3C2semidirect
3. #11723718C2semidirect
4. #11723717C2semidirect
5. #11723715C2semidirect
6. C4⋊#6296C2semidirect
7. C4×#6295C2semidirect
8. #11723667C2semidirect
27. #64147D3semidirect
28. D4×C3⋊Dic3D3semidirect
317. C33#555semidirect
466. C32#5563semidirect
467. C32#5616semidirect
522. C3#10595249semidirect
523. C3#10595253semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
C3×D4×C3⋊Dic32
#117237182
#117237172
#117237152
C4⋊#62962
C4×#62952
#117236672
#105952533
#105952493