Finite Group Databasebeta

← #11025200

Subgroups — #11025200

Order 768.

2328 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5535C211yesmaximal, characteristic, complemented
2C2×#5470C211yesmaximal, characteristic, complemented
3C3×#2290C211yesmaximal, characteristic, complemented
4#82350C212yesmaximal, complemented
5#82265C211yesmaximal, characteristic, complemented
6#82081C211yesmaximal, characteristic, complemented
7#75499C214yesmaximal, complemented
8#74989C214yesmaximal, complemented
9#74916C214yesmaximal, complemented
10#74632C214yesmaximal, complemented
11#73675C212yesmaximal, complemented
12#73507C212yesmaximal, complemented
13#73469C212yesmaximal, complemented
14#73429C211yesmaximal, characteristic, complemented
15#73390C211yesmaximal, characteristic, complemented
16#37710—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×#930C2212yes—
21C2×#930C2212yes—
22C2×#930C2214yes—
23C2×#926C2212yes—
24C2×#924C2214yes—
25#5535—28no—
26C2×#896C2214yes—
27C2×#894C2214yes—
28C2×#885C2214yes—
29C2×#884C2214yes—
30C2×#874C2211yescharacteristic
31C2×C4×D12C2212yes—
32#5470—28no—
33C2×#870C2211yescharacteristic
34C2×#870C2211yescharacteristic
35C2×#869C2212yes—
36C2×C4×Dic6C2214yes—
37C3×#388C2212yes—
38C3×#387C2212yes—
39C3×#386C2212yes—
40C3×#385C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#376C2211yescharacteristic
45#5223C2214yes—
46#5212C2212yes—
47#4994C2214yes—
48#4990C2214yes—
49#4982C2214yes—
50#4961C2214yes—
51#4959C2214yes—
52#4957C2214yes—
53#4957C2214yes—
54#4956C2214yes—
55#4948C2214yes—
56#4939C2212yes—
57#4939C2212yes—
58#4939C2214yes—
59#4939C2212yes—
60#4935C2211yescharacteristic
61C4×C4⋊Dic3C2212yes—
62#4928C2212yes—
63#4928C2212yes—
64#4673C2214yes—
65#4673C2214yes—
66#4672C2214yes—
67#4672C2214yes—
68#4670C2214yes—
69#4669C2214yes—
70#4668C2214yes—
71#4666C2214yes—
72#4655C2214yes—
73#4647C2214yes—
74C2×#540—31no—
75C2×#533—31no—
76#2856—31no—
77#2681—34no—
78#2676—34no—
79#2641—31no—
80#2629—31no—
81#2491—32no—
82#2458—34no—
83#2452—32no—
84#2426—34no—
85#2414—32no—
86#2363—32no—
87#2330—31no—
88#2290D311yescharacteristic, complemented
89C22×C3×D4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2312yes—
92C23×C3×C4C2311yescharacteristic
93C22×#67C2314yes—
94C22×#67C2314yes—
95C22×D12C2312yes—
96C22×D12C2314yes—
97C22×C4×D3C2314yes—
98C22×Dic6C2314yes—
99C3×C4×D4—24no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C42—24no—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×#217C2314yes—
117C2×#217C2314yes—
118#930—28no—
119#930—28no—
120#930—48no—
121C2×(C2×D3)⋊C4C2314yes—
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—
130#926—28no—
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2312yes—
136#924—216no—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143#896—48no—
144#894—216no—
145#885—48no—
146#884—216no—
147#874—24no—
148C4×D12—28no—
149#870—24no—
150#870D418yescomplemented
151#869—28no—
152C4×Dic6—216no—
153C3×#95C2312yes—
154C3×#95C2312yes—
155C3×#95C2312yes—
156C3×#95C2312yes—
157C3×#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#540—68no—
163#533—68no—
164C2×C4⋊Q8—32no—
165C2×#127—31no—
166C2×#125—31no—
167C2×#125—31no—
168C2×#124—34no—
169C2×#123—34no—
170C2×C22⋊Q8—34no—
171C2×C22⋊Q8—34no—
172C2×#121—32no—
173C2×#121—32no—
174C2×#121—34no—
175C2×#121—34no—
176C2×C4×Q8—34no—
177C2×C4×D4—32no—
178C2×C4×D4C2×D311yescharacteristic
179C2×#117—32no—
180C2×#117C2×D311yescharacteristic
181C22×C42C2×D311yescharacteristic
182#398—34no—
183#396—34no—
184#396—34no—
185#394—34no—
186#394—34no—
187#394—34no—
188#394—34no—
189#393—34no—
190#393—34no—
191#392—34no—
192#390—34no—
193#390—34no—
194#390—34no—
195#390—32no—
196#390—32no—
197#390—32no—
