Finite Group Databasebeta

← #13168273

Subgroups — #13168273

Order 1728.

1630 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1113C211yesmaximal, characteristic
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#11723431C212yesmaximal, complemented
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63871C212yesmaximal, complemented
6#11722984C214yesmaximal, complemented
7#11722908C212yesmaximal, complemented
8#11722905C212yesmaximal
9#10595429—31nomaximal
10D3×#897—32nomaximal
11C22×#1113C2211yescharacteristic
12C2×#6308C2211yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic, complemented
16C2×#6294C2211yescharacteristic
17C2×#6294C2212yes—
18C4×#1113—24no—
19C2×#6293C2212yescomplemented
20C2×Dic3×C3⋊D3C2212yes—
21C3⋊D3×Dic6—24no—
22C2×C3×#624C2212yes—
23C32×C4⋊Dic3C2212yescomplemented
24Dic3⋊(C3⋊Dic3)C2214yescomplemented
25#88803C2214yes—
26C3×#3724C2212yescomplemented
27C3×Dic3⋊Dic3C2214yescomplemented
28C3×#3722C2214yes—
29C2×Q8×C3⋊D3—31no—
30C2×C4×D32—32no—
31C2×C4×D32—31no—
32C2×D3×Dic6—32no—
33C2×D3×Dic6—61no—
34#64106—32no—
35C3×#946—32no—
36C3×#897—32no—
37C3×D3×C4⋊C4—32no—
38#63921—32no—
39D3×(C2×D3)⋊C4—34no—
40#63900D311yescharacteristic, complemented
41#63896—34no—
42#63895—34no—
43#63893—34no—
44D3×C4⋊Dic3—34no—
45#63883—32no—
46#63871D312yescomplemented
47#63857—32no—
48#63857—32no—
49#63857—62no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C2×#1113—22no—
53C2×#1113—22no—
54C2×#1113Q814yescomplemented
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)C2312yes—
57#6308—22no—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C3×C4×C3⋊D3—24no—
61C3×C4×C3⋊D3D414yes—
62C2×C32×Dic3C2314yes—
63C2×C32×Dic3C2312yes—
64C32×Dic6—22no—
65#6294—22no—
66#6294—24no—
67#6293—24no—
68Dic3×C3⋊D3—24no—
69C3×#624—24no—
70D3×C22⋊Q8—92no—
71C2×#897—91no—
72C22×D32—32no—
73C22×D32—31no—
74C2×C32×Q8—31no—
75Q8×C3⋊D3—64no—
76C2×C4×C3⋊D3—32no—
77C2×C4×C3⋊D3C2×D312yes—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×#634—32no—
80C2×#634—31no—
81C2×#634—32no—
82C2×#634—61no—
83C2×#634—62no—
84C2×#634—121no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—34no—
89C2×C3×C4×D3—61no—
90C2×C3×C4×D3—62no—
91C2×C3×Dic6—21no—
92C2×C3×Dic6C2×D312yescomplemented
93C2×#627—34no—
94C2×#627—62no—
95C2×#624—31no—
96C2×#624—32no—
97C2×#624C2×D312yes—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—34no—
102C2×D3×Dic3—62no—
103C4×D32—68no—
104C4×D32—64no—
105D3×Dic6—68no—
106D3×Dic6—68no—
107C32×C4⋊C4—32no—
108#3752—34no—
109C4⋊(C3⋊Dic3)—34no—
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×C4⋊Dic3—22no—
113C3×C4⋊Dic3—34no—
114C3×C4⋊Dic3C2×D312yescomplemented
115C3×C4⋊Dic3C2×D312yescomplemented
116C3×Dic3⋊C4—34no—
117C3×Dic3⋊C4—34no—
118#3724C2×D312yescomplemented
119Dic3⋊Dic3—34no—
120Dic3⋊Dic3—34no—
121Dic3⋊Dic3—64no—
122Dic3⋊Dic3C2×D314yescomplemented
123#3722C2×D314yes—
124(C2×D3)⋊Dic3—34no—
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—64no—
127C22×C33C2411yescharacteristic
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3C2×D412yes—
