Finite Group Databasebeta

← #11872229

Subgroups — #11872229

Order 1152.

683 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 683 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594222C211yesmaximal, characteristic, complemented
2#10594181C211yesmaximal, characteristic, complemented
3#10593744C211yesmaximal, characteristic, complemented
4#10593523C211yesmaximal, characteristic, complemented
5#10593519C211yesmaximal, characteristic
6#10593501C211yesmaximal, characteristic, complemented
7#10593494C211yesmaximal, characteristic, complemented
8#10593433C211yesmaximal, characteristic, complemented
9#10593431C211yesmaximal, characteristic, complemented
10#10593428C211yesmaximal, characteristic, complemented
11#10593422C211yesmaximal, characteristic, complemented
12#10593419C211yesmaximal, characteristic, complemented
13C4×#3774C211yesmaximal, characteristic, complemented
14#10593367C211yesmaximal, characteristic, complemented
15#10593359C211yesmaximal, characteristic, complemented
16C22×#625C2211yescharacteristic
17C2×#3775C2211yescharacteristic
18C2×#3775C2211yescharacteristic
19C2×#3774C2211yescharacteristic
20C2×#3774C2211yescharacteristic
21#64086C2211yescharacteristic
22C3×#883—21no—
23C3×D3×C22⋊C4—21no—
24#63982C2211yescharacteristic
25#63980C2211yescharacteristic
26D3×#217—21no—
27#63959—21no—
28#63957C2211yescharacteristic
29#63919C2211yescharacteristic, complemented
30#63915C2211yescharacteristic, complemented
31#63902C2211yescharacteristic
32#63902C2211yescharacteristic
33C4×#625C2211yescharacteristic
34C4×#625C2211yescharacteristic
35C4×#625C2211yescharacteristic, complemented
36#63879—21no—
37#63879—21no—
38#63858—21no—
39#63851—21no—
40#63838C2211yescharacteristic, complemented
41C22⋊#622C2211yescharacteristic
42C22⋊#622C2211yescharacteristic
43#63779C2211yescharacteristic, complemented
44#63777C2211yescharacteristic, complemented
45C4⋊#622C2211yescharacteristic, complemented
46C4×#622C2211yescharacteristic, complemented
47#63743C2211yescharacteristic, complemented
48#63743C2211yescharacteristic, complemented
49#63742C2211yescharacteristic, complemented
50#63742C2211yescharacteristic, complemented
51#63740C2211yescharacteristic, complemented
52#63740C2211yescharacteristic, complemented
53#63739C2211yescharacteristic
54#63739C2211yescharacteristic
55#63739C2211yescharacteristic
56#63739C2211yescharacteristic
57#63739C2211yescharacteristic, complemented
58#63739C2211yescharacteristic, complemented
59#5662—61no—
60#5486—61no—
61C23×C3×D3—21no—
62C22×C3⋊Dic3C2311yescharacteristic
63C2×C3×#67—21no—
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3—21no—
66C2×C3×C4×D3—21no—
67C2×#627—21no—
68C2×#627C2311yescharacteristic
69C2×#625—21no—
70C2×#625—22no—
71C2×#625—22no—
72C2×#625C2311yescharacteristic
73C2×#625C2311yescharacteristic
74C2×#625C2311yescharacteristic
75C2×D3×Dic3—21no—
76C2×D3×Dic3—21no—
77C2×#622C2311yescharacteristic
78C2×#622C2311yescharacteristic
79C2×#622C2311yescharacteristic
80C2×#622C2311yescharacteristic
81#3775—22no—
82#3775—22no—
