Finite Group Databasebeta

← #13177498

Subgroups — #13177498

Order 1728.

2972 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723627C212yesmaximal, complemented
5#11723622C212yesmaximal, complemented
6#11723618C212yesmaximal, complemented
7C2×#88804C212yesmaximal, complemented
8#11723494C211yesmaximal, characteristic, complemented
9#11723484C211yesmaximal, characteristic, complemented
10#11723452C211yesmaximal, characteristic, complemented
11#11723413C211yesmaximal, characteristic, complemented
12#10596016—31nomaximal
13#10595223—34nomaximal
14C23×C3⋊(C3⋊D3)C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yes—
17D4×C3⋊(C3⋊D3)C418yescomplemented
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
20C22×C3×C3⋊Dic3C2212yescomplemented
21C22×C32×Dic3C2212yescomplemented
22C2×#6289C2212yescomplemented
23C2×#6289C2212yescomplemented
24C3×#3757C2212yescomplemented
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
26C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
27C32×#217C2212yescomplemented
28C32×C4⋊Dic3C2211yescharacteristic, complemented
29C32×C4×Dic3C2211yescharacteristic, complemented
30#88806C2211yescharacteristic, complemented
31#88806C2211yescharacteristic, complemented
32#88804C2211yescharacteristic
33#88804C2212yes—
34#88804C2211yescharacteristic
35#88804C2212yescomplemented
36#88804C2212yescomplemented
37Dic3×C3⋊Dic3C2212yescomplemented
38C2×D4×C3⋊D3—31no—
39C2×D4×C3⋊D3—34no—
40C2×D4×C3⋊D3—64no—
41C32×C4×D4—31no—
42D4×C3⋊Dic3D311yescharacteristic, complemented
43C4×#638—32no—
44C2×#3752—32no—
45#64102—31no—
46#64101—31no—
47#64090—32no—
48C22⋊C4×C3⋊D3—32no—
49C4×C4⋊(C3⋊D3)—31no—
50#64081—31no—
51C3×D4×Dic3—34no—
52C3×D4×Dic3D314yescomplemented
53#63974—38no—
54C2×#3722—38no—
55#63952—38no—
56#63952—38no—
57#63894—34no—
58#63894—34no—
59#63874—34no—
60#63849—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)—42no—
63C22×C3⋊(C3⋊D3)—42no—
64C22×C3⋊(C3⋊D3)C2×C418yescomplemented
65C22×C3⋊(C3⋊D3)C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2312yes—
67C33×D4C2×C414yescomplemented
68C2×C33×C4C2311yescharacteristic
69#6312C2×C418yescomplemented
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4⋊(C3⋊(C3⋊D3))C2×C414yescomplemented
72C4×C3⋊(C3⋊D3)C2×C414yescomplemented
73C2×C3×C3⋊Dic3—42no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C2×C32×Dic3—42no—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81#6289—42no—
82#6289—42no—
83#5545—94no—
84#5545—91no—
85C23×C3⋊D3—32no—
86C23×C3⋊D3—38no—
87C23×C3⋊D3—68no—
88C2×C32×D4—24no—
89C2×C32×D4C2×D314yes—
90C2×C32×D4C2×D311yescharacteristic
91C22×C32×C4—32no—
92C2×#638—32no—
93C2×#638—38no—
94C2×#638—68no—
95C22×C3⋊Dic3C2×D312yescomplemented
96D4×C3⋊D3—38no—
97D4×C3⋊D3—332no—
98D4×C3⋊D3—632no—
99C2×C4⋊(C3⋊D3)—31no—
100C2×C4⋊(C3⋊D3)—34no—
101C2×C4⋊(C3⋊D3)—64no—
102C2×C4×C3⋊D3—31no—