131C2×C3×C3⋊D3C2×Q812yes—
132C2×C3×C3⋊D3C2×Q814yescomplemented
133C2×C3×C3⋊D3#4112yes—
134#1113—28no—
135C33×C4C2×D412yes—
136C3⋊(C3⋊Dic3)—22no—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3C2×D412yes—
139C32×Dic3—22no—
140C32×Dic3—24no—
141C2×D3×Q8—92no—
142C2×D3×Q8—181no—
143C22×C4×D3—91no—
144C22×C4×D3—92no—
145C22×Dic6—91no—
146C3×C22⋊Q8—92no—
147C2×C3×C4⋊C4—91no—
148#946—32no—
149C2×(C2×D3)⋊C4—92no—
150C2×C4⋊Dic3—92no—
151#924—92no—
152#897—32no—
153#897—94no—
154#897—92no—
155#897—92no—
156#897—182no—
157#896—94no—
158D3×C4⋊C4—32no—
159D3×C4⋊C4—94no—
160D3×C22⋊C4—94no—
161#878—94no—
162C22×C3⋊D3C22×D312yes—
163C22×C3⋊D3C22×D311yescharacteristic
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—61no—
168C22×C3×D3—62no—
169C2×D32—34no—
170C2×D32—38no—
171C2×D32—64no—
172C2×D32—64no—
173C2×D32—62no—
174C2×D32—62no—
175C32×Q8—62no—
176C2×C32×C4—22no—
177C2×C32×C4—21no—
178C2×C32×C4—32no—
179C2×C32×C4—34no—
180C2×C32×C4—41no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D312yes—
183C2×C3⋊Dic3—32no—
184C2×C3⋊Dic3—34no—
185C2×C3⋊Dic3—62no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3—122no—
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C4×C3⋊D3—24no—
191C4×C3⋊D3—64no—
192C4×C3⋊D3D1214yes—
193#634—62no—
194#634—64no—
195#634—64no—
196#634—64no—
197#634—124no—
198#634—124no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—24no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3C22×D314yes—
207C2×C3×Dic3C22×D314yes—
208C2×C3×Dic3C22×D314yes—
209C3×C4×D3—38no—
210C3×C4×D3—64no—
211C3×C4×D3—68no—
212C3×C4×D3—64no—
213C3×C4×D3—64no—
214C3×C4×D3—124no—
215C3×Dic6—24no—
216C3×Dic6—24no—
217#627—68no—
218#627—68no—
219#624—24no—
220#624—64no—
221#624—62no—
222D3×Dic3—68no—
223D3×Dic3—68no—
224D3×Dic3—64no—
225D3×Dic3—64no—
226D3×Dic3—64no—
227C2×C22⋊Q8—271noSylow
228C2×C33C22×D411yescharacteristic
229C2×C33C22×Q811yescharacteristic
230C2×C33C2×#4111yescharacteristic
231C3×C3⋊D3—24no—
232C3×C3⋊D3C22⋊Q814yescomplemented
233C23×D3—92no—
234C23×D3—91no—
235C2×C3×Q8—32no—
236C2×C3×Q8—61no—
237C22×C3×C4—92no—
238C22×C3×C4—91no—
239C22×C3×C4—92no—
240C22×Dic3—92no—
241C22×Dic3—91no—
242C22×Dic3—92no—
243D3×Q8—188no—
244D3×Q8—188no—
245C2×C4×D3—32no—
246C2×C4×D3—32no—
247C2×C4×D3—32no—
248C2×C4×D3—34no—
249C2×C4×D3—61no—
250C2×C4×D3—62no—
251C2×C4×D3—92no—
252C2×C4×D3—92no—
253C2×C4×D3—92no—
254C2×C4×D3—94no—
255C2×C4×D3—182no—
256C2×C4×D3—188no—
257C2×C4×D3—184no—
258C2×Dic6—94no—
259C2×Dic6—91no—
260C2×Dic6—92no—
261C2×Dic6—92no—
262C2×Dic6—182no—
263C2×Dic6—181no—
264C2×Dic6—182no—
265C2×Dic6—184no—
266C2×Dic6—361no—
267C2×Dic6D3211yescharacteristic, complemented
268C3×C4⋊C4—32no—
269C3×C4⋊C4—32no—
270C3×C4⋊C4—62no—
271C3×C4⋊C4—94no—
272C3×C4⋊C4—92no—
273C3×C22⋊C4—94no—
274#217—94no—
275(C2×D3)⋊C4—34no—
276(C2×D3)⋊C4—34no—
277(C2×D3)⋊C4—94no—
278(C2×D3)⋊C4—94no—
279(C2×D3)⋊C4—184no—
280C4⋊Dic3—34no—
281C4⋊Dic3—94no—
282C4⋊Dic3—94no—
283C4⋊Dic3—184no—
284C4⋊Dic3D3212yescomplemented
285Dic3⋊C4—34no—
286Dic3⋊C4—34no—
