Finite Group Databasebeta

← #11778190

Subgroups — #11778190

Order 1152.

734 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3×#5644C211yesmaximal, characteristic, complemented
2#10592745C211yesmaximal, characteristic, complemented
3#10591395C211yesmaximal, characteristic, complemented
4#10591168C211yesmaximal, characteristic, complemented
5C3×#4538C211yesmaximal, characteristic, complemented
6#10589149C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic, complemented
8#71884—31nomaximal
9#68317—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×#626—21no—
12C32×C22⋊Q8C2211yescharacteristic
13C2×#3755—21no—
14C3×#947—21no—
15C3×#930C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4C2211yescharacteristic
17C3×#898—21no—
18C3×#895—21no—
19C3×#894—21no—
20C3×#882—21no—
21C3×#67⋊C4—21no—
22C2×#3722—21no—
23C2×#3714—21no—
24#63657C2211yescharacteristic
25#63636—21no—
26C3×C3⋊#95C2211yescharacteristic
27C3×#830C2211yescharacteristic
28#63564—21no—
29C3×#538—31no—
30#5644D311yescharacteristic, complemented
31#5046—31no—
32#5046—31no—
33#5046—61no—
34#5036—31no—
35#4960—31no—
36#4732—31no—
37#4671—31no—
38C3×#328—31no—
39#4538D311yescharacteristic, complemented
40#4537—31no—
41#4525—31no—
42#4468—31no—
43#4451—31no—
44C23×C3⋊D3—21no—
45C23×C3×D3D411yescharacteristic
46C2×C32×Q8D411yescharacteristic, complemented
47C22×C32×C4C2311yescharacteristic
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—42no—
51C2×C3×#67—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3D411yescharacteristic
54C2×C3×D12—21no—
55C2×C3×D12—21no—
56C2×C3×D12D411yescharacteristic, complemented
57C2×C3×C4×D3—21no—
58C2×#626—42no—
59C2×#626—42no—
60C2×#626—42no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4D411yescharacteristic, complemented
63C32×C22⋊C4—21no—
64#3755—42no—
65C2×C3⋊(C3⋊C8)—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×(C2×D3)⋊C4—21no—
71C3×(C2×D3)⋊C4—21no—
72C3×C4⋊Dic3D411yescharacteristic, complemented
73C3×Dic3⋊C4—21no—
74C3×C4×Dic3—21no—
75C2×C3×C3⋊C8—21no—
76#3722—42no—
77#3714—42no—
78#1600—91noSylow
79C22×#67—61no—
80C22×D12—31no—
81C22×D12—31no—
82C22×D12—61no—
83C22×D12—61no—
84C3×#126—61no—
85C3×#124—61no—
86C3×#123—61no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8C2×D311yescharacteristic
89C3×C22⋊Q8C2×D311yescharacteristic
90C3×#121—31no—
91C3×#121—61no—
92C3×#120—61no—
93C3×C4×D4—61no—
94C2×C3×C22⋊C4—31no—
95#947—21no—
96C2×#215—62no—
97C2×#215—61no—
98C2×#215—61no—
99C2×#213—61no—
100#930C2×D311yescharacteristic
101C2×(C2×D3)⋊C4—61no—
102C2×(C2×D3)⋊C4—61no—
103C2×(C2×D3)⋊C4C2×D311yescharacteristic
104C2×D24—61no—
105#898—21no—
106#895—21no—
107#894—21no—
108#882—21no—
109#67⋊C4—21no—
110C3×C22⋊C8—31no—
111C3×#95—31no—
112C3⋊#95C2×D311yescharacteristic
113#830—31no—
114#830—31no—
115#830—61no—
116#830C2×D311yescharacteristic
117#821—61no—
118#809—61no—
119#809—62no—
120#809—61no—
121#809—61no—
122C23×C32C2×D411yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—41no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—41no—
128C22×C3⋊D3—41no—
129C22×C3⋊D3—41no—
130C22×C3×D3—21no—
131C22×C3×D3—21no—
132C22×C3×D3—21no—
133C22×C3×D3—21no—
134C22×C3×D3—41no—
135C22×C3×D3—41no—
136C32×Q8—41no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C4⋊(C3⋊D3)—22no—
143C4⋊(C3⋊D3)—22no—
144C4⋊(C3⋊D3)—41no—
145C4⋊(C3⋊D3)—41no—
146C3×#67—41no—
147C3×#67—41no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—41no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3—41no—
154C3×D12—41no—
155C3×D12—81no—
156C3×D12—81no—
157C3×C4×D3—81no—
158#626—48no—
159C3⋊(C3⋊C8)—41no—
160C3×C3⋊C8—41no—
161#538—31no—
162#450—91no—
163#447—91no—
164#392—91no—
165#328—31no—
166#327—91no—
167#323—91no—
168C24×C3—31no—
169C23×D3—61no—
170C23×D3—61no—
171C23×D3—121no—
172C23×D3#6711yescharacteristic
173C2×C3×Q8—21no—
174C2×C3×Q8D1211yescharacteristic, complemented
175C2×C3×Q8#6711yescharacteristic, complemented