103C2×C4×C3⋊D3—34no—
104C2×C4×C3⋊D3—32no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—64no—
107C22×C3×Dic3—38no—
108C22×C3×Dic3C2×D318yescomplemented
109C2×#624—38no—
110C2×#624—38no—
111C32×C4⋊C4—31no—
112C32×C22⋊C4—32no—
113C32×C42—31no—
114#3757C2×D312yescomplemented
115#3752—31no—
116#3752—31no—
117#3752—32no—
118#3752—32no—
119#3752—32no—
120C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
121#3750—31no—
122#3750—31no—
123C4×C3⋊Dic3—32no—
124C4×C3⋊Dic3C2×D311yescharacteristic, complemented
125C3×#217—38no—
126C3×#217C2×D318yescomplemented
127C3×C4⋊Dic3—34no—
128C3×C4⋊Dic3C2×D314yescomplemented
129C3×C4×Dic3—34no—
130C3×C4×Dic3C2×D314yescomplemented
131#3724—34no—
132#3724—34no—
133#3722—38no—
134#3722—34no—
135#3722—34no—
136#3722—38no—
137#3722—38no—
138Dic32—38no—
139C22×C33—22no—
140C22×C33—22no—
141C22×C33C22×C414yes—
142C22×C33C2411yescharacteristic
143C2×C3⋊(C3⋊D3)—21no—
144C2×C3⋊(C3⋊D3)—21no—
145C2×C3⋊(C3⋊D3)—22no—
146C2×C3⋊(C3⋊D3)—22no—
147C2×C3⋊(C3⋊D3)—48no—
148C2×C3⋊(C3⋊D3)C22×C412yes—
149C2×C3⋊(C3⋊D3)C22×C414yes—
150C33×C4C22×C412yes—
151C3⋊(C3⋊Dic3)C22×C412yes—
152C3×C3⋊Dic3—21no—
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3—21no—
155C32×Dic3—21no—
156C32×Dic3—22no—
157C32×Dic3—21no—
158C2×D3×D4—94no—
159C2×D3×D4—91no—
160C2×D3×D4—184no—
161C3×C4×D4—34no—
162C3×C4×D4—91no—
163D4×Dic3—34no—
164D4×Dic3C3⋊D311yescharacteristic, complemented
165C4×#67—98no—
166C4×#67—92no—
167C2×(C2×D3)⋊C4—98no—
168C2×(C2×D3)⋊C4—92no—
169#893—94no—
170#893—91no—
171#892—94no—
172#892—91no—
173#67⋊C4—98no—
174#67⋊C4—92no—
175D3×C22⋊C4—98no—
176D3×C22⋊C4—92no—
177C4×D12—94no—
178C4×D12—91no—
179#872—94no—
180#872—91no—
181C23×C32—28no—
182C23×C32C22×D318yes—
183C23×C32C22×D312yes—
184C22×C3⋊D3—31no—
185C22×C3⋊D3—32no—
186C22×C3⋊D3—38no—
187C22×C3⋊D3—34no—
188C22×C3⋊D3—38no—
189C22×C3⋊D3—332no—
190C22×C3⋊D3—64no—
191C22×C3⋊D3—632no—
192C22×C3⋊D3—68no—
193C22×C3⋊D3—122no—
194C22×C3⋊D3—122no—
195C22×C3⋊D3—128no—
196C22×C3⋊D3—128no—
197C22×C3⋊D3—1216no—
198C22×C3⋊D3—1216no—
199C32×D4—216no—
200C32×D4C4×D3116yescomplemented
201C32×D4C4×D314yescomplemented
202C2×C32×C4—24no—
203C2×C32×C4—32no—
204C2×C32×C4—31no—
205C2×C32×C4—31no—
206C2×C32×C4—122no—
207C2×C32×C4C22×D314yes—
208C2×C32×C4C22×D311yescharacteristic
209#638—38no—
210#638—332no—
211#638—632no—
212C2×C3⋊Dic3—31no—
213C2×C3⋊Dic3—34no—
214C2×C3⋊Dic3—42no—
215C2×C3⋊Dic3—64no—
216C2×C3⋊Dic3C22×D312yes—
217C2×C3⋊Dic3C22×D311yescharacteristic
218C2×C3⋊Dic3C22×D311yescharacteristic
219C4⋊(C3⋊D3)—34no—
220C4⋊(C3⋊D3)—316no—
221C4⋊(C3⋊D3)—616no—
222C4×C3⋊D3—34no—
223C4×C3⋊D3—316no—
224C4×C3⋊D3—616no—
225C4×C3⋊D3—122no—
226C4×C3⋊D3—122no—
227C2×C3×Dic3—38no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3—48no—
231C2×C3×Dic3—128no—
232C2×C3×Dic3C22×D318yes—
233C2×C3×Dic3C22×D314yes—
