Finite Group Databasebeta

← #10896360

Subgroups — #10896360

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74149C211yesmaximal, characteristic
2C4×#888C211yesmaximal, characteristic, complemented
3C3×#1949C211yesmaximal, characteristic, complemented
4#70642C211yesmaximal, characteristic, complemented
5#70641C211yesmaximal, characteristic
6#70590C211yesmaximal, characteristic, complemented
7#70584C211yesmaximal, characteristic
8#70458C211yesmaximal, characteristic
9#70443C211yesmaximal, characteristic, complemented
10#69540C211yesmaximal, characteristic, complemented
11#69488C211yesmaximal, characteristic
12#68910C211yesmaximal, characteristic, complemented
13#68906C211yesmaximal, characteristic, complemented
14#68898C211yesmaximal, characteristic
15#68895C211yesmaximal, characteristic
16#29262—31nomaximal, Sylow
17C2×#893C2211yescharacteristic
18C2×#888C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×C4×Dic6—21no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4988C2211yescharacteristic
32#4984C2211yescharacteristic
33#4975C2211yescharacteristic
34#4973C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43C4×Dic3⋊C4—21no—
44C42×Dic3C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4669—21no—
47#4669—21no—
48#4666—21no—
49#4666—21no—
50#4647—21no—
51#4646—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60C4×C4⋊Q8—31no—
61C4×#124—31no—
62#2096—31no—
63#2093—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983—31no—
68#1949—31no—
69#1949D311yescharacteristic, complemented
70#1864—31no—
71#1864—31no—
72#1863—31no—
73#1863—31no—
74#1831—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×Dic6—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#893—41no—
111#893—41no—
112#888—22no—
113#888—22no—
114#887—41no—
115#887—41no—
116#875—22no—
117#875Q814yescomplemented
118C4×Dic6—44no—
119C3×#107—41no—
120C3×#106—41no—
121C3×C4⋊C8—22no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—22no—
131#821—41no—
132#816—22no—
133#816—41no—
134#810—22no—
135#810—41no—
136#809—22no—
137#809—41no—
138#808—41no—
139#807—41no—
140C4⋊(C3⋊C8)—22no—
141C2×C4⋊Q8—31no—
142C2×#124—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×D4—31no—
146C2×#107—31no—
147C2×#107C2×D311yescharacteristic
148C2×#106—31no—
149C2×#106C2×D311yescharacteristic
150C2×C4⋊C8—31no—
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160#391—31no—
161#391—61no—
162#390—31no—
163#389—31no—
164#389—31no—
165#389—61no—
166#388—61no—
167#388—61no—
168#384—31no—
169#382C2×D311yescharacteristic
170C4×C4⋊C4—31no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4C2×D311yescharacteristic
173C4×C22⋊C4—61no—
174C4×C22⋊C4—61no—
175C43—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#336—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184C23×D3—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12C2×Q812yes—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—24no—
213C2×Dic6—21no—
214C2×Dic6—42no—
215C2×Dic6—42no—
216C2×Dic6C2×Q812yes—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4#4112yes—
231C3×C42—42no—
232C3×C42C2×Q812yes—
233C3×C42#4112yes—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—22no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3—41no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C22×Q8—31no—
266C22×Q8—61no—
267C22×D4—31no—
268C23×C4—61no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4⋊Q8—62no—
274C4⋊Q8—62no—
275#124—34no—
276#124—62no—
277C4×Q8—124no—
278C4×Q8—121no—
279C4×Q8—121no—
280C4×D4—121no—
281C4×D4—121no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—21no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—41no—
310#107—121no—
311#106—41no—
312#106—121no—
313C4⋊C8—22no—
314C4⋊C8—62no—
315#103—62no—
316#103—62no—
317#103—121no—
318#103—121no—
319#102—62no—
320#102—62no—
321#102—121no—
322#102—121no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3C22×D411yescharacteristic
