Finite Group Databasebeta

← #13172158

Subgroups — #13172158

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6290C211yesmaximal, characteristic
2#11723723C211yesmaximal, characteristic, complemented
3#11723598C212yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic, complemented
5C3×D3×#217C212yesmaximal, complemented
6#11722936C214yesmaximal, complemented
7#11722930C214yesmaximal, complemented
8#10595510—31nomaximal
9#10594181—32nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C23×C32×D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×#6294C2212yescomplemented
14C2×#6290C418yescomplemented
15C2×#6290C2212yes—
16C2×#6290C2212yes—
17C2×C3×#624C2212yescomplemented
18C2×C3×D3×Dic3C2214yescomplemented
19C3×#3757C2212yes—
20C32×#217C2212yes—
21#88814C2212yescomplemented
22#88806C2214yescomplemented
23Dic3×C3⋊Dic3C2214yescomplemented
24C3×#3722C2212yescomplemented
25C3×(C2×D3)⋊Dic3C2214yescomplemented
26C22×#638—31no—
27C22×#625—32no—
28C22×#625—61no—
29#64086—32no—
30C2×C3×#217—32no—
31C3×D3×C22⋊C4—32no—
32C3×D3×C22⋊C4—32no—
33#63974—31no—
34#63974D311yescharacteristic, complemented
35D3×#217—32no—
36D3×#217D312yescomplemented
37#63957—32no—
38#63957—32no—
39#63957—62no—
40#63952—34no—
41#63952—34no—
42#63902—34no—
43C4×#625—34no—
44#63879—34no—
45#63879—34no—
46C23×C33C2311yescharacteristic
47C22×C3×C3⋊D3—42no—
48C22×C3×C3⋊D3C2×C414yescomplemented
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C3×C3⋊D3C2311yescharacteristic
51C22×C32×D3—42no—
52C22×C32×D3C2×C414yescomplemented
53C22×C32×D3C2312yes—
54C2×C3⋊(C3⋊Dic3)C2×C414yescomplemented
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C32×Dic3C2314yes—
58#6294—42no—
59#6290C2×C418yescomplemented
60#6290C2×C418yescomplemented
61C3×#624—42no—
62C3×D3×Dic3—44no—
63#5486—91no—
64#5486—92no—
65C24×C32—31no—
66C23×C3⋊D3C2×D311yescharacteristic
67C23×C3×D3—21no—
68C23×C3×D3—32no—
69C23×C3×D3—61no—
70C23×C3×D3C2×D312yes—
71C2×#638—38no—
72C2×#638—32no—
73C2×#638—32no—
74C22×C3⋊Dic3—32no—
75C22×C3⋊Dic3—31no—
76C22×C3⋊Dic3—61no—
77C22×C3⋊Dic3—62no—
78C22×C3⋊Dic3—121no—
79C22×C3×Dic3—34no—
80C2×C3×C4×D3—34no—
81C2×C3×C4×D3—34no—
82C2×#625—316no—
83C2×#625—32no—
84C2×#625—32no—
85C2×#625—32no—
86C2×#625—32no—
87C2×#625—68no—
88C2×#625—62no—
89C2×#625—62no—
90C2×#624—32no—
91C2×#624C2×D312yescomplemented
92C2×D3×Dic3—34no—
93C2×D3×Dic3C2×D314yescomplemented
94C32×C22⋊C4—32no—
95#3757C2×D312yes—
96#3750—34no—
97C4×C3⋊Dic3—34no—
98C3×#217—22no—
99C3×#217—32no—
100C3×#217—32no—
101C3×#217—34no—
102C3×#217—62no—
103C3×#217C2×D312yes—
104C3×#217C2×D312yes—
105C3×(C2×D3)⋊C4—34no—
106C3×(C2×D3)⋊C4—34no—
107#3724—34no—
108#3724—34no—
109#3724—64no—
110#3722—32no—
111#3722C2×D312yescomplemented
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3C2×D314yescomplemented
114Dic32—34no—
115Dic32—34no—
116Dic32—64no—
117C22×C33—22no—
118C22×C33C22×C412yes—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—44no—
123C2×C3×C3⋊D3C22×C412yes—
124C2×C3×C3⋊D3C22×C412yes—
125C2×C32×D3—24no—
126C2×C32×D3—44no—
127C2×C32×D3C22×C414yes—
