Finite Group Databasebeta

← #11807708

Subgroups — #11807708

Order 1152.

1728 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64093C211yesmaximal, characteristic
2C32×#388C211yesmaximal, characteristic, complemented
3#10592688C211yesmaximal, characteristic, complemented
4#10592686C211yesmaximal, characteristic, complemented
5#10592657C211yesmaximal, characteristic, complemented
6#10592657C211yesmaximal, characteristic, complemented
7#10592656C211yesmaximal, characteristic, complemented
8#10592655C211yesmaximal, characteristic, complemented
9#10592649C211yesmaximal, characteristic, complemented
10#10592638C211yesmaximal, characteristic, complemented
11#10592632C211yesmaximal, characteristic, complemented
12#10592369C211yesmaximal, characteristic, complemented
13#10592364C211yesmaximal, characteristic, complemented
14#10592363C211yesmaximal, characteristic, complemented
15#10592354C211yesmaximal, characteristic, complemented
16#75490—34nomaximal
17C22×#638C2211yescharacteristic
18C22×#638C2211yescharacteristic
19C23×C3⋊Dic3C2211yescharacteristic
20C22×C4⋊(C3⋊D3)C2211yescharacteristic
21C22×C4×C3⋊D3C2211yescharacteristic
22C22×C4×C3⋊D3C2211yescharacteristic
23C2×C32×C4⋊C4C2211yescharacteristic
24C2×C32×C22⋊C4C2211yescharacteristic
25C2×C32×C22⋊C4C2211yescharacteristic
26C2×C32×C22⋊C4C2211yescharacteristic
27C2×C32×C42C2211yescharacteristic
28C2×#3757C2211yescharacteristic
29C2×#3752C2211yescharacteristic
30C2×#3752C2211yescharacteristic
31C2×#3752C2211yescharacteristic
32C2×#3752C2211yescharacteristic
33C2×#3752C2211yescharacteristic
34C2×#3752C2211yescharacteristic
35C2×#3752C2211yescharacteristic
36C2×#3752C2211yescharacteristic
37C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
38C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
39C2×#3750C2211yescharacteristic
40C2×#3750C2211yescharacteristic
41C2×C4×C3⋊Dic3C2211yescharacteristic
42#64093C418yescomplemented
43C32×#95C2211yescharacteristic
44C32×#95C2211yescharacteristic
45#63658C2211yescharacteristic
46#63658C2211yescharacteristic
47#63658C2211yescharacteristic
48#63658C2211yescharacteristic
49#63658C2211yescharacteristic
50#63658C2211yescharacteristic
51#63658C2211yescharacteristic
52#63658C2211yescharacteristic
53C2×#884—34no—
54C3×#388D314yescomplemented
55#4989—34no—
56#4987—34no—
57#4958—34no—
58#4958—34no—
59#4957—34no—
60#4956—34no—
61#4950—34no—
62#4939—34no—
63#4933—34no—
64#4670—34no—
65#4665—34no—
66#4664—34no—
67#4655—34no—
68C24×C32C2311yescharacteristic
69C23×C3⋊D3C2311yescharacteristic
70C23×C3⋊D3C2311yescharacteristic
71C22×C32×C4C2311yescharacteristic
72C22×C32×C4C2311yescharacteristic
73C22×C32×C4C2311yescharacteristic
74C22×C32×C4C2311yescharacteristic
75C22×C32×C4C2311yescharacteristic
76C22×C32×C4C2311yescharacteristic
77C2×#638—41no—
78C2×#638—41no—
79C2×#638—41no—
80C2×#638—41no—
81C2×#638C2×C414yes—
