Finite Group Databasebeta

← #11921985

Subgroups — #11921985

Order 1152.

712 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#64259C211yesmaximal, characteristic, complemented
2C2×Dic3×S4C211yesmaximal, characteristic
3A4×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C211yesmaximal, characteristic
5(C2×S4)⋊Dic3C211yesmaximal, characteristic
6SL(2,ℤ/4)⋊Dic3C211yesmaximal, characteristic, complemented
7#10593238C211yesmaximal, characteristic, complemented
8#85311—31nomaximal
9#74709—31nomaximal
10C22×C3×S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×Dic3×A4C2211yescharacteristic
13C2×Dic3×A4C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16#64259—22no—
17C2×C3×SL(2,ℤ/4)C2211yescharacteristic
18Dic3×S4—22no—
19#63896—41nomaximal
20C2×C4×S4—31no—
21C2×A4⋊Q8—31no—
22C2×A4⋊Q8—61no—
23C2×D4×Dic3—31no—
24C2×#924—31no—
25C22×Dic3⋊C4D311yescharacteristic, complemented
26C2×#878—31no—
27A4×C4⋊C4—31no—
28(C2×S4)⋊C4—31no—
29(C2×S4)⋊C4D311yescharacteristic
30SL(2,ℤ/4)⋊C4—31no—
31SL(2,ℤ/4)⋊C4—31no—
32C3×#386—31no—
33#5220—31no—
34#4950—31no—
35#4949—31no—
36#4948—31no—
37#4946—31no—
38#4946—31no—
39#4657—31no—
40#4656—31no—
41#4655—31no—
42#4647—31no—
43C22×C3×A4C2311yescharacteristic
44C2×C3×S4—21no—
45C2×C3×S4—21no—
46C2×C3×S4Q812yescomplemented
47C2×#634—41no—
48C3×C4×A4—21no—
49C2×D3×Dic3—41no—
50Dic3×A4—21no—
51Dic3×A4—21no—
52C3⋊SL(2,ℤ/4)—21no—
53C3⋊SL(2,ℤ/4)D412yes—
54C3×SL(2,ℤ/4)—21no—
55C3×(C2×D3)⋊C4—41no—
56C3×Dic3⋊C4—41no—
57#3724—41no—
58Dic3⋊Dic3—41no—
59(C2×D3)⋊Dic3—41no—
60#2449—91noSylow
61C22×S4C2×D311yescharacteristic
62C22×C3×D4—31no—
63C23×C3×C4C2×D311yescharacteristic
64C23×Dic3—31no—
65C23×Dic3C2×D311yescharacteristic
66C23×Dic3C2×D311yescharacteristic
67C22×Dic6—31no—
68C2×C4×A4—21no—
69C2×C4×A4—31no—
70C2×C4×A4—31no—
71C2×C4×A4C2×D311yescharacteristic
72C2×SL(2,ℤ/4)—31no—
73C2×SL(2,ℤ/4)—31no—
74C2×SL(2,ℤ/4)—61no—
75C2×SL(2,ℤ/4)—61no—
76C2×SL(2,ℤ/4)C2×D311yescharacteristic
77C4×S4—62no—
78A4⋊Q8—62no—
79A4⋊Q8—64no—
80C2×C3×C22⋊C4—31no—
81C2×C3×C22⋊C4—31no—
82C2×C3×C22⋊C4—31no—
83C2×#217—31no—
84C2×#217—31no—
85C2×#217—31no—
86C2×#217—31no—
87C2×#217—31no—
88D4×Dic3—62no—
89D4×Dic3—62no—
90C2×C4⋊Dic3—31no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—31no—
94#924—62no—
95#924—62no—
96C2×Dic3⋊C4—31no—
97C2×Dic3⋊C4—31no—
98C2×Dic3⋊C4—31no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—31no—
101C2×Dic3⋊C4—31no—
102C2×Dic3⋊C4—31no—
103C2×Dic3⋊C4—31no—
104C2×C4×Dic3—31no—
105#896—121no—
106#878—61no—
107#878—61no—
108#878—61no—
109#878—61no—
110C3×#95—31no—
111C3×#95—31no—
112C3⋊#95—31no—
113C3⋊#95—31no—
114C3⋊#95—31no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C3⋊#95—31no—
118C22×C3×D3—41no—
119C2×C3×A4C2×D411yescharacteristic
120C2×C3×A4C2×Q811yescharacteristic
121C2×C3×A4#4111yescharacteristic
122C3×S4—22no—
123C2×C32×C4—41no—
124C2×C3⋊Dic3—41no—
125C2×C3⋊Dic3—41no—
126#634—82no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130D3×Dic3—82no—
131C2×C22⋊Q8—91no—
132C2×C22⋊Q8—91no—
133C2×C4×D4—91no—
134C22×C4⋊C4—31no—
135#399—91no—
136#398—91no—
137#396—91no—
138#395—91no—
139#389—91no—
140#387—91no—
141#386—31no—
142#386—91no—
143#385—91no—
144#385—91no—
145#385—91no—
146C24×C3—31no—
147C24×C3C22×D311yescharacteristic
148C22×A4—21no—
149C22×A4C22×D311yescharacteristic
150C2×S4—21no—
151C2×S4—21no—
152C2×S4Dic612yescomplemented
153C2×C3×D4—32no—
154C2×C3×D4—32no—
155C2×C3×D4—61no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C22×C3×C4—21no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—61no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—31no—
