Finite Group Databasebeta

← #11806919

Subgroups — #11806919

Order 1152.

1896 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64096C211yesmaximal, characteristic, complemented
2C2×#64094C211yesmaximal, characteristic
3C2×#64090C211yesmaximal, characteristic
4C32×#388C211yesmaximal, characteristic, complemented
5#10592688C211yesmaximal, characteristic, complemented
6#10592656C211yesmaximal, characteristic
7#10592645C211yesmaximal, characteristic, complemented
8#10592643C211yesmaximal, characteristic, complemented
9C22⋊C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
10C4×#3752C211yesmaximal, characteristic, complemented
11C4×#3750C211yesmaximal, characteristic, complemented
12#10592368C211yesmaximal, characteristic, complemented
13#10592368C211yesmaximal, characteristic, complemented
14#10592348C211yesmaximal, characteristic, complemented
15#10592345C211yesmaximal, characteristic, complemented
16#75481—34nomaximal
17C22×#638C2211yescharacteristic
18C23×C3⋊Dic3C2211yescharacteristic
19C22×C4×C3⋊D3C2211yescharacteristic
20C22×#634C2211yescharacteristic
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C22⋊C4C2211yescharacteristic
23C2×C32×C22⋊C4C2211yescharacteristic
24C2×C32×C22⋊C4C2211yescharacteristic
25C2×C32×C42C2211yescharacteristic
26C2×#3757C2211yescharacteristic
27C2×#3757C2211yescharacteristic
28C2×#3752C2211yescharacteristic
29C2×#3752C2211yescharacteristic
30C2×#3752C2211yescharacteristic
31C2×#3752C2211yescharacteristic
32C2×#3752C2211yescharacteristic
33C2×#3750C2211yescharacteristic
34C2×#3750C2211yescharacteristic
35C2×#3750C2211yescharacteristic
36C2×#3750C2211yescharacteristic
37C2×#3750C2211yescharacteristic
38C2×C4×C3⋊Dic3C2211yescharacteristic
39C2×C4×C3⋊Dic3C2211yescharacteristic
40C2×C4×C3⋊Dic3C2211yescharacteristic
41C2×C4×C3⋊Dic3C2211yescharacteristic
42C2×C4×C3⋊Dic3C2211yescharacteristic
43C2×C4×C3⋊Dic3C2211yescharacteristic
44#64096—24no—
45#64094C418yescomplemented
46#64090—24no—
47C32×#95C2211yescharacteristic
48C32×#95C2211yescharacteristic
49#63658C2211yescharacteristic
50#63658C2211yescharacteristic
51#63658C2211yescharacteristic
52#63658C2211yescharacteristic
53#63658C2211yescharacteristic
54#63658C2211yescharacteristic
55C2×#887—34no—
56C2×#885—34no—
57C2×#67⋊C4—34no—
58C3×#388D314yescomplemented
59#4989—34no—
60#4957—34no—
61#4946—34no—
62#4944—34no—
63Dic3×C22⋊C4—34no—
64C4×(C2×D3)⋊C4—34no—
65C4×Dic3⋊C4—34no—
66#4669—34no—
67#4669—34no—
68#4649—34no—
69#4646—34no—
70C24×C32C2311yescharacteristic
71C23×C3⋊D3C2311yescharacteristic
72C22×C32×C4C2311yescharacteristic
73C22×C32×C4C2311yescharacteristic
74C22×C32×C4C2311yescharacteristic
75C22×C32×C4C2311yescharacteristic