176C2×C3×D4—31no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C2×C3×D4—61no—
180C22×C3×C4—21no—
181C22×C3×C4—31no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C22×D311yescharacteristic
186C2×#67—21no—
187C2×#67—122no—
188C2×#67—122no—
189C2×#67—122no—
190C22×Dic3—21no—
191C22×Dic3#6711yescharacteristic
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—62no—
195C2×D12—61no—
196C2×D12—61no—
197C2×D12—61no—
198C2×D12—61no—
199C2×D12—121no—
200C2×D12—122no—
201C2×D12—122no—
202C2×D12—124no—
203C2×D12—122no—
204C2×D12—122no—
205C2×D12—122no—
206C2×D12#6711yescharacteristic, complemented
207C2×C4×D3—21no—
208C2×C3×C8—61no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4D1211yescharacteristic, complemented
216C3×C4⋊C4#6711yescharacteristic, complemented
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—41no—
220C3×C22⋊C4—61no—
221C3×C22⋊C4—61no—
222C3×C22⋊C4—61no—
223C3×C22⋊C4—61no—
224C3×C22⋊C4—61no—
225C3×C22⋊C4—61no—
226C3×C42—61no—
227#215—122no—
228#215—122no—
229#215—124no—
230#213—122no—
231(C2×D3)⋊C4—21no—
232(C2×D3)⋊C4—21no—
233(C2×D3)⋊C4—21no—
234(C2×D3)⋊C4—21no—
235(C2×D3)⋊C4—21no—
236(C2×D3)⋊C4—21no—
237(C2×D3)⋊C4—122no—
238(C2×D3)⋊C4—122no—
239C4⋊Dic3#6711yescharacteristic, complemented
240Dic3⋊C4—21no—
241C4×Dic3—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—62no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—61no—
246D24—122no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32#12011yescharacteristic
250C22×C32#13511yescharacteristic
251C22×C32AGL(1,ℤ/8)11yescharacteristic
252C2×C3⋊D3—42no—
253C2×C3⋊D3—42no—
254C2×C3⋊D3—42no—
255C2×C3⋊D3—81no—
256C2×C3⋊D3—81no—
257C2×C3⋊D3—81no—
258C2×C3⋊D3—81no—
259C2×C3×D3—42no—
260C2×C3×D3—41no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C2×C3×D3—41no—
264C2×C3×D3—41no—
265C2×C3×D3—41no—
266C2×C3×D3—81no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—41no—
270C32×C4—41no—
271C3×Dic3—42no—
272C3×Dic3—41no—
273C3×Dic3—41no—
274C22×D4—91no—
275C22×D4—181no—
276C2×SD16—181no—
277C2×D8—181no—
278#126—61no—
279#124—61no—
280#123—61no—
281C22⋊Q8D3211yescharacteristic
282#121—31no—
283#121—61no—
284#120—61no—
285C4×D4—61no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—181no—
288#102—181no—
289#102—181no—
290C22⋊C8—31no—
291C22⋊C8—91no—
292#95—31no—
293C23×C3—21no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—121no—
299C23×C3—121no—
300C23×C3D3×D411yescharacteristic
301C23×C3C2×#6711yescharacteristic
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—122no—
311C22×D3—121no—
312C22×D3—121no—
313C22×D3—122no—
314C22×D3—121no—
315C22×D3—121no—
316C22×D3—121no—
317C22×D3—121no—
318C22×D3—122no—
319C22×D3—121no—
320C22×D3—121no—
321C22×D3—121no—
322C22×D3—121no—
323C22×D3—121no—
324C22×D3—121no—
325C22×D3—241no—
326C22×D3—241no—
327C22×D3—241no—
328C22×D3—241no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×Q8—42no—
332C3×D4—121no—
333C3×D4—121no—
334C3×D4—121no—
335C3×D4—241no—
336C3×D4—241no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—62no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—121no—
352C2×C3×C4—121no—
353C2×C3×C4—121no—
354C2×C3×C4—241no—
355C2×C3×C4C2×D1211yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4C2×#6711yescharacteristic
358C2×C3×C4C2×#6711yescharacteristic
359#67—41no—
360#67—41no—
361#67—128no—
362C2×Dic3—22no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368D12—41no—
369D12—62no—
370D12—62no—
371D12—64no—
372D12—62no—
373D12—62no—
374D12—81no—
375D12—81no—
376D12—122no—
377D12—122no—
378D12—121no—
379D12—121no—
380D12—121no—
381D12—121no—
382D12—122no—
383D12—128no—
384C4×D3—81no—
385C3×C8—121no—
386C3⋊C8—41no—
387C3⋊C8—122no—
388C3⋊C8—121no—
389C3⋊C8—121no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32#44911yescharacteristic
393C2×C32#45311yescharacteristic
394C2×C32B(4,𝔽2)11yescharacteristic
395C3⋊D3—82no—
396C3×D3—81no—
397C3×D3—81no—
398C24—31no—
399C24—181no—
400C2×Q8#62611yescharacteristic, complemented