82C2×#638C2×C414yescomplemented
83C22×C3⋊Dic3—41no—
84C22×C3⋊Dic3—41no—
85C22×C3⋊Dic3—41no—
86C22×C3⋊Dic3C2311yescharacteristic
87C22×C3⋊Dic3C2311yescharacteristic
88C22×C3⋊Dic3C2311yescharacteristic
89C22×C3⋊Dic3C2311yescharacteristic
90C22×C3⋊Dic3C2311yescharacteristic
91C22×C3⋊Dic3C2311yescharacteristic
92C2×C4⋊(C3⋊D3)—41no—
93C2×C4⋊(C3⋊D3)—41no—
94C2×C4⋊(C3⋊D3)C2×C414yescomplemented
95C2×C4×C3⋊D3—41no—
96C2×C4×C3⋊D3—41no—
97C2×C4×C3⋊D3—41no—
98C2×C4×C3⋊D3—41no—
99C2×C4×C3⋊D3—41no—
100C2×C4×C3⋊D3C2×C414yescomplemented
101C32×C4⋊C4—41no—
102C32×C22⋊C4—41no—
103C32×C22⋊C4—41no—
104C32×C22⋊C4C2×C414yescomplemented
105C32×C42—41no—
106#3757—41no—
107#3752—41no—
108#3752—41no—
109#3752—41no—
110#3752—41no—
111#3752—41no—
112#3752—41no—
113#3752—41no—
114#3752C2×C414yescomplemented
115C4⋊(C3⋊Dic3)—41no—
116C4⋊(C3⋊Dic3)—41no—
117#3750—41no—
118#3750C2×C414yescomplemented
119C4×C3⋊Dic3—41no—
120#2377—91noSylow
121C22×#67—34no—
122C22×#67—34no—
123C23×Dic3—34no—
124C22×D12—34no—
125C22×C4×D3—34no—
126C22×C4×D3—34no—
127C2×C3×C4⋊C4C2×D314yes—
128C2×C3×C22⋊C4C2×D314yes—
129C2×C3×C22⋊C4C2×D314yes—
130C2×C3×C22⋊C4C2×D314yes—
131C2×C3×C42C2×D314yes—
132C2×#217—34no—
133C2×(C2×D3)⋊C4—34no—
134C2×(C2×D3)⋊C4—34no—
135C2×(C2×D3)⋊C4—34no—
136C2×(C2×D3)⋊C4—34no—
137C2×(C2×D3)⋊C4—34no—
138C2×(C2×D3)⋊C4—34no—
139C2×(C2×D3)⋊C4—34no—
140C2×(C2×D3)⋊C4—34no—
141C2×C4⋊Dic3—34no—
142C2×C4⋊Dic3—34no—
143C2×Dic3⋊C4—34no—
144C2×Dic3⋊C4—34no—
145C2×C4×Dic3—34no—
146#884—332no—
147C3×#95C2×D314yes—
148C3×#95C2×D314yes—
149C3⋊#95—34no—
150C3⋊#95—34no—
151C3⋊#95—34no—
152C3⋊#95—34no—
153C3⋊#95—34no—
154C3⋊#95—34no—
155C3⋊#95—34no—
156C3⋊#95—34no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32—21no—
163C23×C32C22×C412yes—
164C23×C32C2411yescharacteristic
165C22×C3⋊D3—21no—
166C22×C3⋊D3—21no—
167C22×C3⋊D3—21no—
168C22×C3⋊D3—21no—
169C22×C3⋊D3—21no—
170C22×C3⋊D3—21no—
171C22×C3⋊D3—21no—
172C22×C3⋊D3—21no—
173C22×C3⋊D3—21no—
174C22×C3⋊D3—21no—
175C22×C3⋊D3—21no—
176C22×C3⋊D3—21no—
177C22×C3⋊D3C22×C412yes—
178C22×C3⋊D3C22×C412yes—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4C22×C412yes—
196C2×C32×C4C22×C412yes—
197#638—44no—
198#638—44no—
199C2×C3⋊Dic3—21no—
200C2×C3⋊Dic3—21no—
201C2×C3⋊Dic3—21no—
202C2×C3⋊Dic3—21no—
203C2×C3⋊Dic3—21no—
204C2×C3⋊Dic3—21no—
205C2×C3⋊Dic3—21no—
206C2×C3⋊Dic3—21no—
207C2×C3⋊Dic3—21no—
208C2×C3⋊Dic3—21no—
209C2×C3⋊Dic3—21no—
210C2×C3⋊Dic3—21no—
211C2×C3⋊Dic3—21no—
212C2×C3⋊Dic3—21no—
213C2×C3⋊Dic3—21no—
214C2×C3⋊Dic3—21no—
215C2×C3⋊Dic3—44no—
216C2×C3⋊Dic3C22×C412yes—
217C2×C3⋊Dic3C22×C412yes—
218C4⋊(C3⋊D3)—44no—
219C4×C3⋊D3—44no—
220C4×C3⋊D3—44no—
221C2×#121—91no—
222#390—91no—
223#390—91no—
224#388—91no—
225#388C3⋊D311yescharacteristic, complemented
226#387—91no—
227#387—91no—
228#386—91no—
229#386—91no—
230#386—91no—
231#386—91no—
232#385—91no—
233#385—91no—
234#381—91no—
235#380—91no—
236C24×C3C22×D314yes—
