Finite Group Databasebeta

← #11885793

Subgroups — #11885793

Order 1152.

1578 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#546C211yesmaximal, characteristic, complemented
2C2×#64159C211yesmaximal, characteristic, complemented
3C22×#3753C211yesmaximal, characteristic, complemented
4#10596021C211yesmaximal, characteristic, complemented
5#10592937C211yesmaximal, characteristic, complemented
6#10592917C211yesmaximal, characteristic, complemented
7#10592758C211yesmaximal, characteristic, complemented
8#10592756C211yesmaximal, characteristic, complemented
9#10592750C211yesmaximal, characteristic, complemented
10#10592745C211yesmaximal, characteristic, complemented
11#10592737C211yesmaximal, characteristic, complemented
12#10592736C211yesmaximal, characteristic, complemented
13#10592715C211yesmaximal, characteristic, complemented
14C4×#3753C211yesmaximal, characteristic, complemented
15D4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
16#83505—34nomaximal
17C2×C32×#41C2211yescharacteristic, complemented
18C22×C32×D4C2211yescharacteristic
19C2×#3828C2211yescharacteristic, complemented
20C22×C4⋊(C3⋊D3)C2211yescharacteristic
21C32×#124C2211yescharacteristic, complemented
22C32×#123—21no—
23C32×C22⋊Q8C2211yescharacteristic
24C32×#121—21no—
25C32×#121C2211yescharacteristic, complemented
26C32×#120—21no—
27C32×C4×D4C2211yescharacteristic
28C32×C4×D4C2211yescharacteristic, complemented
29C2×C32×C22⋊C4—21no—
30#64159—24no—
31C2×#3756C2211yescharacteristic, complemented
32C2×#3755C2211yescharacteristic, complemented
33C2×#3755C2211yescharacteristic, complemented
34#64146—21no—
35C2×#3754C2211yescharacteristic, complemented
36C2×#3754C2211yescharacteristic, complemented
37C2×#3753—24no—
38C2×#3753C2211yescharacteristic
39C2×#3753C2211yescharacteristic
40C2×#3753C2211yescharacteristic
41C2×#3753C2211yescharacteristic
42#64139C2211yescharacteristic, complemented
43#64133C2211yescharacteristic
44C22×C3⋊(C3⋊C8)C2211yescharacteristic
45C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
46#64103—21no—
47#64093—21no—
48C22⋊C4×C3⋊D3—21no—
49#64084C2211yescharacteristic, complemented
50C4×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
51#63662C2211yescharacteristic, complemented
52#63659C2211yescharacteristic, complemented
53#63659C2211yescharacteristic, complemented
54#63657C2211yescharacteristic
55#63657C2211yescharacteristic, complemented
56#63637C2211yescharacteristic, complemented
57#63636C2211yescharacteristic, complemented
58#63636C2211yescharacteristic, complemented
59#63634C2211yescharacteristic, complemented
60C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
61C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
62C3×#546D314yescomplemented
63C2×#950—34no—
64C22×#213—34no—
65#5550—34no—
66#5238—34no—
67#5218—34no—
68#5059—34no—
69#5057—34no—
70#5051—34no—
71#5046—34no—
72#5038—34no—
73#5037—34no—
74#5016—34no—
75C4×#213—34no—
76D4×C3⋊C8—34no—
77C24×C32—21no—
78C23×C3⋊D3—21no—
79C32×#41—22no—
80C32×#41—41no—
81C2×C32×Q8C2311yescharacteristic
82C2×C32×D4—22no—
83C2×C32×D4—21no—
