Finite Group Databasebeta

← #13169304

Subgroups — #13169304

Order 1728.

1726 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1113C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723526C212yesmaximal, complemented
4C3×#63871C212yesmaximal, complemented
5C3×#63849C211yesmaximal, characteristic, complemented
6#11722999C212yesmaximal, complemented
7#11722952C212yesmaximal, complemented
8#11722937C214yesmaximal, complemented
9#10595430—31nomaximal
10#10594627—32nomaximal
11C22×#1113C2211yescharacteristic
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic, complemented
16C2×#6294C2212yes—
17C2×#6294C2211yescharacteristic
18C4×#1113—24no—
19C2×#6292C2214yes—
20C2×#6292C2212yes—
21C2×#6289C2212yescomplemented
22#89018—24no—
23C32×C4⋊Dic3C2212yescomplemented
24#88804C2214yescomplemented
25C3×Dic3⋊Dic3C2214yescomplemented
26C3×#3722C2212yes—
27C3×#3722C2214yes—
28C3×Dic32C2212yescomplemented
29C2×#3832—31no—
30C2×C4×D32—32no—
31C2×C4×D32—31no—
32C2×#3799—32no—
33C2×#3799—61no—
34#64104—32no—
35C3×#946—32no—
36C3×#897—32no—
37C3×#892—32no—
38#63923—34no—
39#63923—32no—
40#63909—32no—
41#63909—32no—
42#63909—62no—
43#63897—32no—
44#63894—34no—
45#63890—34no—
46#63881—34no—
47#63880—34no—
48#63871D312yescomplemented
49#63849D311yescharacteristic, complemented
50C22×C3⋊(C3⋊D3)C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C22×C3×C3⋊D3C2312yes—
53C2×#1113—22no—
54C2×#1113—24no—
55C2×C33×C4C2311yescharacteristic
56C4⋊(C3⋊(C3⋊D3))—22no—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C3×C4×C3⋊D3—24no—
60C3×C4×C3⋊D3D414yescomplemented
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2314yes—
63C32×Dic6—22no—
64#6294—22no—
65#6294—24no—
66#6292—24no—
67#6292—28no—
68#6289—24no—
69#5627—92no—
70#5524—91no—
71C22×D32—31no—
72C22×D32—32no—
73C2×C32×Q8—31no—
74#3832—64no—
75C2×C4⋊(C3⋊D3)—34no—
76C2×C4⋊(C3⋊D3)—32no—
77C2×C4⋊(C3⋊D3)—32no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C4⋊(C3⋊D3)—61no—
80C2×C4⋊(C3⋊D3)—62no—
81C2×C4⋊(C3⋊D3)—121no—
82C2×C4×C3⋊D3—32no—
83C2×C4×C3⋊D3C2×D312yes—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—61no—
89C2×C3×C4×D3—62no—
90C2×C3×Dic6—21no—
91C2×C3×Dic6C2×D312yescomplemented
92C2×#626—34no—
93C2×#626—34no—
94C2×#626—34no—
95C2×#626—62no—
96C2×#626—64no—
97C2×#624—32no—
98C2×#624—31no—
99C2×#624—34no—
100C2×#624—62no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—32no—
104C4×D32—68no—
105C4×D32—64no—
106#3799—68no—
107#3799—68no—
108C32×C4⋊C4—32no—
109#3752—34no—
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—34no—
113C3×C4⋊Dic3—22no—
114C3×C4⋊Dic3—34no—
115C3×C4⋊Dic3C2×D312yescomplemented
116C3×C4⋊Dic3C2×D312yescomplemented
117C3×Dic3⋊C4—34no—
118C3×C4×Dic3—34no—
119Dic3⋊Dic3C2×D314yescomplemented
120#3722—34no—
121#3722—34no—
122#3722—64no—
123#3722C2×D312yes—
124#3722C2×D314yes—
125Dic32C2×D312yescomplemented
126C22×C33C2411yescharacteristic
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—24no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3C2×D412yes—