128C3⋊(C3⋊Dic3)C22×C414yes—
129C3×C3⋊Dic3—24no—
130C32×Dic3—24no—
131C22×#67—92no—
132C22×#67—91no—
133C22×#67—181no—
134C2×C3×C22⋊C4—91no—
135C2×C3×C22⋊C4—92no—
136C2×#217—32no—
137C2×#217—91no—
138C4×#67—94no—
139C4×#67—94no—
140C4×#67—94no—
141#67⋊C4—94no—
142#67⋊C4—94no—
143#67⋊C4—94no—
144D3×C22⋊C4—32no—
145D3×C22⋊C4—32no—
146D3×C22⋊C4—92no—
147#877—92no—
148#877—92no—
149#877—92no—
150#877—182no—
151C23×C32—21no—
152C23×C32—22no—
153C23×C32—32no—
154C23×C32—34no—
155C23×C32—41no—
156C23×C32—122no—
157C23×C32C22×D312yes—
158C23×C32C22×D311yescharacteristic
159C22×C3⋊D3—42no—
160C22×C3⋊D3C4×D314yescomplemented
161C22×C3⋊D3C22×D311yescharacteristic
162C22×C3⋊D3C22×D311yescharacteristic
163C22×C3×D3—24no—
164C22×C3×D3—22no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—38no—
168C22×C3×D3—44no—
169C22×C3×D3—44no—
170C22×C3×D3—64no—
171C22×C3×D3—61no—
172C22×C3×D3—61no—
173C22×C3×D3—124no—
174C22×C3×D3—124no—
175C22×C3×D3C4×D318yescomplemented
176C22×C3×D3C22×D312yes—
177C22×C3×D3C22×D312yes—
178C2×C32×C4—34no—
179#638—38no—
180#638—38no—
181C2×C3⋊Dic3—38no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—32no—
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—62no—
187C2×C3⋊Dic3—62no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3—62no—
191C2×C3⋊Dic3—124no—
192C2×C3⋊Dic3—122no—
193C2×C3⋊Dic3C22×D314yes—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—64no—
198C2×C3×Dic3—124no—
199C2×C3×Dic3C22×D314yes—
200C2×C3×Dic3C22×D314yes—
201C3×C4×D3—124no—
202C3×C4×D3—124no—
203#625—38no—
204#625—38no—
205#625—38no—
206#625—38no—
207#625—68no—
208#625—68no—
209#624—42no—
210#624—122no—
211D3×Dic3—44no—
212D3×Dic3—124no—
213#518—271noSylow
214C2×C33—22no—
215C2×C33C23×C411yescharacteristic
216C2×C33H(5,𝔽2)12yes—
217C3×C3⋊D3—22no—
218C3×C3⋊D3—22no—
219C32×D3—24no—
220C24×C3—32no—
221C24×C3—61no—
222C24×C3—91no—
223C23×D3—32no—
224C23×D3—61no—
225C23×D3D3211yescharacteristic
226C22×C3×C4—92no—
227C22×C3×C4—94no—
228C2×#67—92no—
229C2×#67—92no—
230C2×#67—92no—
231C2×#67—92no—
232C2×#67—916no—
233C2×#67—98no—
234C2×#67—92no—
235C2×#67—92no—
236C2×#67—182no—
237C2×#67—182no—
238C2×#67—188no—
239C22×Dic3—34no—
240C22×Dic3—92no—
241C22×Dic3—92no—
242C22×Dic3—91no—
243C22×Dic3—182no—
244C22×Dic3—181no—
245C22×Dic3—361no—
246C2×C4×D3—34no—
247C2×C4×D3—34no—
248C2×C4×D3—94no—
249C3×C22⋊C4—32no—
250C3×C22⋊C4—32no—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4—94no—
254C3×C22⋊C4—92no—
255#217—32no—
256#217—32no—
257#217—34no—
258#217—62no—
259#217—92no—
260#217D3212yes—
261(C2×D3)⋊C4—34no—
262(C2×D3)⋊C4—34no—
263(C2×D3)⋊C4—94no—
264Dic3⋊C4—94no—
265Dic3⋊C4—94no—
266Dic3⋊C4—94no—
267Dic3⋊C4—184no—
268C4×Dic3—94no—
269C4×Dic3—94no—
270C4×Dic3—94no—
271C4×Dic3—184no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—24no—
278C22×C32—22no—
279C22×C32—34no—
280C22×C32—42no—
281C22×C32—41no—
282C22×C32—44no—
283C22×C32—44no—
284C22×C32—64no—
285C22×C32—124no—
286C22×C32C2×C4×D312yes—
287C22×C32C2×C4×D314yes—
288C22×C32C23×D311yescharacteristic
