Finite Group Databasebeta

← #11787451

Subgroups — #11787451

Order 1152.

905 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63894C211yesmaximal, characteristic
2C22×#3717C211yesmaximal, characteristic
3C2×#63636C211yesmaximal, characteristic
4C3×#4973C211yesmaximal, characteristic, complemented
5C2×#63565C211yesmaximal, characteristic, complemented
6C3×#4529C211yesmaximal, characteristic, complemented
7#10589155C211yesmaximal, characteristic, complemented
8#69687—31nomaximal
9#68896—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C22×#624—21no—
13C2×C32×C4⋊C4C2211yescharacteristic
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×C4×Dic3C2211yescharacteristic
17C22×C3×C3⋊C8C2211yescharacteristic
18C2×#3722—21no—
19#63894—22no—
20#63894—22no—
21C2×#3717—22no—
22C2×#3717—22no—
23C2×#3717C414yescomplemented
24#63636—22no—
25C3×C3⋊#95—21no—
26C3×C3⋊#95—21no—
27#63565—22no—
28C22×#215—31no—
29C22×#204—31no—
30C2×#893—31no—
31C2×C4×D12—31no—
32C3×#391—31no—
33C2×#835—31no—
34#4973D311yescharacteristic, complemented
35C2×#810—31no—
36C2×#809—31no—
37C2×#809—31no—
38C2×#809—61no—
39C3×#340—31no—
40#4550—31no—
41#4529D311yescharacteristic, complemented
42#4524—31no—
43C23×C3⋊D3—21no—
44C22×C32×C4—21no—
45C22×C32×C4C2311yescharacteristic
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)C2×C412yes—
50C2×C4⋊(C3⋊D3)D412yes—
51C2×C4⋊(C3⋊D3)D412yes—
52C22×C3×Dic3—21no—
53C22×C3×Dic3—21no—
54C22×C3×Dic3D411yescharacteristic
55C22×C3×Dic3D411yescharacteristic
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6C2×C412yescomplemented
62C2×#624—24no—
63C2×#624—42no—
64C2×#624—42no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C2×C3⋊(C3⋊C8)—21no—
68C2×C3⋊(C3⋊C8)—21no—
69C2×C3⋊(C3⋊C8)—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3D412yes—
72C3×C4×Dic3—21no—
73C3×C4×Dic3—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8C2×C412yescomplemented
77#3722—24no—
78#3717—28no—
79#1877—91noSylow
80C22×C3×Q8—31no—
81C22×D12—31no—
82C22×D12—31no—
83C22×D12—61no—
84C22×C4×D3—61no—
85C22×C4×D3—61no—
86C22×Dic6C2×D311yescharacteristic
87C22×C3×C8—31no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—31no—
90C2×C3×C4⋊C4C2×D311yescharacteristic
91C2×C3×C4⋊C4C2×D311yescharacteristic
92C2×C3×C42—31no—
93C2×#215—34no—
94C2×#215—62no—
95C2×#215—62no—
96C2×(C2×D3)⋊C4—61no—
97C2×(C2×D3)⋊C4—61no—
98C2×C4⋊Dic3C2×D311yescharacteristic
99C2×C4×Dic3C2×D311yescharacteristic
100C22×C3⋊C8—31no—
101C22×C3⋊C8—31no—
102C22×C3⋊C8—61no—
103C22×C3⋊C8C2×D311yescharacteristic
104C2×#204—34no—
105C2×#204—62no—
106C2×#204—62no—
107#893—62no—
108#893—62no—
109C4×D12—62no—
110C4×D12—62no—
111C3×#95—61no—
112C3×#95—61no—
113#835—62no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116#810—62no—
117#809—62no—
118#809—62no—
119#809—64no—
120C23×C32C2×D411yescharacteristic
121C22×C3⋊D3—22no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3—41no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—41no—
128C2×C32×C4—41no—
129C2×C32×C4—41no—
130C2×C32×C4—41no—
131C2×C32×C4C22×C411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4#4111yescharacteristic
136C2×C32×C4#4111yescharacteristic
137C4⋊(C3⋊D3)—22no—
138C4⋊(C3⋊D3)—22no—
139C4⋊(C3⋊D3)—22no—
140C4⋊(C3⋊D3)C22⋊C414yes—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C2×C3×Dic3SD1612yescomplemented
153C3×Dic6—22no—
154C3×Dic6—22no—
155C3×Dic6—41no—
156C3×Dic6—41no—
157#624—48no—
158C3⋊(C3⋊C8)—41no—
159C3×C3⋊C8—22no—
160C22×SD16—91no—
161C2×C4×D4—91no—
162C2×#103—91no—
163C2×#102—91no—
164#391—31no—
165#340—31no—
166#336—91no—
167C23×D3—61no—
168C23×D3—61no—
169C23×D3—121no—
170C2×C3×Q8—32no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8—61no—
174C2×C3×Q8—61no—
175C2×C3×Q8—61no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—41no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C22×D311yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3#6711yescharacteristic
189C22×Dic3#6711yescharacteristic
190C2×D12—32no—
191C2×D12—32no—
192C2×D12—32no—
193C2×D12—32no—
194C2×D12—32no—
195C2×D12—32no—
196C2×D12—62no—
197C2×D12—62no—