76C22×C32×C4C2311yescharacteristic
77C22×C32×C4C2311yescharacteristic
78C2×#638—22no—
79C2×#638—22no—
80C2×#638C2×C414yes—
81C22×C3⋊Dic3—22no—
82C22×C3⋊Dic3—22no—
83C22×C3⋊Dic3—41no—
84C22×C3⋊Dic3C2311yescharacteristic
85C22×C3⋊Dic3C2311yescharacteristic
86C22×C3⋊Dic3C2311yescharacteristic
87C22×C3⋊Dic3C2311yescharacteristic
88C22×C3⋊Dic3C2311yescharacteristic
89C22×C3⋊Dic3C2311yescharacteristic
90C22×C3⋊Dic3C2311yescharacteristic
91C2×C4×C3⋊D3—22no—
92C2×C4×C3⋊D3—22no—
93C2×C4×C3⋊D3—41no—
94C2×#634—22no—
95C2×#634—22no—
96C2×#634C2×C414yescomplemented
97C32×C4⋊C4—22no—
98C32×C22⋊C4—22no—
99C32×C22⋊C4—41no—
100C32×C22⋊C4C2×C414yescomplemented
101C32×C42—41no—
102#3757—22no—
103#3757C2×C414yescomplemented
104#3752—22no—
105#3752—22no—
106#3752—41no—
107#3752C2×C414yescomplemented
108#3752C2×C414yescomplemented
109#3750—22no—
110#3750—22no—
111#3750—22no—
112#3750—22no—
113#3750D414yes—
114C4×C3⋊Dic3—22no—
115C4×C3⋊Dic3—22no—
116C4×C3⋊Dic3—22no—
117C4×C3⋊Dic3—22no—
118C4×C3⋊Dic3—41no—
119C4×C3⋊Dic3C2×C414yescomplemented
120#2367—91noSylow
121C22×#67—34no—
122C23×Dic3—34no—
123C22×C4×D3—34no—
124C22×Dic6—34no—
125C2×C3×C4⋊C4C2×D314yes—
126C2×C3×C22⋊C4C2×D314yes—
127C2×C3×C22⋊C4C2×D314yes—
128C2×C3×C22⋊C4C2×D314yes—
129C2×C3×C42C2×D314yes—
130C2×#217—34no—
131C2×#217—34no—
132C2×(C2×D3)⋊C4—34no—
133C2×(C2×D3)⋊C4—34no—
134C2×(C2×D3)⋊C4—34no—
135C2×(C2×D3)⋊C4—34no—
136C2×(C2×D3)⋊C4—34no—
137C2×Dic3⋊C4—34no—
138C2×Dic3⋊C4—34no—
139C2×Dic3⋊C4—34no—
140C2×Dic3⋊C4—34no—
141C2×Dic3⋊C4—34no—
142C2×C4×Dic3—34no—
143C2×C4×Dic3—34no—
144C2×C4×Dic3—34no—
145C2×C4×Dic3—34no—
146C2×C4×Dic3—34no—
147C2×C4×Dic3—34no—
148#887—616no—
149#885—332no—
150#67⋊C4—616no—
151C3×#95C2×D314yes—
152C3×#95C2×D314yes—
153C3⋊#95—34no—
154C3⋊#95—34no—
155C3⋊#95—34no—
156C3⋊#95—34no—
157C3⋊#95—34no—
158C3⋊#95—34no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32—21no—
163C23×C32—21no—
164C23×C32—21no—
165C23×C32C22×C412yes—
166C23×C32C2411yescharacteristic
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⋊D3C22×C412yes—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
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×C4C22×C412yes—
190C2×C32×C4C22×C412yes—
191C2×C32×C4C2×D412yes—
192#638—28no—
193C2×C3⋊Dic3—21no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—21no—
196C2×C3⋊Dic3—21no—
197C2×C3⋊Dic3—21no—
198C2×C3⋊Dic3—21no—
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—44no—
207C2×C3⋊Dic3C22×C412yes—
208C2×C3⋊Dic3C22×C412yes—
209C2×C3⋊Dic3C22×C412yes—
210C2×C3⋊Dic3C2×D412yes—
211C2×C3⋊Dic3C2×D412yes—
212C2×C3⋊Dic3#4112yes—
213C2×C3⋊Dic3#4112yes—
214C2×C3⋊Dic3#4112yes—
215C4×C3⋊D3—44no—
216#634—28no—
217C2×#124—91no—
218C2×C4×Q8—91no—
