Finite Group Databasebeta

← #11794029

Subgroups — #11794029

Order 1152.

849 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3714C211yesmaximal, characteristic
2#10592638C211yesmaximal, characteristic, complemented
3C3×#4939C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C2×#63564C211yesmaximal, characteristic, complemented
6C2×#63563C211yesmaximal, characteristic, complemented
7#10589164C211yesmaximal, characteristic, complemented
8#70481—31nomaximal
9#69650—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C2×C32×C42C2211yescharacteristic
13C2×#3752—21no—
14C2×#3752—21no—
15C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C2×#3714—22no—
22C2×#3714—22no—
23C2×#3714—22no—
24C3×C4⋊(C3⋊C8)—22no—
25#63564—22no—
26#63563—22no—
27C22×#213—31no—
28C22×D24—31no—
29C2×C3×C4⋊C8—31no—
30C3×#390—31no—
31C2×#832—31no—
32C2×#821—31no—
33C2×#821—31no—
34C2×#809—31no—
35#4939—31no—
36#4939—31no—
37#4939—61no—
38#4939D311yescharacteristic, complemented
39C2×C4⋊(C3⋊C8)D311yescharacteristic, complemented
40#4551—31no—
41#4529—31no—
42C23×C3⋊D3—21no—
43C23×C3×D3—21no—
44C22×C32×C4D411yescharacteristic
45C22×C32×C4D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—21no—
51C2×C4⋊(C3⋊D3)—21no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)D412yes—
54C22×C3×Dic3—21no—
55C2×C3×D12—21no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12—21no—
59C2×C3×D12—21no—
60C2×C3×D12D412yes—
61C32×C42—21no—
62C32×C42—21no—
63#3752—42no—
64#3752—42no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)—21no—
67C3×(C2×D3)⋊C4—42no—
68C3×(C2×D3)⋊C4—42no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8D412yes—
77#3714—28no—
78#2054—91noSylow
79C22×C3×D4—31no—
80C22×D12—31no—
81C22×D12—31no—
82C22×D12—61no—
83C22×D12C2×D311yescharacteristic
84C22×C3×C8—31no—
85C22×C3×C8—31no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C22⋊C4—61no—
88C2×C3×C22⋊C4—61no—
89C2×C3×C42—21no—
90C2×C3×C42C2×D311yescharacteristic
91C2×C3×C42C2×D311yescharacteristic
92C2×#213—62no—
93C2×#213—62no—
94C2×#213—62no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—62no—
98C2×(C2×D3)⋊C4—62no—
99C2×(C2×D3)⋊C4—61no—
100C2×(C2×D3)⋊C4—61no—
101C2×(C2×D3)⋊C4—61no—
102C2×(C2×D3)⋊C4—61no—
103C2×C4⋊Dic3—31no—
104C2×C4⋊Dic3—31no—
105C2×C4⋊Dic3—61no—
106C2×C4⋊Dic3C2×D311yescharacteristic
107C22×C3⋊C8C2×D311yescharacteristic
108C22×C3⋊C8C2×D311yescharacteristic
109C2×D24—62no—
110C2×D24—62no—
111C2×D24—62no—
112C3×C4⋊C8—62no—
113#832—62no—
114#821—62no—
115#821—62no—
116#809—62no—
117C4⋊(C3⋊C8)—22no—
118C23×C32C2×D411yescharacteristic
119C22×C3⋊D3—22no—
120C22×C3⋊D3—41no—
121C22×C3⋊D3—41no—
122C22×C3⋊D3—41no—
123C22×C3⋊D3—41no—
124C22×C3⋊D3—41no—
125C22×C3⋊D3—41no—
126C22×C3×D3—22no—
127C22×C3×D3—41no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—41no—
131C22×C3×D3—41no—
132C22×C3×D3—41no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4D812yescomplemented
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C32×C4#4111yescharacteristic
144C2×C32×C4#4111yescharacteristic
145C2×C3⋊Dic3—41no—
146C2×C3⋊Dic3—41no—
147C2×C3⋊Dic3—41no—
148C4⋊(C3⋊D3)—22no—
149C4⋊(C3⋊D3)—22no—
150C4⋊(C3⋊D3)—41no—
151C4⋊(C3⋊D3)—41no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3—41no—