198C2×D12—62no—
199C2×D12—61no—
200C2×D12—61no—
201C2×D12—61no—
202C2×D12—61no—
203C2×D12—61no—
204C2×D12—61no—
205C2×D12—121no—
206C2×D12—121no—
207C2×D12—121no—
208C2×C4×D3—64no—
209C2×C4×D3—64no—
210C2×C4×D3—122no—
211C2×C4×D3—122no—
212C2×C4×D3—122no—
213C2×C4×D3—122no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6C4×D312yescomplemented
220C2×C3×C8—32no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—61no—
231C3×C42—61no—
232C3×C42—61no—
233#215—68no—
234(C2×D3)⋊C4—64no—
235(C2×D3)⋊C4—64no—
236C4⋊Dic3—21no—
237C4⋊Dic3D1212yes—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—62no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8—61no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8—121no—
250C2×C3⋊C8—121no—
251C2×C3⋊C8C4×D312yescomplemented
252#204—68no—
253C22×C32C4×D411yescharacteristic
254C22×C32#12011yescharacteristic
255C22×C32#12111yescharacteristic
256C22×C32C2×SD1611yescharacteristic
257C22×C32#13511yescharacteristic
258C22×C32AGL(1,ℤ/8)11yescharacteristic
259C22×C32#13711yescharacteristic
260C2×C3⋊D3—24no—
261C2×C3⋊D3—42no—
262C2×C3⋊D3—42no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3—42no—
265C2×C3⋊D3—42no—
266C2×C3⋊D3—42no—
267C32×C4—81no—
268C32×C4C2×C22⋊C411yescharacteristic
269C32×C4C4×D411yescharacteristic
270C32×C4#12111yescharacteristic
271C32×C4#12311yescharacteristic
272C3×Dic3—22no—
273C3×Dic3—42no—
274C3×Dic3—42no—
275C3×Dic3—41no—
276C22×Q8—31no—
277C22×D4—91no—
278C23×C4—181no—
279C2×SD16—94no—
280C2×SD16—182no—
281C2×SD16—182no—
282C22×C8—31no—
283C22×C8—91no—
284C4×D4—182no—
285C4×D4—182no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4D3211yescharacteristic
288C2×C22⋊C4—181no—
289C2×C42—31no—
290#103—182no—
291#102—182no—
292#95—61no—
293#95—61no—
294C23×C3—21no—
295C23×C3D3×D411yescharacteristic
296C23×C3C2×#6711yescharacteristic
297C22×D3—62no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—62no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—122no—
304C22×D3—122no—
305C22×D3—122no—
306C22×D3—121no—
307C22×D3—121no—
308C22×D3—121no—
309C22×D3—121no—
310C22×D3—121no—
311C22×D3—121no—
312C22×D3—121no—
313C22×D3—121no—
314C22×D3—241no—
315C22×D3—241no—
316C22×D3—241no—
317C22×D3—241no—
318C3×Q8—62no—
319C3×Q8—62no—
320C3×Q8—121no—
321C3×Q8—121no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—32no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—62no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—62no—
343C2×C3×C4—81no—
344C2×C3×C4—81no—
345C2×C3×C4—121no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4—121no—
349C2×C3×C4C2×C4×D311yescharacteristic
350C2×C3×C4C2×C4×D311yescharacteristic
351C2×C3×C4C2×D1211yescharacteristic
352C2×C3×C4#23511yescharacteristic
353C2×C3×C4#23511yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4D3×D411yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4#23711yescharacteristic
360C2×C3×C4#23911yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3#21512yescomplemented
373D12—34no—
374D12—34no—
375D12—64no—
376D12—62no—
377D12—62no—
378D12—62no—
379D12—62no—
380D12—62no—
381D12—62no—
382D12—122no—
383D12—122no—
384D12—122no—
385C4×D3—128no—
386C4×D3—128no—
387Dic6—22no—
388Dic6—22no—
389Dic6—41no—
390Dic6—41no—
391C3×C8—62no—
392C3⋊C8—22no—
393C3⋊C8—122no—
394C3⋊C8—121no—
395C3⋊C8—121no—
396C2×C32#38611yescharacteristic
397C2×C32C4×SD1611yescharacteristic
398C2×C32#44011yescharacteristic
399C2×C32#45011yescharacteristic
400C2×C32#45211yescharacteristic
401C2×C32#46111yescharacteristic
402C2×C32#46311yescharacteristic
403C3⋊D3—44no—
404C24—181no—
405C2×Q8—32no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×D4—92no—
412C2×D4—92no—
413C2×D4—92no—
414C2×D4—181no—
415C2×D4—181no—
416C2×D4—181no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—184no—
423C22×C4—362no—
424C22×C4—362no—
425C22×C4C2×D3211yescharacteristic
426SD16—188no—
427C2×C8—32no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—181no—
431C2×C8—181no—
432C2×C8—181no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—32no—
436C4⋊C4—61no—
437C22⋊C4—184no—
438C42—61no—