237C23×D3—34no—
238C23×D3—34no—
239C22×C3×C4C22×D314yes—
240C22×C3×C4C22×D314yes—
241C22×C3×C4C22×D314yes—
242C22×C3×C4C22×D314yes—
243C22×C3×C4C22×D314yes—
244C22×C3×C4C22×D314yes—
245C2×#67—316no—
246C2×#67—316no—
247C2×#67—124no—
248C2×#67—124no—
249C2×#67—124no—
250C2×#67—124no—
251C22×Dic3—34no—
252C22×Dic3—34no—
253C22×Dic3—34no—
254C22×Dic3—34no—
255C22×Dic3—34no—
256C22×Dic3—34no—
257C22×Dic3—124no—
258C22×Dic3—124no—
259C22×Dic3—124no—
260C2×D12—316no—
261C2×D12—124no—
262C2×D12—124no—
263C2×C4×D3—316no—
264C2×C4×D3—124no—
265C2×C4×D3—124no—
266C2×C4×D3—124no—
267C2×C4×D3—124no—
268C2×C4×D3—124no—
269C3×C4⋊C4—44no—
270C3×C22⋊C4—44no—
271C3×C22⋊C4—44no—
272C3×C22⋊C4C4×D3116yescomplemented
273C3×C42—44no—
274#217—124no—
275(C2×D3)⋊C4—316no—
276(C2×D3)⋊C4—124no—
277(C2×D3)⋊C4—124no—
278(C2×D3)⋊C4—124no—
279(C2×D3)⋊C4—124no—
280(C2×D3)⋊C4—124no—
281(C2×D3)⋊C4—124no—
282(C2×D3)⋊C4—124no—
283C4⋊Dic3—124no—
284C4⋊Dic3—124no—
285Dic3⋊C4—316no—
286Dic3⋊C4—124no—
287C4×Dic3—124no—
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—22no—
291C22×C32—41no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32C23×C411yescharacteristic
296C22×C32H(5,𝔽2)11yescharacteristic
297C22×C32H(5,𝔽2)11yescharacteristic
298C22×C32H(5,𝔽2)11yescharacteristic
299C22×C32H(5,𝔽2)11yescharacteristic
300C22×C32H(5,𝔽2)11yescharacteristic
301C22×C322−1+411yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—41no—
309C2×C3⋊D3—41no—
310C2×C3⋊D3—41no—
311C2×C3⋊D3—41no—
312C2×C3⋊D3—41no—
313C2×C3⋊D3—41no—
314C2×C3⋊D3—41no—
315C2×C3⋊D3—41no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—41no—
322C3⋊Dic3—22no—
323C3⋊Dic3—22no—
324C3⋊Dic3—41no—
325C3⋊Dic3—41no—
326C3⋊Dic3—41no—
327C3⋊Dic3—41no—
328C22×D4—91no—
329C22×D4—91no—
330C22×D4—91no—
331C23×C4—91no—
332C23×C4—91no—
333C23×C4—91no—
334#121—98no—
335C2×C4⋊C4—91no—
336C2×C4⋊C4—91no—
337C2×C4⋊C4—91no—
338C2×C4⋊C4—91no—
339C2×C4⋊C4C2×C3⋊D311yescharacteristic
340C2×C22⋊C4—91no—
341C2×C22⋊C4—91no—
342C2×C22⋊C4—91no—
343C2×C22⋊C4—91no—
344C2×C22⋊C4—91no—
345C2×C22⋊C4—91no—
346C2×C22⋊C4—91no—
347C2×C22⋊C4—91no—
348C2×C22⋊C4—91no—
349C2×C22⋊C4C2×C3⋊D311yescharacteristic
350C2×C22⋊C4C2×C3⋊D311yescharacteristic
351C2×C22⋊C4C2×C3⋊D311yescharacteristic
352C2×C42—91no—
353C2×C42C2×C3⋊D311yescharacteristic
354#95—91no—
355#95—91no—
356#95—91no—
357#95—91no—
358#95—91no—
359#95—91no—
360#95—91no—
361#95—91no—
362#95C2×C3⋊D311yescharacteristic
363#95C2×C3⋊D311yescharacteristic
364C23×C3—24no—
365C23×C3—24no—
366C23×C3—24no—
367C23×C3—24no—
368C23×C3—24no—
369C23×C3—24no—
370C23×C3C2×C4×D318yes—
371C23×C3C23×D314yes—
372C22×D3—38no—
373C22×D3—38no—
374C22×D3—64no—
375C22×D3—64no—
376C22×D3—64no—
377C22×D3—64no—
378C22×D3—64no—
379C22×D3—64no—
380C22×D3—64no—
381C22×D3—64no—
382C22×D3—64no—
383C22×D3—64no—
384C22×D3—64no—
385C22×D3—64no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—24no—
