Finite Group Databasebeta

← #11859501

Subgroups — #11859501

Order 1152.

1067 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63805C212yesmaximal, complemented
2#10593968C211yesmaximal, characteristic
3#10592533C211yesmaximal, characteristic, complemented
4C3×#4838C211yesmaximal, characteristic, complemented
5C3×#4837C211yesmaximal, characteristic, complemented
6#10590955C212yesmaximal, complemented
7#10590906C212yesmaximal, complemented
8#10590731C212yesmaximal, complemented
9#10590338C211yesmaximal, characteristic, complemented
10#10590329C211yesmaximal, characteristic, complemented
11C4×D3×C3⋊C8C211yesmaximal, characteristic, complemented
12#79538—31nomaximal
13#79041—31nomaximal
14C2×#3798C2212yes—
15C2×C8×C3⋊D3C2212yes—
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×ΓL(1,𝔽25)C2212yes—
18C2×C3×ΓL(1,𝔽25)C2212yes—
19C3×C4×D12C2211yescharacteristic
20C3×C42×D3C2211yescharacteristic
21#63918C411yescharacteristic
22#63911C411yescharacteristic
23#63909C411yescharacteristic
24#63908C411yescharacteristic
25#63906C412yes—
26C4×#626C2212yes—
27#63852C412yes—
28C2×#3712C2212yes—
29C2×D3×C3⋊C8C2211yescharacteristic
30C2×D3×C3⋊C8C2212yes—
31C2×D3×C3⋊C8C2211yescharacteristic
32#63805—28no—
33C32×C4⋊C8C2211yescharacteristic
34#63647C2212yes—
35C4×C3⋊(C3⋊C8)C2211yescharacteristic
36C3×(C2×D3)⋊C8C2212yes—
37C3×Dic3⋊C8C2212yes—
38C3×C4⋊(C3⋊C8)C2211yescharacteristic
39C3×C4×C3⋊C8C2211yescharacteristic
40#63557C2212yes—
41(C2×D3)⋊(C3⋊C8)C2212yes—
42Dic3×C3⋊C8C2211yescharacteristic
43Dic3×C3⋊C8C2211yescharacteristic
44C2×#907—32no—
45C2×#907—32no—
46#5548—31no—
47#5477—31no—
48C3×#436—31no—
49C3×#423—31no—
50#5129—32no—
51#5128—32no—
52D4×C3⋊C8—31no—
53C2×C4×C3⋊C8—31no—
54#4838—32no—
55#4838—31no—
56#4838D311yescharacteristic, complemented
57#4837D311yescharacteristic, complemented
58#4834—31no—
59#4834—31no—
60#4834—31no—
61#4834—32no—
62#4834—61no—
63#4727—32no—
64#67⋊C8—32no—
65#4688—31no—
66C4×D3×C8—31no—
67C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
68C2×C4×C3⋊D3C2312yes—
69C2×C3×D12C2×C411yescharacteristic
70C2×C3×C4×D3C2×C411yescharacteristic
71C2×C3×C4×D3C2×C411yescharacteristic
72C2×C3×C4×D3C2312yes—
73C2×C3×C4×D3C2311yescharacteristic
74C2×#627C2×C412yes—
75C2×#626C2×C412yes—
76C2×#626C2×C412yes—
77C2×#626C2×C412yes—
78C2×#625C2×C412yes—
79#3798—24no—
80#3798—24no—
81C2×C32×C8C2312yes—
82C32×C42C2311yescharacteristic
83#3752C2×C412yes—
84C4⋊(C3⋊Dic3)C2×C411yescharacteristic
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2311yescharacteristic
87C8×C3⋊D3—24no—
88C3×(C2×D3)⋊C4C2×C412yes—
89C3×C4⋊Dic3C2×C411yescharacteristic
90C3×C4×Dic3C2×C411yescharacteristic
91C3×C4×Dic3C2×C411yescharacteristic
92C3×C4×Dic3C2311yescharacteristic
93C2×C3×C3⋊C8C2312yes—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2311yescharacteristic
96C3×ΓL(1,𝔽25)—24no—
97C3×ΓL(1,𝔽25)—24no—
98Dic3⋊Dic3C2×C412yes—
99#3722C2×C412yes—
100(C2×D3)⋊Dic3C2×C412yes—
101#3712—24no—
102D3×C3⋊C8—22no—
103D3×C3⋊C8—24no—
104D3×C3⋊C8D414yescomplemented
105#2931—91noSylow
106C2×#235—32no—
107C2×#235—32no—
108C2×C3×M16—32no—
109C2×C3×M16—32no—
110C3×C4×D4—31no—
111C2×C3×C42—31no—
112#940—31no—
113#938—31no—
114#930—31no—
115#930—32no—
116#930—31no—
117C4×#67—32no—
118C22×C3⋊C8—31no—
119C22×C3⋊C8—31no—
120C22×C3⋊C8—32no—