401C2×D4—31no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—181no—
406C2×D4—181no—
407C2×D4—362no—
408C2×D4—362no—
409C2×D4—362no—
410C2×D4—362no—
411C22×C4—31no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4C2×D3211yescharacteristic
415SD16—362no—
416D8—362no—
417C2×C8—61no—
418C2×C8—181no—
419C4⋊C4—21no—
420C4⋊C4—31no—
421C4⋊C4—61no—
422C4⋊C4#62611yescharacteristic, complemented
423C22⋊C4—21no—
424C22⋊C4—61no—
425C22⋊C4—61no—
426C22⋊C4—61no—
427C22⋊C4—61no—
428C22⋊C4—61no—
429C22⋊C4—61no—
430C22⋊C4—362no—
431C42—61no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—122no—
442C22×C3—121no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—241no—
449C22×C3#88211yescharacteristic
450C22×C3#91411yescharacteristic
451C22×C3#91911yescharacteristic
452C22×C3#94911yescharacteristic
453C22×C3#95011yescharacteristic
454C22×C3#95311yescharacteristic
455C2×D3—42no—
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—81no—
463C2×D3—124no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—242no—
471C2×D3—242no—
472C2×D3—241no—
473C2×D3—241no—
474C2×D3—241no—
475C2×D3—241no—
476C2×D3—242no—
477C2×D3—242no—
478C2×D3—242no—
479C2×D3—242no—
480C2×D3—241no—
481C2×D3—241no—
482C2×D3—241no—
483C2×D3—241no—
484C3×C4—22no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—122no—
497C3×C4—121no—
498C3×C4—121no—
499Dic3—42no—
500Dic3—41no—
501Dic3—41no—
502C32#160011yescharacteristic, complemented, Sylow
503C23—61no—
504C23—61no—
505C23—61no—
506C23—61no—
507C23—121no—
508C23—121no—
509C23—182no—
510C23—361no—
511C23—361no—
512C23—361no—
513C23—361no—
514C23—361no—
515C23—361no—
516C23D3×#6711yescharacteristic
517Q8—41no—
518D4—121no—
519D4—121no—
520D4—121no—
521D4—182no—
522D4—182no—
523D4—241no—
524D4—241no—
525D4—361no—
526D4—361no—
527D4—368no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—62no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—241no—
538C2×C4C2×#62611yescharacteristic
539C2×C4D3×#6711yescharacteristic
540C8—121no—
541C8—361no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—241no—
552C2×C3—241no—
553C2×C3#474111yescharacteristic
554C2×C3#481011yescharacteristic
555C2×C3#482211yescharacteristic
556C2×C3#523811yescharacteristic
557C2×C3#524111yescharacteristic
558C2×C3#524411yescharacteristic
559D3—81no—
560D3—81no—
561D3—244no—
562D3—242no—
563D3—242no—
564C22—21no—
565C22—21no—
566C22—122no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22—121no—
571C22—121no—
572C22—121no—
573C22—241no—
574C22—362no—
575C22—362no—
576C22—362no—
577C22—721no—
578C22—721no—
579C22—721no—
580C22—721no—
581C22#6393911yescharacteristic
582C22#6394911yescharacteristic
583C22#6397711yescharacteristic
584C4—21no—
585C4—21no—
586C4—41no—
587C4—41no—
588C4—122no—
589C4—121no—
590C4—121no—
591C3—21no—
592C3#6831711yescharacteristic, complemented
593C3#7188411yescharacteristic, complemented
594C2—21no—
595C2—21no—
596C2—241no—
597C2—241no—
598C2—722no—
599C2#1059153111yescharacteristic
600C2#1059158811yescharacteristic
601C2#1059166611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
hypercenterC22⋊Q8D32
hypocenterC32#1600
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5644C2semidirect
2. #10592745C2semidirect
3. #10591395C2semidirect
4. #10591168C2semidirect
5. C3×#4538C2semidirect
6. #10589149C2semidirect
7. #10589145C2semidirect
30. #5644D3semidirect
39. #4538D3semidirect
46. C2×C32×Q8D4semidirect
56. C2×C3×D12D4semidirect
62. C32×C4⋊C4D4semidirect
72. C3×C4⋊Dic3D4semidirect
174. C2×C3×Q8D12semidirect
175. C2×C3×Q8#67semidirect
206. C2×D12#67semidirect
215. C3×C4⋊C4D12semidirect
216. C3×C4⋊C4#67semidirect
239. C4⋊Dic3#67semidirect
400. C2×Q8#626semidirect
422. C4⋊C4#626semidirect
502. C32#1600semidirect
592. C3#68317semidirect
593. C3#71884semidirect

Maximal subgroups

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

subgroupindex
C3×#56442
#105927452
#105913952
#105911682
C3×#45382
#105891492
#105891452
#718843
#683173