332C22×D3—22no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4C22⋊Q812yes—
356C2×C3×C4C22⋊Q812yes—
357C2×C3×C4C22×Q811yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3#13512yes—
382C2×Dic3AGL(1,ℤ/8)12yes—
383C2×Dic3#13712yes—
384D12—22no—
385D12—22no—
386D12—41no—
387D12—41no—
388C4×D3—84no—
389Dic6—22no—
390Dic6—22no—
391Dic6—48no—
392Dic6—41no—
393Dic6—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—61no—
399C2×Q8—32no—
400C2×Q8—64no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—122no—
407C2×Q8—122no—
408C2×D4—32no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4C23×D311yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—32no—
451C4⋊C4—64no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4#23912yes—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—122no—
466C42—42no—
467C42—62no—
468C42—62no—
469C42—62no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42—61no—
474C42—121no—
475C42—122no—
476C42—122no—
477C42#23512yes—
478C42D3×Q812yes—
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3#52511yescharacteristic
481C22×C3#53511yescharacteristic
482C22×C3C2×#13511yescharacteristic
483C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
484C22×C3C2×#13711yescharacteristic
485C22×C3#57511yescharacteristic
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#55411yescharacteristic
500C3×C4#55411yescharacteristic
501C3×C4#55911yescharacteristic
502C3×C4#55911yescharacteristic
503Dic3—22no—
504Dic3—22no—
505Dic3—22no—
506Dic3—42no—
507Dic3—42no—
508Dic3—41no—
509Dic3—81no—
510C23—62no—
511C23—121no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23C2×D3×D411yescharacteristic
518Q8—62no—
519Q8—62no—
520Q8—128no—
521Q8—121no—
522Q8—121no—
523D4—62no—
524D4—62no—
525D4—121no—
526D4—121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—244no—
565C2×C4#89612yes—
566C2×C4#89612yes—
567C2×C4C2×#23511yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4C2×D3×Q811yescharacteristic
570C2×C4C2×#23911yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4#100911yescharacteristic
573C8—41no—
574C8—41no—
575C8—121no—
576C8—121no—
577C2×C3#258011yescharacteristic
578C2×C3#307611yescharacteristic
579C2×C3#307911yescharacteristic
580C2×C3#310811yescharacteristic
581C2×C3#311411yescharacteristic
582C2×C3#320711yescharacteristic
583C2×C3#321511yescharacteristic
584D3—82no—
585C22—122no—
586C22—122no—
587C22—122no—
588C22—241no—
589C22—241no—
590C22—241no—
591C22—241no—
592C22C2×#89611yescharacteristic
593C22#560711yescharacteristic
594C22#562711yescharacteristic
595C22D3×#13511yescharacteristic
596C22#576811yescharacteristic
597C22D3×AGL(1,ℤ/8)11yescharacteristic
598C22D3×#13711yescharacteristic
599C4—22no—
600C4—22no—
601C4—42no—
602C4—62no—
603C4—62no—
604C4—62no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—122no—
609C4—121no—
610C4—122no—
611C4—241no—
612C4#555911yescharacteristic
613C4#558511yescharacteristic
614C4#569011yescharacteristic
615C4#573011yescharacteristic
616C3#2926211yescharacteristic, complemented, Sylow
617C2—242no—
618C2#7607511yescharacteristic
619C2#7919611yescharacteristic
620C2#7924011yescharacteristic
621C2#8054011yescharacteristic
622C2#8060611yescharacteristic
623C2#8114311yescharacteristic
624C2#8120511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1949C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1949D3
hypocenterC3#29262
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. C4×#888C2semidirect
3. C3×#1949C2semidirect
4. #70642C2semidirect
6. #70590C2semidirect
9. #70443C2semidirect
10. #69540C2semidirect
12. #68910C2semidirect
13. #68906C2semidirect
69. #1949D3semidirect
117. #875Q8semidirect
616. C3#29262semidirect

Maximal subgroups

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

subgroupindex
#741492
C4×#8882
C3×#19492
#706422
#706412
#705902
#705842
#704582
#704432
#695402
#694882
#689102
#689062
#688982
#688952
#292623