Finite Group Databasebeta

← #10698057

Subgroups — #10698057

Order 768.

988 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 103 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#84032C211yesmaximal, characteristic
2#84031C211yesmaximal, characteristic, complemented
3#78612C211yesmaximal, characteristic, complemented
4C3×#1779C211yesmaximal, characteristic, complemented
5#69946C211yesmaximal, characteristic, complemented
6#69944C211yesmaximal, characteristic, complemented
7#67676C211yesmaximal, characteristic, complemented
8#10322—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×#953—21no—
11C2×#930—21no—
12C2×#930C2211yescharacteristic
13C2×#924—21no—
14C2×#924C2211yescharacteristic
15C2×C3×C22⋊C8—22no—
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×C22⋊C8C2211yescharacteristic
18#5247—21no—
19#5213—21no—
20#5213—22no—
21#5212—21no—
22#5212—21no—
23#5212C2211yescharacteristic
24#5212C2211yescharacteristic
25#5210—21no—
26#5210—22no—
27#5114—22no—
28#4738—21no—
29#4735—21no—
30#4731—21no—
31#4721—21no—
32#4470—22no—
33#4467—22no—
34#3187—31no—
35#2848—31no—
36#2847—31no—
37#1779D311yescharacteristic, complemented
38#1520—31no—
39#1519—31no—
40#1289—31no—
41C25×C3D411yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4—44no—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×#67—22no—
52C22×#67—21no—
53C22×D12D411yescharacteristic
54C22×Dic6D411yescharacteristic, complemented
55C22×C3×C8—21no—
56C22×C3×C8D412yescomplemented
57#953—42no—
58#953—42no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68#930—21no—
69#930—44no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×C4⋊Dic3D411yescharacteristic, complemented
76#924—21no—
77#924—21no—
78#924—21no—
79#924—21no—
80#924—44no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—21no—
83C2×#204—22no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3×C22⋊C8—21no—
87C3×C22⋊C8—21no—
88C3×C22⋊C8—44no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95D412yes—
92C3⋊#95D412yescomplemented
93#821—22no—
94#817—22no—
95#816—22no—
96C24×C4C2×D311yescharacteristic
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8—61no—
99C2×#121—31no—
100C2×#121—61no—
101C2×#120—61no—
102#483—61no—
103#481—61no—
104#465—62no—
105#450—61no—
106#448—61no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8C2×D311yescharacteristic
109C2×C22⋊C8C2×D311yescharacteristic
110#386—62no—
111#386—61no—
112#385—62no—
113#385—61no—
114#380—31no—
115#380—31no—
116#380—61no—
117#380—61no—
118#379—61no—
119#328—62no—
120#327—62no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—21no—
126C24×C3—21no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3SD1612yescomplemented
130C24×C3C2×D411yescharacteristic
131C23×D3—21no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×#67—44no—
160C2×#67—44no—
161C2×#67—44no—
162C2×#67—41no—
163C2×#67—41no—
164C2×#67—41no—
165C2×#67—41no—
166C2×#67—41no—
167C2×#67—41no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—42no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—42no—
179C2×C3×C8—22no—
180C2×C3×C8—42no—
181C2×C3×C8—42no—
182C2×C3×C8—41no—
183#217—44no—
184#217—44no—
185#217—44no—
186#217—41no—
187#217—41no—
188#217—41no—
189#217—41no—
190#217—41no—
191#217—41no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—41no—
194(C2×D3)⋊C4—41no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—42no—
200C4⋊Dic3—42no—
201Dic3⋊C4—44no—
202Dic3⋊C4—41no—
203Dic3⋊C4—41no—
204#204—82no—
205C25D1211yescharacteristic
206C22×Q8—31no—
207C22×D4—31no—
208C22×D4—62no—
209C22×D4—61no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—44no—
217C23×C4D1211yescharacteristic
218C23×C4C22×D311yescharacteristic
219C2×SD16—62no—
220C22×C8—21no—
221C22×C8#6712yescomplemented
222C22⋊Q8—61no—
223C22⋊Q8—61no—
224C22⋊Q8—61no—
225C22⋊Q8—61no—
226C22⋊Q8—124no—
227#121—61no—
228#121—61no—
229#121—61no—
230#121—61no—
231#121—124no—
232#120—122no—
233#120—122no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248#106—62no—
249#103—62no—
250#102—62no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—44no—
256#95—32no—
257#95—32no—
258#95—62no—
259#95—62no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—41no—
270C23×C3—44no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3#12011yescharacteristic