84C2×C32×D4—21no—
85C2×C32×D4—21no—
86C2×C32×D4—41no—
87C2×C32×D4D411yescharacteristic
88C2×C32×D4C2311yescharacteristic
89C2×C32×D4C2311yescharacteristic
90C22×C32×C4—21no—
91C22×C32×C4—21no—
92C22×C32×C4C2311yescharacteristic
93C22×C32×C4C2311yescharacteristic
94C2×#638—21no—
95C2×#638—21no—
96#3828—22no—
97#3828—41no—
98C2×C4⋊(C3⋊D3)—22no—
99C2×C4⋊(C3⋊D3)—21no—
100C2×C4⋊(C3⋊D3)—41no—
101C2×C4⋊(C3⋊D3)C2311yescharacteristic
102C2×C4⋊(C3⋊D3)C2311yescharacteristic
103C2×C4×C3⋊D3—21no—
104C2×C4×C3⋊D3—21no—
105C2×#634C2311yescharacteristic
106C32×C4⋊C4—21no—
107C32×C4⋊C4D411yescharacteristic
108C32×C4⋊C4C2311yescharacteristic
109C32×C22⋊C4—21no—
110C32×C22⋊C4—21no—
111C32×C22⋊C4—21no—
112C32×C22⋊C4—21no—
113C32×C22⋊C4—21no—
114C32×C22⋊C4D411yescharacteristic
115C32×C22⋊C4D411yescharacteristic
116C32×C42C2311yescharacteristic
117#3757—21no—
118#3756—22no—
119#3755—22no—
120#3755—41no—
121#3754—22no—
122#3754—41no—
123#3753—22no—
124#3753—22no—
125#3753—22no—
126#3753D414yescomplemented
127#3752—21no—
128#3752—21no—
129#3752—21no—
130#3752—21no—
131C4⋊(C3⋊Dic3)C2311yescharacteristic
132#3750—21no—
133C2×C3⋊(C3⋊C8)—22no—
134C2×C3⋊(C3⋊C8)—22no—
135C2×C3⋊(C3⋊C8)C2311yescharacteristic
136C2×C3⋊(C3⋊C8)C2311yescharacteristic
137C2×C3⋊(C3⋊C8)C2311yescharacteristic
138C2×C3⋊(C3⋊C8)C2311yescharacteristic
139#3280—91noSylow
140C2×C3×#41C2×D314yescomplemented
141C22×C3×D4C2×D314yes—
142C2×#235—34no—
143C22×D12—34no—
144C3×#124C2×D314yescomplemented
145C3×#123—24no—
146C3×C22⋊Q8C2×D314yes—
147C3×#121—24no—
148C3×#121C2×D314yescomplemented
149C3×#120—24no—
150C3×C4×D4C2×D314yes—
151C3×C4×D4C2×D314yescomplemented
152C2×C3×C22⋊C4—24no—
153#950—616no—
154C2×#216—34no—
155C2×#215—34no—
156C2×#215—34no—
157#937—64no—
158C2×#214—34no—
159C2×#214—34no—
160C2×#213—34no—
161C2×#213—34no—
162C2×#213—34no—
163C2×#213—34no—
164C2×#213—616no—
165#930—34no—
166#924—34no—
167C22×C3⋊C8—34no—
168C22×C3⋊C8—34no—
169#894—64no—
170#884—64no—
171D3×C22⋊C4—64no—
172#875—34no—
173C4×D12—34no—
174#835—34no—
175#832—34no—
176#832—34no—
177#830—34no—
178#830—34no—
179#810—34no—
180#809—34no—
181#809—34no—
182#807—34no—
183C4⋊(C3⋊C8)—34no—
184C4×C3⋊C8—34no—
185C23×C32—21no—
186C23×C32—21no—
187C23×C32—42no—
188C23×C32—42no—
189C23×C32—81no—
190C23×C32C2×D411yescharacteristic
191C23×C32C2×D411yescharacteristic
192C22×C3⋊D3—21no—
193C22×C3⋊D3—21no—
194C22×C3⋊D3—42no—
195C22×C3⋊D3—42no—
196C22×C3⋊D3—81no—
197C32×Q8—21no—
198C32×Q8—21no—
199C32×D4—22no—
200C32×D4—21no—
201C32×D4—21no—
202C32×D4—21no—
203C32×D4—42no—
204C32×D4—42no—
205C32×D4—81no—
206C32×D4C2×D412yes—
207C2×C32×C4—21no—
208C2×C32×C4—21no—
209C2×C32×C4—21no—
210C2×C32×C4—21no—
211C2×C32×C4—21no—
212C2×C32×C4—21no—
213C2×C32×C4—42no—
214C2×C32×C4—81no—
215C2×C32×C4C2×D411yescharacteristic
216C2×C32×C4C2×D411yescharacteristic
217C2×C32×C4C2×D411yescharacteristic
218C2×C32×C4C2×D411yescharacteristic
219C2×C32×C4C2411yescharacteristic
220#638—42no—
221#638—81no—
222C2×C3⋊Dic3—21no—
223C2×C3⋊Dic3—21no—
224C4⋊(C3⋊D3)—21no—
225C4⋊(C3⋊D3)—21no—
226C4⋊(C3⋊D3)—21no—
227C4⋊(C3⋊D3)—42no—