83#3774—22no—
84#3774—22no—
85C32×C22⋊C4D411yescharacteristic
86#3757D411yescharacteristic
87#3750D411yescharacteristic
88#3750D411yescharacteristic
89C4×C3⋊Dic3C2311yescharacteristic
90C4×C3⋊Dic3C2311yescharacteristic
91C3×#217—21no—
92C3×#217—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×(C2×D3)⋊C4—21no—
95C3×(C2×D3)⋊C4—21no—
96C3×C4⋊Dic3—21no—
97C3×Dic3⋊C4—21no—
98#3724C2311yescharacteristic
99#3724C2311yescharacteristic
100#3724C2311yescharacteristic
101#3724C2311yescharacteristic
102Dic3⋊Dic3—21no—
103(C2×D3)⋊Dic3—21no—
104(C2×D3)⋊Dic3—21no—
105(C2×D3)⋊Dic3—21no—
106(C2×D3)⋊Dic3—21no—
107(C2×D3)⋊Dic3—21no—
108Dic32—21no—
109Dic32—21no—
110#3118—91nomaximal, Sylow
111C22×#67—61no—
112C3×#123—61no—
113C2×C3×C22⋊C4—61no—
114C2×#217—61no—
115#939—61no—
116#936—61no—
117C4×#67—61no—
118C4×#67—61no—
119C4×#67—61no—
120C4×#67—61no—
121C4×#67—61no—
122#924—61no—
123#898—61no—
124#896—61no—
125#888—61no—
126#887—61no—
127#885—61no—
128#885—61no—
129#883—21no—
130#67⋊C4—61no—
131#67⋊C4—61no—
132#67⋊C4—61no—
133#67⋊C4—61no—
134D3×C22⋊C4—21no—
135D3×C22⋊C4—61no—
136#879—61no—
137#878—61no—
138#877—61no—
139#877—61no—
140#877—61no—
141C23×C32C2×D411yescharacteristic
142C22×C3×D3—21no—
143C22×C3×D3—21no—
144C22×C3×D3—24no—
145C22×C3×D3—21no—
146C22×C3×D3—81no—
147C22×C3×D3—81no—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3C2×D411yescharacteristic
153C2×C3⋊Dic3C2×D411yescharacteristic
154C2×C3⋊Dic3C2×D411yescharacteristic
155C2×C3⋊Dic3C2411yescharacteristic
156C3×#67—81no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C3×C4×D3—81no—
163C3×C4×D3—81no—
164C3×C4×D3—81no—
165#627—21no—
166#627—21no—
167#627—81no—
168#625—24no—
169#625—21no—
170#625—21no—
171#625—21no—
172#625—21no—
173#625#4112yes—
174#625#4112yes—
175D3×Dic3—81no—
176D3×Dic3—81no—
177#622—21no—
178#622—21no—
179#622—21no—
180#622—21no—
181#542—91no—
182#518—91no—
183#485—91no—
184#482—91no—
185#476—91no—
186#474—91no—
187#463—91no—
188#463—91no—
189#450—91no—
190#447—91no—
191#442—91no—
192#440—91no—
193C4×SD16—91no—
194C4×D8—91no—
195#433—91no—
196C24×C3—61no—
197C23×D3—21no—
198C2×C3×D4—61no—
199C22×C3×C4—61no—
200C22×C3×C4—61no—
201C22×C3×C4—61no—
202C2×#67—21no—
203C2×#67—61no—
204C2×#67—61no—
205C2×#67—61no—
206C2×#67—61no—
207C2×#67—61no—
208C2×#67—64no—
209C2×#67—64no—
210C22×Dic3—61no—
211C22×Dic3—61no—
212C22×Dic3—61no—
213C22×Dic3—61no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—61no—
218C2×C4×D3—61no—
219C2×Dic6—61no—
220C2×Dic6—61no—
221C2×Dic6—61no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—61no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—61no—
227C3×C22⋊C4—61no—
228C3×C22⋊C4—61no—
229C3×C22⋊C4—61no—
230C3×C22⋊C4—61no—
231#217—21no—
232#217—21no—
233#217—61no—
234#217—61no—
235#217—61no—
236#217—61no—
237#217—61no—
238#217—61no—
239#217—61no—
240(C2×D3)⋊C4—21no—
241(C2×D3)⋊C4—21no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—61no—
244(C2×D3)⋊C4—61no—
245(C2×D3)⋊C4—61no—
246(C2×D3)⋊C4—61no—
247(C2×D3)⋊C4—61no—
248C4⋊Dic3—21no—
249C4⋊Dic3—61no—
250Dic3⋊C4—21no—