234C2×C3×Dic3C22×D314yes—
235#624—128no—
236#624—128no—
237#518—271noSylow
238C2×C33—24no—
239C2×C33C23×C411yescharacteristic
240C2×C33H(5,𝔽2)11yescharacteristic
241C2×C33H(5,𝔽2)11yescharacteristic
242C3⋊(C3⋊D3)—22no—
243C3⋊(C3⋊D3)—24no—
244C23×D3—98no—
245C23×D3—92no—
246C23×D3—188no—
247C2×C3×D4—24no—
248C2×C3×D4D3214yes—
249C2×C3×D4C2×C3⋊D311yescharacteristic
250C22×C3×C4—38no—
251C22×C3×C4—92no—
252C2×#67—98no—
253C2×#67—92no—
254C2×#67—188no—
255C22×Dic3—38no—
256C22×Dic3C2×C3⋊D312yescomplemented
257D3×D4—932no—
258D3×D4—98no—
259D3×D4—1832no—
260C2×D12—94no—
261C2×D12—91no—
262C2×D12—184no—
263C2×C4×D3—94no—
264C2×C4×D3—91no—
265C2×C4×D3—98no—
266C2×C4×D3—98no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—184no—
270C3×C4⋊C4—34no—
271C3×C4⋊C4—91no—
272C3×C22⋊C4—38no—
273C3×C22⋊C4—92no—
274C3×C42—34no—
275C3×C42—91no—
276#217—38no—
277#217C2×C3⋊D312yescomplemented
278(C2×D3)⋊C4—94no—
279(C2×D3)⋊C4—94no—
280(C2×D3)⋊C4—98no—
281(C2×D3)⋊C4—98no—
282(C2×D3)⋊C4—98no—
283(C2×D3)⋊C4—91no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288C4⋊Dic3—34no—
289C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
290Dic3⋊C4—94no—
291Dic3⋊C4—94no—
292Dic3⋊C4—91no—
293Dic3⋊C4—91no—
294C4×Dic3—34no—
295C4×Dic3—98no—
296C4×Dic3—92no—
297C4×Dic3C2×C3⋊D311yescharacteristic, complemented
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—24no—
301C22×C32—216no—
302C22×C32—28no—
303C22×C32—28no—
304C22×C32—48no—
305C22×C32—48no—
306C22×C32C2×C4×D3116yes—
307C22×C32C2×C4×D314yes—
308C22×C32C23×D314yes—
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—32no—
311C2×C3⋊D3—34no—
312C2×C3⋊D3—38no—
313C2×C3⋊D3—316no—
314C2×C3⋊D3—68no—
315C2×C3⋊D3—616no—
316C2×C3⋊D3—68no—
317C2×C3⋊D3—68no—
318C2×C3⋊D3—62no—
319C2×C3⋊D3—64no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—64no—
323C2×C3⋊D3—61no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—128no—
327C2×C3⋊D3—128no—
328C2×C3⋊D3—128no—
329C2×C3⋊D3—1232no—
330C2×C3⋊D3—1264no—
331C32×C4—28no—
332C32×C4—61no—
333C32×C4—62no—
334C32×C4—61no—
335C32×C4C2×C4×D318yes—
336C32×C4C2×C4×D312yes—
337C3⋊Dic3—22no—
338C3⋊Dic3—21no—
339C3⋊Dic3—21no—
340C3⋊Dic3—32no—
341C3⋊Dic3—38no—
342C3⋊Dic3—68no—
343C3×Dic3—28no—
344C3×Dic3—24no—
345C3×Dic3—24no—
346C3×Dic3—64no—
347C3×Dic3—68no—
348C3×Dic3—64no—
349C22×D4—271no—
350C4×D4—31no—
351C4×D4—91no—
352C4×D4—271no—
353C4×D4—271no—
354C4×D4—272no—
355C4×D4—272no—
356#117—271no—
357#117—271no—
358C2×C22⋊C4—272no—
359C2×C22⋊C4—272no—
360C33#51811yescharacteristic, complemented, Sylow
361C23×C3—28no—
362C23×C3C2×D3218yes—
363C23×C3C22×C3⋊D312yes—
364C22×D3—94no—
365C22×D3—98no—
366C22×D3—91no—
367C22×D3—92no—
368C22×D3—932no—
369C22×D3—98no—
370C22×D3—184no—
371C22×D3—188no—
372C22×D3—1832no—
373C22×D3—368no—
374C22×D3—368no—
375C22×D3—362no—
376C22×D3—362no—
377C22×D3—3616no—
378C22×D3—3616no—