289C22×C32C23×D312yes—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—44no—
293C2×C3⋊D3C2×C4×D312yes—
294C2×C3⋊D3C2×C4×D312yes—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—34no—
299C2×C3×D3—34no—
300C2×C3×D3—44no—
301C2×C3×D3—48no—
302C2×C3×D3—48no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—64no—
306C2×C3×D3—64no—
307C2×C3×D3—122no—
308C2×C3×D3—122no—
309C2×C3×D3—128no—
310C2×C3×D3—128no—
311C2×C3×D3C2×C4×D314yes—
312C2×C3×D3C2×C4×D314yes—
313C32×C4—64no—
314C3⋊Dic3—24no—
315C3⋊Dic3—34no—
316C3⋊Dic3—34no—
317C3⋊Dic3—34no—
318C3⋊Dic3—64no—
319C3⋊Dic3—64no—
320C3⋊Dic3—64no—
321C3⋊Dic3—124no—
322C3×Dic3—24no—
323C3×Dic3—24no—
324C3×Dic3—44no—
325C3×Dic3—64no—
326C3×Dic3—64no—
327C3×Dic3—124no—
328C22×D4—271no—
329C4×D4—274no—
330C4×D4—274no—
331#117—272no—
332C2×C22⋊C4—91no—
333C2×C22⋊C4—92no—
334C2×C22⋊C4—271no—
335C33#51811yescharacteristic, complemented, Sylow
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—32no—
339C23×C3—32no—
340C23×C3—38no—
341C23×C3—41no—
342C23×C3—62no—
343C23×C3—64no—
344C23×C3—91no—
345C23×C3—91no—
346C23×C3—94no—
347C23×C3—124no—
348C23×C3—124no—
349C23×C3—362no—
350C23×C3C2×D3212yes—
351C23×C3C2×D3211yescharacteristic
352C22×D3—32no—
353C22×D3—32no—
354C22×D3—38no—
355C22×D3—42no—
356C22×D3—64no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—124no—
360C22×D3—124no—
361C22×D3#62414yescomplemented
362C22×D3C2×D3212yes—
363C2×C3×C4—34no—
364C2×C3×C4—34no—
365C2×C3×C4—64no—
366C2×C3×C4—94no—
367C2×C3×C4—362no—
368C2×C3×C4—364no—
369#67—98no—
370#67—98no—
371#67—98no—
372#67—98no—
373#67—98no—
374#67—98no—
375#67—188no—
376#67—188no—
377C2×Dic3—34no—
378C2×Dic3—34no—
379C2×Dic3—64no—
380C2×Dic3—92no—
381C2×Dic3—92no—
382C2×Dic3—92no—
383C2×Dic3—98no—
384C2×Dic3—94no—
385C2×Dic3—124no—
386C2×Dic3—182no—
387C2×Dic3—182no—
388C2×Dic3—188no—
389C2×Dic3—182no—
390C2×Dic3—184no—
391C2×Dic3—362no—
392C2×Dic3—364no—
393C2×Dic3—362no—
394C2×Dic3C2×D3214yes—
395C4×D3—124no—
396C4×D3—124no—
397C4×D3—364no—
398C2×C32—22no—
399C2×C32—24no—
400C2×C32—24no—
401C2×C32—22no—
402C2×C32—24no—
403C2×C32—21no—
404C2×C32—22no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32—44no—
408C2×C32—44no—
409C2×C32—64no—
410C2×C32C22×C4×D311yescharacteristic
411C2×C32C22×C4×D312yes—
412C2×C32#100412yes—
413C2×C32#100414yes—
414C3⋊D3—22no—
415C3⋊D3—22no—
416C3×D3—24no—
417C3×D3—24no—
418C3×D3—44no—
419C3×D3—64no—
420C3×D3—64no—
421C3×D3—122no—
422C3×D3—122no—
423C24—31no—
424C24—91no—
425C2×D4—272no—
426C2×D4—272no—
427C2×D4—278no—
428C22×C4—92no—
429C22×C4—94no—
430C22×C4—272no—
431C22×C4—271no—
432C4⋊C4—274no—
433C22⋊C4—32no—
434C22⋊C4—92no—
435C22⋊C4—94no—
436C22⋊C4—92no—
437C22⋊C4—272no—
438C42—274no—
439C22×C3—24no—
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—22no—
445C22×C3—24no—
446C22×C3—34no—
447C22×C3—34no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—41no—
451C22×C3—42no—
452C22×C3—64no—
453C22×C3—64no—
454C22×C3—64no—
455C22×C3—92no—
456C22×C3—92no—
457C22×C3—124no—
458C22×C3—128no—
459C22×C3—128no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—364no—