287Dic3⋊C4—94no—
288C22×C32—22no—
289C22×C32—21no—
290C22×C32—41no—
291C22×C32C23×D311yescharacteristic
292C22×C32C23×D312yes—
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3C2×D1212yes—
296C2×C3⋊D3#23712yes—
297C2×C3⋊D3D3×Q812yes—
298C2×C3⋊D3D3×Q814yescomplemented
299C2×C3×D3—34no—
300C2×C3×D3—34no—
301C2×C3×D3—34no—
302C2×C3×D3—34no—
303C2×C3×D3—34no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—64no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314D32—616no—
315D32—68no—
316C32×C4—24no—
317C32×C4—22no—
318C32×C4—42no—
319C32×C4—62no—
320C32×C4—64no—
321C32×C4C2×D1212yes—
322C32×C4D3×D414yes—
323C3⋊Dic3—22no—
324C3⋊Dic3—62no—
325C3⋊Dic3—64no—
326C3⋊Dic3—64no—
327C3⋊Dic3—124no—
328C3⋊Dic3—124no—
329C3⋊Dic3C2×D1212yes—
330C3×Dic3—24no—
331C3×Dic3—24no—
332C3×Dic3—24no—
333C3×Dic3—24no—
334C3×Dic3—34no—
335C3×Dic3—44no—
336C3×Dic3—62no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—122no—
340C22×Q8—271no—
341C23×C4—271no—
342C22⋊Q8—92no—
343C22⋊Q8—274no—
344C22⋊Q8—272no—
345C2×C4⋊C4—91no—
346C2×C4⋊C4—272no—
347C2×C22⋊C4—272no—
348C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
349C23×C3—91no—
350C23×C3—92no—
351C22×D3—32no—
352C22×D3—32no—
353C22×D3—32no—
354C22×D3—61no—
355C22×D3—62no—
356C22×D3—98no—
357C22×D3—94no—
358C22×D3—184no—
359C22×D3—184no—
360C22×D3—182no—
361C22×D3—182no—
362C3×Q8—64no—
363C3×Q8—64no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—34no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
371C2×C3×C4—64no—
372C2×C3×C4—91no—
373C2×C3×C4—94no—
374C2×C3×C4—92no—
375C2×C3×C4—184no—
376C2×C3×C4—184no—
377C2×C3×C4C2×D3212yes—
378C2×C3×C4C2×D3211yescharacteristic
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—61no—
383C2×Dic3—62no—
384C2×Dic3—94no—
385C2×Dic3—92no—
386C2×Dic3—182no—
387C2×Dic3—184no—
388C2×Dic3—184no—
389C2×Dic3—184no—
390C2×Dic3—182no—
391C2×Dic3—362no—
392C2×Dic3C2×D3214yes—
393C2×Dic3C2×D3212yes—
394C4×D3—38no—
395C4×D3—64no—
396C4×D3—68no—
397C4×D3—64no—
398C4×D3—64no—
399C4×D3—124no—
400C4×D3—184no—
401C4×D3—184no—
402C4×D3—184no—
403C4×D3—188no—
404C4×D3—188no—
405Dic6—22no—
406Dic6—184no—
407Dic6—188no—
408Dic6—188no—
409Dic6—184no—
410Dic6—184no—
411Dic6—182no—
412Dic6—364no—
413Dic6—364no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—22no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32C22×D1211yescharacteristic
424C2×C32C2×D3×D412yes—
425C2×C32C2×#23711yescharacteristic
426C2×C32C2×D3×Q811yescharacteristic
427C2×C32C2×D3×Q812yes—
428C2×C32D3×#4112yes—
429C3⋊D3—24no—
430C3⋊D3#89714yescomplemented
431C3×D3—38no—
432C3×D3—64no—
433C3×D3—64no—
434C3×D3—64no—
435C3×D3—124no—
436C24—271no—
437C2×Q8—31no—
438C2×Q8—272no—
439C2×Q8—271no—
440C2×Q8—544no—
441C22×C4—92no—
442C22×C4—91no—
443C22×C4—92no—
444C22×C4—272no—
445C22×C4—271no—
446C22×C4—272no—
447C22×C4—544no—
448C4⋊C4—32no—
449C4⋊C4—92no—
450C4⋊C4—94no—
451C4⋊C4—274no—
452C22⋊C4—94no—
453C22⋊C4—274no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—41no—
457C22×C3—92no—
458C22×C3—92no—
459C22×C3—92no—