379C3×D4—216no—
380C3×D4#624116yescomplemented
381C3×D4C4×C3⋊D314yescomplemented
382C2×C3×C4—24no—
383C2×C3×C4—38no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—92no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—128no—
390C2×C3×C4—362no—
391C2×C3×C4C2×D3214yes—
392C2×C3×C4C22×C3⋊D311yescharacteristic
393#67—932no—
394#67—98no—
395#67—1832no—
396C2×Dic3—34no—
397C2×Dic3—34no—
398C2×Dic3—38no—
399C2×Dic3—42no—
400C2×Dic3—94no—
401C2×Dic3—91no—
402C2×Dic3—128no—
403C2×Dic3—184no—
404C2×Dic3C22×C3⋊D312yes—
405C2×Dic3C22×C3⋊D311yescharacteristic
406C2×Dic3C22×C3⋊D311yescharacteristic
407D12—916no—
408D12—94no—
409D12—1816no—
410C4×D3—916no—
411C4×D3—94no—
412C4×D3—1816no—
413C4×D3—362no—
414C4×D3—362no—
415C4×D3—368no—
416C4×D3—368no—
417C2×C32—24no—
418C2×C32—24no—
419C2×C32—24no—
420C2×C32—24no—
421C2×C32—216no—
422C2×C32—416no—
423C2×C32C22×C4×D314yes—
424C2×C32C22×C4×D311yescharacteristic
425C2×C32#100414yes—
426C2×C32#100411yescharacteristic
427C2×C32#100914yes—
428C2×C32#100911yescharacteristic
429C3⋊D3—616no—
430C3⋊D3—64no—
431C3⋊D3—68no—
432C3⋊D3—62no—
433C3⋊D3—616no—
434C3⋊D3—1216no—
435C24—272no—
436C2×D4—272no—
437C2×D4—278no—
438C2×D4—271no—
439C2×D4D3×C3⋊D311yescharacteristic
440C22×C4—32no—
441C22×C4—92no—
442C22×C4—272no—
443C22×C4—272no—
444C22×C4—271no—
445C4⋊C4—31no—
446C4⋊C4—91no—
447C4⋊C4—271no—
448C4⋊C4—271no—
449C22⋊C4—32no—
450C22⋊C4—92no—
451C22⋊C4—271no—
452C22⋊C4—271no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—272no—
456C42—31no—
457C42—91no—
458C42—272no—
459C22×C3—28no—
460C22×C3—28no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—24no—
464C22×C3—216no—
465C22×C3—48no—
466C22×C3—48no—
467C22×C3C2×#624116yes—
468C22×C3C2×C4×C3⋊D314yes—
469C22×C3C22×D3214yes—
470C22×C3C23×C3⋊D311yescharacteristic
471C2×D3—98no—
472C2×D3—916no—
473C2×D3—92no—
474C2×D3—94no—
475C2×D3—188no—
476C2×D3—1816no—
477C2×D3—188no—
478C2×D3—188no—
479C2×D3—188no—
480C2×D3—188no—
481C2×D3—184no—
482C2×D3—184no—
483C2×D3—182no—
484C2×D3—181no—
485C2×D3—182no—
486C2×D3—181no—
487C2×D3—368no—
488C2×D3—368no—
489C2×D3—3632no—
490C2×D3—368no—
491C2×D3—3664no—
492C3×C4—28no—
493C3×C4—64no—
494C3×C4—68no—
495C3×C4—64no—
496C3×C4—181no—
497C3×C4—182no—
498C3×C4—181no—
499C3×C4C2×#62418yes—
500C3×C4C2×C4×C3⋊D312yes—
501Dic3—22no—
502Dic3—21no—
503Dic3—21no—
504Dic3—64no—
505Dic3—68no—
506Dic3—64no—
507Dic3—98no—
508Dic3—92no—
509Dic3—188no—
510C32—24no—
511C32#554514yescomplemented
512C32#554511yescharacteristic, complemented
513C23—271no—
514C23—272no—
515C23—278no—
516C23—1082no—
517C23—1082no—
518C23C2×D3×C3⋊D312yes—
519D4—278no—
520D4—274no—
521D4#628914yescomplemented
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—32no—
525C2×C4—91no—
526C2×C4—91no—
527C2×C4—92no—
528C2×C4—122no—
529C2×C4—271no—
530C2×C4—274no—
531C2×C4—362no—