228C4⋊(C3⋊D3)C2×D412yes—
229C4×C3⋊D3—42no—
230C4×C3⋊D3—81no—
231#634—21no—
232#634—21no—
233C3⋊(C3⋊C8)—21no—
234C3⋊(C3⋊C8)—21no—
235C3⋊(C3⋊C8)—21no—
236C3⋊(C3⋊C8)C2×D412yes—
237C2×#135—91no—
238C22×D8—91no—
239#546—91no—
240#546C3⋊D311yescharacteristic, complemented
241#495—91no—
242#467—91no—
243#466—91no—
244#463—91no—
245#463—91no—
246#450—91no—
247#450—91no—
248#449—91no—
249#449—91no—
250C4×D8—91no—
251C8×D4—91no—
252C24×C3—24no—
253C23×D3—64no—
254C3×#41—28no—
255C3×#41—44no—
256C2×C3×Q8C22×D314yes—
257C2×C3×D4—28no—
258C2×C3×D4—24no—
259C2×C3×D4—24no—
260C2×C3×D4—24no—
261C2×C3×D4—44no—
262C2×C3×D4#6714yes—
263C2×C3×D4C22×D314yes—
264C2×C3×D4C22×D314yes—
265C22×C3×C4—24no—
266C22×C3×C4—24no—
267C22×C3×C4C22×D314yes—
268C22×C3×C4C22×D314yes—
269C2×#67—64no—
270C2×#67—64no—
271#235—68no—
272#235—124no—
273C2×D12—34no—
274C2×D12—34no—
275C2×D12—68no—
276C2×D12—64no—
277C2×D12—124no—
278C2×C4×D3—64no—
279C2×C4×D3—64no—
280C2×Dic6—34no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4#6714yes—
283C3×C4⋊C4C22×D314yes—
284C3×C22⋊C4—24no—
285C3×C22⋊C4—24no—
286C3×C22⋊C4—24no—
287C3×C22⋊C4—24no—
288C3×C22⋊C4—24no—
289C3×C22⋊C4#6714yes—
290C3×C22⋊C4#6714yes—
291C3×C42C22×D314yes—
292#217—64no—
293#216—68no—
294#215—68no—
295#215—124no—
296#214—68no—
297#214—124no—
298#213—316no—
299#213—68no—
300#213—68no—
301#213—68no—
302(C2×D3)⋊C4—64no—
303(C2×D3)⋊C4—64no—
304(C2×D3)⋊C4—64no—
305(C2×D3)⋊C4—64no—
306C4⋊Dic3—34no—
307Dic3⋊C4—64no—
308C2×C3⋊C8—34no—
309C2×C3⋊C8—34no—
310C2×C3⋊C8—34no—
311C2×C3⋊C8—34no—
312C2×C3⋊C8—68no—
313C2×C3⋊C8—68no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—22no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32—41no—
321C22×C32—41no—
322C22×C32—41no—
323C22×C32—41no—
324C22×C32—41no—
325C22×C32—84no—
326C22×C32#13512yescomplemented
327C22×C32C22×D411yescharacteristic
328C2×C3⋊D3—22no—
329C2×C3⋊D3—41no—
330C2×C3⋊D3—41no—
331C2×C3⋊D3—41no—
332C2×C3⋊D3—41no—
333C2×C3⋊D3—41no—
334C2×C3⋊D3—84no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4C22×D411yescharacteristic
342C32×C4H(5,𝔽2)11yescharacteristic
343C3⋊Dic3—41no—
344C3⋊Dic3—41no—
345C2×#41—91no—
346C2×#41C2×C3⋊D311yescharacteristic, complemented
347C22×D4—91no—
348C22×D4C2×C3⋊D311yescharacteristic
349#135—184no—
350C2×Q16—91no—
351C2×SD16—91no—
352C2×SD16—91no—
353C2×SD16—91no—
354C2×SD16—91no—
355C2×D8—91no—
356C2×D8—91no—
357C2×D8—91no—
358C2×D8—91no—
359C2×D8—184no—
360C22×C8—91no—
361C22×C8—91no—
362#124—91no—
363#124C2×C3⋊D311yescharacteristic, complemented
364#123—21no—
365#123—181no—
366C22⋊Q8—91no—
367C22⋊Q8C2×C3⋊D311yescharacteristic
368#121—21no—
369#121—91no—
370#121—181no—
371#121C2×C3⋊D311yescharacteristic, complemented
372#120—21no—
373#120—181no—
374C4×D4—91no—
375C4×D4C2×C3⋊D311yescharacteristic
376C4×D4C2×C3⋊D311yescharacteristic, complemented
377C2×C22⋊C4—21no—
378C2×C22⋊C4—181no—
379#107—91no—
380C4⋊C8—91no—
381#103—91no—
382#103—91no—
383#102—91no—
384#102—91no—
385#102—91no—
386#102—91no—
387C22⋊C8—91no—
388C22⋊C8—91no—
389C4×C8—91no—
390C23×C3—24no—
391C23×C3—24no—
392C23×C3—48no—