219C2×C4×D4—91no—
220#391—91no—
221#390—91no—
222#388—91no—
223#388—91no—
224#388—91no—
225#388C3⋊D311yescharacteristic, complemented
226#385—91no—
227#385—91no—
228C4×C4⋊C4—91no—
229C4×C4⋊C4—91no—
230C4×C22⋊C4—91no—
231C4×C22⋊C4—91no—
232C24×C3C22×D314yes—
233C23×D3—34no—
234C22×C3×C4C22×D314yes—
235C22×C3×C4C22×D314yes—
236C22×C3×C4C22×D314yes—
237C22×C3×C4C22×D314yes—
238C22×C3×C4C22×D314yes—
239C22×C3×C4C22×D314yes—
240C2×#67—316no—
241C2×#67—68no—
242C2×#67—68no—
243C22×Dic3—34no—
244C22×Dic3—34no—
245C22×Dic3—34no—
246C22×Dic3—34no—
247C22×Dic3—34no—
248C22×Dic3—34no—
249C22×Dic3—34no—
250C22×Dic3—68no—
251C22×Dic3—68no—
252C22×Dic3—124no—
253C2×C4×D3—68no—
254C2×C4×D3—68no—
255C2×C4×D3—124no—
256C2×Dic6—316no—
257C2×Dic6—68no—
258C2×Dic6—68no—
259C3×C4⋊C4—28no—
260C3×C22⋊C4—28no—
261C3×C22⋊C4—44no—
262C3×C22⋊C4C4×D3116yescomplemented
263C3×C42—44no—
264#217—316no—
265#217—68no—
266(C2×D3)⋊C4—316no—
267(C2×D3)⋊C4—316no—
268(C2×D3)⋊C4—68no—
269(C2×D3)⋊C4—68no—
270(C2×D3)⋊C4—124no—
271Dic3⋊C4—316no—
272Dic3⋊C4—68no—
273Dic3⋊C4—68no—
274Dic3⋊C4—68no—
275Dic3⋊C4—68no—
276C4×Dic3—316no—
277C4×Dic3—68no—
278C4×Dic3—68no—
279C4×Dic3—68no—
280C4×Dic3—68no—
281C4×Dic3—124no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32C23×C411yescharacteristic
290C22×C32C22×D411yescharacteristic
291C22×C32C2×#4111yescharacteristic
292C22×C32C2×#4111yescharacteristic
293C22×C32C2×#4111yescharacteristic
294C22×C32H(5,𝔽2)11yescharacteristic
295C22×C322−1+411yescharacteristic
296C2×C3⋊D3—22no—
297C2×C3⋊D3—22no—
298C2×C3⋊D3—22no—
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3⋊D3—41no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—22no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—41no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—22no—
312C3⋊Dic3—22no—
313C3⋊Dic3—22no—
314C3⋊Dic3—41no—
315C3⋊Dic3C4×D414yes—
316C22×Q8—91no—
317C22×D4—91no—
318C23×C4—91no—
319C23×C4—91no—
320#124—98no—
321C4×Q8—184no—
322C4×D4—184no—
323C2×C4⋊C4—91no—
324C2×C4⋊C4—91no—
325C2×C4⋊C4—91no—
326C2×C4⋊C4—91no—
327C2×C4⋊C4—91no—
328C2×C4⋊C4C2×C3⋊D311yescharacteristic
329C2×C22⋊C4—91no—
330C2×C22⋊C4—91no—
331C2×C22⋊C4—91no—
332C2×C22⋊C4—91no—
333C2×C22⋊C4—91no—
334C2×C22⋊C4—91no—
335C2×C22⋊C4—91no—
336C2×C22⋊C4C2×C3⋊D311yescharacteristic
337C2×C22⋊C4C2×C3⋊D311yescharacteristic
338C2×C22⋊C4C2×C3⋊D311yescharacteristic
339C2×C42—91no—
340C2×C42—91no—
341C2×C42—91no—
342C2×C42—91no—
343C2×C42—91no—
344C2×C42—91no—
345C2×C42C2×C3⋊D311yescharacteristic
346#95—91no—
347#95—91no—
348#95—91no—
349#95—91no—
350#95—91no—
351#95—91no—
352#95C2×C3⋊D311yescharacteristic
353#95C2×C3⋊D311yescharacteristic
354C23×C3—24no—
355C23×C3—24no—
356C23×C3—24no—
357C23×C3—24no—
