Finite Group Databasebeta

← #11069614

Subgroups — #11069614

Order 768.

2206 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5535C211yesmaximal, characteristic, complemented
2C2×#5518C211yesmaximal, characteristic, complemented
3C3×#2293C211yesmaximal, characteristic, complemented
4#82377C212yesmaximal, complemented
5#82286C211yesmaximal, characteristic, complemented
6#82262C211yesmaximal, characteristic, complemented
7#76395C214yesmaximal, complemented
8#76110C214yesmaximal, complemented
9#75596C214yesmaximal, complemented
10#75545C214yesmaximal, complemented
11#73655C212yesmaximal, complemented
12#73486C212yesmaximal, complemented
13#73485C212yesmaximal, complemented
14#73313C211yesmaximal, characteristic, complemented
15C4×#870C211yesmaximal, characteristic, complemented
16#39382—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#930C2212yes—
21C2×#930C2212yes—
22C2×#924C2212yes—
23C2×#924C2212yes—
24#5535—28no—
25#5518—28no—
26C2×#894C2214yes—
27C2×#889C2214yes—
28C2×#884C2214yes—
29C2×#879C2214yes—
30C2×#875C2212yes—
31C2×#875C2212yes—
32C2×#874C2211yescharacteristic
33C2×C4×D12C2212yes—
34C2×#870C2211yescharacteristic
35C2×#869C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#388C2212yes—
40C3×#388C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×C43C2211yescharacteristic
45#5223C2214yes—
46#4993C2214yes—
47#4991C2214yes—
48#4990C2214yes—
49#4989C2214yes—
50#4981C2214yes—
51#4960C2214yes—
52#4959C2214yes—
53#4959C2214yes—
54#4957C2214yes—
55#4957C2214yes—
56#4957C2214yes—
57#4956C2214yes—
58#4953C2214yes—
59#4939C2212yes—
60#4939C2214yes—
61#4939C2214yes—
62#4936C2211yescharacteristic
63C4×C4⋊Dic3C2212yes—
64#4933C2214yes—
65C4×Dic3⋊C4C2214yes—
66#4929C2212yes—
67#4928C2212yes—
68#4676C2214yes—
69#4675C2214yes—
70#4669C2214yes—
71#4669C2214yes—
72#4667C2214yes—
73#4666C2214yes—
74C2×#541—31no—
75C2×#540—31no—
76#2858—31no—
77#2714—34no—
78#2694—34no—
79#2621—31no—
80#2619—31no—
81#2554—34no—
82#2540—32no—
83#2490—34no—
84#2427—32no—
85#2426—32no—
86#2367—32no—
87C4×#125—31no—
88#2293D311yescharacteristic, complemented
89C22×C3×D4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C22×#67C2314yes—
93C22×#67C2314yes—
94C22×D12C2312yes—
95C22×D12C2314yes—
96C22×C4×D3C2314yes—
97C22×Dic6C2312yes—
98C3×C4×D4—24no—
99C3×#117—24no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C22⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2312yes—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×#217C2314yes—
119C2×#217C2314yes—
120#930—28no—
121#930—44no—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
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×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134#924—28no—
135#924—44no—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143#894—216no—
144#889—216no—
145#884—216no—
146#879—216no—
147#875—28no—
148#875—44no—
149#874—24no—
150C4×D12—28no—
151#870D418yescomplemented
152#869—24no—
153C4×Dic6—28no—
154C3×#95C2312yes—
155C3×#95C2312yes—
156C3×#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#541—68no—
163#540—68no—
164C2×C4⋊Q8—31no—
165C2×#127—31no—
166C2×#126—34no—
167C2×#125—34no—
168C2×#125—31no—
169C2×#124—32no—
170C2×#124—32no—
171C2×#123—34no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—32no—
174C2×#121—32no—
175C2×#121—32no—
176C2×#121—34no—
177C2×C4×Q8—32no—
178C2×C4×D4—32no—
179C2×C4×D4C2×D311yescharacteristic
180C2×#117C2×D311yescharacteristic
181C2×#117C2×D311yescharacteristic
182#399—34no—
183#398—34no—
184#397—34no—
185#397—34no—
186#396—34no—
187#394—34no—
188#394—34no—
189#394—34no—
190#392—34no—
191#392—34no—