121#907—68no—
122#907—68no—
123C2×ΓL(1,𝔽25)—32no—
124C2×ΓL(1,𝔽25)—32no—
125C2×ΓL(1,𝔽25)C2×D312yes—
126C2×ΓL(1,𝔽25)C2×D312yes—
127C2×D3×C8—31no—
128C2×D3×C8—31no—
129C2×D3×C8—32no—
130C2×D3×C8—32no—
131C2×D3×C8—32no—
132C2×D3×C8—62no—
133#895—31no—
134#888—31no—
135#885—32no—
136#884—32no—
137#875—32no—
138#874—31no—
139C4×D12—32no—
140C4×D12—31no—
141C4×D12—31no—
142C4×D12—61no—
143C4×D12C2×D311yescharacteristic
144C42×D3C2×D311yescharacteristic
145#869—31no—
146C3×C4⋊C8—21no—
147C3×C4⋊C8—32no—
148C3×C4⋊C8—31no—
149C3×C4⋊C8C2×D311yescharacteristic
150C3×C4⋊C8C2×D311yescharacteristic
151C3×C22⋊C8—32no—
152C3×C4×C8—31no—
153#830—32no—
154(C2×D3)⋊C8—32no—
155(C2×D3)⋊C8—32no—
156(C2×D3)⋊C8—32no—
157(C2×D3)⋊C8—32no—
158(C2×D3)⋊C8—32no—
159(C2×D3)⋊C8—62no—
160(C2×D3)⋊C8C2×D312yes—
161Dic3⋊C8C2×D312yes—
162C8×Dic3—31no—
163C8×Dic3—31no—
164C4⋊(C3⋊C8)C2×D311yescharacteristic
165C4×C3⋊C8—31no—
166C4×C3⋊C8—31no—
167C4×C3⋊C8—31no—
168C4×C3⋊C8—31no—
169C4×C3⋊C8—61no—
170C4×C3⋊C8C2×D311yescharacteristic
171C22×C3⋊D3C22×C412yes—
172C22×C3×D3C22×C412yes—
173C22×C3×D3C22×C411yescharacteristic
174C2×C32×C4C22×C411yescharacteristic
175C2×C32×C4C22×C411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C2×C3⋊Dic3C22×C412yes—
178C4⋊(C3⋊D3)—22no—
179C4×C3⋊D3—22no—
180C4×C3⋊D3—22no—
181C2×C3×Dic3C22×C411yescharacteristic
182C2×C3×Dic3C22×C412yes—
183C2×C3×Dic3C22×C411yescharacteristic
184C2×C3×Dic3C22×C411yescharacteristic
185C3×D12—22no—
186C3×C4×D3—22no—
187C3×C4×D3—22no—
188C3×C4×D3—22no—
189C3×C4×D3—22no—
190C3×C4×D3C2×D412yes—
191C3×C4×D3#4112yes—
192#627—24no—
193#626—24no—
194#626—24no—
195#626—24no—
196#625—24no—
197C32×C8—22no—
198C3⋊(C3⋊C8)—21no—
199C3⋊(C3⋊C8)C2×D412yes—
200C3×C3⋊C8—22no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8C2×D412yes—
203C2×#131—92no—
204#532—91no—
205#436—31no—
206#436—91no—
207#436—92no—
208C8×D4—92no—
209C8×D4—91no—
210C8×D4—91no—
211#423—31no—
212#408—92no—
213C2×C4×C8—91no—
214C2×C3×D4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—32no—
219C2×#67—32no—
220C2×#67—32no—
221C2×#67—32no—
222C2×#67—32no—
223C2×#67—32no—
224#235—64no—
225#235—64no—
226#235—64no—
227#235—64no—
228C2×D12—32no—
229C2×D12—32no—
230C2×D12—32no—
231C2×D12—31no—
232C2×D12—31no—
233C2×D12—61no—
234C2×D12C4×D311yescharacteristic
235C2×C4×D3—32no—
236C2×C4×D3—32no—
237C2×C4×D3—62no—
238C2×C4×D3C4×D311yescharacteristic
239C2×C4×D3C4×D311yescharacteristic
240C2×C4×D3C22×D312yes—
241C2×C4×D3C22×D311yescharacteristic
242C2×Dic6—32no—
243C2×Dic6—32no—
244C3×M16—64no—
245C3×M16—64no—
246C2×C3×C8—22no—
247C2×C3×C8—31no—
248C2×C3×C8—31no—
249C2×C3×C8—32no—
250C2×C3×C8C22×D312yes—
251C2×C3×C8C22×D312yes—
252C3×C4⋊C4—31no—
253C3×C22⋊C4—32no—
254C3×C42—21no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—31no—
258C3×C42C22×D311yescharacteristic
259C3×C42C22×D311yescharacteristic
260#217—32no—
261(C2×D3)⋊C4—32no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264(C2×D3)⋊C4—32no—
265(C2×D3)⋊C4—32no—
266(C2×D3)⋊C4—62no—
267(C2×D3)⋊C4C4×D312yes—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—32no—
271C4⋊Dic3—61no—
272C4⋊Dic3C4×D311yescharacteristic
273Dic3⋊C4—32no—
274C4×Dic3C4×D311yescharacteristic
275C4×Dic3C4×D311yescharacteristic
276C4×Dic3C22×D311yescharacteristic
