Finite Group Databasebeta

← #11233295

Subgroups — #11233295

Order 768.

2282 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5726C211yesmaximal, characteristic
3#83255C212yesmaximal, complemented
4#83254C212yesmaximal, complemented
5#83227C211yesmaximal, characteristic, complemented
6#83070C214yesmaximal, complemented
7#83046C214yesmaximal, complemented
8#83032C214yesmaximal, complemented
9#82501C211yesmaximal, characteristic, complemented
10#82499C211yesmaximal, characteristic, complemented
11#82497C214yesmaximal, complemented
12#82494C212yesmaximal, complemented
13#82492C211yesmaximal, characteristic, complemented
14#82457C212yesmaximal, complemented
15#82455C211yesmaximal, characteristic, complemented
16#45335—31nomaximal, Sylow
17C22×#239C2212yes—
18C3×#557—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#125C2214yes—
22C2×C3×#125C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2212yes—
27C2×#947C2211yescharacteristic
28C2×#946C2212yes—
29C2×#946C2212yes—
30C2×Q8×Dic3C2212yes—
31#5726—28no—
32C2×#895C2212yes—
33C2×#894C2214yes—
34C2×D3×C4⋊C4C2212yes—
35C2×C4×D12C2211yescharacteristic
36C2×C4×D12C2211yescharacteristic
37#5231C2212yes—
38#5231C2212yes—
39#5229C2211yescharacteristic
40#5229C2212yes—
41#4994C2214yes—
42#4993C2214yes—
43#4993C2212yes—
44#4992C2212yes—
45#4991C2214yes—
46#4991C2214yes—
47#4991C2212yes—
48#4991C2212yes—
49#4990C2214yes—
50#4989C2214yes—
51#4989C2214yes—
52#4989C2214yes—
53#4989C2214yes—
54#4988C2211yescharacteristic
55#4988C2214yes—
56#4987C2212yes—
57#4987C2214yes—
58#4986C2212yes—
59#4983C2214yes—
60#4982C2214yes—
61#4981C2212yes—
62#4980C2214yes—
63#4980C2214yes—
64#4976C2214yes—
65#4975C2211yescharacteristic
66#4940C2214yes—
67#4940C2214yes—
68#4940C2214yes—
69#4939C2212yes—
70#4939C2214yes—
71C4×(C2×D3)⋊C4C2212yes—
72C4×(C2×D3)⋊C4C2214yes—
73C4×(C2×D3)⋊C4C2212yes—
74#4936C2211yescharacteristic
75#4933C2214yes—
76#4932C2211yescharacteristic
77#4928C2212yes—
78C2×#557D311yescharacteristic, complemented
79C2×#547—31no—
80#2711—34no—
81#2694—34no—
82#2580—31no—
83#2536—31no—
84#2473—34no—
85#2458—34no—
86#2432—32no—
87#2429—32no—
88#2423—32no—
89#2398—31no—
90#2341—32no—
91#2327—31no—
92#2326—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×Q8C2311yescharacteristic
95C22×C3×Q8C2312yes—
96C2×#239—28no—
97C2×#239—28no—
98C2×#239—44no—
99C22×D12C2314yes—
100C22×D12C2311yescharacteristic
101C22×D12C2311yescharacteristic
102C22×C4×D3C2312yes—
103C22×C4×D3C2314yes—
104C22×C4×D3C2312yes—
105C3×C4⋊Q8—28no—
106C3×C4⋊Q8D418yes—
107C3×#125—216no—
108C3×#125—28no—
109C3×C4×Q8—28no—
110C3×C4×Q8—28no—
111C3×C4×Q8—24no—
112C3×C4×Q8—24no—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C42C2312yes—
123C2×C3×C42C2314yes—
124C2×C3×C42C2312yes—
125C2×C3×C42C2311yescharacteristic
126#947—24no—
127#946—28no—
128#946—44no—
129Q8×Dic3—44no—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2312yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2312yes—
139C2×(C2×D3)⋊C4C2312yes—
140C2×(C2×D3)⋊C4C2314yes—
141C2×(C2×D3)⋊C4C2314yes—
142C2×(C2×D3)⋊C4C2314yes—
143C2×C4⋊Dic3C2311yescharacteristic
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2311yescharacteristic
146C2×C4⋊Dic3C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×C4×Dic3C2311yescharacteristic
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2314yes—
154#895—28no—
155#894—216no—
156D3×C4⋊C4—28no—
157C4×D12—24no—
158C4×D12—24no—
159C3⋊#95C2314yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2312yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C22×#41—32no—
173#557—28no—
174#547—68no—
175C2×C4⋊Q8C2×D312yes—
176C2×C4⋊Q8C2×D311yescharacteristic
177C2×#125C2×D312yes—
178C2×#125C2×D314yes—
179C2×#124—31no—
180C2×#123—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—32no—
183C2×#121—32no—
184C2×C4×Q8—32no—
185C2×C4×Q8C2×D311yescharacteristic
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×Q8C2×D312yes—
