Finite Group Databasebeta

← #11069181

Subgroups — #11069181

Order 768.

1940 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5583C212yesmaximal
2C3×#2365C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic
4#83239C211yesmaximal, characteristic, complemented
5#82471C212yesmaximal, complemented
6#76642C212yesmaximal
7#76502C212yesmaximal, complemented
8#76238C212yesmaximal
9#76030C212yesmaximal, complemented
10#75173C214yesmaximal
11#75168C214yesmaximal, complemented
12#73792C212yesmaximal
13C4×#897C211yesmaximal, characteristic
14#73753C211yesmaximal, characteristic
15#73717C211yesmaximal, characteristic, complemented
16#73716C212yesmaximal, complemented
17C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
18#39353—31nomaximal, Sylow
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2212yes—
21C2×#946C2212yes—
22C2×#944C2212yes—
23#5583—216no—
24C2×#898C2214yes—
25C2×#897C2211yescharacteristic
26C2×#890C2212yes—
27C2×#889C2214yes—
28C2×#876C2214yes—
29C2×#872C2212yes—
30C2×C42×D3C2211yescharacteristic
31C2×#870C2214yes—
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C3×#391C2212yes—
35C3×#389C2212yes—
36C3×#387C2214yes—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C4⋊C4C2212yes—
40#5229C2212yes—
41#5229C2212yes—
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4987C2211yescharacteristic
45#4987C2212yes—
46#4987C2212yes—
47#4986C2214yes—
48#4984C2211yescharacteristic
49#4984C2212yes—
50#4983C2214yes—
51#4982C2214yes—
52#4975C2211yescharacteristic
53#4975C2211yescharacteristic
54#4975C2212yes—
55Dic3×C4⋊C4C2212yes—
56#4973C2211yescharacteristic
57#4972C2212yes—
58#4972C2212yes—
59#4972C2212yes—
60#4972C2212yes—
61C4×(C2×D3)⋊C4C2212yes—
62#4935C2211yescharacteristic
63#4935C2212yes—
64C4×C4⋊Dic3C2211yescharacteristic
65C4×C4⋊Dic3C2212yes—
66#4933C2214yes—
67C4×Dic3⋊C4C2212yes—
68#4928C2212yes—
69#4928C2214yes—
70#4928C2212yes—
71#4675C2214yes—
72#4674C2214yes—
73#4668C2212yes—
74#4667C2212yes—
75#4661C2214yes—
76#4658C2214yes—
77#4658C2214yes—
78#4657C2214yes—
79#4657C2214yes—
80#4654C2214yes—
81#4653C2212yes—
82#4652C2214yes—
83#4650C2212yes—
84#4647C2214yes—
85C2×#541—32no—
86#2857—31no—
87#2722—34no—
88#2719—34no—
89#2625—32no—
90#2552—32no—
91#2552—31no—
92#2546—31no—
93#2506—32no—
94#2496—32no—
95#2496—32no—
96#2453—32no—
97#2446—32no—
98#2365D311yescharacteristic, complemented
99Q8×C4⋊C4—31no—
100C4×C22⋊Q8—31no—
101#2285—31no—
102C22×C3×Q8C2312yes—
103C22×C4×D3C2311yescharacteristic
104C22×C4×D3C2311yescharacteristic
105C22×C4×D3C2311yescharacteristic
106C22×Dic6C2311yescharacteristic
107C3×C4⋊Q8—24no—
108C3×C4×Q8—28no—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C42C2312yes—
117C2×C3×C42C2314yes—
118C2×C3×C42C2312yes—
119C2×C3×C42C2311yescharacteristic
120#946—28no—
121#944—28no—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×C4⋊Dic3C2314yes—
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×C4×Dic3C2314yes—
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2311yescharacteristic
151C2×C4×Dic3C2312yes—
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2312yes—
154#898—216no—