251Dic3⋊C4—61no—
252Dic3⋊C4—61no—
253Dic3⋊C4—61no—
254Dic3⋊C4—61no—
255Dic3⋊C4—61no—
256Dic3⋊C4—61no—
257Dic3⋊C4—61no—
258Dic3⋊C4—61no—
259Dic3⋊C4—61no—
260C4×Dic3—61no—
261C4×Dic3—61no—
262C4×Dic3—61no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3—61no—
266C4×Dic3—61no—
267C4×Dic3—61no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32C22×D411yescharacteristic
272C2×C3×D3—22no—
273C2×C3×D3—22no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C2×C3×D3—84no—
282C32×C4—21no—
283C32×C4—21no—
284C3⋊Dic3—21no—
285C3⋊Dic3—21no—
286C3⋊Dic3—21no—
287C3⋊Dic3C2×#4111yescharacteristic
288C3⋊Dic3C2×#4111yescharacteristic
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C22×D4—91no—
295C2×SD16—91no—
296C2×SD16—91no—
297C2×D8—91no—
298C2×D8—91no—
299C4⋊Q8—91no—
300#126—181no—
301#124—91no—
302#124—181no—
303#123—61no—
304C22⋊Q8—91no—
305C22⋊Q8—181no—
306#121—91no—
307C4×Q8—91no—
308C4×D4—91no—
309C4×D4—91no—
310C4×D4—91no—
311C4×D4—91no—
312C4×D4—91no—
313C4×D4—181no—
314C4×D4—181no—
315#117—91no—
316#117—91no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—181no—
319#107—91no—
320#107—91no—
321#106—91no—
322#106—91no—
323C4⋊C8—91no—
324C4⋊C8—91no—
325#103—91no—
326#103—91no—
327#102—91no—
328#102—91no—
329#102—91no—
330#102—91no—
331#102—91no—
332#102—91no—
333C22⋊C8—91no—
334C22⋊C8—91no—
335#97—91no—
336C4×C8—91no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—64no—
342C23×C3—61no—
343C23×C3—241no—
344C23×C3—241no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—24no—
348C22×D3—21no—
349C22×D3—81no—
350C22×D3—81no—
351C3×D4—241no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—241no—
362C2×C3×C4—241no—
363C2×C3×C4—241no—
364#67—62no—
365#67—62no—
366#67—62no—
367#67—62no—
368#67—64no—
369#67—64no—
370#67—81no—
371#67—121no—
372#67—121no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—121no—
389C2×Dic3—121no—
390C2×Dic3—241no—
391C2×Dic3—241no—
392C4×D3—81no—
393C4×D3—81no—
394C4×D3—81no—
395C4×D3—241no—
396C4×D3—241no—
397Dic6—121no—
398Dic6—121no—
399Dic6—241no—
400Dic6—241no—
401C2×C32—41no—
402C2×C32#52511yescharacteristic
403C2×C32#57611yescharacteristic
404C2×C32#57711yescharacteristic
405C3×D3—42no—
406C3×D3—42no—
407C3×D3—81no—
408C24—61no—
409C2×Q8—91no—
410C2×Q8—181no—
411C2×D4—61no—
412C2×D4—91no—
413C2×D4—91no—
414C2×D4—91no—
415C2×D4—181no—
416C2×D4—182no—
417C2×D4—182no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—91no—
422C22×C4—181no—
423C22×C4—181no—
424SD16—182no—
425SD16—182no—
426D8—182no—
427D8—182no—
428C2×C8—91no—
429C2×C8—91no—
430C2×C8—91no—
431C2×C8—91no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—91no—
435C4⋊C4—91no—
436C4⋊C4—91no—
437C4⋊C4—91no—
438C4⋊C4—91no—
439C4⋊C4—91no—
440C4⋊C4—181no—
441C22⋊C4—61no—
442C22⋊C4—61no—
443C22⋊C4—61no—
444C22⋊C4—61no—
445C22⋊C4—61no—
446C22⋊C4—91no—
447C22⋊C4—181no—
448C22⋊C4—181no—
449C22⋊C4—181no—
450C22⋊C4—181no—
451C22⋊C4—181no—
452C22⋊C4GO+(4,𝔽2)11yescharacteristic
453C42—91no—
454C42—91no—
455C42—181no—
456C42—181no—