393C23×C3—48no—
394C23×C3—84no—
395C23×C3C2×#6714yes—
396C23×C3C2×#6714yes—
397C22×D3—64no—
398C22×D3—64no—
399C22×D3—128no—
400C22×D3—128no—
401C22×D3—244no—
402C3×Q8—24no—
403C3×Q8—24no—
404C3×D4—28no—
405C3×D4—24no—
406C3×D4—24no—
407C3×D4—24no—
408C3×D4—48no—
409C3×D4—48no—
410C3×D4—84no—
411C3×D4D3×D418yes—
412C2×C3×C4—24no—
413C2×C3×C4—24no—
414C2×C3×C4—24no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—24no—
418C2×C3×C4—48no—
419C2×C3×C4—84no—
420C2×C3×C4C2×#6714yes—
421C2×C3×C4C2×#6714yes—
422C2×C3×C4C2×#6714yes—
423C2×C3×C4C2×#6714yes—
424C2×C3×C4C23×D314yes—
425#67—128no—
426#67—244no—
427C2×Dic3—64no—
428C2×Dic3—64no—
429D12—38no—
430D12—64no—
431D12—64no—
432D12—64no—
433D12—128no—
434C4×D3—128no—
435C4×D3—244no—
436Dic6—64no—
437Dic6—64no—
438C3⋊C8—38no—
439C3⋊C8—64no—
440C3⋊C8—64no—
441C3⋊C8—64no—
442C2×C32—22no—
443C2×C32—41no—
444C2×C32—42no—
445C2×C32—81no—
446C2×C32D4211yescharacteristic
447C2×C32C2×#13511yescharacteristic
448C2×C32#57611yescharacteristic
449C3⋊D3—42no—
450C3⋊D3—81no—
451C24—21no—
452C24—181no—
453#41—22no—
454#41—41no—
455#41—182no—
456#41—361no—
457C2×Q8—91no—
458C2×Q8C22×C3⋊D311yescharacteristic
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—41no—
464C2×D4—91no—
465C2×D4—91no—
466C2×D4—182no—
467C2×D4—181no—
468C2×D4—181no—
469C2×D4—181no—
470C2×D4—361no—
471C2×D4#63811yescharacteristic
472C2×D4C22×C3⋊D311yescharacteristic
473C2×D4C22×C3⋊D311yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—181no—
477C22×C4—181no—
478C22×C4C22×C3⋊D311yescharacteristic
479C22×C4C22×C3⋊D311yescharacteristic
480Q16—182no—
481SD16—182no—
482SD16—182no—
483SD16—361no—
484SD16—361no—
485D8—94no—
486D8—182no—
487D8—182no—
488D8—182no—
489C2×C8—91no—
490C2×C8—91no—
491C2×C8—91no—
492C2×C8—91no—
493C2×C8—182no—
494C2×C8—182no—
495C4⋊C4—21no—
496C4⋊C4—91no—
497C4⋊C4—181no—
498C4⋊C4#63811yescharacteristic
499C4⋊C4C22×C3⋊D311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—181no—
506C22⋊C4—181no—
507C22⋊C4—181no—
508C22⋊C4—181no—
509C22⋊C4—181no—
510C22⋊C4#63811yescharacteristic
511C22⋊C4#63811yescharacteristic
512C42C22×C3⋊D311yescharacteristic
513C22×C3—24no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3—28no—
517C22×C3—24no—
518C22×C3—24no—
519C22×C3—44no—
520C22×C3—44no—
521C22×C3—44no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3—816no—
525C22×C3#95018yescomplemented
526C22×C3C22×#6714yes—
527C2×D3—68no—
528C2×D3—124no—
529C2×D3—124no—
530C2×D3—124no—
531C2×D3—124no—
532C2×D3—124no—
533C2×D3—2416no—
534C3×C4—24no—
535C3×C4—24no—
536C3×C4—24no—
537C3×C4—24no—
538C3×C4—44no—
539C3×C4—44no—
540C3×C4C2×D3×D414yes—
541C3×C4#100414yes—
542Dic3—124no—
543Dic3—124no—
544C32#328011yescharacteristic, complemented, Sylow
545C23—21no—
546C23—21no—
547C23—42no—
548C23—42no—
549C23—81no—
550C23—181no—
551C23—181no—
552C23—362no—
553C23—362no—
554C23—721no—
555C23C2×#63811yescharacteristic
556C23C2×#63811yescharacteristic
557Q8—21no—
558Q8—21no—
559Q8—181no—
560Q8—181no—
561D4—22no—
562D4—21no—
563D4—21no—
564D4—21no—
565D4—42no—
566D4—42no—
567D4—81no—