155#897D418yes—
156#890—28no—
157#889—216no—
158#876—216no—
159#872—28no—
160C42×D3—24no—
161#870—216no—
162C4×Dic6—28no—
163C4×Dic6—24no—
164C3×#95C2312yes—
165C3×#95C2314yes—
166C3×#95C2312yes—
167C3×#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2312yes—
172C3⋊#95C2312yes—
173C3⋊#95C2314yes—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2312yes—
177C3⋊#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181#541—616no—
182C2×C4⋊Q8—32no—
183C2×C4⋊Q8C2×D311yescharacteristic
184C2×#126—34no—
185C2×#126—34no—
186C2×#125—34no—
187C2×#125—32no—
188C2×#125—34no—
189C2×C22⋊Q8—32no—
190C2×C22⋊Q8—31no—
191C2×C4×Q8—32no—
192C2×C4×Q8—31no—
193C2×C4×Q8C2×D312yes—
194C2×#117—32no—
195C22×C42—31no—
196#400—34no—
197#400—34no—
198#400—34no—
199#398—34no—
200#398—34no—
201#398—34no—
202#398—34no—
203#398—34no—
204#397—34no—
205#397—34no—
206#391—32no—
207#391—32no—
208#391—31no—
209#391C2×D312yes—
210#389—32no—
211#389—32no—
212#389—32no—
213#389—32no—
214#389—32no—
215#389—31no—
216#389—32no—
217#389—34no—
218#389—32no—
219#389—34no—
220#389C2×D312yes—
221#387—34no—
222#387—34no—
223#387—32no—
224#387—34no—
225#387C2×D314yes—
226#385—32no—
227#384—32no—
228#384—31no—
229#384—31no—
230#384—31no—
231#384—32no—
232#382—32no—
233#380—32no—
234#380—32no—
235#380—31no—
236#380—31no—
237#380—31no—
238#380—32no—
239C4×C4⋊C4—32no—
240C4×C4⋊C4—32no—
241C4×C4⋊C4—32no—
242C4×C4⋊C4—31no—
243C4×C4⋊C4C2×D311yescharacteristic
244C4×C4⋊C4C2×D312yes—
245C4×C4⋊C4C2×D311yescharacteristic
246C4×C22⋊C4—32no—
247C23×D3C2411yescharacteristic
248C2×C3×Q8—24no—
249C2×C3×Q8—24no—
250C2×C3×Q8—24no—
251C22×C3×C4C2412yes—
252C22×C3×C4C2414yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2412yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C22×Dic3C2411yescharacteristic
260C22×Dic3C2412yes—
261C22×Dic3C2412yes—
262C22×Dic3C2414yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2411yescharacteristic
266C22×Dic3C2411yescharacteristic
267C2×C4×D3—22no—
268C2×C4×D3—22no—
269C2×C4×D3—22no—
270C2×C4×D3—22no—
271C2×C4×D3—22no—
272C2×C4×D3—22no—
273C2×C4×D3—22no—
274C2×C4×D3—41no—
275C2×C4×D3C2×D414yes—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C2×Dic6C2×D414yes—
279C3×C4⋊C4—28no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4—22no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4C2×D414yes—
286C3×C42—24no—
287C3×C42—28no—
288C3×C42—24no—
289C3×C42#4114yes—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—28no—
293(C2×D3)⋊C4—42no—
294(C2×D3)⋊C4C2×D418yes—
295C4⋊Dic3—24no—
296C4⋊Dic3—28no—
297C4⋊Dic3—22no—
298C4⋊Dic3—24no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3—24no—
302C4⋊Dic3—42no—
303C4⋊Dic3—44no—
304C4⋊Dic3C2×D418yes—
305Dic3⋊C4—28no—
306Dic3⋊C4—24no—
307Dic3⋊C4—28no—
308Dic3⋊C4—24no—
309Dic3⋊C4—28no—
310Dic3⋊C4—28no—
311Dic3⋊C4—24no—
312Dic3⋊C4—28no—
313Dic3⋊C4—24no—
314Dic3⋊C4—42no—
315Dic3⋊C4—44no—
316C4×Dic3—24no—
317C4×Dic3—24no—
318C4×Dic3—22no—
319C4×Dic3—22no—
320C4×Dic3—22no—
321C4×Dic3—28no—
322C22×Q8—31no—
323C22×Q8C22×D312yes—
324C23×C4—31no—
325C23×C4—31no—
326C23×C4—31no—
327C4⋊Q8—24no—
328C4⋊Q8—68no—