460C22×C3—94no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3C22×D3212yes—
464C22×C3C22×D3211yescharacteristic
465C2×D3—34no—
466C2×D3—34no—
467C2×D3—34no—
468C2×D3—34no—
469C2×D3—34no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—64no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—1816no—
481C2×D3—188no—
482C3×C4—22no—
483C3×C4—24no—
484C3×C4—42no—
485C3×C4—64no—
486C3×C4—64no—
487C3×C4—64no—
488C3×C4—64no—
489C3×C4—92no—
490C3×C4—124no—
491C3×C4—182no—
492C3×C4D3×D1214yes—
493C3×C4#381112yes—
494Dic3—24no—
495Dic3—22no—
496Dic3—34no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—122no—
501Dic3—184no—
502Dic3—184no—
503Dic3—182no—
504Dic3—364no—
505Dic3—364no—
506C32—22no—
507C32—21no—
508C32—41no—
509C32C2×#89711yescharacteristic, complemented
510C32D3×C22⋊Q812yescomplemented
511C23—91no—
512C23—92no—
513C23—274no—
514C23—542no—
515C23—542no—
516Q8—62no—
517Q8—544no—
518Q8—542no—
519C2×C4—32no—
520C2×C4—34no—
521C2×C4—91no—
522C2×C4—94no—
523C2×C4—92no—
524C2×C4—184no—
525C2×C4—184no—
526C2×C4—272no—
527C2×C4—544no—
528C2×C4—544no—
529C2×C4—542no—
530C2×C4D3311yescharacteristic
531C2×C3—22no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—94no—
541C2×C3—184no—
542C2×C3C2×D3×D1212yes—
543C2×C3D3×#23712yes—
544C2×C3D32×Q812yes—
545C2×C3D32×Q811yescharacteristic
546C2×C3#6431911yescharacteristic
547C2×C3C2×#381111yescharacteristic
548D3—38no—
549D3—64no—
550D3—64no—
551D3—64no—
552D3—124no—
553C22—92no—
554C22—92no—
555C22—92no—
556C22—94no—
557C22—182no—
558C22—182no—
559C22—548no—
560C22C2×D3311yescharacteristic
561C4—64no—
562C4—62no—
563C4—92no—
564C4—182no—
565C4—542no—
566C4#8895512yes—
567C3—21no—
568C3—22no—
569C3—41no—
570C3D3×#89712yescomplemented
571C3#1059542911yescharacteristic, complemented
572C2—94no—
573C2—184no—
574C2#1172437411yescharacteristic
575C2#1172437811yescharacteristic
576C2C2×#8895511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×#1113C22
hypercenterC2×C4D33
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×Dic6C2semidirect
3. #11723431C2semidirect
4. C3×#63900C2semidirect
5. C3×#63871C2semidirect
6. #11722984C2semidirect
7. #11722908C2semidirect
12. C2×#6308C22semidirect
15. C2×C32×Dic6C22semidirect
19. C2×#6293C22semidirect
23. C32×C4⋊Dic3C22semidirect
24. Dic3⋊(C3⋊Dic3)C22semidirect
26. C3×#3724C22semidirect
27. C3×Dic3⋊Dic3C22semidirect
40. #63900D3semidirect
46. #63871D3semidirect
54. C2×#1113Q8semidirect
92. C2×C3×Dic6C2×D3semidirect
114. C3×C4⋊Dic3C2×D3semidirect
115. C3×C4⋊Dic3C2×D3semidirect
118. #3724C2×D3semidirect
122. Dic3⋊Dic3C2×D3semidirect
132. C2×C3×C3⋊D3C2×Q8semidirect
232. C3×C3⋊D3C22⋊Q8semidirect
267. C2×Dic6D32semidirect
284. C4⋊Dic3D32semidirect
298. C2×C3⋊D3D3×Q8semidirect
348. C33C2×C22⋊Q8semidirect
430. C3⋊D3#897semidirect
509. C32C2×#897semidirect
510. C32D3×C22⋊Q8semidirect
570. C3D3×#897semidirect
571. C3#10595429semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×C3⋊D3×Dic62
#117234312
C3×#639002
C3×#638712
#117229842
#117229082
#117229052
#105954293
D3×#8973