286C23×C3#12011yescharacteristic
287C23×C3#12111yescharacteristic
288C23×C3C2×SD1611yescharacteristic
289C23×C3#13511yescharacteristic
290C23×C3AGL(1,ℤ/8)11yescharacteristic
291C23×C3#13711yescharacteristic
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4—44no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—44no—
314C2×C3×C4#12011yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12711yescharacteristic
318#67—88no—
319#67—81no—
320#67—81no—
321#67—81no—
322#67—81no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336D12—42no—
337D12—81no—
338Dic6—42no—
339Dic6—81no—
340C3×C8—42no—
341C3×C8—81no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—42no—
349C24—42no—
350C24—61no—
351C24#20412yescomplemented
352C24C2×D1211yescharacteristic
353C2×Q8—61no—
354C2×Q8—61no—
355C2×Q8—122no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—122no—
359C2×D4—124no—
360C2×D4—124no—
361C2×D4—124no—
362C2×D4—121no—
363C2×D4—121no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—121no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4#23511yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C2×#6711yescharacteristic
400SD16—242no—
401C2×C8—22no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—41no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—122no—
410C4⋊C4—124no—
411C4⋊C4—122no—
412C4⋊C4—121no—
413C4⋊C4—121no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—121no—
419C22⋊C4—121no—
420C22⋊C4—121no—
421C22⋊C4—121no—
422C22⋊C4—121no—
423C22⋊C4—121no—
424C22⋊C4—121no—
425C22⋊C4—121no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3#39211yescharacteristic
452C22×C3#44811yescharacteristic
453C22×C3#44911yescharacteristic
454C22×C3#45011yescharacteristic
455C22×C3#45211yescharacteristic
456C22×C3#46511yescharacteristic
457C22×C3#46811yescharacteristic
458C2×D3—82no—
459C2×D3—82no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4B(4,𝔽2)12yes—
468Dic3—82no—
469Dic3—82no—
470Dic3—81no—
471Dic3—81no—
472Dic3—81no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—41no—
483C23—44no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—122no—
499C23—122no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23#88211yescharacteristic
504C23#88211yescharacteristic
505C23C2×#20411yescharacteristic
506C23#90411yescharacteristic
507C23#90811yescharacteristic
508C23#90911yescharacteristic
509C23#93011yescharacteristic
510Q8—122no—
511Q8—241no—
512D4—122no—
513D4—241no—
514D4—248no—
515D4—241no—
516D4—241no—
517D4—241no—
518D4—241no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—122no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—122no—
549C2×C4#88411yescharacteristic
550C2×C4#88411yescharacteristic
551C2×C4#93711yescharacteristic
552C2×C4#94011yescharacteristic
553C8—42no—
554C8—81no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—44no—
562C2×C3#200011yescharacteristic
563C2×C3#201011yescharacteristic
564C2×C3#202111yescharacteristic
565D3—161no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—242no—
592C22—242no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22#473111yescharacteristic
597C22#473311yescharacteristic
598C22#473411yescharacteristic
599C22#473811yescharacteristic
600C22#496011yescharacteristic
601C22#511411yescharacteristic
602C22#513411yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—242no—
608C4—242no—
609C4—241no—
610C4—241no—
611C4—241no—
612C4#503212yes—
613C3#1032211yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—44no—
621C2—481no—
622C2#6950011yescharacteristic
623C2#6996211yescharacteristic
624C2#6999311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC22×C3#392
prime radical#84032C2
hypercenter#1779D3
hypocenterC3#10322
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #84031C2semidirect
3. #78612C2semidirect
4. C3×#1779C2semidirect
5. #69946C2semidirect
6. #69944C2semidirect
7. #67676C2semidirect
37. #1779D3semidirect
54. C22×Dic6D4semidirect
56. C22×C3×C8D4semidirect
75. C2×C4⋊Dic3D4semidirect
92. C3⋊#95D4semidirect
129. C24×C3SD16semidirect
221. C22×C8#67semidirect
351. C24#204semidirect
613. C3#10322semidirect

Maximal subgroups

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

subgroupindex
#840322
#840312
#786122
C3×#17792
#699462
#699442
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