Finite Group Databasebeta

← #11778027

Subgroups — #11778027

Order 1152.

672 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 88 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×#5634C211yesmaximal, characteristic, complemented
2C3×#5046C211yesmaximal, characteristic, complemented
3#10591389C211yesmaximal, characteristic
4#10591169C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589152C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic, complemented
8#71892—31nomaximal
9#70976—31nomaximal
10C22×C3×D12C2211yescharacteristic
11C22×#625—21no—
12C32×C22⋊Q8C2211yescharacteristic
13C3×#946—21no—
14C2×C3×#215—21no—
15C2×C3×(C2×D3)⋊C4C2211yescharacteristic
16C3×#924C2211yescharacteristic
17C3×#897—21no—
18C3×#896—21no—
19C3×#896—21no—
20C3×D3×C22⋊C4—21no—
21C3×#878—21no—
22C2×(C2×D3)⋊Dic3—21no—
23C2×#3715—21no—
24#63658C2211yescharacteristic
25#63657C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#809—21no—
28#63562—21no—
29C3×#543—31no—
30#5634D311yescharacteristic, complemented
31C3×#450—31no—
32#5223—31no—
33#5218—31no—
34#5046—31no—
35#5046D311yescharacteristic, complemented
36#4960—31no—
37#4538—31no—
38#4538—31no—
39#4538—31no—
40#4538—61no—
41#4525—31no—
42#4468—31no—
43#4467—31no—
44C23×C3×D3—21no—
45C23×C3×D3D411yescharacteristic
46C2×C32×Q8D411yescharacteristic, complemented
47C22×C32×C4C2311yescharacteristic
48C22×C3⋊Dic3—21no—
49C22×C3×Dic3D411yescharacteristic
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—42no—
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6D411yescharacteristic, complemented
57C2×#625—42no—
58C2×#625—42no—
59C2×#625—42no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4D411yescharacteristic, complemented
62C32×C22⋊C4—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C3×#217—21no—
65C3×#215—42no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3D411yescharacteristic, complemented
72C3×Dic3⋊C4—21no—
73C3×Dic3⋊C4—21no—
74C3×Dic3⋊C4—21no—
75C2×C3×C3⋊C8—21no—
76(C2×D3)⋊Dic3—42no—
77#3715—42no—
78#1596—91noSylow
79C22×C3×D4—31no—
80C22×#67—61no—
81C22×#67—61no—
82C22×D12C2×D311yescharacteristic
83C2×C3×SD16—61no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—31no—
86C3×C22⋊Q8—61no—
87C3×C22⋊Q8—61no—
88C3×C22⋊Q8—61no—
89C3×C22⋊Q8—61no—
90C3×C22⋊Q8—61no—
91C3×C22⋊Q8C2×D311yescharacteristic
92C3×C22⋊Q8C2×D311yescharacteristic
93C2×C3×C22⋊C4—31no—
94C2×C3×C22⋊C4—61no—
95C2×#217—61no—
96#946—21no—
97C2×#215—21no—
98C2×#215—61no—
99C2×#214—61no—
100C2×(C2×D3)⋊C4—61no—
101C2×(C2×D3)⋊C4C2×D311yescharacteristic
102#924C2×D311yescharacteristic
103#897—21no—
104#896—21no—
105#896—21no—
106D3×C22⋊C4—21no—
107#878—21no—
108C3×#102—61no—
109C3×C22⋊C8—31no—
110#832—61no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C3⋊#95—61no—
114#830—31no—
115#830—31no—
116#830—61no—
117#830C2×D311yescharacteristic
118#809—21no—
119#809—61no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3—21no—
122C22×C3×D3—21no—
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—41no—
131C22×C3×D3—41no—
132C22×C3×D3SD1612yescomplemented
133C32×Q8—41no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—41no—
141C2×C3⋊Dic3—41no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—41no—
146C3×D12—22no—
147C3×D12—22no—
148C3×D12—41no—
149C3×D12—41no—
150C3×C4×D3—81no—
151C3×C4×D3—81no—
152C3×Dic6—41no—
153C3×Dic6—81no—
154#625—48no—
155C3⋊(C3⋊C8)—41no—
156C3×C3⋊C8—41no—
157#543—31no—
158#450—31no—
159#450—91no—
160#392—91no—
161#328—91no—
162#328—91no—
163#323—91no—
164C24×C3—31no—
165C24×C3—61no—
166C23×D3—21no—
167C23×D3#6711yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8—31no—
170C2×C3×Q8—61no—
171C2×C3×Q8#6711yescharacteristic, complemented
172C2×C3×Q8#6711yescharacteristic, complemented
173C2×C3×D4—61no—
174C2×C3×D4—61no—