133C2×C3×C3⋊D3#4114yescomplemented
134#1113—28no—
135C33×C4C2×D412yes—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3C2×D412yes—
138C32×Dic3—22no—
139C32×Dic3—24no—
140C2×#239—92no—
141C2×#239—181no—
142C2×#235—91no—
143C22×C4×D3—91no—
144C22×C4×D3—92no—
145C3×C22⋊Q8—92no—
146C3×#117—91no—
147#946—32no—
148#930—92no—
149#897—32no—
150#895—92no—
151#895—92no—
152#895—182no—
153#894—94no—
154#893—94no—
155#892—32no—
156#886—92no—
157#882—92no—
158#882—94no—
159#67⋊C4—94no—
160C4×D12—94no—
161C22×C3⋊D3—32no—
162C22×C3⋊D3—34no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—64no—
165C22×C3⋊D3—122no—
166C22×C3⋊D3C22×D311yescharacteristic
167C22×C3⋊D3C22×D312yes—
168C22×C3×D3—32no—
169C22×C3×D3—32no—
170C22×C3×D3—32no—
171C22×C3×D3—61no—
172C22×C3×D3—62no—
173C2×D32—64no—
174C2×D32—64no—
175C2×D32—64no—
176C2×D32—62no—
177C2×D32—64no—
178C32×Q8—62no—
179C2×C32×C4—22no—
180C2×C32×C4—21no—
181C2×C32×C4—34no—
182C2×C32×C4—32no—
183C2×C32×C4—41no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C4⋊(C3⋊D3)—64no—
189C4⋊(C3⋊D3)—62no—
190C4⋊(C3⋊D3)—64no—
191C4⋊(C3⋊D3)—68no—
192C4⋊(C3⋊D3)—64no—
193C4⋊(C3⋊D3)—124no—
194C4⋊(C3⋊D3)—124no—
195C4×C3⋊D3—24no—
196C4×C3⋊D3—64no—
197C4×C3⋊D3D1214yescomplemented
198C2×C3×Dic3—22no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3C22×D314yes—
206C2×C3×Dic3C22×D314yes—
207C2×C3×Dic3C22×D314yes—
208C3×C4×D3—38no—
209C3×C4×D3—64no—
210C3×C4×D3—64no—
211C3×C4×D3—64no—
212C3×C4×D3—124no—
213C3×Dic6—24no—
214C3×Dic6—24no—
215#626—68no—
216#626—68no—
217#626—68no—
218#626—68no—
219#626—128no—
220#624—68no—
221#624—68no—
222#624—64no—
223#624—62no—
224D3×Dic3—64no—
225D3×Dic3—64no—
226D3×Dic3—64no—
227#525—271noSylow
228C2×C33C22×D411yescharacteristic
229C2×C33C2×#4112yes—
230C3⋊(C3⋊D3)—42no—
231C3×C3⋊D3—22no—
232C3×C3⋊D3—24no—
233C23×D3—92no—
234C23×D3—91no—
235C2×C3×Q8—32no—
236C2×C3×Q8—61no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C2×#67—94no—
240C2×#67—92no—
241C22×Dic3—92no—
242C22×Dic3—91no—
243#239—188no—
244#239—188no—
245#235—184no—
246C2×D12—94no—
247C2×D12—94no—
248C2×D12—94no—
249C2×D12—91no—
250C2×D12—92no—
251C2×D12—181no—
252C2×D12—182no—
253C2×D12—182no—
254C2×D12—184no—
255C2×D12—361no—
256C2×C4×D3—32no—
257C2×C4×D3—32no—
258C2×C4×D3—32no—
259C2×C4×D3—61no—
260C2×C4×D3—62no—
261C2×C4×D3—94no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—182no—
267C2×C4×D3—188no—
268C2×C4×D3—184no—
269C2×Dic6D3211yescharacteristic, complemented
270C3×C4⋊C4—32no—
271C3×C4⋊C4—32no—
272C3×C4⋊C4—62no—
273C3×C4⋊C4—94no—
274C3×C22⋊C4—94no—
275C3×C22⋊C4—92no—
276C3×C42—92no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—34no—
279(C2×D3)⋊C4—34no—
280(C2×D3)⋊C4—94no—
281(C2×D3)⋊C4—94no—
282(C2×D3)⋊C4—94no—
283(C2×D3)⋊C4—184no—
284C4⋊Dic3—34no—
285C4⋊Dic3D3212yescomplemented
286Dic3⋊C4—34no—
287C4×Dic3—34no—
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⋊D3—22no—
296C2×C3⋊D3—62no—