532C2×C4—1082no—
533C2×C4—1082no—
534C2×C4C2×D3×C3⋊D311yescharacteristic
535C2×C3—216no—
536C2×C3—24no—
537C2×C3—24no—
538C2×C3—24no—
539C2×C3—24no—
540C2×C3—416no—
541C2×C3#6431414yes—
542C2×C3#6431414yes—
543C2×C3C22×#62414yes—
544C2×C3C22×C4×C3⋊D311yescharacteristic
545C2×C3#6436111yescharacteristic
546C2×C3#6436611yescharacteristic
547D3—1816no—
548D3—188no—
549D3—1816no—
550D3—182no—
551D3—184no—
552D3—3616no—
553C22—22no—
554C22—22no—
555C22—272no—
556C22—274no—
557C22—542no—
558C22—541no—
559C22—542no—
560C22—541no—
561C22—1088no—
562C22C2×#628914yes—
563C22C22×D3×C3⋊D311yescharacteristic
564C4—61no—
565C4—62no—
566C4—61no—
567C4—181no—
568C4—181no—
569C4—182no—
570C4—272no—
571C4C2×#628912yes—
572C3—24no—
573C3#1059522314yescomplemented
574C3#1059601611yescharacteristic, complemented
575C2—24no—
576C2—542no—
577C2—544no—
578C2#1172458811yescharacteristic
579C2#1172458911yescharacteristic
580C2C22×#628911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×D4×C3⋊(C3⋊D3)C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×D4×C3⋊Dic3C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723627C2semidirect
5. #11723622C2semidirect
6. #11723618C2semidirect
7. C2×#88804C2semidirect
8. #11723494C2semidirect
9. #11723484C2semidirect
10. #11723452C2semidirect
11. #11723413C2semidirect
17. D4×C3⋊(C3⋊D3)C4semidirect
20. C22×C3×C3⋊Dic3C22semidirect
21. C22×C32×Dic3C22semidirect
22. C2×#6289C22semidirect
23. C2×#6289C22semidirect
24. C3×#3757C22semidirect
25. C3×C4⋊(C3⋊Dic3)C22semidirect
26. C3×C4×C3⋊Dic3C22semidirect
27. C32×#217C22semidirect
28. C32×C4⋊Dic3C22semidirect
29. C32×C4×Dic3C22semidirect
30. #88806C22semidirect
31. #88806C22semidirect
35. #88804C22semidirect
36. #88804C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
42. D4×C3⋊Dic3D3semidirect
52. C3×D4×Dic3D3semidirect
64. C22×C3⋊(C3⋊D3)C2×C4semidirect
67. C33×D4C2×C4semidirect
69. #6312C2×C4semidirect
71. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
72. C4×C3⋊(C3⋊D3)C2×C4semidirect
95. C22×C3⋊Dic3C2×D3semidirect
108. C22×C3×Dic3C2×D3semidirect
114. #3757C2×D3semidirect
120. C4⋊(C3⋊Dic3)C2×D3semidirect
124. C4×C3⋊Dic3C2×D3semidirect
126. C3×#217C2×D3semidirect
128. C3×C4⋊Dic3C2×D3semidirect
130. C3×C4×Dic3C2×D3semidirect
164. D4×Dic3C3⋊D3semidirect
200. C32×D4C4×D3semidirect
201. C32×D4C4×D3semidirect
256. C22×Dic3C2×C3⋊D3semidirect
277. #217C2×C3⋊D3semidirect
289. C4⋊Dic3C2×C3⋊D3semidirect
297. C4×Dic3C2×C3⋊D3semidirect
360. C33#518semidirect
380. C3×D4#624semidirect
381. C3×D4C4×C3⋊D3semidirect
511. C32#5545semidirect
512. C32#5545semidirect
521. D4#6289semidirect
573. C3#10595223semidirect
574. C3#10596016semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C3×D4×C3⋊Dic32
C32×D4×Dic32
#117236272
#117236222
#117236182
C2×#888042
#117234942
#117234842
#117234522
#117234132
#105960163
#105952233