329#126—616no—
330#126—616no—
331#125—616no—
332#125—616no—
333#125—68no—
334C22⋊Q8—38no—
335C22⋊Q8—68no—
336C4×Q8—28no—
337C4×Q8—68no—
338C4×Q8—64no—
339#117—68no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—31no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—34no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4C22×D314yes—
362C2×C4⋊C4C22×D312yes—
363C2×C4⋊C4C22×D312yes—
364C2×C4⋊C4C22×D311yescharacteristic
365C2×C4⋊C4C22×D311yescharacteristic
366C2×C4⋊C4C22×D311yescharacteristic
367C2×C4⋊C4C22×D311yescharacteristic
368C2×C22⋊C4—32no—
369C2×C22⋊C4—32no—
370C2×C22⋊C4—32no—
371C2×C22⋊C4—34no—
372C2×C22⋊C4—32no—
373C2×C42—32no—
374C2×C42—34no—
375C2×C42—32no—
376C2×C42—31no—
377C2×C42—31no—
378C2×C42—31no—
379C2×C42—64no—
380C2×C42C22×D312yes—
381C2×C42C22×D314yes—
382C2×C42C22×D312yes—
383C2×C42C22×D311yescharacteristic
384#95—32no—
385#95—32no—
386#95—32no—
387#95—32no—
388#95—32no—
389#95—32no—
390#95—32no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95—34no—
395#95—34no—
396#95—34no—
397#95C22×D312yes—
398#95C22×D312yes—
399#95C22×D314yes—
400#95C22×D314yes—
401C23×C3C2511yescharacteristic
402C22×D3—21no—
403C22×D3—21no—
404C22×D3—21no—
405C22×D3—21no—
406C22×D3—21no—
407C22×D3—21no—
408C22×D3C22×D412yes—
409C3×Q8—48no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—24no—
417C2×C3×C4—24no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—24no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4C22×D414yes—
427C2×C3×C4C22×D412yes—
428C2×C3×C4C2×#4112yes—
429C2×C3×C4C2×#4112yes—
430C2×C3×C4C2×#4112yes—
431C2×C3×C42−1+412yes—
432C2×C3×C42−1+412yes—
433C2×C3×C42−1+412yes—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—22no—
441C2×Dic3—22no—
442C2×Dic3—24no—
443C2×Dic3—21no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—24no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—21no—
451C2×Dic3—21no—
452C2×Dic3—22no—
453C2×Dic3—22no—
454C2×Dic3—24no—
455C2×Dic3C22×D414yes—
456C2×Dic3C22×D412yes—
457C2×Dic32−1+412yes—
458C4×D3—28no—
459C4×D3—44no—
460C4×D3—44no—
461Dic6—28no—
462C24—31no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—34no—
467C2×Q8—62no—
468C2×Q8—62no—
469C22×C4—31no—
470C22×C4—32no—
471C22×C4—31no—
472C22×C4—34no—
473C22×C4—31no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—34no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—121no—
486C22×C4C23×D312yes—
487C22×C4C23×D314yes—
488C22×C4C23×D312yes—
489C22×C4C23×D312yes—
490C22×C4C23×D311yescharacteristic
491C22×C4C23×D312yes—
492C22×C4C23×D311yescharacteristic
493C22×C4C23×D311yescharacteristic
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—38no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—68no—
504C4⋊C4—68no—
505C4⋊C4—68no—
506C4⋊C4—68no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—62no—
511C4⋊C4—64no—
512C4⋊C4—68no—
513C4⋊C4—68no—
514C4⋊C4—64no—
515C4⋊C4—62no—
516C4⋊C4—62no—
517C4⋊C4—122no—
518C4⋊C4—124no—
519C4⋊C4—122no—
520C4⋊C4—124no—
521C4⋊C4D3×D414yes—
522C22⋊C4—38no—
523C22⋊C4—64no—
524C22⋊C4—64no—
525C22⋊C4—68no—
526C22⋊C4—122no—
527C42—24no—
528C42—28no—
529C42—24no—
