Finite Group Databasebeta

← #11025026

Subgroups — #11025026

Order 768.

2420 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5535C211yesmaximal, characteristic, complemented
2C2×#5477C211yesmaximal, characteristic, complemented
3C3×#2290C211yesmaximal, characteristic, complemented
4#82402C212yesmaximal, complemented
5#82262C211yesmaximal, characteristic, complemented
6#82074C211yesmaximal, characteristic, complemented
7#75849C214yesmaximal, complemented
8#75315C214yesmaximal, complemented
9#74921C214yesmaximal, complemented
10#74639C214yesmaximal, complemented
11#73781C212yesmaximal, complemented
12#73573C212yesmaximal, complemented
13#73486C212yesmaximal, complemented
14#73407C211yesmaximal, characteristic, complemented
15#73389C211yesmaximal, characteristic, complemented
16#37700—31nomaximal, Sylow
17C22×#235C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C2×#930C2212yes—
22C2×#930C2214yes—
23C2×#924C2212yes—
24#5535—28no—
25C2×#894C2214yes—
26C2×#884C2214yes—
27#5477—28no—
28C2×#875C2212yes—
29C2×#874C2211yescharacteristic
30C2×C4×D12C2214yes—
31C2×C4×D12C2212yes—
32C2×C4×D12C2212yes—
33C2×#870C2211yescharacteristic
34C2×#869C2211yescharacteristic
35C3×#388C2212yes—
36C3×#387C2212yes—
37C3×#386C2212yes—
38C3×#385C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×C4×C22⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4C2211yescharacteristic
42C3×#376C2211yescharacteristic
43#5223C2214yes—
44#5212C2212yes—
45#5210C2214yes—
46#4992C2214yes—
47#4989C2214yes—
48#4989C2214yes—
49#4981C2214yes—
50#4959C2214yes—
51#4959C2214yes—
52#4956C2214yes—
53#4956C2214yes—
54#4956C2214yes—
55#4956C2214yes—
56#4953C2214yes—
57#4939C2212yes—
58#4936C2211yescharacteristic
59#4933C2214yes—
60#4929C2212yes—
61#4929C2212yes—
62#4929C2212yes—
63#4928C2212yes—
64#4675C2214yes—
65#4675C2214yes—
66#4667C2214yes—
67#4665C2214yes—
68#4665C2214yes—
69#4664C2214yes—
70#4664C2214yes—
71D3×#95C2214yes—
72#4659C2214yes—
73#4659C2214yes—
74C2×#540—31no—
75C2×#532—31no—
76#2851—31no—
77#2708—34no—
78#2688—34no—
79#2642—31no—
80#2621—31no—
81#2447—32no—
82#2444—34no—
83#2427—32no—
84#2419—34no—
85#2407—32no—
86#2339—32no—
87#2328—31no—
88#2290D311yescharacteristic, complemented
89C22×C3×D4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C22×#67C2314yes—
94C22×#67C2314yes—
95C2×#235—28no—
96C2×#235—28no—
97C2×#235—44no—
98C22×D12C2312yes—
99C22×D12C2314yes—
100C22×C4×D3C2314yes—
101C22×C4×D3C2314yes—
102C22×C4×D3C2314yes—
103C22×Dic6C2312yes—
104C3×C4×D4—24no—
105C3×#117—24no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C42—24no—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2312yes—
120C2×C3×C42C2311yescharacteristic
121C2×#217C2314yes—
122C2×#217C2314yes—
123#930—216no—
124#930—28no—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135#924—44no—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139#894—216no—
140#884—216no—
141#875—28no—
142#874D418yescomplemented
143C4×D12—216no—
144C4×D12—28no—
145C4×D12—44no—
146#870—24no—
147#869—24no—
148C3×#95C2312yes—
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3×#95C2312yes—
152C3×#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C22×#41—32no—
163#540—68no—
164#532—68no—
165C2×C4⋊Q8—31no—
166C2×#127—31no—
167C2×#125—31no—
168C2×#124—32no—
169C2×#123—34no—
170C2×C22⋊Q8—32no—
171C2×#121—32no—
172C2×#121—34no—
173C2×#121—34no—
174C2×C4×D4—32no—
175C2×C4×D4—34no—
176C2×C4×D4—32no—
177C2×C4×D4C2×D311yescharacteristic
178C2×#117C2×D311yescharacteristic
179C22×C42C2×D311yescharacteristic
180#400—34no—
181#400—34no—
182#398—34no—
183#397—34no—
184#397—34no—
185#397—34no—
186#397—34no—
187#394—34no—
188#394—34no—
189#392—34no—
190#391—32no—
191#391—32no—
192#391—32no—
193#390—32no—
194#389—32no—
195#388C2×D312yes—
196#387—34no—
197#387C2×D312yes—
198#386—34no—
199#386—34no—
200#386—34no—
201#386—34no—
202#386—34no—
203#386—34no—
204#386C2×D312yes—