175C2×C3×D4—122no—
176C22×C3×C4—21no—
177C22×C3×C4—31no—
178C22×C3×C4—61no—
179C22×C3×C4—61no—
180C22×C3×C4C22×D311yescharacteristic
181C22×C3×C4C22×D311yescharacteristic
182C2×#67—122no—
183C2×#67—122no—
184C2×#67—122no—
185C2×#67—122no—
186C2×#67—122no—
187C2×#67—122no—
188C22×Dic3—61no—
189C22×Dic3—61no—
190C22×Dic3—121no—
191C22×Dic3#6711yescharacteristic
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—42no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×Dic6—21no—
198C2×Dic6#6711yescharacteristic, complemented
199C3×SD16—122no—
200C2×C3×C8—61no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—31no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4#6711yescharacteristic, complemented
211C3×C4⋊C4#6711yescharacteristic, complemented
212C3×C22⋊C4—21no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—41no—
215C3×C22⋊C4—61no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4—61no—
218C3×C22⋊C4—61no—
219C3×C22⋊C4—61no—
220#217—21no—
221#217—122no—
222#215—42no—
223#215—122no—
224#214—122no—
225(C2×D3)⋊C4—21no—
226(C2×D3)⋊C4—21no—
227(C2×D3)⋊C4—21no—
228(C2×D3)⋊C4—21no—
229(C2×D3)⋊C4—122no—
230C4⋊Dic3—21no—
231C4⋊Dic3#6711yescharacteristic, complemented
232Dic3⋊C4—21no—
233Dic3⋊C4—21no—
234Dic3⋊C4—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—61no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8—121no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32#12011yescharacteristic
242C22×C32C2×SD1611yescharacteristic
243C22×C32#13711yescharacteristic
244C2×C3×D3—22no—
245C2×C3×D3—41no—
246C2×C3×D3—41no—
247C2×C3×D3—42no—
248C2×C3×D3—42no—
249C2×C3×D3—42no—
250C2×C3×D3—42no—
251C2×C3×D3—81no—
252C2×C3×D3—81no—
253C2×C3×D3—81no—
254C2×C3×D3—81no—
255C2×C3×D3—81no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—41no—
259C32×C4—41no—
260C3⋊Dic3—42no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—41no—
264C22×D4—31no—
265C22×D4—181no—
266C2×SD16—61no—
267C2×SD16—181no—
268C22⋊Q8—31no—
269C22⋊Q8—61no—
270C22⋊Q8—61no—
271C22⋊Q8—61no—
272C22⋊Q8—61no—
273C22⋊Q8—61no—
274C22⋊Q8D3211yescharacteristic
275C2×C22⋊C4—31no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—181no—
278#102—61no—
279#102—181no—
280C22⋊C8—31no—
281C22⋊C8—91no—
282#95—91no—
283C23×C3—21no—
284C23×C3—32no—
285C23×C3—62no—
286C23×C3—61no—
287C23×C3—61no—
288C23×C3—121no—
289C23×C3—121no—
290C23×C3—121no—
291C23×C3—121no—
292C23×C3—121no—
293C23×C3—121no—
294C23×C3—121no—
295C23×C3—121no—
296C23×C3D3×D411yescharacteristic
297C23×C3C2×#6711yescharacteristic
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—22no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3#21512yescomplemented
310C3×Q8—41no—
311C3×Q8—41no—
312C3×Q8—81no—
313C3×Q8—121no—
314C3×Q8—241no—
315C3×D4—62no—
316C3×D4—62no—
317C3×D4—121no—
318C3×D4—121no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—81no—
333C2×C3×C4—121no—
334C2×C3×C4—241no—
335C2×C3×C4—241no—
336C2×C3×C4D3×D411yescharacteristic
337C2×C3×C4C2×#6711yescharacteristic
338C2×C3×C4C2×#6711yescharacteristic
339C2×C3×C4C2×#6711yescharacteristic
340#67—128no—
341#67—128no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—41no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—122no—
349C2×Dic3—121no—
350C2×Dic3—121no—
351C2×Dic3—121no—
352C2×Dic3—121no—
353C2×Dic3—241no—
354C2×Dic3—241no—
355D12—22no—
356D12—22no—
357D12—41no—
358D12—41no—
359C4×D3—81no—
360C4×D3—81no—
361Dic6—41no—
362Dic6—81no—
363C3×C8—121no—
364C3⋊C8—41no—
365C3⋊C8—121no—
366C3⋊C8—121no—
367C3⋊C8—241no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32#44811yescharacteristic
371C2×C32#45311yescharacteristic
372C2×C32B(4,𝔽2)11yescharacteristic
373C3×D3—42no—
374C3×D3—82no—
375C24—31no—
376C24—61no—
377C2×Q8—31no—
378C2×Q8—61no—
379C2×Q8#62511yescharacteristic, complemented
