Finite Group Databasebeta

← #11233305

Subgroups — #11233305

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×#5570C211yesmaximal, characteristic, complemented
2C2×#5551C211yesmaximal, characteristic
3C3×#2398C211yesmaximal, characteristic, complemented
4#82501C211yesmaximal, characteristic, complemented
5#82375C211yesmaximal, characteristic, complemented
6#78141C214yesmaximal, complemented
7#77575C214yesmaximal, complemented
8#77033C214yesmaximal, complemented
9#76785C214yesmaximal, complemented
10#73489C212yesmaximal, complemented
11#73488C212yesmaximal, complemented
12#73484C212yesmaximal, complemented
13#73454C212yesmaximal, complemented
14#73388C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic, complemented
16#45335—31nomaximal, Sylow
17C22×#235C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×Q8×Dic3C2212yes—
21C2×#938C2212yes—
22C2×#924C2212yes—
23C2×#924C2212yes—
24#5570—28no—
25#5551—28no—
26C2×D3×C4⋊C4C2212yes—
27C2×#890C2212yes—
28C2×#890C2214yes—
29C2×#883C2214yes—
30C2×#875C2211yescharacteristic
31C2×C4×D12C2211yescharacteristic
32C2×#869C2212yes—
33C2×#869C2211yescharacteristic
34C2×C4×Dic6C2212yes—
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#400C2212yes—
38C3×#397C2212yes—
39C3×#394C2212yes—
40C3×#393C2212yes—
41C3×C4×C22⋊C4C2212yes—
42C3×C4×C22⋊C4C2212yes—
43C3×#376C2211yescharacteristic
44#5231C2212yes—
45#5222C2212yes—
46#5212C2212yes—
47#4993C2212yes—
48#4989C2214yes—
49#4983C2212yes—
50#4961C2214yes—
51#4961C2214yes—
52#4957C2214yes—
53#4957C2214yes—
54#4951C2214yes—
55#4951C2214yes—
56#4949C2214yes—
57#4948C2214yes—
58#4944C2214yes—
59#4939C2212yes—
60#4939C2211yescharacteristic
61#4936C2211yescharacteristic
62#4935C2211yescharacteristic
63#4929C2212yes—
64#4929C2211yescharacteristic
65#4928C2212yes—
66#4670C2214yes—
67#4668C2214yes—
68#4668C2214yes—
69#4668C2214yes—
70#4667C2214yes—
71#4666C2214yes—
72#4657C2214yes—
73#4657C2214yes—
74#4654C2214yes—
75#4653C2214yes—
76#4653C2214yes—
77#4647C2214yes—
78C2×#557—31no—
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#2398D311yescharacteristic, complemented
90#2341—32no—
91#2327—31no—
92#2326—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C2×#235—28no—
98C2×#235—28no—
99C2×#235—44no—
100C22×D12C2311yescharacteristic
101C22×C4×D3C2314yes—
102C22×C4×D3C2312yes—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2312yes—
105C3×C4×Q8—24no—
106C3×C4×D4—24no—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×#217C2314yes—
120C2×#217C2314yes—
121C2×#217C2314yes—
122Q8×Dic3—28no—
123#938—28no—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137#924—28no—
138#924—44no—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2312yes—
147C2×C4×Dic3C2314yes—
148D3×C4⋊C4—28no—
149#890—28no—
150#890—216no—
151#883—216no—
152#875—24no—
153C4×D12—24no—
154#869—28no—
155#869D418yes—
156C4×Dic6—28no—
157C4×Dic6—24no—
158C4×Dic6—44no—
159C3×#95C2312yes—
160C3×#95C2312yes—
161C3×#95C2312yes—
162C3×#95C2312yes—
163C3×#95C2312yes—
164C3×#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C22×#41—32no—
173#557—68no—
174#547—68no—
175C2×C4⋊Q8—31no—
176C2×C4⋊Q8—32no—
177C2×#125—32no—
178C2×#125—34no—
179C2×#124—31no—
180C2×#123—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—32no—
183C2×#121—32no—
184C2×C4×Q8—32no—
185C2×C4×Q8—31no—
186C2×C4×Q8—32no—
187C2×C4×Q8—32no—
188C2×C4×Q8C2×D311yescharacteristic
189C2×C4×D4—31no—
190C2×C4×D4C2×D311yescharacteristic
191C22×C4⋊C4—32no—
192#400C2×D312yes—
193#399—32no—
194#399—34no—
195#398—32no—
196#398—34no—
197#398—34no—
198#397C2×D312yes—