169C22×Dic3—31no—
170C22×Dic3—31no—
171C22×Dic3—31no—
172C22×Dic3—31no—
173C22×Dic3—31no—
174C22×Dic3—31no—
175C22×Dic3—31no—
176C22×Dic3—61no—
177C22×Dic3—61no—
178C22×Dic3—62no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181C22×Dic3—121no—
182C2×C4×D3—121no—
183C2×Dic6—62no—
184C2×Dic6—61no—
185C2×Dic6—61no—
186C2×Dic6—121no—
187C2×Dic6—121no—
188C2×Dic6—241no—
189C4×A4—22no—
190C4×A4—21no—
191C4×A4—61no—
192C4×A4—61no—
193SL(2,ℤ/4)—21no—
194SL(2,ℤ/4)—32no—
195SL(2,ℤ/4)—62no—
196SL(2,ℤ/4)—62no—
197SL(2,ℤ/4)—61no—
198C3×C4⋊C4—121no—
199C3×C22⋊C4—61no—
200C3×C22⋊C4—61no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—121no—
204#217—32no—
205#217—32no—
206#217—61no—
207#217—62no—
208#217—62no—
209#217—61no—
210#217—61no—
211#217—61no—
212(C2×D3)⋊C4—41no—
213(C2×D3)⋊C4—121no—
214C4⋊Dic3—61no—
215C4⋊Dic3—62no—
216C4⋊Dic3—62no—
217C4⋊Dic3—61no—
218C4⋊Dic3—121no—
219Dic3⋊C4—31no—
220Dic3⋊C4—31no—
221Dic3⋊C4—31no—
222Dic3⋊C4—61no—
223Dic3⋊C4—61no—
224Dic3⋊C4—62no—
225Dic3⋊C4—62no—
226Dic3⋊C4—62no—
227Dic3⋊C4—61no—
228Dic3⋊C4—121no—
229Dic3⋊C4—121no—
230Dic3⋊C4S411yescharacteristic, complemented
231C4×Dic3—62no—
232C22×C32—41no—
233C2×C3×D3—42no—
234C2×C3×D3—81no—
235C2×C3×D3—81no—
236C3×A4C22⋊Q811yescharacteristic, complemented
237C32×C4—81no—
238C3⋊Dic3—42no—
239C3⋊Dic3—81no—
240C3×Dic3—81no—
241C3×Dic3—81no—
242C3×Dic3—81no—
243C22×Q8—91no—
244C22×D4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—91no—
248C23×C4D3211yescharacteristic
249C22⋊Q8—181no—
250C22⋊Q8—181no—
251C22⋊Q8—181no—
252C22⋊Q8—181no—
253C22⋊Q8—182no—
254C22⋊Q8—182no—
255C4×D4—182no—
256C4×D4—182no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—91no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—91no—
262C2×C4⋊C4—91no—
263C2×C4⋊C4—91no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C22⋊C4—31no—
270C2×C22⋊C4—31no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—91no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—91no—
276C2×C22⋊C4—91no—
277C2×C42—91no—
278#95—31no—
279#95—31no—
280#95—91no—
281#95—91no—
282#95—91no—
283#95—91no—
284#95—91no—
285#95—91no—
286C23×C3—31no—
287C23×C3—31no—
288C23×C3—31no—
289C23×C3—31no—
290C23×C3—32no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3#23511yescharacteristic
298C23×C3D3×D411yescharacteristic
299C23×C3D3×Q811yescharacteristic
300C22×D3—41no—
301C2×A4—21no—
302C2×A4—21no—
303C2×A4—21no—
304C2×A4C2×Dic611yescharacteristic
305C2×A4D3×D411yescharacteristic
306C2×A4#23711yescharacteristic
307S4—22no—
308C3×D4—62no—
309C3×D4—62no—
310C3×D4—62no—
311C3×D4—62no—
312C2×C3×C4—32no—
313C2×C3×C4—31no—
314C2×C3×C4—41no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—81no—
328C2×C3×C4—121no—
329C2×C3×C4—121no—
330C2×C3×C4C2×S411yescharacteristic
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—31no—
335C2×Dic3—31no—
336C2×Dic3—41no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—121no—
355C2×Dic3—121no—
356C2×Dic3—122no—
357C2×Dic3—122no—
358C2×Dic3—241no—
359C2×Dic3—241no—
360C2×Dic3C2×S411yescharacteristic
361C2×Dic3C2×S411yescharacteristic
362C4×D3—242no—
363Dic6—122no—
364Dic6—122no—
365Dic6—242no—
366Dic6—244no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C3×D3—82no—
371C24—31no—
372C24C2×D3211yescharacteristic
373C2×Q8—181no—
374C2×Q8—182no—
375C2×Q8—181no—
376C2×Q8—361no—
377C2×D4—32no—
378C2×D4—32no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C22×C4—21no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—91no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—91no—