205#385—34no—
206#385—34no—
207#385—34no—
208#385C2×D312yes—
209#382—31no—
210#381—34no—
211#381—34no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4C2×D311yescharacteristic
215C4×C22⋊C4C2×D311yescharacteristic
216C4×C22⋊C4C2×D311yescharacteristic
217#376C2×D311yescharacteristic
218C2×#95—34no—
219C24×C3C2411yescharacteristic
220C24×C3C2411yescharacteristic
221C23×D3C2414yes—
222C2×C3×D4—22no—
223C2×C3×D4—42no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—24no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—42no—
231C22×C3×C4—42no—
232C22×C3×C4#4114yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2412yes—
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2412yes—
237C22×C3×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2412yes—
240C22×C3×C4C2411yescharacteristic
241C22×C3×C4C2411yescharacteristic
242C2×#67—28no—
243C2×#67—28no—
244C2×#67—28no—
245C2×#67—44no—
246C2×#67—44no—
247C2×#67—44no—
248C22×Dic3C2414yes—
249C22×Dic3C2414yes—
250C22×Dic3C2414yes—
251#235—432no—
252C2×D12—24no—
253C2×D12—24no—
254C2×D12—44no—
255C2×D12—44no—
256C2×D12C2×D418yes—
257C2×D12C2×D4116yes—
258C2×C4×D3—28no—
259C2×C4×D3—28no—
260C2×C4×D3—28no—
261C2×C4×D3—44no—
262C2×C4×D3—44no—
263C2×C4×D3—44no—
264C2×C4×D3—44no—
265C2×C4×D3—44no—
266C2×C4×D3—44no—
267C2×Dic6—24no—
268C2×Dic6—24no—
269C2×Dic6—24no—
270C3×C4⋊C4—24no—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—42no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—24no—
275C3×C22⋊C4—22no—
276C3×C22⋊C4—42no—
277C3×C22⋊C4—42no—
278C3×C42—24no—
279C3×C42—22no—
280C3×C42—22no—
281C3×C42—22no—
282C3×C42—22no—
283C3×C42C2×D414yes—
284#217—44no—
285#217—44no—
286(C2×D3)⋊C4—28no—
287(C2×D3)⋊C4—28no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—28no—
290(C2×D3)⋊C4—44no—
291(C2×D3)⋊C4—44no—
292(C2×D3)⋊C4—44no—
293C4⋊Dic3—28no—
294C4⋊Dic3—24no—
295C4⋊Dic3—24no—
296Dic3⋊C4—28no—
297Dic3⋊C4—44no—
298Dic3⋊C4—44no—
299C2×#41—68no—
300C2×#41—68no—
301C2×#41—124no—
302C22×Q8—32no—
303C22×D4—34no—
304C22×D4—34no—
305C22×D4—32no—
306C22×D4—34no—
307C22×D4C22×D311yescharacteristic
308C23×C4—34no—
309C23×C4—34no—
310C23×C4—34no—
311C23×C4C22×D311yescharacteristic
312C23×C4C22×D311yescharacteristic
313C23×C4C22×D312yes—
314C4⋊Q8—64no—
315#127—38no—
316#125—64no—
317#124—68no—
318#123—616no—
319C22⋊Q8—124no—
320#121—616no—
321#121—616no—
322#121—68no—
323C4×D4—24no—
324C4×D4—616no—
325C4×D4—68no—
326C4×D4—124no—
327#117—24no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D312yes—
337C2×C22⋊C4—34no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4C22×D312yes—
347C2×C22⋊C4C22×D311yescharacteristic
348C2×C22⋊C4C22×D311yescharacteristic
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D312yes—
351C2×C42—24no—
352C2×C42C22×D311yescharacteristic
353C2×C42C22×D311yescharacteristic
354C2×C42C22×D312yes—
355C2×C42C22×D311yescharacteristic
356C2×C42C22×D311yescharacteristic
357C2×C42C22×D311yescharacteristic
358#95—34no—
359#95—34no—
360#95—34no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95C22×D312yes—
368#95C22×D312yes—
369#95C22×D312yes—
370#95C22×D312yes—
371#95C22×D312yes—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3C2×#4112yes—
382C23×C3C2511yescharacteristic
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—24no—
389C22×D3C22×D418yes—
390C3×D4—44no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—28no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—48no—
413C2×C3×C4—44no—
414C2×C3×C4C22×D414yes—
415C2×C3×C4C22×D412yes—
416C2×C3×C4C2×#4112yes—
417C2×C3×C4C2×#4112yes—
418C2×C3×C4H(5,𝔽2)12yes—
419#67—416no—
420#67—88no—
421C2×Dic3—24no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430D12—216no—
431D12—416no—
432C4×D3—416no—
433C4×D3—88no—
434C4×D3—88no—
435Dic6—48no—
436C24—34no—
437C24C23×D311yescharacteristic
438C24C23×D311yescharacteristic
439#41—1232no—
440C2×Q8—64no—
441C2×Q8—64no—