297C2×C3⋊D3—64no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—64no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—64no—
302C2×C3⋊D3—124no—
303C2×C3⋊D3—124no—
304C2×C3⋊D3—124no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—124no—
307C2×C3⋊D3—244no—
308C2×C3⋊D3C2×D1212yes—
309C2×C3⋊D3#23914yescomplemented
310C2×C3×D3—34no—
311C2×C3×D3—34no—
312C2×C3×D3—34no—
313C2×C3×D3—62no—
314C2×C3×D3—64no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—122no—
322C2×C3×D3—122no—
323C2×C3×D3—122no—
324D32—68no—
325D32—616no—
326C32×C4—24no—
327C32×C4—22no—
328C32×C4—42no—
329C32×C4—62no—
330C32×C4—64no—
331C32×C4C2×D1212yes—
332C32×C4D3×D414yes—
333C3⋊Dic3—22no—
334C3⋊Dic3C2×D1212yes—
335C3×Dic3—24no—
336C3×Dic3—24no—
337C3×Dic3—24no—
338C3×Dic3—24no—
339C3×Dic3—34no—
340C3×Dic3—44no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—122no—
345C2×#41—271no—
346C23×C4—271no—
347#123—272no—
348C22⋊Q8—92no—
349#121—272no—
350#120—272no—
351C4×D4—274no—
352#117—91no—
353C33#52511yescharacteristic, complemented, Sylow
354C23×C3—92no—
355C23×C3—91no—
356C22×D3—32no—
357C22×D3—32no—
358C22×D3—32no—
359C22×D3—61no—
360C22×D3—62no—
361C22×D3—94no—
362C22×D3—92no—
363C22×D3—182no—
364C22×D3—184no—
365C22×D3—184no—
366C22×D3—184no—
367C22×D3—184no—
368C22×D3—182no—
369C22×D3—184no—
370C22×D3—362no—
371C3×Q8—64no—
372C3×Q8—64no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—34no—
376C2×C3×C4—34no—
377C2×C3×C4—34no—
378C2×C3×C4—41no—
379C2×C3×C4—62no—
380C2×C3×C4—64no—
381C2×C3×C4—91no—
382C2×C3×C4—94no—
383C2×C3×C4—92no—
384C2×C3×C4—184no—
385C2×C3×C4C2×D3212yes—
386C2×C3×C4C2×D3211yescharacteristic
387#67—184no—
388#67—188no—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—61no—
393C2×Dic3—62no—
394C2×Dic3—184no—
395C2×Dic3—182no—
396C2×Dic3C2×D3214yes—
397C2×Dic3C2×D3212yes—
398D12—184no—
399D12—188no—
400D12—188no—
401D12—188no—
402D12—188no—
403D12—184no—
404D12—182no—
405D12—364no—
406D12—364no—
407D12—368no—
408C4×D3—38no—
409C4×D3—64no—
410C4×D3—64no—
411C4×D3—64no—
412C4×D3—124no—
413C4×D3—184no—
414C4×D3—188no—
415C4×D3—184no—
416C4×D3—188no—
417C4×D3—184no—
418C4×D3—184no—
419Dic6—22no—
420C2×C32—22no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—41no—
426C2×C32—42no—
427C2×C32C22×D1211yescharacteristic
428C2×C32C2×D3×D412yes—
429C2×C32C2×#23912yes—
430C2×C32D3×#4112yes—
431C2×C32D3×#4112yes—
432C3⋊D3—22no—
433C3⋊D3—24no—
434C3⋊D3—122no—
435C3⋊D3—124no—
436C3⋊D3—124no—
437C3⋊D3—244no—
438C3⋊D3—244no—
439C3×D3—64no—
440C3×D3—64no—
441C3×D3—64no—
442C3×D3—124no—
443C3×D3—122no—
444C24—271no—
445#41—544no—
446C2×Q8—31no—
447C2×D4—274no—
448C2×D4—272no—
449C2×D4—271no—
450C22×C4—91no—
451C22×C4—92no—
452C22×C4—272no—
453C22×C4—272no—
454C22×C4—271no—
455C22×C4—544no—
456C4⋊C4—32no—
457C4⋊C4—94no—
458C22⋊C4—92no—
459C22⋊C4—94no—
460C22⋊C4—274no—
461C42—92no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—41no—
465C22×C3—94no—
466C22×C3—92no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3C22×D3212yes—
471C22×C3C22×D3211yescharacteristic