389C2×C3×C4—24no—
390C2×C3×C4—24no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4—24no—
395C2×C3×C4—24no—
396C2×C3×C4—24no—
397C2×C3×C4—24no—
398C2×C3×C4—24no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4C2×C4×D318yes—
403C2×C3×C4C2×C4×D318yes—
404#67—1216no—
405#67—1216no—
406C2×Dic3—38no—
407C2×Dic3—38no—
408C2×Dic3—64no—
409C2×Dic3—64no—
410C2×Dic3—64no—
411C2×Dic3—64no—
412C2×Dic3—64no—
413C2×Dic3—64no—
414C2×Dic3—64no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3—64no—
418C2×Dic3—64no—
419C2×Dic3—64no—
420C2×Dic3—64no—
421C2×Dic3—64no—
422C2×Dic3—64no—
423C2×Dic3—64no—
424C2×Dic3—1216no—
425D12—1216no—
426C4×D3—1216no—
427C4×D3—1216no—
428C2×C32—42no—
429C2×C32#51811yescharacteristic
430C2×C32#51811yescharacteristic
431C2×C32#52011yescharacteristic
432C2×C32#56011yescharacteristic
433C2×C32#56011yescharacteristic
434C2×C32#56211yescharacteristic
435C2×C32#56211yescharacteristic
436C3⋊D3—42no—
437C3⋊D3—42no—
438C24—91no—
439C24—91no—
440C24C22×C3⋊D311yescharacteristic
441C2×D4—94no—
442C2×D4—94no—
443C2×D4—94no—
444C2×D4—361no—
445C2×D4—361no—
446C2×D4—361no—
447C2×D4—361no—
448C2×D4—361no—
449C2×D4—361no—
450C22×C4—91no—
451C22×C4—91no—
452C22×C4—94no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—91no—
456C22×C4—91no—
457C22×C4—361no—
458C22×C4—361no—
459C22×C4—361no—
460C22×C4—361no—
461C22×C4—361no—
462C22×C4—361no—
463C22×C4—361no—
464C22×C4—361no—
465C22×C4C22×C3⋊D311yescharacteristic
466C22×C4C22×C3⋊D311yescharacteristic
467C22×C4C22×C3⋊D311yescharacteristic
468C22×C4C22×C3⋊D311yescharacteristic
469C22×C4C22×C3⋊D311yescharacteristic
470C22×C4C22×C3⋊D311yescharacteristic
471C4⋊C4—41no—
472C4⋊C4—94no—
473C4⋊C4—361no—
474C4⋊C4—361no—
475C4⋊C4—361no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—94no—
479C22⋊C4—361no—
480C22⋊C4—361no—
481C22⋊C4—361no—
482C22⋊C4—361no—
483C22⋊C4—361no—
484C22⋊C4—361no—
485C22⋊C4—361no—
486C22⋊C4—361no—
487C22⋊C4C4×C3⋊D314yescomplemented
488C42—41no—
489C42—361no—
490C22×C3—28no—
491C22×C3—28no—
492C22×C3—28no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—44no—
496C22×C3—44no—
497C22×C3C22×C4×D314yes—
498C22×C3#100414yes—
499C22×C3#100414yes—
500C22×C3#100914yes—
501C22×C3#100914yes—
502C22×C3#100914yes—
503C22×C3#101014yes—
504C2×D3—68no—
505C2×D3—68no—
506C2×D3—68no—
507C2×D3—68no—
508C2×D3—68no—
509C2×D3—68no—
510C2×D3—124no—
511C2×D3—124no—
512C2×D3—124no—
513C2×D3—124no—
514C2×D3—124no—
515C2×D3—124no—
516C2×D3—124no—
517C2×D3—124no—
518C3×C4—28no—
519C3×C4—28no—
520C3×C4—44no—
521C3×C4—44no—
522C3×C4—44no—
523C3×C4—44no—
524Dic3—68no—
525Dic3—68no—
526Dic3—124no—
527Dic3—124no—
528Dic3—124no—
529Dic3—124no—
530C32#237711yescharacteristic, complemented, Sylow
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—92no—
538C23—92no—
539C23—181no—
540C23—181no—
541C23—181no—
542C23—181no—
543C23—181no—
544C23—181no—