358C23×C3—24no—
359C23×C3—24no—
360C23×C3C2×C4×D318yes—
361C23×C3C23×D314yes—
362C22×D3—38no—
363C22×D3—64no—
364C22×D3—64no—
365C22×D3—64no—
366C22×D3—64no—
367C22×D3—64no—
368C22×D3—64no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—24no—
384C2×C3×C4C2×C4×D318yes—
385C2×C3×C4C2×C4×D318yes—
386C2×C3×C4D3×D418yes—
387#67—632no—
388C2×Dic3—38no—
389C2×Dic3—38no—
390C2×Dic3—38no—
391C2×Dic3—38no—
392C2×Dic3—38no—
393C2×Dic3—38no—
394C2×Dic3—38no—
395C2×Dic3—38no—
396C2×Dic3—64no—
397C2×Dic3—64no—
398C2×Dic3—64no—
399C2×Dic3—64no—
400C2×Dic3—64no—
401C2×Dic3—64no—
402C2×Dic3—64no—
403C2×Dic3—64no—
404C2×Dic3—64no—
405C2×Dic3—64no—
406C2×Dic3—64no—
407C2×Dic3—64no—
408C2×Dic3—64no—
409C2×Dic3—1216no—
410C4×D3—1216no—
411Dic6—632no—
412C2×C32—42no—
413C2×C32C2×C4×D411yescharacteristic
414C2×C32C4×#4111yescharacteristic
415C2×C32#52011yescharacteristic
416C2×C32#54611yescharacteristic
417C2×C32#54811yescharacteristic
418C2×C32#55611yescharacteristic
419C2×C32#55911yescharacteristic
420C3⋊D3—42no—
421C24—91no—
422C24C22×C3⋊D311yescharacteristic
423C2×Q8—94no—
424C2×Q8—182no—
425C2×Q8—182no—
426C2×D4—94no—
427C2×D4—182no—
428C2×D4—182no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—91no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—91no—
435C22×C4—91no—
436C22×C4—182no—
437C22×C4—182no—
438C22×C4—182no—
439C22×C4—182no—
440C22×C4—361no—
441C22×C4—361no—
442C22×C4C22×C3⋊D311yescharacteristic
443C22×C4C22×C3⋊D311yescharacteristic
444C22×C4C22×C3⋊D311yescharacteristic
445C22×C4C22×C3⋊D311yescharacteristic
446C22×C4C22×C3⋊D311yescharacteristic
447C22×C4C22×C3⋊D311yescharacteristic
448C4⋊C4—22no—
449C4⋊C4—94no—
450C4⋊C4—182no—
451C4⋊C4—182no—
452C4⋊C4—182no—
453C4⋊C4—182no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—94no—
457C22⋊C4—94no—
458C22⋊C4—94no—
459C22⋊C4—182no—
460C22⋊C4—182no—
461C22⋊C4—182no—
462C22⋊C4—361no—
463C22⋊C4C4×C3⋊D314yescomplemented
464C42—41no—
465C42—94no—
466C42—182no—
467C42—182no—
468C42—182no—
469C42—182no—
470C42—361no—
471C22×C3—28no—
472C22×C3—28no—
473C22×C3—28no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3C22×C4×D314yes—
479C22×C3C2×D3×D414yes—
480C22×C3#100414yes—
481C22×C3D3×#4114yes—
482C22×C3D3×#4114yes—
483C22×C3D3×#4114yes—
484C22×C3#101014yes—
485C2×D3—68no—
486C2×D3—68no—
487C2×D3—68no—
488C2×D3—124no—
489C2×D3—124no—
490C2×D3—124no—
491C2×D3—124no—
492C3×C4—28no—
493C3×C4—28no—
494C3×C4—28no—
495C3×C4—44no—
496C3×C4—44no—
497C3×C4—44no—
498Dic3—316no—
499Dic3—68no—
500Dic3—68no—
501Dic3—68no—
502Dic3—68no—
503Dic3—68no—
504Dic3—124no—
505C32#236711yescharacteristic, complemented, Sylow
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—92no—