442C2×Q8—64no—
443C2×D4—22no—
444C2×D4—38no—
445C2×D4—316no—
446C2×D4—42no—
447C2×D4—68no—
448C2×D4—68no—
449C2×D4—68no—
450C2×D4—64no—
451C2×D4—64no—
452C2×D4—124no—
453C2×D4—124no—
454C2×D4—124no—
455C2×D4—124no—
456C2×D4—124no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—34no—
464C22×C4—34no—
465C22×C4—34no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—68no—
469C22×C4—68no—
470C22×C4—68no—
471C22×C4—124no—
472C22×C4—124no—
473C22×C4—124no—
474C22×C4—124no—
475C22×C4—124no—
476C22×C4—124no—
477C22×C4#23514yes—
478C22×C4C23×D311yescharacteristic
479C22×C4C23×D312yes—
480C22×C4C23×D311yescharacteristic
481C22×C4C23×D312yes—
482C22×C4C23×D312yes—
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D311yescharacteristic
485C22×C4C23×D312yes—
486C22×C4C23×D311yescharacteristic
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—64no—
491C4⋊C4—68no—
492C4⋊C4—68no—
493C4⋊C4—64no—
494C4⋊C4—124no—
495C4⋊C4—124no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—68no—
502C22⋊C4—68no—
503C22⋊C4—68no—
504C22⋊C4—68no—
505C22⋊C4—124no—
506C22⋊C4—124no—
507C22⋊C4—124no—
508C22⋊C4—124no—
509C22⋊C4—124no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42D3×D414yes—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—42no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3C23×D411yescharacteristic
527C22×C3C22×#4111yescharacteristic
528C22×C3C2×H(5,𝔽2)11yescharacteristic
529C22×C3#58512yes—
530C22×C3#58512yes—
531C2×D3—28no—
532C2×D3—28no—
533C2×D3—28no—
534C2×D3—44no—
535C2×D3—44no—
536C2×D3—44no—
537C2×D3—44no—
538C3×C4—22no—
539C3×C4—24no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—41no—
545C3×C4—42no—
546C3×C4#54614yes—
547Dic3—44no—
548Dic3—44no—
549Dic3—44no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—38no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23C2×#23512yes—
567C23C24×D311yescharacteristic
568Q8—128no—
569D4—44no—
570D4—616no—
571D4—1216no—
572D4—1216no—
573D4—248no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—1216no—
607C2×C4—248no—
608C2×C4—248no—
609C2×C4C2×#23512yes—
610C2×C4C2×#23512yes—
611C2×C4C2×D3×D414yes—
612C2×C4C2×D3×D412yes—
613C2×C4#100912yes—
614C2×C3—42no—
615C2×C3—81no—
616C2×C3#345511yescharacteristic
617C2×C3C2×#54611yescharacteristic
618C2×C3#348512yes—
619C2×C3#351211yescharacteristic
620C2×C3#356212yes—
621D3—48no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—42no—
626C22—42no—
627C22—41no—
628C22—42no—
629C22—41no—
630C22—42no—
631C22—41no—
632C22—68no—
633C22—68no—
634C22—68no—
635C22—124no—
636C22—124no—
637C22—124no—
638C22—124no—
639C22C22×#23511yescharacteristic
640C22C22×D3×D411yescharacteristic
641C22C2×#100911yescharacteristic
642C22#596412yes—
643C22#596412yes—
644C4—22no—
645C4—24no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—124no—
653C4—124no—
654C4—124no—
655C4#555014yes—
656C3#3770011yescharacteristic, complemented, Sylow
657C2—42no—
658C2—81no—
659C2—128no—
660C2#8573811yescharacteristic
661C2C2×#555011yescharacteristic
662C2#8585111yescharacteristic
663C2#8604012yes—
664C2#8617912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2290C2
hypercenter#2290D3
hypocenterC3#37700
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2290

Complements

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

NH ≅ G/Ntype
1. C2×#5535C2semidirect
2. C2×#5477C2semidirect
3. C3×#2290C2semidirect
4. #82402C2semidirect
5. #82262C2semidirect
6. #82074C2semidirect
7. #75849C2semidirect
8. #75315C2semidirect
9. #74921C2semidirect
10. #74639C2semidirect
11. #73781C2semidirect
12. #73573C2semidirect
13. #73486C2semidirect
14. #73407C2semidirect
15. #73389C2semidirect
88. #2290D3semidirect
142. #874D4semidirect
656. C3#37700semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C2×#54772
C3×#22902
#824022
#822622
#820742
#758492
#753152
#749212
#746392
#737812
#735732
#734862
#734072
#733892
#377003