545C23—181no—
546C23—181no—
547C23—181no—
548C23—181no—
549C23—181no—
550C23—181no—
551C23C2×C4×C3⋊D312yes—
552C23C23×C3⋊D311yescharacteristic
553D4—364no—
554D4—364no—
555D4—364no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—92no—
573C2×C4—92no—
574C2×C4—181no—
575C2×C4—181no—
576C2×C4—181no—
577C2×C4—181no—
578C2×C4—181no—
579C2×C4—181no—
580C2×C4—181no—
581C2×C4—181no—
582C2×C4—181no—
583C2×C4—181no—
584C2×C4—181no—
585C2×C4—181no—
586C2×C4—181no—
587C2×C4—181no—
588C2×C4—181no—
589C2×C4—181no—
590C2×C4—364no—
591C2×C4—364no—
592C2×C4—364no—
593C2×C4C2×C4×C3⋊D312yes—
594C2×C4C2×C4×C3⋊D312yes—
595C2×C3—48no—
596C2×C3#552314yes—
597C2×C3#554514yes—
598C2×C3#554514yes—
599C2×C3#562014yes—
600C2×C3#566714yes—
601C2×C3#568414yes—
602C2×C3#571114yes—
603D3—128no—
604D3—128no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—182no—
613C22—182no—
614C22—182no—
615C22—182no—
616C22—182no—
617C22—182no—
618C22—361no—
619C22—361no—
620C22—361no—
621C22—361no—
622C22—361no—
623C22—361no—
624C22—361no—
625C22—361no—
626C22C22×C4×C3⋊D311yescharacteristic
627C22#6436111yescharacteristic
628C22#6436111yescharacteristic
629C22#6436611yescharacteristic
630C22#6436611yescharacteristic
631C22#6436611yescharacteristic
632C22#6436711yescharacteristic
633C4—22no—
634C4—22no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—182no—
640C4—182no—
641C4—361no—
642C4—361no—
643C4—361no—
644C4—361no—
645C3#7549014yescomplemented
646C2—42no—
647C2—362no—
648C2—362no—
649C2#1059599411yescharacteristic
650C2#1059601611yescharacteristic
651C2#1059601611yescharacteristic
652C2#1059609111yescharacteristic
653C2#1059613811yescharacteristic
654C2#1059615511yescharacteristic
655C2#1059618211yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×#388C2
socle, cosocleC23×C32C24
prime radicalC2×#64093C2
hypercenter#388C3⋊D3
hypocenterC32#2377
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C32×#388C2semidirect
3. #10592688C2semidirect
4. #10592686C2semidirect
5. #10592657C2semidirect
6. #10592657C2semidirect
7. #10592656C2semidirect
8. #10592655C2semidirect
9. #10592649C2semidirect
10. #10592638C2semidirect
11. #10592632C2semidirect
12. #10592369C2semidirect
13. #10592364C2semidirect
14. #10592363C2semidirect
15. #10592354C2semidirect
42. #64093C4semidirect
54. C3×#388D3semidirect
82. C2×#638C2×C4semidirect
94. C2×C4⋊(C3⋊D3)C2×C4semidirect
100. C2×C4×C3⋊D3C2×C4semidirect
104. C32×C22⋊C4C2×C4semidirect
114. #3752C2×C4semidirect
118. #3750C2×C4semidirect
225. #388C3⋊D3semidirect
272. C3×C22⋊C4C4×D3semidirect
487. C22⋊C4C4×C3⋊D3semidirect
530. C32#2377semidirect
645. C3#75490semidirect

Maximal subgroups

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

subgroupindex
C2×#640932
C32×#3882
#105926882
#105926862
#105926572
#105926562
#105926552
#105926492
#105926382
#105926322
#105923692
#105923642
#105923632
#105923542
#754903