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—34no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8—64no—
285C2×C3⋊C8—62no—
286C2×C3⋊C8C22×D311yescharacteristic
287C2×C3⋊C8C22×D312yes—
288C2×C3⋊C8C22×D311yescharacteristic
289ΓL(1,𝔽25)—24no—
290ΓL(1,𝔽25)—24no—
291ΓL(1,𝔽25)—64no—
292ΓL(1,𝔽25)—64no—
293D3×C8—34no—
294D3×C8—64no—
295D3×C8—64no—
296D3×C8—64no—
297D3×C8—62no—
298D3×C8—124no—
299C22×C32C23×C411yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3—22no—
303C2×C3×D3—21no—
304C2×C3×D3—21no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3—22no—
308C2×C3×D3C4×D412yes—
309C32×C4—21no—
310C32×C4#13112yescomplemented
311C32×C4C22×D411yescharacteristic
312C32×C4C2×#4111yescharacteristic
313C3⋊Dic3—22no—
314C3×Dic3—22no—
315C3×Dic3—21no—
316C3×Dic3C4×D412yes—
317C3×Dic3#13112yescomplemented
318C2×#41—92no—
319#131—188no—
320C2×M16—32no—
321C2×M16—32no—
322C2×M16—92no—
323C22×C8—91no—
324C22×C8—92no—
325C22×C8—91no—
326C22×C8—92no—
327C4⋊Q8—91no—
328#127—91no—
329#124—92no—
330#121—92no—
331#121—91no—
332#121—91no—
333C4×D4—31no—
334C4×D4—92no—
335C4×D4—91no—
336C2×C42—31no—
337C4⋊C8—32no—
338C4⋊C8—31no—
339C4⋊C8D3211yescharacteristic
340C22⋊C8—32no—
341C22⋊C8—92no—
342C22⋊C8—92no—
343C22⋊C8—92no—
344C4×C8—31no—
345C4×C8—91no—
346C4×C8—91no—
347C4×C8—91no—
348C23×C3—31no—
349C23×C3—32no—
350C22×D3—32no—
351C22×D3—32no—
352C22×D3—62no—
353C22×D3C2×C4×D312yes—
354C22×D3C2×C4×D311yescharacteristic
355C3×D4—62no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—32no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—31no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4C2×C4×D311yescharacteristic
370C2×C3×C4C2×C4×D311yescharacteristic
371C2×C3×C4C2×C4×D311yescharacteristic
372C2×C3×C4C2×C4×D311yescharacteristic
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C23×D311yescharacteristic
375#67—64no—
376#67—64no—
377#67—64no—
378#67—64no—
379#67—64no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—62no—
383C2×Dic3C2×C4×D311yescharacteristic
384C2×Dic3C2×C4×D312yes—
385C2×Dic3C2×C4×D311yescharacteristic
386C2×Dic3C2×C4×D311yescharacteristic
387D12—22no—
388D12—64no—
389D12—64no—
390D12—64no—
391D12—64no—
392D12—62no—
393D12—62no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—62no—
399C4×D3—62no—
400C4×D3—62no—
401C4×D3—62no—
402C4×D3—122no—
403C4×D3—122no—
404C4×D3D3×D412yes—
405C4×D3#23912yes—
406Dic6—64no—
407Dic6—64no—
408C3×C8—22no—
409C3×C8—22no—
410C3×C8—32no—
411C3×C8—42no—
412C3×C8—62no—
413C3×C8—61no—
414C3⋊C8—22no—
415C3⋊C8—21no—
416C3⋊C8—32no—
417C3⋊C8—32no—
418C3⋊C8—62no—
419C3⋊C8—61no—
420C3⋊C8—61no—
421C3⋊C8—121no—
422C3⋊C8D3×D412yes—
423C2×C32C2×C4×D411yescharacteristic
424C2×C32C2×#13111yescharacteristic
425C2×C32C2×#13111yescharacteristic
426C3⋊D3—42no—
427C3×D3—22no—
428C3×D3—42no—
429#41—184no—
430#41—184no—
431C2×Q8—92no—
432C2×D4—31no—
433C2×D4—92no—
434C2×D4—92no—
435C2×D4—92no—
436C2×D4—92no—
437C2×D4—91no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—92no—
443M16—64no—
444M16—64no—
445M16—184no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—32no—
449C2×C8—94no—
450C2×C8—91no—
451C2×C8—91no—
452C2×C8—184no—
453C2×C8—184no—
454C2×C8—182no—
455C2×C8C2×D3212yes—
456C4⋊C4—31no—
457C4⋊C4—92no—
458C4⋊C4—91no—
459C22⋊C4—32no—
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—92no—