154C2×C3×Dic3—41no—
155C3×D12—22no—
156C3×D12—22no—
157C3×D12—41no—
158C3×D12—41no—
159C3×C3⋊C8—22no—
160C3×C3⋊C8—41no—
161C22×D8—91no—
162C2×C4⋊C8—31no—
163C2×#102—91no—
164C2×#102—91no—
165#390—31no—
166#390—91no—
167#340—91no—
168C24×C3—61no—
169C23×D3—21no—
170C23×D3—62no—
171C23×D3—61no—
172C23×D3—61no—
173C2×C3×D4—32no—
174C2×C3×D4—61no—
175C2×C3×D4—61no—
176C2×C3×D4—61no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—61no—
183C22×C3×C4D1211yescharacteristic
184C22×C3×C4D1211yescharacteristic
185C22×C3×C4#6711yescharacteristic
186C22×C3×C4#6711yescharacteristic
187C22×C3×C4C22×D311yescharacteristic
188C22×C3×C4C22×D311yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—62no—
191C22×Dic3—61no—
192C22×Dic3—61no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—32no—
199C2×D12—32no—
200C2×D12—62no—
201C2×D12—61no—
202C2×D12—61no—
203C2×D12—61no—
204C2×D12—61no—
205C2×D12—61no—
206C2×D12—61no—
207C2×D12—61no—
208C2×D12—61no—
209C2×D12—62no—
210C2×D12—61no—
211C2×D12—61no—
212C2×D12—121no—
213C2×D12—121no—
214C2×D12—121no—
215C2×D12—121no—
216C2×D12D1212yes—
217C2×C3×C8—32no—
218C2×C3×C8—61no—
219C2×C3×C8—61no—
220C2×C3×C8—61no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—61no—
225C3×C22⋊C4—122no—
226C3×C22⋊C4—122no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—41no—
232C3×C42—41no—
233#213—68no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—124no—
237(C2×D3)⋊C4—122no—
238(C2×D3)⋊C4—122no—
239(C2×D3)⋊C4—124no—
240(C2×D3)⋊C4—122no—
241(C2×D3)⋊C4—122no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—61no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3—61no—
248C4⋊Dic3—121no—
249C4⋊Dic3—121no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8#6712yes—
256D24—68no—
257C22×C32#12111yescharacteristic
258C22×C32#12111yescharacteristic
259C22×C32#12111yescharacteristic
260C22×C32C2×D811yescharacteristic
261C22×C32#13511yescharacteristic
262C22×C32AGL(1,ℤ/8)11yescharacteristic
263C22×C32AGL(1,ℤ/8)11yescharacteristic
264C2×C3⋊D3—42no—
265C2×C3⋊D3—42no—
266C2×C3⋊D3—42no—
267C2×C3⋊D3—81no—
268C2×C3⋊D3—81no—
269C2×C3⋊D3—81no—
270C2×C3⋊D3—81no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—42no—
274C2×C3×D3—81no—
275C2×C3×D3—81no—
276C2×C3×D3—81no—
277C2×C3×D3—81no—
278C32×C4—22no—
279C32×C4—41no—
280C32×C4#12011yescharacteristic
281C32×C4#12311yescharacteristic
282C32×C4#12311yescharacteristic
283C32×C4#12411yescharacteristic
284C3⋊Dic3—81no—
285C3×Dic3—81no—
286C22×D4—31no—
287C22×D4—91no—
288C2×D8—182no—
289C2×D8—182no—
290C2×D8—182no—
291C22×C8—31no—
292C22×C8—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—91no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—181no—
298C2×C22⋊C4—181no—
299C2×C42D3211yescharacteristic
300C4⋊C8—62no—
301#102—182no—
302#102—182no—
303C23×C3—21no—
304C23×C3—62no—
305C23×C3—121no—
306C23×C3—121no—
307C23×C3—121no—
308C23×C3—121no—
309C23×C3—121no—
310C23×C3—121no—
311C23×C3C2×D1211yescharacteristic
312C23×C3C2×#6711yescharacteristic
313C22×D3—22no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—64no—
321C22×D3—62no—
322C22×D3—62no—
323C22×D3—122no—
324C22×D3—122no—
325C22×D3—122no—
326C22×D3—122no—
327C22×D3—122no—
328C22×D3—122no—
329C22×D3—121no—