192#391—32no—
193#390—34no—
194#390—34no—
195#390—34no—
196#390—34no—
197#390—34no—
198#390—32no—
199#390C2×D312yes—
200#389—32no—
201#389C2×D312yes—
202#388—34no—
203#388—34no—
204#388—34no—
205#388C2×D312yes—
206#388C2×D312yes—
207#387—34no—
208#386—34no—
209#385—34no—
210#382—31no—
211#380—34no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—34no—
214C4×C4⋊C4C2×D311yescharacteristic
215C4×C22⋊C4—34no—
216C4×C22⋊C4C2×D311yescharacteristic
217C4×C22⋊C4C2×D311yescharacteristic
218C43C2×D311yescharacteristic
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—22no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2412yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2411yescharacteristic
239C2×#67—28no—
240C2×#67—44no—
241C2×#67—44no—
242C2×#67—44no—
243C2×#67—44no—
244C2×#67—44no—
245C22×Dic3C2414yes—
246C22×Dic3C2414yes—
247C22×Dic3C2414yes—
248C2×D12—24no—
249C2×D12—24no—
250C2×D12—24no—
251C2×D12—28no—
252C2×D12—44no—
253C2×D12—44no—
254C2×C4×D3—28no—
255C2×C4×D3—44no—
256C2×C4×D3—44no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—24no—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—42no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—42no—
269C3×C22⋊C4—42no—
270C3×C42—22no—
271C3×C42—22no—
272C3×C42—22no—
273C3×C42—42no—
274C3×C42—42no—
275C3×C42C2×D414yes—
276C3×C42#4114yes—
277#217—28no—
278#217—44no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—28no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—28no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288C4⋊Dic3—28no—
289C4⋊Dic3—24no—
290C4⋊Dic3C2×D418yes—
291Dic3⋊C4—28no—
292Dic3⋊C4—28no—
293Dic3⋊C4—44no—
294Dic3⋊C4—44no—
295Dic3⋊C4C2×D4116yes—
296C4×Dic3—28no—
297C4×Dic3—44no—
298C22×Q8—32no—
299C22×D4—34no—
300C22×D4—34no—
301C22×D4—32no—
302C22×D4—34no—
303C22×D4C22×D311yescharacteristic
304C23×C4—34no—
305C23×C4C22×D311yescharacteristic
306C23×C4C22×D311yescharacteristic
307C4⋊Q8—64no—
308#127—64no—
309#126—616no—
310#125—38no—
311#125—616no—
312#124—68no—
313#124—124no—
314#123—616no—
315C22⋊Q8—68no—
316C22⋊Q8—124no—
317#121—616no—
318#121—68no—
319#121—124no—
320C4×Q8—68no—
321C4×D4—24no—
322C4×D4—68no—
323#117—24no—
324#117—24no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4C22×D312yes—
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⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4C22×D312yes—
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D311yescharacteristic
351C2×C22⋊C4C22×D311yescharacteristic
352C2×C22⋊C4C22×D312yes—
353C2×C22⋊C4C22×D312yes—
354C2×C42—34no—
355C2×C42—34no—
356C2×C42C22×D311yescharacteristic
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359C2×C42C22×D312yes—
360C2×C42C22×D312yes—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—34no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95—34no—
368#95C22×D312yes—
369#95C22×D312yes—
370#95C22×D312yes—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3C2511yescharacteristic
382C22×D3—24no—
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—24no—
389C3×D4—44no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
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—44no—
410C2×C3×C4—44no—
411C2×C3×C4C22×D414yes—
412C2×C3×C4C22×D412yes—
413C2×C3×C4C2×#4112yes—
414C2×C3×C4C2×#4112yes—
415C2×C3×C4C2×#4112yes—
416C2×C3×C4H(5,𝔽2)12yes—
417#67—88no—
418#67—88no—
419C2×Dic3—24no—
420C2×Dic3—24no—
421C2×Dic3—24no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3C22×D418yes—
428D12—48no—
429D12—88no—
430C4×D3—88no—
431Dic6—48no—
432C24—34no—
433C24C23×D311yescharacteristic
434C24C23×D311yescharacteristic