188C2×C4×Q8C2×D312yes—
189C2×C4×D4—31no—
190C2×C4×D4—31no—
191C22×C4⋊C4—32no—
192#400—32no—
193#399—32no—
194#399—34no—
195#398—32no—
196#398—34no—
197#398—34no—
198#397—32no—
199#396—34no—
200#394—32no—
201#394—34no—
202#394—34no—
203#394—32no—
204#393—32no—
205#393—32no—
206#391—31no—
207#391—32no—
208#390—31no—
209#390—34no—
210#390—32no—
211#390—34no—
212#389—34no—
213#389—32no—
214#388—34no—
215#388—34no—
216#388—34no—
217#388—34no—
218#387—34no—
219#387—34no—
220#387—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382—31no—
227#380—32no—
228#380—34no—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—34no—
231C4×C22⋊C4—32no—
232#376—31no—
233C23×D3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—22no—
236C2×C3×Q8—22no—
237C2×C3×Q8—24no—
238C2×C3×Q8—22no—
239C2×C3×Q8—24no—
240C2×C3×Q8—22no—
241C2×C3×Q8—24no—
242C2×C3×Q8C2×D418yes—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2414yes—
249C22×C3×C4C2412yes—
250C22×C3×C4C2411yescharacteristic
251C22×Dic3C2412yes—
252C22×Dic3C2412yes—
253C22×Dic3C2414yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256#239—432no—
257C2×D12—22no—
258C2×D12—22no—
259C2×D12—28no—
260C2×D12—22no—
261C2×D12—28no—
262C2×D12—42no—
263C2×D12—44no—
264C2×D12#4114yes—
265C2×C4×D3—28no—
266C2×C4×D3—24no—
267C2×C4×D3—24no—
268C2×C4×D3—28no—
269C2×C4×D3—24no—
270C2×C4×D3—24no—
271C2×C4×D3—44no—
272C2×C4×D3—44no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—28no—
276C3×C4⋊C4—24no—
277C3×C4⋊C4—22no—
278C3×C4⋊C4—28no—
279C3×C4⋊C4—28no—
280C3×C4⋊C4C2×D418yes—
281C3×C4⋊C4C2×D418yes—
282C3×C42—28no—
283C3×C42—24no—
284C3×C42—24no—
285C3×C42C2×D414yes—
286(C2×D3)⋊C4—28no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—24no—
289(C2×D3)⋊C4—28no—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—42no—
294(C2×D3)⋊C4—44no—
295(C2×D3)⋊C4—44no—
296(C2×D3)⋊C4—42no—
297(C2×D3)⋊C4—44no—
298(C2×D3)⋊C4—44no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3—24no—
302C4⋊Dic3—44no—
303Dic3⋊C4—28no—
304Dic3⋊C4—44no—
305Dic3⋊C4—44no—
306Dic3⋊C4—44no—
307C4×Dic3—22no—
308C4×Dic3—22no—
309C4×Dic3—44no—
310C2×#41—68no—
311C2×#41—68no—
312C2×#41—124no—
313C22×Q8C22×D311yescharacteristic
314C22×Q8C22×D311yescharacteristic
315C22×Q8C22×D312yes—
316C22×D4—31no—
317C22×D4—34no—
318C22×D4—31no—
319C23×C4—32no—
320C23×C4—34no—
321C23×C4—32no—
322C4⋊Q8—28no—
323C4⋊Q8#6718yes—
324#125—216no—
325#125—28no—
326#124—64no—
327#123—616no—
328C22⋊Q8—68no—
329C22⋊Q8—124no—
330#121—68no—
331C4×Q8—28no—
332C4×Q8—28no—
333C4×Q8—24no—
334C4×Q8—24no—
335C4×Q8—124no—
336C4×D4—64no—
337C4×D4—64no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—68no—
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C4⋊C4C22×D314yes—
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D312yes—
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D314yes—
353C2×C4⋊C4C22×D314yes—
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D312yes—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—32no—
366C2×C22⋊C4—32no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C42—31no—
370C2×C42—31no—
371C2×C42—34no—
372C2×C42C22×D314yes—
373C2×C42C22×D312yes—
374C2×C42C22×D312yes—
375C2×C42C22×D311yescharacteristic
376#95—32no—
377#95—32no—
378#95—32no—
379#95—34no—
380#95—32no—
381#95—32no—
382#95—32no—
383#95—32no—
384#95—32no—
385#95—34no—
386#95—34no—
387#95—34no—
388#95—34no—
389C23×C3C2511yescharacteristic
390C22×D3—22no—
391C22×D3—22no—
392C22×D3—24no—
393C22×D3—24no—
394C22×D3—22no—
395C22×D3—22no—
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—22no—
401C22×D3C2×#4114yes—
402C3×Q8—28no—
403C3×Q8—216no—
404C3×Q8—44no—
405C2×C3×C4—22no—
406C2×C3×C4—24no—
407C2×C3×C4—24no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—24no—
418C2×C3×C4—22no—
419C2×C3×C4C22×D418yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×D412yes—
423C2×C3×C4C2×#4112yes—
424C2×C3×C4H(5,𝔽2)12yes—