463C42—31no—
464C42—31no—
465C42—31no—
466C42C2×D3211yescharacteristic
467C22×C3—21no—
468C22×C3—32no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—62no—
474C22×C3C22×C4×D311yescharacteristic
475C22×C3C22×C4×D311yescharacteristic
476C2×D3—21no—
477C2×D3—21no—
478C2×D3—22no—
479C2×D3—22no—
480C2×D3—22no—
481C2×D3—64no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3#89312yes—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—32no—
497C3×C4—32no—
498C3×C4—41no—
499C3×C4—61no—
500C3×C4—62no—
501C3×C4#90712yescomplemented
502C3×C4#90712yescomplemented
503C3×C4C2×D3×D411yescharacteristic
504C3×C4C2×D3×D411yescharacteristic
505C3×C4C2×#23911yescharacteristic
506C3×C4D3×#4111yescharacteristic
507Dic3—21no—
508Dic3—22no—
509Dic3—62no—
510Dic3—62no—
511Dic3—122no—
512Dic3#89312yes—
513Dic3#90712yescomplemented
514C32#293111yescharacteristic, complemented, Sylow
515C23—31no—
516C23—32no—
517C23—92no—
518Q8—184no—
519D4—62no—
520D4—184no—
521D4—184no—
522D4—184no—
523D4—184no—
524D4—182no—
525C2×C4—31no—
526C2×C4—31no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—31no—
530C2×C4—32no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—92no—
536C2×C4—182no—
537C2×C4—182no—
538C2×C4C4×D3211yescharacteristic
539C2×C4C4×D3211yescharacteristic
540C2×C4C22×D3211yescharacteristic
541C8—22no—
542C8—32no—
543C8—62no—
544C8—61no—
545C8—92no—
546C8—181no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—62no—
551C2×C3—122no—
552C2×C3C2×#89311yescharacteristic
553C2×C3C4×D3×D411yescharacteristic
554C2×C3C2×#90711yescharacteristic
555C2×C3C2×#90711yescharacteristic
556C2×C3C2×#90711yescharacteristic
557C2×C3D3×#13111yescharacteristic
558D3—22no—
559D3—42no—
560D3—124no—
561D3—122no—
562D3—122no—
563C22—32no—
564C22—61no—
565C22—61no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—182no—
570C22—182no—
571C22—182no—
572C22C2×C4×D3211yescharacteristic
573C4—21no—
574C4—32no—
575C4—32no—
576C4—62no—
577C4—61no—
578C4—182no—
579C4#6380512yescomplemented
580C4D32×D411yescharacteristic
581C4D3×#23911yescharacteristic
582C3—21no—
583C3#7904111yescharacteristic, complemented
584C3#7953811yescharacteristic, complemented
585C2—62no—
586C2—122no—
587C2—362no—
588C2D3×#89311yescharacteristic
589C2C2×#6380511yescharacteristic
590C2D3×#90711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC32×C4⋊C8C22
prime radical#10593968C2
cosocleC2×C32×C4C24
hypercenterC4⋊C8D32
hypocenterC32#2931
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#63805C2semidirect
3. #10592533C2semidirect
4. C3×#4838C2semidirect
5. C3×#4837C2semidirect
6. #10590955C2semidirect
7. #10590906C2semidirect
8. #10590731C2semidirect
9. #10590338C2semidirect
10. #10590329C2semidirect
11. C4×D3×C3⋊C8C2semidirect
56. #4838D3semidirect
57. #4837D3semidirect
104. D3×C3⋊C8D4semidirect
310. C32×C4#131semidirect
317. C3×Dic3#131semidirect
501. C3×C4#907semidirect
502. C3×C4#907semidirect
513. Dic3#907semidirect
514. C32#2931semidirect
579. C4#63805semidirect
583. C3#79041semidirect
584. C3#79538semidirect

Maximal subgroups

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

subgroupindex
C2×#638052
#105939682
#105925332
C3×#48382
C3×#48372
#105909552
#105909062
#105907312
#105903382
#105903292
C4×D3×C3⋊C82
#795383
#790413