439C42—61no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3#88211yescharacteristic
448C22×C3#88411yescharacteristic
449C22×C3#89311yescharacteristic
450C22×C3#91111yescharacteristic
451C22×C3D3×SD1611yescharacteristic
452C22×C3#91611yescharacteristic
453C22×C3#91811yescharacteristic
454C22×C3C4×#6711yescharacteristic
455C22×C3#93711yescharacteristic
456C22×C3#93811yescharacteristic
457C22×C3C2×#21511yescharacteristic
458C22×C3#94211yescharacteristic
459C22×C3#94911yescharacteristic
460C22×C3#95011yescharacteristic
461C2×D3—64no—
462C2×D3—64no—
463C2×D3—124no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—122no—
471C2×D3—122no—
472C2×D3—122no—
473C2×D3—122no—
474C2×D3—122no—
475C2×D3—122no—
476C2×D3—242no—
477C2×D3—242no—
478C2×D3—242no—
479C2×D3—242no—
480C2×D3—242no—
481C2×D3—242no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—62no—
487C3×C4—82no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—122no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4C4×D1211yescharacteristic
494C3×C4D3×C22⋊C411yescharacteristic
495C3×C4D3×C22⋊C411yescharacteristic
496C3×C4#67⋊C411yescharacteristic
497C3×C4#88311yescharacteristic
498C3×C4#88411yescharacteristic
499C3×C4#89411yescharacteristic
500C3×C4#89511yescharacteristic
501Dic3—22no—
502Dic3—42no—
503Dic3—42no—
504Dic3—41no—
505C32#187711yescharacteristic, complemented, Sylow
506C23—182no—
507C23—182no—
508C23—182no—
509C23—361no—
510C23—361no—
511C23—361no—
512C23—361no—
513C23D3×#6711yescharacteristic
514Q8—62no—
515Q8—62no—
516Q8—121no—
517Q8—121no—
518D4—94no—
519D4—182no—
520D4—182no—
521D4—182no—
522C2×C4—32no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—62no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—368no—
538C2×C4#379911yescharacteristic
539C2×C4C4×D3211yescharacteristic
540C2×C4D3×D1211yescharacteristic
541C2×C4#380211yescharacteristic
542C2×C4#380411yescharacteristic
543C2×C4#381111yescharacteristic
544C8—62no—
545C8—361no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3#466511yescharacteristic
554C2×C3#477611yescharacteristic
555C2×C3#478711yescharacteristic
556C2×C3#480811yescharacteristic
557C2×C3#481811yescharacteristic
558C2×C3#485211yescharacteristic
559C2×C3#488911yescharacteristic
560C2×C3#495611yescharacteristic
561C2×C3C4×#21511yescharacteristic
562C2×C3#502611yescharacteristic
563C2×C3#504611yescharacteristic
564C2×C3#504711yescharacteristic
565C2×C3#505711yescharacteristic
566C2×C3#508311yescharacteristic
567D3—124no—
568D3—124no—
569D3—244no—
570C22—184no—
571C22—362no—
572C22—362no—
573C22—362no—
574C22—362no—
575C22—362no—
576C22—362no—
577C22#6390411yescharacteristic
578C22#6392211yescharacteristic
579C22#6392511yescharacteristic
580C22#6393511yescharacteristic
581C22D3×#21511yescharacteristic
582C22#6394811yescharacteristic
583C22#6395811yescharacteristic
584C4—62no—
585C4—81no—
586C4—122no—
587C4—122no—
588C4—121no—
589C4#6385411yescharacteristic
590C4#6388011yescharacteristic
591C4#6391011yescharacteristic
592C4#6392111yescharacteristic
593C3—21no—
594C3#6889611yescharacteristic, complemented
595C3#6968711yescharacteristic, complemented
596C2—364no—
597C2#1059045711yescharacteristic
598C2#1059045811yescharacteristic
599C2#1059059611yescharacteristic
600C2#1059061611yescharacteristic
601C2#1059127511yescharacteristic
602C2#1059143211yescharacteristic
603C2#1059156511yescharacteristic
subgroupG/H
centerC23D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1877
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#4973C2semidirect
5. C2×#63565C2semidirect
6. C3×#4529C2semidirect
7. #10589155C2semidirect
23. C2×#3717C4semidirect
34. #4973D3semidirect
41. #4529D3semidirect
61. C2×C3×Dic6C2×C4semidirect
76. C2×C3×C3⋊C8C2×C4semidirect
152. C2×C3×Dic3SD16semidirect
219. C2×Dic6C4×D3semidirect
251. C2×C3⋊C8C4×D3semidirect
372. C2×Dic3#215semidirect
505. C32#1877semidirect
594. C3#68896semidirect
595. C3#69687semidirect

Maximal subgroups

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

subgroupindex
C2×#638942
C22×#37172
C2×#636362
C3×#49732
C2×#635652
C3×#45292
#105891552
#696873
#688963