398C22×C4—91no—
399C22×C4—91no—
400C22×C4—181no—
401C22×C4—181no—
402C22×C4—182no—
403C22×C4—181no—
404C22×C4—181no—
405C22×C4—361no—
406C4⋊C4—31no—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4—181no—
411C4⋊C4—181no—
412C4⋊C4—181no—
413C4⋊C4—182no—
414C4⋊C4—182no—
415C4⋊C4—182no—
416C4⋊C4—181no—
417C4⋊C4—182no—
418C4⋊C4—181no—
419C4⋊C4—182no—
420C4⋊C4—361no—
421C22⋊C4—61no—
422C22⋊C4—61no—
423C22⋊C4—61no—
424C22⋊C4—61no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4—121no—
428C22⋊C4—181no—
429C22⋊C4—182no—
430C22⋊C4—181no—
431C22⋊C4—182no—
432C22⋊C4—181no—
433C22⋊C4—181no—
434C42—182no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—41no—
445C22×C3—61no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3—81no—
450C22×C3—121no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3—121no—
454C22×C3#89611yescharacteristic, complemented
455C22×C3C22×S411yescharacteristic
456C2×D3—42no—
457C2×D3—81no—
458C2×D3—81no—
459A4—21no—
460A4#87811yescharacteristic, complemented
461C3×C4—21no—
462C3×C4—62no—
463C3×C4—61no—
464C3×C4—81no—
465C3×C4—82no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—241no—
469C3×C4—241no—
470Dic3—21no—
471Dic3—21no—
472Dic3—62no—
473Dic3—61no—
474Dic3—62no—
475Dic3—61no—
476Dic3—81no—
477Dic3—122no—
478Dic3—121no—
479Dic3—121no—
480Dic3—242no—
481Dic3—241no—
482Dic3—242no—
483C32—41noSylow
484C23—31no—
485C23—31no—
486C23—31no—
487C23—31no—
488C23—32no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23D3×Dic611yescharacteristic
496C23#379511yescharacteristic
497C23#381111yescharacteristic
498Q8—362no—
499Q8—362no—
500D4—62no—
501D4—62no—
502D4—62no—
503D4—62no—
504C2×C4—32no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—91no—
524C2×C4—91no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C4—181no—
532C2×C4—181no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4—362no—
543C2×C4—362no—
544C2×C4D3×S411yescharacteristic
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—41no—
552C2×C3—81no—
553C2×C3—81no—
554C2×C3—81no—
555C2×C3—122no—
556C2×C3#591211yescharacteristic
557C2×C3D4×S411yescharacteristic
558C2×C3Q8×S411yescharacteristic
559D3—82no—
560C22—31no—
561C22—31no—
562C22—31no—
563C22—31no—
564C22—31no—
565C22—31no—
566C22—31no—
567C22—31no—
568C22—31no—
569C22—61no—
570C22—62no—
571C22—62no—
572C22—62no—
573C22—121no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22#6389611yescharacteristic, complemented
578C22C2×D3×S411yescharacteristic
579C4—21no—
580C4—62no—
581C4—61no—
582C4—61no—
583C4—61no—
584C4—121no—
585C4—121no—
586C4—182no—
587C4—181no—
588C4—181no—
589C4—182no—
590C4—361no—
591C4—361no—
592C3—41no—
593C3—81no—
594C3#8531111yescharacteristic, complemented
595C2—31no—
596C2—31no—
597C2—31no—
598C2—31no—
599C2—122no—
600C2Dic6×S411yescharacteristic
601C2#1059642711yescharacteristic
602C2#1059647311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC2×C4D3×S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#64259C2semidirect
3. A4×Dic3⋊C4C2semidirect
6. SL(2,ℤ/4)⋊Dic3C2semidirect
7. #10593238C2semidirect
25. C22×Dic3⋊C4D3semidirect
46. C2×C3×S4Q8semidirect
152. C2×S4Dic6semidirect
230. Dic3⋊C4S4semidirect
236. C3×A4C22⋊Q8semidirect
454. C22×C3#896semidirect
460. A4#878semidirect
577. C22#63896semidirect
594. C3#85311semidirect

Maximal subgroups

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

subgroupindex
C2×#642592
C2×Dic3×S42
A4×Dic3⋊C42
C3×(C2×S4)⋊C42
(C2×S4)⋊Dic32
SL(2,ℤ/4)⋊Dic32
#105932382
#853113
#747093
#638964