330C22×D3—121no—
331C22×D3—121no—
332C22×D3—121no—
333C22×D3—121no—
334C22×D3—121no—
335C22×D3—121no—
336C22×D3—121no—
337C22×D3—121no—
338C22×D3—121no—
339C22×D3—121no—
340C22×D3—121no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—121no—
344C3×D4—121no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—121no—
368C2×C3×C4—121no—
369C2×C3×C4—121no—
370C2×C3×C4D2412yescomplemented
371C2×C3×C4#21312yescomplemented
372C2×C3×C4C2×D1211yescharacteristic
373C2×C3×C4#23511yescharacteristic
374C2×C3×C4#23511yescharacteristic
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4D3×D411yescharacteristic
379C2×C3×C4#23711yescharacteristic
380C2×C3×C4#23711yescharacteristic
381C2×C3×C4#23911yescharacteristic
382C2×C3×C4#23911yescharacteristic
383C2×C3×C4C2×#6711yescharacteristic
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—122no—
388C2×Dic3—122no—
389C2×Dic3—122no—
390C2×Dic3—121no—
391C2×Dic3—121no—
392C2×Dic3—121no—
393C2×Dic3—121no—
394C2×Dic3—121no—
395C2×Dic3—121no—
396D12—22no—
397D12—22no—
398D12—41no—
399D12—41no—
400D12—62no—
401D12—62no—
402D12—62no—
403D12—62no—
404D12—122no—
405D12—122no—
406D12—122no—
407D12—122no—
408D12—121no—
409D12—121no—
410D12—121no—
411D12—121no—
412C3×C8—62no—
413C3×C8—121no—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C2×C32#39411yescharacteristic
417C2×C32#45911yescharacteristic
418C2×C32#45911yescharacteristic
419C2×C32#46311yescharacteristic
420C2×C32#46311yescharacteristic
421C2×C32#46611yescharacteristic
422C2×C32#46911yescharacteristic
423C3⋊D3—82no—
424C3×D3—82no—
425C24—61no—
426C24—181no—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—92no—
434C2×D4—181no—
435C2×D4—181no—
436C2×D4—181no—
437C2×D4—181no—
438C2×D4—181no—
439C22×C4—61no—
440C22×C4—181no—
441C22×C4#62611yescharacteristic
442C22×C4#62611yescharacteristic
443C22×C4C2×D3211yescharacteristic
444D8—188no—
445C2×C8—32no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—181no—
454C4⋊C4—181no—
455C22⋊C4—122no—
456C22⋊C4—122no—
457C22⋊C4—362no—
458C22⋊C4—362no—
459C42—21no—
460C42—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—241no—
472C22×C3—241no—
473C22×C3—241no—
474C22×C3—241no—
475C22×C3#89511yescharacteristic
476C22×C3#89511yescharacteristic
477C22×C3C2×D2411yescharacteristic
478C22×C3#90411yescharacteristic
479C22×C3#90811yescharacteristic
480C22×C3#90811yescharacteristic
481C22×C3#93011yescharacteristic
482C22×C3#93011yescharacteristic
483C22×C3C2×#21311yescharacteristic
484C22×C3#93211yescharacteristic
485C22×C3#93811yescharacteristic
486C22×C3#93911yescharacteristic
487C22×C3#94911yescharacteristic
488C22×C3#95011yescharacteristic
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—124no—
497C2×D3—124no—
498C2×D3—124no—
499C2×D3—122no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—122no—
503C2×D3—122no—
504C2×D3—122no—
505C2×D3—242no—
506C2×D3—242no—
507C2×D3—242no—
508C2×D3—242no—
509C2×D3—241no—
510C2×D3—241no—
511C2×D3—241no—
512C2×D3—241no—
513C2×D3—241no—
514C2×D3—241no—
515C2×D3—241no—
516C2×D3—241no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—241no—
528C3×C4#88211yescharacteristic
529C3×C4#88311yescharacteristic
530C3×C4#88511yescharacteristic
531C3×C4#88611yescharacteristic