463C22×C3C4×D3214yes—
464C22×C3C2×#62412yes—
465C22×C3C22×D3212yes—
466C22×C3C22×D3211yescharacteristic
467C2×D3—24no—
468C2×D3—34no—
469C2×D3—34no—
470C2×D3—44no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—64no—
474C2×D3—64no—
475C2×D3—122no—
476C2×D3—122no—
477C2×D3—128no—
478C2×D3—128no—
479C2×D3C2×#62414yes—
480C3×C4—64no—
481C3×C4—64no—
482C3×C4—124no—
483C3×C4—184no—
484Dic3—24no—
485Dic3—64no—
486Dic3—64no—
487Dic3—94no—
488Dic3—94no—
489Dic3—94no—
490Dic3—124no—
491Dic3—184no—
492Dic3—184no—
493Dic3—184no—
494Dic3—364no—
495C32—22no—
496C32—21no—
497C32—41no—
498C32#548611yescharacteristic, complemented
499C32#548612yescomplemented
500C23—32no—
501C23—34no—
502C23—91no—
503C23—91no—
504C23—94no—
505C23—122no—
506C23—362no—
507C23D3311yescharacteristic
508D4—278no—
509D4—278no—
510C2×C4—34no—
511C2×C4—94no—
512C2×C4—272no—
513C2×C4—274no—
514C2×C4—362no—
515C2×C4—364no—
516C2×C4—1082no—
517C2×C3—24no—
518C2×C3—24no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—24no—
523C2×C3—22no—
524C2×C3—44no—
525C2×C3—44no—
526C2×C3—41no—
527C2×C3—42no—
528C2×C3—64no—
529C2×C3—64no—
530C2×C3—124no—
531C2×C3—182no—
532C2×C3—182no—
533C2×C3C2×C4×D3212yes—
534C2×C3#6431314yes—
535C2×C3C22×#62411yescharacteristic
536C2×C3#6433012yes—
537D3—24no—
538D3—64no—
539D3—64no—
540D3—122no—
541D3—122no—
542C22—22no—
543C22—34no—
544C22—64no—
545C22—92no—
546C22—92no—
547C22—124no—
548C22—182no—
549C22—182no—
550C22—364no—
551C22D3×#62412yes—
552C22C2×D3311yescharacteristic
553C4—64no—
554C4—184no—
555C4—274no—
556C3—21no—
557C3—22no—
558C3—41no—
559C3#1059418112yescomplemented
560C3#1059551011yescharacteristic, complemented
561C2—22no—
562C2—64no—
563C2—182no—
564C2—182no—
565C2C2×D3×#62411yescharacteristic
566C2#1172441512yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC22×#6290C2
hypercenterC23D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723723C2semidirect
3. #11723598C2semidirect
4. C3×#63974C2semidirect
5. C3×D3×#217C2semidirect
6. #11722936C2semidirect
7. #11722930C2semidirect
13. C2×#6294C22semidirect
14. C2×#6290C4semidirect
17. C2×C3×#624C22semidirect
18. C2×C3×D3×Dic3C22semidirect
21. #88814C22semidirect
22. #88806C22semidirect
23. Dic3×C3⋊Dic3C22semidirect
24. C3×#3722C22semidirect
25. C3×(C2×D3)⋊Dic3C22semidirect
34. #63974D3semidirect
36. D3×#217D3semidirect
48. C22×C3×C3⋊D3C2×C4semidirect
52. C22×C32×D3C2×C4semidirect
54. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
59. #6290C2×C4semidirect
60. #6290C2×C4semidirect
91. C2×#624C2×D3semidirect
93. C2×D3×Dic3C2×D3semidirect
111. #3722C2×D3semidirect
113. (C2×D3)⋊Dic3C2×D3semidirect
160. C22×C3⋊D3C4×D3semidirect
175. C22×C3×D3C4×D3semidirect
335. C33#518semidirect
361. C22×D3#624semidirect
498. C32#5486semidirect
499. C32#5486semidirect
559. C3#10594181semidirect
560. C3#10595510semidirect

Maximal subgroups

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

subgroupindex
C22×#62902
#117237232
#117235982
C3×#639742
C3×D3×#2172
#117229362
#117229302
#105955103
#105941813