425C2×C3×C42−1+414yes—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—24no—
436C2×Dic3—22no—
437C2×Dic3—24no—
438C2×Dic3—24no—
439C2×Dic3—22no—
440D12—28no—
441D12—416no—
442D12—44no—
443C4×D3—416no—
444C4×D3—48no—
445C4×D3—48no—
446C24—32no—
447C24—32no—
448#41—1232no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8C2×#6718yes—
457C2×D4—34no—
458C2×D4—68no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—62no—
462C2×D4—68no—
463C2×D4—124no—
464C2×D4—122no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—34no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—68no—
471C22×C4—64no—
472C22×C4—64no—
473C22×C4—64no—
474C22×C4—68no—
475C22×C4—64no—
476C22×C4—124no—
477C22×C4—124no—
478C22×C4C23×D312yes—
479C22×C4C23×D312yes—
480C22×C4C23×D312yes—
481C22×C4C23×D312yes—
482C22×C4C23×D311yescharacteristic
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D314yes—
485C22×C4C23×D311yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—64no—
494C4⋊C4—62no—
495C4⋊C4—68no—
496C4⋊C4—62no—
497C4⋊C4—124no—
498C4⋊C4—124no—
499C4⋊C4—124no—
500C4⋊C4—124no—
501C4⋊C4C2×#6718yes—
502C4⋊C4C2×#6718yes—
503C22⋊C4—68no—
504C22⋊C4—68no—
505C22⋊C4—64no—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—68no—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—122no—
513C22⋊C4—124no—
514C22⋊C4—124no—
515C22⋊C4—122no—
516C42—28no—
517C42—24no—
518C42—24no—
519C42—62no—
520C42—62no—
521C42—124no—
522C42C2×#6714yes—
523C22×C3C23×D411yescharacteristic
524C22×C3C22×#4111yescharacteristic
525C22×C3C2×H(5,𝔽2)11yescharacteristic
526C22×C3C2×2−1+412yes—
527C22×C3#58512yes—
528C2×D3—24no—
529C2×D3—28no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—44no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C3×C4—28no—
541C3×C4—22no—
542C3×C4—24no—
543C3×C4—22no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4#54614yes—
548Dic3—42no—
549Dic3—42no—
550Dic3—42no—
551Dic3—42no—
552Dic3—44no—
553C23—34no—
554C23—64no—
555C23—62no—
556C23—64no—
557C23—62no—
558C23—62no—
559C23—62no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—62no—
565C23C24×D311yescharacteristic
566Q8—28no—
567Q8—216no—
568Q8—44no—
569D4—68no—
570D4—1216no—
571D4—124no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—1216no—
601C2×C4—128no—
602C2×C4—128no—
603C2×C4#100714yes—
604C2×C4D3×#4112yes—
605C2×C4#100912yes—
606C2×C4C22×#6712yes—
607C2×C4C22×#6718yes—
608C2×C4C22×#6712yes—
609C2×C4C22×#6712yes—
610C2×C3C2×#54611yescharacteristic
611C2×C3#348611yescharacteristic
612C2×C3#348711yescharacteristic
613C2×C3#351512yes—
614C2×C3#355412yes—
615D3—44no—
616D3—82no—
617C22—64no—
618C22—68no—
619C22—122no—
620C22—122no—
621C22—124no—
622C22—122no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22—122no—
629C22C2×#100712yes—
630C22C2×D3×#4111yescharacteristic
631C22C2×#100911yescharacteristic
632C22#596812yes—
633C22C23×#6711yescharacteristic
634C4—24no—
635C4—22no—
636C4—28no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4—122no—
645C4—124no—
646C4#582914yes—
647C3#4533511yescharacteristic, complemented, Sylow
648C2—124no—
649C2—242no—
650C2#8702112yes—
651C2#8703512yes—
652C2#8725111yescharacteristic
653C2C2×#582911yescharacteristic
654C2#8729011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#557C2
prime radicalC2×#5726C2
hypercenterC2×#557D3
hypocenterC3#45335
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
3. #83255C2semidirect
4. #83254C2semidirect
5. #83227C2semidirect
6. #83070C2semidirect
7. #83046C2semidirect
8. #83032C2semidirect
9. #82501C2semidirect
10. #82499C2semidirect
11. #82497C2semidirect
12. #82494C2semidirect
13. #82492C2semidirect
14. #82457C2semidirect
15. #82455C2semidirect
78. C2×#557D3semidirect
647. C3#45335semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#57262
#832552
#832542
#832272
#830702
#830462
#830322
#825012
#824992
#824972
#824942
#824922
#824572
#824552
#453353