472C2×D3—34no—
473C2×D3—34no—
474C2×D3—34no—
475C2×D3—62no—
476C2×D3—64no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—122no—
484C2×D3—122no—
485C2×D3—122no—
486C2×D3—1816no—
487C2×D3—188no—
488C2×D3—184no—
489C2×D3—184no—
490C2×D3—184no—
491C2×D3—184no—
492C2×D3—184no—
493C2×D3—182no—
494C2×D3—364no—
495C2×D3—364no—
496C2×D3—364no—
497C2×D3—364no—
498C2×D3—364no—
499C2×D3—724no—
500C3×C4—22no—
501C3×C4—24no—
502C3×C4—42no—
503C3×C4—64no—
504C3×C4—64no—
505C3×C4—64no—
506C3×C4—64no—
507C3×C4—92no—
508C3×C4—124no—
509C3×C4—182no—
510C3×C4D3×D1214yes—
511C3×C4#381112yes—
512Dic3—24no—
513Dic3—22no—
514Dic3—34no—
515Dic3—62no—
516Dic3—62no—
517Dic3—62no—
518Dic3—122no—
519C32—22no—
520C32—21no—
521C32—41no—
522C32#552411yescharacteristic, complemented
523C32#562712yescomplemented
524C23—92no—
525C23—91no—
526C23—272no—
527C23—542no—
528C23—544no—
529Q8—62no—
530D4—544no—
531D4—548no—
532D4—542no—
533C2×C4—34no—
534C2×C4—32no—
535C2×C4—91no—
536C2×C4—94no—
537C2×C4—92no—
538C2×C4—184no—
539C2×C4—544no—
540C2×C4—544no—
541C2×C4—542no—
542C2×C4D3311yescharacteristic
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—22no—
548C2×C3—42no—
549C2×C3—41no—
550C2×C3—184no—
551C2×C3—182no—
552C2×C3C2×D3×D1212yes—
553C2×C3#6431212yes—
554C2×C3D3×#23912yes—
555C2×C3D3×#23912yes—
556C2×C3C2×#381111yescharacteristic
557D3—64no—
558D3—64no—
559D3—64no—
560D3—122no—
561D3—124no—
562D3—364no—
563D3—364no—
564D3—362no—
565D3—724no—
566D3—724no—
567C22—94no—
568C22—92no—
569C22—182no—
570C22—182no—
571C22—182no—
572C22—548no—
573C22—544no—
574C22—542no—
575C22C2×D3311yescharacteristic
576C4—64no—
577C4—62no—
578C4—92no—
579C4—182no—
580C4#8895512yes—
581C3—21no—
582C3—22no—
583C3—41no—
584C3#1059462712yescomplemented
585C3#1059543011yescharacteristic, complemented
586C2—184no—
587C2—182no—
588C2—1082no—
589C2#1172437712yes—
590C2C2×#8895511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×#1113C2semidirect
2. C2×#89018C2semidirect
3. #11723526C2semidirect
4. C3×#63871C2semidirect
5. C3×#63849C2semidirect
6. #11722999C2semidirect
7. #11722952C2semidirect
8. #11722937C2semidirect
12. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C2×C32×Dic6C22semidirect
21. C2×#6289C22semidirect
23. C32×C4⋊Dic3C22semidirect
24. #88804C22semidirect
25. C3×Dic3⋊Dic3C22semidirect
28. C3×Dic32C22semidirect
48. #63871D3semidirect
49. #63849D3semidirect
60. C3×C4×C3⋊D3D4semidirect
91. C2×C3×Dic6C2×D3semidirect
115. C3×C4⋊Dic3C2×D3semidirect
116. C3×C4⋊Dic3C2×D3semidirect
119. Dic3⋊Dic3C2×D3semidirect
125. Dic32C2×D3semidirect
133. C2×C3×C3⋊D3#41semidirect
197. C4×C3⋊D3D12semidirect
269. C2×Dic6D32semidirect
285. C4⋊Dic3D32semidirect
309. C2×C3⋊D3#239semidirect
353. C33#525semidirect
522. C32#5524semidirect
523. C32#5627semidirect
584. C3#10594627semidirect
585. C3#10595430semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×#890182
#117235262
C3×#638712
C3×#638492
#117229992
#117229522
#117229372
#105954303
#105946273