435C2×Q8—64no—
436C2×Q8—64no—
437C2×Q8—64no—
438C2×D4—22no—
439C2×D4—42no—
440C2×D4—68no—
441C2×D4—64no—
442C2×D4—64no—
443C2×D4—68no—
444C2×D4—64no—
445C2×D4—124no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—124no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—34no—
459C22×C4—34no—
460C22×C4—34no—
461C22×C4—68no—
462C22×C4—124no—
463C22×C4—124no—
464C22×C4C23×D312yes—
465C22×C4C23×D312yes—
466C22×C4C23×D312yes—
467C22×C4C23×D311yescharacteristic
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D311yescharacteristic
470C22×C4C23×D311yescharacteristic
471C22×C4C23×D312yes—
472C22×C4C23×D311yescharacteristic
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—38no—
477C4⋊C4—316no—
478C4⋊C4—42no—
479C4⋊C4—68no—
480C4⋊C4—68no—
481C4⋊C4—68no—
482C4⋊C4—64no—
483C4⋊C4—124no—
484C4⋊C4—124no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—68no—
492C22⋊C4—68no—
493C22⋊C4—68no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—124no—
497C22⋊C4—124no—
498C22⋊C4—124no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—42no—
506C42—42no—
507C42—68no—
508C42—124no—
509C42#23514yes—
510C42D3×D414yes—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3C23×D411yescharacteristic
522C22×C3C22×#4111yescharacteristic
523C22×C3C2×H(5,𝔽2)11yescharacteristic
524C22×C3#58512yes—
525C22×C3#58512yes—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—44no—
533C3×C4—22no—
534C3×C4—24no—
535C3×C4—22no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—42no—
539C3×C4—42no—
540C3×C4—42no—
541C3×C4#55014yes—
542Dic3—28no—
543Dic3—44no—
544Dic3—44no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—64no—
556C23—64no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23C24×D311yescharacteristic
563Q8—128no—
564D4—44no—
565D4—128no—
566D4—248no—
567D4—248no—
568D4—248no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—38no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—248no—
600C2×C4C2×#23512yes—
601C2×C4C2×#23512yes—
602C2×C4C2×#23512yes—
603C2×C4C2×D3×D414yes—
604C2×C4C2×D3×D412yes—
605C2×C4#100912yes—
606C2×C3—81no—
607C2×C3—81no—
608C2×C3#345511yescharacteristic
609C2×C3C2×#55011yescharacteristic
610C2×C3#350012yes—
611C2×C3#351211yescharacteristic
612C2×C3#356212yes—
613D3—84no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—124no—
625C22—124no—
626C22—124no—
627C22—124no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22C22×#23511yescharacteristic
632C22C22×D3×D411yescharacteristic
633C22C2×#100911yescharacteristic
634C22#596412yes—
635C22#596412yes—
636C4—22no—
637C4—24no—
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—68no—
645C4—124no—
646C4—124no—
647C4#555314yes—
648C3#3938211yescharacteristic, complemented, Sylow
649C2—81no—
650C2—81no—
651C2—244no—
652C2#8573811yescharacteristic
653C2C2×#555311yescharacteristic
654C2#8585111yescharacteristic
655C2#8617912yes—
656C2#8624412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2293C2
hypercenter#2293D3
hypocenterC3#39382
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. C2×#5535C2semidirect
2. C2×#5518C2semidirect
3. C3×#2293C2semidirect
4. #82377C2semidirect
5. #82286C2semidirect
6. #82262C2semidirect
7. #76395C2semidirect
8. #76110C2semidirect
9. #75596C2semidirect
10. #75545C2semidirect
11. #73655C2semidirect
12. #73486C2semidirect
13. #73485C2semidirect
14. #73313C2semidirect
15. C4×#870C2semidirect
88. #2293D3semidirect
151. #870D4semidirect
648. C3#39382semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C2×#55182
C3×#22932
#823772
#822862
#822622
#763952
#761102
#755962
#755452
#736552
#734862
#734852
#733132
C4×#8702
#393823