199#396—34no—
200#394—32no—
201#394—34no—
202#394—34no—
203#394C2×D312yes—
204#393—32no—
205#393C2×D312yes—
206#391—31no—
207#391—32no—
208#390—34no—
209#390—34no—
210#390—32no—
211#390—31no—
212#389—32no—
213#389—34no—
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—34no—
230C4×C22⋊C4C2×D312yes—
231C4×C22⋊C4C2×D312yes—
232#376C2×D311yescharacteristic
233C24×C3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—22no—
236C2×C3×Q8—24no—
237C2×C3×D4—22no—
238C2×C3×D4—42no—
239C22×C3×C4—24no—
240C22×C3×C4—24no—
241C22×C3×C4—24no—
242C22×C3×C4C2412yes—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2412yes—
249C22×C3×C4C2411yescharacteristic
250C2×#67—28no—
251C2×#67—28no—
252C2×#67—44no—
253C22×Dic3C2414yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C22×Dic3C2414yes—
258#235—432no—
259C2×D12—22no—
260C2×D12—22no—
261C2×D12#4114yes—
262C2×C4×D3—28no—
263C2×C4×D3—24no—
264C2×C4×D3—28no—
265C2×C4×D3—44no—
266C2×C4×D3—44no—
267C2×Dic6—24no—
268C2×Dic6—24no—
269C2×Dic6—22no—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6C2×D418yes—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—22no—
276C3×C22⋊C4—24no—
277C3×C22⋊C4—24no—
278C3×C22⋊C4—24no—
279C3×C22⋊C4—42no—
280C3×C22⋊C4—42no—
281C3×C42—24no—
282C3×C42—22no—
283C3×C42—22no—
284C3×C42C2×D414yes—
285#217—28no—
286#217—44no—
287#217—44no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—44no—
292(C2×D3)⋊C4—44no—
293C4⋊Dic3—22no—
294C4⋊Dic3—22no—
295C4⋊Dic3—24no—
296C4⋊Dic3—28no—
297C4⋊Dic3—28no—
298C4⋊Dic3—44no—
299C4⋊Dic3C2×D418yes—
300C4⋊Dic3C2×D418yes—
301Dic3⋊C4—24no—
302Dic3⋊C4—28no—
303Dic3⋊C4—28no—
304Dic3⋊C4—44no—
305Dic3⋊C4—44no—
306Dic3⋊C4—44no—
307C4×Dic3—24no—
308C4×Dic3—28no—
309C4×Dic3—44no—
310C2×#41—68no—
311C2×#41—68no—
312C2×#41—124no—
313C22×Q8—32no—
314C22×Q8—31no—
315C22×Q8C22×D311yescharacteristic
316C22×D4—34no—
317C22×D4—31no—
318C22×D4C22×D311yescharacteristic
319C23×C4—34no—
320C23×C4—32no—
321C23×C4C22×D312yes—
322C4⋊Q8—38no—
323C4⋊Q8—68no—
324#125—616no—
325#125—68no—
326#124—64no—
327#123—616no—
328C22⋊Q8—68no—
329C22⋊Q8—124no—
330#121—68no—
331C4×Q8—24no—
332C4×Q8—68no—
333C4×Q8—68no—
334C4×Q8—64no—
335C4×Q8—124no—
336C4×D4—24no—
337C4×D4—64no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—68no—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C22⋊C4—34no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4C22×D312yes—
365C2×C22⋊C4C22×D312yes—
366C2×C22⋊C4C22×D312yes—
367C2×C22⋊C4C22×D312yes—
368C2×C22⋊C4C22×D312yes—
369C2×C42—34no—
370C2×C42—32no—
371C2×C42—34no—
372C2×C42C22×D311yescharacteristic
373C2×C42C22×D312yes—
374C2×C42C22×D311yescharacteristic
375C2×C42C22×D311yescharacteristic
376#95—34no—
377#95—32no—
378#95—32no—
379#95—34no—
380#95—34no—
381#95—34no—
382#95—34no—
383#95C22×D312yes—
384#95C22×D312yes—
385#95C22×D312yes—
386#95C22×D312yes—
387#95C22×D312yes—
388#95C22×D312yes—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—22no—
396C23×C3C2511yescharacteristic
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—24no—
400C22×D3—24no—
401C22×D3C2×#4114yes—
402C3×Q8—28no—
403C3×D4—44no—
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—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—48no—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×D412yes—
423C2×C3×C4C22×D412yes—
424C2×C3×C4C2×#4112yes—
425C2×C3×C4H(5,𝔽2)12yes—
426C2×C3×C42−1+414yes—
427#67—416no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—24no—
435C2×Dic3—22no—
436C2×Dic3—24no—
437C2×Dic3—24no—
438C2×Dic3—24no—
439C2×Dic3—24no—
440C2×Dic3C22×D418yes—
441D12—28no—
442C4×D3—416no—