198#389—32no—
199#389—32no—
200#389—34no—
201#388—34no—
202#388—34no—
203#388—34no—
204#388C2×D312yes—
205#387—34no—
206#387C2×D312yes—
207#386—34no—
208#386C2×D312yes—
209#385—34no—
210#385C2×D312yes—
211#384—31no—
212#380—32no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4C2×D311yescharacteristic
215C4×C22⋊C4—34no—
216C4×C22⋊C4C2×D311yescharacteristic
217C4×C22⋊C4C2×D311yescharacteristic
218#376C2×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—24no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—24no—
230C22×C3×C4—22no—
231C22×C3×C4—42no—
232C22×C3×C4#4114yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2412yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2412yes—
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2412yes—
241C22×C3×C4C2411yescharacteristic
242C2×#67—28no—
243C2×#67—28no—
244C2×#67—44no—
245C2×#67—44no—
246C2×#67—44no—
247C2×#67—44no—
248C22×Dic3C2414yes—
249C22×Dic3C2414yes—
250C22×Dic3C2414yes—
251C2×D12—24no—
252C2×D12—24no—
253C2×D12—24no—
254C2×D12—28no—
255C2×D12—44no—
256C2×D12—44no—
257C2×C4×D3—28no—
258C2×C4×D3—44no—
259C2×C4×D3—44no—
260C2×Dic6—28no—
261C2×Dic6—28no—
262C2×Dic6—44no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—22no—
265C3×C4⋊C4—24no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—24no—
270C3×C22⋊C4—42no—
271C3×C42—24no—
272C3×C42—22no—
273C3×C42—22no—
274C3×C42—22no—
275C3×C42—22no—
276C3×C42C2×D414yes—
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—28no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288C4⋊Dic3—24no—
289C4⋊Dic3—24no—
290C4⋊Dic3—28no—
291C4⋊Dic3—44no—
292C4⋊Dic3C2×D418yes—
293Dic3⋊C4—28no—
294Dic3⋊C4—44no—
295Dic3⋊C4—44no—
296Dic3⋊C4C2×D4116yes—
297C4×Dic3—28no—
298C4×Dic3—44no—
299C22×Q8—34no—
300C22×D4—34no—
301C22×D4—34no—
302C22×D4—32no—
303C22×D4—34no—
304C22×D4C22×D311yescharacteristic
305C23×C4—34no—
306C23×C4C22×D311yescharacteristic
307C23×C4C22×D312yes—
308C23×C4C22×D311yescharacteristic
309C4⋊Q8—68no—
310#127—64no—
311#125—38no—
312#125—64no—
313#124—128no—
314#123—616no—
315C22⋊Q8—616no—
316C22⋊Q8—128no—
317#121—616no—
318#121—68no—
319#121—68no—
320#121—128no—
321C4×Q8—616no—
322C4×D4—24no—
323C4×D4—68no—
324#117—24no—
325#117—68no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D312yes—
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⋊C4—34no—
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D311yescharacteristic
351C2×C22⋊C4C22×D311yescharacteristic
352C2×C22⋊C4C22×D312yes—
353C2×C22⋊C4C22×D312yes—
354C2×C42—24no—
355C2×C42—34no—
356C2×C42—34no—
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359C2×C42C22×D312yes—
360C2×C42C22×D311yescharacteristic
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
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—22no—
374C23×C3—21no—
375C23×C3—22no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—22no—
379C23×C3—21no—
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×D3—24no—
390C3×D4—44no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—28no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
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—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—48no—
412C2×C3×C4—44no—
413C2×C3×C4C22×D414yes—
414C2×C3×C4C22×D412yes—
415C2×C3×C4C2×#4112yes—
416C2×C3×C4C2×#4112yes—
417C2×C3×C4H(5,𝔽2)14yes—
418C2×C3×C42−1+412yes—
419#67—88no—
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×Dic3C22×D418yes—
430D12—48no—
431D12—88no—
432C4×D3—88no—
433Dic6—416no—
434C24—34no—
435C24C23×D311yescharacteristic
436C24C23×D311yescharacteristic
437C2×Q8—68no—
438C2×Q8—68no—
439C2×Q8—124no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—68no—
443C2×D4—68no—
444C2×D4—64no—