513C23—181no—
514C23—181no—
515C23—181no—
516C23—181no—
517C23—181no—
518C23—181no—
519C23C2×C4×C3⋊D312yes—
520C23C23×C3⋊D311yescharacteristic
521Q8—188no—
522D4—188no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—92no—
539C2×C4—92no—
540C2×C4—92no—
541C2×C4—92no—
542C2×C4—92no—
543C2×C4—92no—
544C2×C4—92no—
545C2×C4—92no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4—181no—
551C2×C4—181no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—181no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—364no—
560C2×C4—364no—
561C2×C4C2×C4×C3⋊D312yes—
562C2×C4C2×C4×C3⋊D312yes—
563C2×C4D4×C3⋊D312yes—
564C2×C3—48no—
565C2×C3#552214yes—
566C2×C3C4×D3×D414yes—
567C2×C3#554614yes—
568C2×C3#561014yes—
569C2×C3#566414yes—
570C2×C3#567714yes—
571C2×C3#570214yes—
572D3—128no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—182no—
581C22—182no—
582C22—182no—
583C22—361no—
584C22—361no—
585C22—361no—
586C22—361no—
587C22C22×C4×C3⋊D311yescharacteristic
588C22C2×D4×C3⋊D311yescharacteristic
589C22#6436111yescharacteristic
590C22#41×C3⋊D311yescharacteristic
591C22#41×C3⋊D311yescharacteristic
592C22#41×C3⋊D311yescharacteristic
593C22#6436711yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—94no—
601C4—182no—
602C4—182no—
603C4—182no—
604C4—182no—
605C4—182no—
606C4—361no—
607C3#7548114yescomplemented
608C2—42no—
609C2—362no—
610C2#1059599311yescharacteristic
611C2C4×D4×C3⋊D311yescharacteristic
612C2#1059601711yescharacteristic
613C2#1059608111yescharacteristic
614C2#1059613511yescharacteristic
615C2#1059614811yescharacteristic
616C2#1059617311yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×#388C2
socle, cosocleC23×C32C24
prime radicalC22×#638C22
hypercenter#388C3⋊D3
hypocenterC32#2367
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#64096C2semidirect
4. C32×#388C2semidirect
5. #10592688C2semidirect
7. #10592645C2semidirect
8. #10592643C2semidirect
9. C22⋊C4×C3⋊Dic3C2semidirect
10. C4×#3752C2semidirect
11. C4×#3750C2semidirect
12. #10592368C2semidirect
13. #10592368C2semidirect
14. #10592348C2semidirect
15. #10592345C2semidirect
45. #64094C4semidirect
58. C3×#388D3semidirect
96. C2×#634C2×C4semidirect
100. C32×C22⋊C4C2×C4semidirect
103. #3757C2×C4semidirect
107. #3752C2×C4semidirect
108. #3752C2×C4semidirect
119. C4×C3⋊Dic3C2×C4semidirect
225. #388C3⋊D3semidirect
262. C3×C22⋊C4C4×D3semidirect
463. C22⋊C4C4×C3⋊D3semidirect
505. C32#2367semidirect
607. C3#75481semidirect

Maximal subgroups

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

subgroupindex
C2×#640962
C2×#640942
C2×#640902
C32×#3882
#105926882
#105926562
#105926452
#105926432
C22⋊C4×C3⋊Dic32
C4×#37522
C4×#37502
#105923682
#105923482
#105923452
#754813