530C42—68no—
531C42—62no—
532C42—62no—
533C42—62no—
534C42—64no—
535C42—64no—
536C42#23514yes—
537C22×C3C23×D411yescharacteristic
538C22×C3C22×#4111yescharacteristic
539C22×C3C2×2−1+411yescharacteristic
540C22×C3C2×2−1+411yescharacteristic
541C22×C3C2×2−1+411yescharacteristic
542C22×C3C2×2−1+411yescharacteristic
543C22×C3#58511yescharacteristic
544C2×D3—22no—
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C3×C4—22no—
552C3×C4—24no—
553C3×C4—44no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—42no—
557C3×C4#54114yescomplemented
558C3×C4#54714yes—
559Dic3—24no—
560Dic3—22no—
561Dic3—22no—
562Dic3—41no—
563Dic3—42no—
564Dic3—42no—
565Dic3—44no—
566Dic3—42no—
567C23—32no—
568C23—61no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23C24×D311yescharacteristic
575Q8—48no—
576Q8—68no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—34no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—62no—
599C2×C4—61no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—68no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—61no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—64no—
618C2×C4—124no—
619C2×C4—124no—
620C2×C4C2×#23512yes—
621C2×C4C2×#23512yes—
622C2×C4C2×#23512yes—
623C2×C4C2×D3×D414yes—
624C2×C4C2×D3×D412yes—
625C2×C4#100712yes—
626C2×C4#101012yes—
627C2×C4#101012yes—
628C2×C3C2×#54111yescharacteristic
629C2×C3C2×#54711yescharacteristic
630C2×C3#349511yescharacteristic
631C2×C3#349711yescharacteristic
632C2×C3#349911yescharacteristic
633C2×C3#355511yescharacteristic
634C2×C3#355611yescharacteristic
635D3—42no—
636C22—62no—
637C22—62no—
638C22—62no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22C22×#23511yescharacteristic
644C22C22×D3×D411yescharacteristic
645C22C2×#100711yescharacteristic
646C22C2×#101011yescharacteristic
647C22C2×#101011yescharacteristic
648C22D3×2−1+411yescharacteristic
649C22#596811yescharacteristic
650C4—22no—
651C4—24no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—62no—
657C4—64no—
658C4—62no—
659C4—121no—
660C4—122no—
661C4—122no—
662C4—124no—
663C4—122no—
664C4#555114yes—
665C4#558314yescomplemented
666C3#3935311yescharacteristic, complemented, Sylow
667C2—122no—
668C2C2×#555111yescharacteristic
669C2C2×#558311yescharacteristic
670C2#8588211yescharacteristic
671C2#8667911yescharacteristic
672C2#8676011yescharacteristic
673C2#8701111yescharacteristic
674C2#8702711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2365C2
prime radicalC23×D3C24
hypercenter#2365D3
hypocenterC3#39353
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. C3×#2365C2semidirect
4. #83239C2semidirect
5. #82471C2semidirect
7. #76502C2semidirect
9. #76030C2semidirect
11. #75168C2semidirect
15. #73717C2semidirect
16. #73716C2semidirect
17. C4⋊C4×Dic6C2semidirect
98. #2365D3semidirect
557. C3×C4#541semidirect
665. C4#5583semidirect
666. C3#39353semidirect

Maximal subgroups

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

subgroupindex
C2×#55832
C3×#23652
#832692
#832392
#824712
#766422
#765022
#762382
#760302
#751732
#751682
#737922
C4×#8972
#737532
#737172
#737162
C4⋊C4×Dic62
#393533