532C3×C4#89411yescharacteristic
533C3×C4#89411yescharacteristic
534C3×C4#93711yescharacteristic
535C3×C4#94711yescharacteristic
536Dic3—81no—
537Dic3—242no—
538Dic3—241no—
539Dic3—241no—
540C32#205411yescharacteristic, complemented, Sylow
541C23—62no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—182no—
549C23—361no—
550C23—361no—
551C23—361no—
552C23—361no—
553C23—361no—
554C23—361no—
555C23C2×#62611yescharacteristic
556D4—62no—
557D4—62no—
558D4—121no—
559D4—121no—
560D4—182no—
561D4—182no—
562D4—361no—
563D4—361no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—361no—
573C2×C4—361no—
574C2×C4—361no—
575C2×C4#371412yescomplemented
576C2×C4#379611yescharacteristic
577C2×C4#379911yescharacteristic
578C2×C4D3×D1211yescharacteristic
579C2×C4#380211yescharacteristic
580C2×C4#380411yescharacteristic
581C2×C4D3×#6711yescharacteristic
582C8—62no—
583C8—121no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—242no—
592C2×C3#484311yescharacteristic
593C2×C3#484311yescharacteristic
594C2×C3#484411yescharacteristic
595C2×C3#484411yescharacteristic
596C2×C3#496111yescharacteristic
597C2×C3#499111yescharacteristic
598C2×C3#501511yescharacteristic
599C2×C3#501611yescharacteristic
600C2×C3#503811yescharacteristic
601C2×C3#503911yescharacteristic
602C2×C3#505711yescharacteristic
603C2×C3#507211yescharacteristic
604C2×C3#511511yescharacteristic
605C2×C3#513511yescharacteristic
606D3—82no—
607D3—244no—
608D3—242no—
609D3—242no—
610C22—122no—
611C22—122no—
612C22—122no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—362no—
618C22—362no—
619C22—362no—
620C22—721no—
621C22—721no—
622C22—721no—
623C22—721no—
624C22C2×#371411yescharacteristic
625C22#6382511yescharacteristic
626C22#6382911yescharacteristic
627C22#6383011yescharacteristic
628C22#6390911yescharacteristic
629C22#6391611yescharacteristic
630C22#6397811yescharacteristic
631C4—22no—
632C4—41no—
633C4—241no—
634C4—721no—
635C4#6385411yescharacteristic
636C4#6389011yescharacteristic
637C4#6390711yescharacteristic
638C4#6392211yescharacteristic
639C3—21no—
640C3#6965011yescharacteristic, complemented
641C3#7048111yescharacteristic, complemented
642C2—242no—
643C2—722no—
644C2#1059034911yescharacteristic
645C2#1059035511yescharacteristic
646C2#1059035611yescharacteristic
647C2#1059036611yescharacteristic
648C2#1059116411yescharacteristic
649C2#1059116511yescharacteristic
650C2#1059139211yescharacteristic
subgroupG/H
centerC23C2×#626
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC23×C32C2×D4
prime radicalC22×#3714C2
hypercenterC2×C42D32
hypocenterC32#2054
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. #10592638C2semidirect
3. C3×#4939C2semidirect
4. C2×C3×C4⋊(C3⋊C8)C2semidirect
5. C2×#63564C2semidirect
6. C2×#63563C2semidirect
7. #10589164C2semidirect
38. #4939D3semidirect
39. C2×C4⋊(C3⋊C8)D3semidirect
138. C2×C32×C4D8semidirect
370. C2×C3×C4D24semidirect
371. C2×C3×C4#213semidirect
540. C32#2054semidirect
575. C2×C4#3714semidirect
640. C3#69650semidirect
641. C3#70481semidirect

Maximal subgroups

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

subgroupindex
C22×#37142
#105926382
C3×#49392
C2×C3×C4⋊(C3⋊C8)2
C2×#635642
C2×#635632
#105891642
#704813
#696503