457C22×C3—22no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—121no—
468C22×C3—121no—
469C22×C3—121no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—244no—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—84no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—121no—
494Dic3—41no—
495Dic3—41no—
496Dic3—41no—
497Dic3—41no—
498Dic3—41no—
499Dic3—61no—
500Dic3—61no—
501Dic3—61no—
502Dic3—61no—
503Dic3—62no—
504Dic3—121no—
505Dic3—121no—
506Dic3—121no—
507Dic3—121no—
508Dic3—121no—
509C32#311811yescharacteristic, complemented, Sylow
510C23—64no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—241no—
515C23—241no—
516C23C2×GO+(4,𝔽2)11yescharacteristic
517Q8—181no—
518Q8—181no—
519Q8—721no—
520D4—92no—
521D4—92no—
522D4—181no—
523D4—181no—
524D4—181no—
525D4—181no—
526D4—184no—
527D4—241no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—91no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—241no—
540C2×C4—241no—
541C2×C4—241no—
542C2×C4—721no—
543C2×C4—721no—
544C2×C4C2×GO+(4,𝔽2)11yescharacteristic
545C2×C4C2×GO+(4,𝔽2)11yescharacteristic
546C8—181no—
547C8—181no—
548C8—181no—
549C8—181no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—42no—
557C2×C3—81no—
558C2×C3—122no—
559C2×C3—122no—
560C2×C3—241no—
561D3—42no—
562D3—42no—
563D3—81no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—62no—
568C22—62no—
569C22—121no—
570C22—121no—
571C22—121no—
572C22—121no—
573C22—121no—
574C22—121no—
575C22—121no—
576C22—244no—
577C22C22×GO+(4,𝔽2)11yescharacteristic
578C4—21no—
579C4—21no—
580C4—91no—
581C4—91no—
582C4—121no—
583C4—121no—
584C4—121no—
585C4—121no—
586C4—121no—
587C4—181no—
588C4—181no—
589C4—181no—
590C3—21no—
591C3—21no—
592C2—41no—
593C2—122no—
594C2—122no—
595C2—241no—
596C2#1059651011yescharacteristic
597C2#1059651411yescharacteristic
598C2#1059651511yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C24
FittingC32×C22⋊C4D4
socleC22×C32C22×D4
prime radical#10593519C2
hypercenterC22⋊C4GO+(4,𝔽2)
hypocenterC32#3118
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #10594222C2semidirect
2. #10594181C2semidirect
3. #10593744C2semidirect
4. #10593523C2semidirect
6. #10593501C2semidirect
7. #10593494C2semidirect
8. #10593433C2semidirect
9. #10593431C2semidirect
10. #10593428C2semidirect
11. #10593422C2semidirect
12. #10593419C2semidirect
13. C4×#3774C2semidirect
14. #10593367C2semidirect
15. #10593359C2semidirect
29. #63919C22semidirect
30. #63915C22semidirect
35. C4×#625C22semidirect
40. #63838C22semidirect
43. #63779C22semidirect
44. #63777C22semidirect
45. C4⋊#622C22semidirect
46. C4×#622C22semidirect
47. #63743C22semidirect
48. #63743C22semidirect
49. #63742C22semidirect
50. #63742C22semidirect
51. #63740C22semidirect
52. #63740C22semidirect
57. #63739C22semidirect
58. #63739C22semidirect
509. C32#3118semidirect

Maximal subgroups

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

subgroupindex
#105942222
#105941812
#105937442
#105935232
#105935192
#105935012
#105934942
#105934332
#105934312
#105934282
#105934222
#105934192
C4×#37742
#105933672
#105933592
#31189