443C4×D3—48no—
444Dic6—216no—
445Dic6—44no—
446C24—32no—
447C24C23×D312yes—
448#41—1232no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—38no—
452C2×Q8—62no—
453C2×Q8—64no—
454C2×Q8—62no—
455C2×Q8—64no—
456C2×Q8—62no—
457C2×D4—22no—
458C2×D4—34no—
459C2×D4—42no—
460C2×D4—68no—
461C2×D4—68no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—124no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—34no—
469C22×C4—34no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—68no—
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×D312yes—
485C22×C4C23×D311yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—38no—
490C4⋊C4—38no—
491C4⋊C4—68no—
492C4⋊C4—68no—
493C4⋊C4—68no—
494C4⋊C4—68no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—124no—
500C4⋊C4—124no—
501C4⋊C4—124no—
502C4⋊C4—124no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—68no—
509C22⋊C4—68no—
510C22⋊C4—64no—
511C22⋊C4—68no—
512C22⋊C4—124no—
513C22⋊C4—124no—
514C22⋊C4—124no—
515C22⋊C4—124no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—68no—
520C42—64no—
521C42—124no—
522C42D3×D414yes—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3C23×D411yescharacteristic
531C22×C3C22×#4111yescharacteristic
532C22×C3C2×H(5,𝔽2)11yescharacteristic
533C22×C3C2×2−1+412yes—
534C22×C3#58512yes—
535C2×D3—24no—
536C2×D3—28no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—44no—
540C3×C4—22no—
541C3×C4—24no—
542C3×C4—22no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4#54614yes—
548Dic3—28no—
549Dic3—42no—
550Dic3—42no—
551Dic3—42no—
552Dic3—44no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—34no—
561C23—62no—
562C23—62no—
563C23—64no—
564C23—64no—
565C23C24×D311yescharacteristic
566Q8—28no—
567Q8—616no—
568Q8—124no—
569D4—44no—
570D4—68no—
571D4—1216no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—38no—
589C2×C4—48no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—128no—
603C2×C4—1216no—
604C2×C4C2×D3×D412yes—
605C2×C4C2×D3×D412yes—
606C2×C4C2×D3×D412yes—
607C2×C4D3×#4112yes—
608C2×C4#100912yes—
609C2×C4#101014yes—
610C2×C3—82no—
611C2×C3C2×#54611yescharacteristic
612C2×C3#348611yescharacteristic
613C2×C3#348711yescharacteristic
614C2×C3#351512yes—
615C2×C3#355412yes—
616D3—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—64no—
625C22—68no—
626C22—122no—
627C22—122no—
628C22—124no—
629C22C22×D3×D411yescharacteristic
630C22C2×D3×#4111yescharacteristic
631C22C2×#100911yescharacteristic
632C22C2×#101012yes—
633C22#596412yes—
634C4—22no—
635C4—24no—
636C4—22no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—68no—
642C4—122no—
643C4—122no—
644C4—122no—
645C4—124no—
646C4#567614yes—
647C3#4533511yescharacteristic, complemented, Sylow
648C2—82no—
649C2—124no—
650C2#8602211yescharacteristic
651C2#8602311yescharacteristic
652C2C2×#567611yescharacteristic
653C2#8613712yes—
654C2#8616512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2398C2
prime radicalC2×#5551C2
hypercenter#2398D3
hypocenterC3#45335
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. C2×#5570C2semidirect
3. C3×#2398C2semidirect
4. #82501C2semidirect
5. #82375C2semidirect
6. #78141C2semidirect
7. #77575C2semidirect
8. #77033C2semidirect
9. #76785C2semidirect
10. #73489C2semidirect
11. #73488C2semidirect
12. #73484C2semidirect
13. #73454C2semidirect
14. #73388C2semidirect
15. #73387C2semidirect
89. #2398D3semidirect
647. C3#45335semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
C2×#55512
C3×#23982
#825012
#823752
#781412
#775752
#770332
#767852
#734892
#734882
#734842
#734542
#733882
#733872
#453353