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—122no—
383C2×D4—362no—
384C2×D4—362no—
385C2×D4—362no—
386C22×C4—31no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—181no—
390C22×C4C2×D3211yescharacteristic
391SD16—122no—
392SD16—362no—
393C2×C8—61no—
394C2×C8—181no—
395C4⋊C4—21no—
396C4⋊C4—31no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4#62511yescharacteristic, complemented
402C22⋊C4—21no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C22⋊C4—61no—
406C22⋊C4—61no—
407C22⋊C4—61no—
408C22⋊C4—362no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—62no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—122no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—122no—
425C22×C3—241no—
426C22×C3—241no—
427C22×C3—241no—
428C22×C3—241no—
429C22×C3—241no—
430C22×C3D3×SD1611yescharacteristic
431C22×C3#91511yescharacteristic
432C22×C3#93711yescharacteristic
433C22×C3C2×#21511yescharacteristic
434C22×C3#94211yescharacteristic
435C22×C3#95311yescharacteristic
436C2×D3—22no—
437C2×D3—41no—
438C2×D3—41no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—81no—
444C2×D3—81no—
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—121no—
463Dic3—41no—
464Dic3—41no—
465Dic3—41no—
466Dic3—122no—
467Dic3—122no—
468Dic3—242no—
469C32#159611yescharacteristic, complemented, Sylow
470C23—32no—
471C23—62no—
472C23—61no—
473C23—61no—
474C23—121no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—121no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23D3×#6711yescharacteristic
483Q8—41no—
484Q8—121no—
485Q8—241no—
486D4—62no—
487D4—62no—
488D4—121no—
489D4—121no—
490D4—368no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—121no—
498C2×C4—182no—
499C2×C4—241no—
500C2×C4—241no—
501C2×C4—361no—
502C2×C4—361no—
503C2×C4C2×#62511yescharacteristic
504C2×C4D3×#6711yescharacteristic
505C8—121no—
506C8—361no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—122no—
517C2×C3—242no—
518C2×C3#503211yescharacteristic
519C2×C3#517011yescharacteristic
520C2×C3#519811yescharacteristic
521C2×C3#522711yescharacteristic
522C2×C3#524111yescharacteristic
523C2×C3#524411yescharacteristic
524D3—42no—
525D3—82no—
526C22—21no—
527C22—21no—
528C22—62no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—121no—
533C22—121no—
534C22—122no—
535C22—241no—
536C22—241no—
537C22—241no—
538C22—241no—
539C22—241no—
540C22D3×#21511yescharacteristic
541C22#6394611yescharacteristic
542C22#6397611yescharacteristic
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—121no—
548C4—121no—
549C4—121no—
550C4—362no—
551C3—21no—
552C3#7097611yescharacteristic, complemented
553C3#7189211yescharacteristic, complemented
554C2—21no—
555C2—21no—
556C2—122no—
557C2—242no—
558C2#1059153011yescharacteristic
559C2#1059158311yescharacteristic
560C2#1059166511yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radical#10591389C2
hypercenterC22⋊Q8D32
hypocenterC32#1596
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×#5634C2semidirect
2. C3×#5046C2semidirect
4. #10591169C2semidirect
5. #10589404C2semidirect
6. #10589152C2semidirect
7. #10589145C2semidirect
30. #5634D3semidirect
35. #5046D3semidirect
46. C2×C32×Q8D4semidirect
56. C2×C3×Dic6D4semidirect
61. C32×C4⋊C4D4semidirect
71. C3×C4⋊Dic3D4semidirect
132. C22×C3×D3SD16semidirect
171. C2×C3×Q8#67semidirect
172. C2×C3×Q8#67semidirect
198. C2×Dic6#67semidirect
210. C3×C4⋊C4#67semidirect
211. C3×C4⋊C4#67semidirect
231. C4⋊Dic3#67semidirect
309. C22×D3#215semidirect
379. C2×Q8#625semidirect
401. C4⋊C4#625semidirect
469. C32#1596semidirect
552. C3#70976semidirect
553. C3#71892semidirect

Maximal subgroups

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

subgroupindex
C3×#56342
C3×#50462
#105913892
#105911692
#105894042
#105891522
#105891452
#718923
#709763