445C2×D4—64no—
446C2×D4—68no—
447C2×D4—64no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—124no—
452C2×D4—124no—
453C2×D4—124no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—34no—
462C22×C4—34no—
463C22×C4—34no—
464C22×C4—42no—
465C22×C4—68no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4#23514yes—
469C22×C4C23×D311yescharacteristic
470C22×C4C23×D311yescharacteristic
471C22×C4C23×D312yes—
472C22×C4C23×D312yes—
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D312yes—
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—38no—
482C4⋊C4—316no—
483C4⋊C4—68no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—68no—
487C4⋊C4—124no—
488C4⋊C4—124no—
489C4⋊C4—124no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—68no—
496C22⋊C4—68no—
497C22⋊C4—68no—
498C22⋊C4—68no—
499C22⋊C4—68no—
500C22⋊C4—68no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—124no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—68no—
512C42—124no—
513C42D3×D414yes—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—42no—
522C22×C3—41no—
523C22×C3—42no—
524C22×C3C23×D411yescharacteristic
525C22×C3C22×#4111yescharacteristic
526C22×C3C2×H(5,𝔽2)12yes—
527C22×C3C2×2−1+411yescharacteristic
528C22×C3#58512yes—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—44no—
535C2×D3—44no—
536C3×C4—22no—
537C3×C4—24no—
538C3×C4—24no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—42no—
542C3×C4—41no—
543C3×C4—42no—
544C3×C4#54814yes—
545Dic3—28no—
546Dic3—44no—
547Dic3—44no—
548C23—21no—
549C23—22no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—21no—
556C23—22no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23C2×#23512yes—
565C23C24×D311yescharacteristic
566Q8—1216no—
567D4—44no—
568D4—128no—
569D4—248no—
570D4—248no—
571D4—248no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
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—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—38no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—248no—
604C2×C4C2×#23512yes—
605C2×C4C2×#23512yes—
606C2×C4C2×D3×D414yes—
607C2×C4C2×D3×D412yes—
608C2×C4#100914yes—
609C2×C4#101012yes—
610C2×C3—42no—
611C2×C3—81no—
612C2×C3#345511yescharacteristic
613C2×C3C2×#54811yescharacteristic
614C2×C3#348812yes—
615C2×C3#350811yescharacteristic
616C2×C3#354712yes—
617D3—84no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—41no—
627C22—41no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22—124no—
632C22—124no—
633C22—124no—
634C22—124no—
635C22C22×#23511yescharacteristic
636C22C22×D3×D411yescharacteristic
637C22C2×#100912yes—
638C22C2×#101011yescharacteristic
639C22#596412yes—
640C4—24no—
641C4—24no—
642C4—22no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—68no—
649C4—124no—
650C4—124no—
651C4#555214yes—
652C3#3771011yescharacteristic, complemented, Sylow
653C2—42no—
654C2—81no—
655C2—244no—
656C2#8573811yescharacteristic
657C2C2×#555211yescharacteristic
658C2#8584811yescharacteristic
659C2#8604112yes—
660C2#8617512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2290C2
hypercenter#2290D3
hypocenterC3#37710
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×#5470C2semidirect
3. C3×#2290C2semidirect
4. #82350C2semidirect
5. #82265C2semidirect
6. #82081C2semidirect
7. #75499C2semidirect
8. #74989C2semidirect
9. #74916C2semidirect
10. #74632C2semidirect
11. #73675C2semidirect
12. #73507C2semidirect
13. #73469C2semidirect
14. #73429C2semidirect
15. #73390C2semidirect
88. #2290D3semidirect
150. #870D4semidirect
652. C3#37710semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C2×#54702
C3×#22902
#823502
#822652
#820812
#754992
#749892
#749162
#746322
#736752
#735072
#734692
#734292
#733902
#377103