568D4—92no—
569D4—181no—
570D4—181no—
571D4—181no—
572D4—362no—
573D4—362no—
574D4—721no—
575D4D4×C3⋊D312yes—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—42no—
583C2×C4—81no—
584C2×C4—181no—
585C2×C4—181no—
586C2×C4—362no—
587C2×C4—721no—
588C2×C4C2×#63811yescharacteristic
589C2×C4C2×#63811yescharacteristic
590C2×C4C2×#63811yescharacteristic
591C2×C4C2×#63811yescharacteristic
592C2×C4C23×C3⋊D311yescharacteristic
593C8—92no—
594C8—181no—
595C8—181no—
596C8—181no—
597C2×C3—28no—
598C2×C3—44no—
599C2×C3—48no—
600C2×C3—84no—
601C2×C3D4×#6714yes—
602C2×C3C2×#95014yes—
603C2×C3#583514yes—
604D3—128no—
605D3—244no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—84no—
618C22—182no—
619C22—361no—
620C22—361no—
621C22—361no—
622C22—361no—
623C22—361no—
624C22—724no—
625C22#6415912yescomplemented
626C22C22×#63811yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—41no—
632C4—41no—
633C4—361no—
634C4—361no—
635C4C2×D4×C3⋊D311yescharacteristic
636C4#6436111yescharacteristic
637C3#8350514yescomplemented
638C2—22no—
639C2—41no—
640C2—42no—
641C2—81no—
642C2—362no—
643C2—721no—
644C2D4×#63811yescharacteristic
645C2C2×#6415911yescharacteristic
646C2#1059630611yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#546C2
socleC22×C32C22×D4
hypercenter#546C3⋊D3
hypocenterC32#3280
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C32×#546C2semidirect
2. C2×#64159C2semidirect
3. C22×#3753C2semidirect
4. #10596021C2semidirect
5. #10592937C2semidirect
6. #10592917C2semidirect
7. #10592758C2semidirect
8. #10592756C2semidirect
9. #10592750C2semidirect
10. #10592745C2semidirect
11. #10592737C2semidirect
12. #10592736C2semidirect
13. #10592715C2semidirect
14. C4×#3753C2semidirect
15. D4×C3⋊(C3⋊C8)C2semidirect
17. C2×C32×#41C22semidirect
19. C2×#3828C22semidirect
21. C32×#124C22semidirect
25. C32×#121C22semidirect
28. C32×C4×D4C22semidirect
31. C2×#3756C22semidirect
32. C2×#3755C22semidirect
33. C2×#3755C22semidirect
35. C2×#3754C22semidirect
36. C2×#3754C22semidirect
42. #64139C22semidirect
45. C22×C3⋊(C3⋊C8)C22semidirect
49. #64084C22semidirect
50. C4×C4⋊(C3⋊D3)C22semidirect
51. #63662C22semidirect
52. #63659C22semidirect
53. #63659C22semidirect
55. #63657C22semidirect
56. #63637C22semidirect
57. #63636C22semidirect
58. #63636C22semidirect
59. #63634C22semidirect
60. C4⋊(C3⋊(C3⋊C8))C22semidirect
61. C4×C3⋊(C3⋊C8)C22semidirect
62. C3×#546D3semidirect
126. #3753D4semidirect
140. C2×C3×#41C2×D3semidirect
144. C3×#124C2×D3semidirect
148. C3×#121C2×D3semidirect
151. C3×C4×D4C2×D3semidirect
240. #546C3⋊D3semidirect
326. C22×C32#135semidirect
346. C2×#41C2×C3⋊D3semidirect
363. #124C2×C3⋊D3semidirect
371. #121C2×C3⋊D3semidirect
376. C4×D4C2×C3⋊D3semidirect
525. C22×C3#950semidirect
544. C32#3280semidirect
625. C22#64159semidirect
637. C3#83505semidirect

Maximal subgroups

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

subgroupindex
C32×#5462
C2×#641592
C22×#37532
#105960212
#105929372
#105929172
#105927582
#105927562
#105927502
#105927452
#105927372
#105927362
#105927152
C4×#37532
D4×C3⋊(C3⋊C8)2
#835053