Finite Group Databasebeta

← #11233315

Subgroups — #11233315

Order 768.

2218 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D12C211yesmaximal, characteristic
2C2×#5570C211yesmaximal, characteristic, complemented
3C3×#2581C211yesmaximal, characteristic, complemented
4#82492C211yesmaximal, characteristic
5#82456C211yesmaximal, characteristic, complemented
6#78237C214yesmaximal, complemented
7#77179C214yesmaximal, complemented
8#76614C212yesmaximal, complemented
9#76499C212yesmaximal, complemented
10#76221C212yesmaximal, complemented
11#76214C212yesmaximal, complemented
12#75158C214yesmaximal, complemented
13#74992C214yesmaximal
14#74240C211yesmaximal, characteristic
15#74152C211yesmaximal, characteristic, complemented
16#45336—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×Q8×Dic3C2212yes—
21C2×#944C2211yescharacteristic
22Q8×D12—28no—
23#5570—28no—
24C2×#897C2214yes—
25C2×#897C2212yes—
26C2×#897C2212yes—
27C2×#890C2214yes—
28C2×#890C2212yes—
29C2×#887C2212yes—
30C2×C4×D12C2211yescharacteristic
31C2×C4×D12C2212yes—
32C2×#869C2212yes—
33C2×#869C2212yes—
34C2×#869C2211yescharacteristic
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#400C2212yes—
38C3×#395C2212yes—
39C3×#391C2212yes—
40C3×#389C2212yes—
41C3×#387C2214yes—
42C3×#382C2211yescharacteristic
43#5231C2212yes—
44#5229C2211yescharacteristic
45#4994C2214yes—
46#4992C2212yes—
47#4991C2212yes—
48#4990C2214yes—
49#4987C2212yes—
50#4986C2212yes—
51#4984C2214yes—
52#4980C2214yes—
53#4978C2214yes—
54#4977C2211yescharacteristic
55#4973C2211yescharacteristic
56#4973C2212yes—
57C4×(C2×D3)⋊C4C2212yes—
58C4×(C2×D3)⋊C4C2212yes—
59#4933C2214yes—
60#4932C2211yescharacteristic
61#4675C2214yes—
62#4672C2212yes—
63#4670C2214yes—
64#4670C2214yes—
65#4668C2214yes—
66#4668C2214yes—
67#4667C2214yes—
68#4666C2214yes—
69#4661C2212yes—
70#4661C2214yes—
71#4660C2214yes—
72#4656C2214yes—
73#4656C2214yes—
74#4654C2214yes—
75#4654C2214yes—
76#4646C2214yes—
77#4646C2214yes—
78C2×#557—31no—
79C2×D4×Q8—31no—
80#2713—34no—
81#2690—34no—
82#2581D311yescharacteristic, complemented
83#2536—31no—
84#2465—34no—
85#2460—34no—
86#2434—32no—
87#2429—32no—
88#2421—32no—
89#2398—31no—
90Q8×C22⋊C4—32no—
91#2327—31no—
92#2327—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×Q8C2311yescharacteristic
95C2×D3×Q8—28no—
96C2×D3×Q8—28no—
97C2×D3×Q8—44no—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C22×C4×D3C2314yes—
101C22×Dic6C2311yescharacteristic
102C22×Dic6C2314yes—
103C22×Dic6C2312yes—
104C3×C4×Q8—24no—
105C3×C4×Q8—24no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C42C2312yes—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115Q8×Dic3—28no—
116#944—24no—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×C4⋊Dic3C2314yes—
126C2×C4⋊Dic3C2314yes—
127C2×C4⋊Dic3C2314yes—
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2314yes—
137C2×C4×Dic3C2314yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2314yes—
142#897—28no—
143#897—216no—
144#897—44no—
145#890—28no—
146#890—216no—
147#887—44no—
148C4×D12—24no—
149C4×D12—28no—
150#869—28no—
151#869—28no—
152#869D418yes—
153C4×Dic6—28no—
154C4×Dic6—24no—
155C3×#95C2314yes—
156C3×#95C2314yes—
157C3×#95C2312yes—
158C3×#95C2312yes—
159C3×#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C3⋊#95C2314yes—
163C3⋊#95C2312yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C23×Q8—32no—
173#557—68no—
174D4×Q8—68no—
175C2×C4⋊Q8—31no—
176C2×C4⋊Q8—31no—
177C2×C4⋊Q8—32no—
178C2×C4⋊Q8—32no—
179C2×#125—32no—
180C2×#125—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—34no—
183C2×C22⋊Q8—32no—
184C2×C4×Q8—32no—
185C2×C4×Q8—32no—
186C2×C4×Q8—31no—
187C2×C4×Q8—32no—
188C2×C4×Q8C2×D311yescharacteristic
189C2×C4×Q8C2×D311yescharacteristic
190C2×C4×D4—31no—
191C2×C4×D4—32no—
192#400—32no—
193#400—34no—
194#400C2×D312yes—
195#398—34no—
196#398—32no—
197#397—32no—
198#397—34no—
199#396—34no—
200#395—34no—
201#395—34no—
202#395C2×D312yes—
203#394—32no—
204#394—32no—
205#393—32no—
206#391—31no—
207#391—32no—
208#391—31no—
209#391—34no—
210#391—34no—
211#391C2×D312yes—
212#389—34no—
213#389C2×D312yes—
214#388—34no—
215#388—34no—
216#388—34no—
217#387—34no—
218#387—34no—
219#387—34no—
220#387—34no—
221#387—34no—
222#387C2×D314yes—
223#385—34no—
224#385—34no—
225#382—31no—
226#382C2×D311yescharacteristic
227#380—32no—
228#380—34no—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—34no—
232#376—31no—
233C23×D3C2412yes—
234C2×C3×Q8—22no—
235C2×C3×Q8—22no—
236C2×C3×Q8—22no—
237C2×C3×Q8—24no—
238C2×C3×Q8#4114yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2414yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2411yescharacteristic
247C22×Dic3C2412yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253D3×Q8—432no—
254C2×D12—24no—
255C2×D12—22no—
256C2×C4×D3—24no—
257C2×C4×D3—28no—
258C2×C4×D3—24no—
259C2×C4×D3—24no—
260C2×C4×D3—28no—
261C2×C4×D3—44no—
262C2×Dic6—24no—
263C2×Dic6—24no—
264C2×Dic6—22no—
265C2×Dic6—22no—
266C2×Dic6—28no—
267C2×Dic6—22no—
268C2×Dic6—28no—
269C2×Dic6—44no—
270C2×Dic6C2×D418yes—
271C3×C4⋊C4—24no—
272C3×C4⋊C4—22no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—24no—
276C3×C4⋊C4—28no—
277C3×C42—24no—
278C3×C42—24no—
279C3×C42C2×D414yes—
280(C2×D3)⋊C4—28no—
281(C2×D3)⋊C4—24no—
282(C2×D3)⋊C4—24no—
283(C2×D3)⋊C4—24no—
284(C2×D3)⋊C4—24no—
285(C2×D3)⋊C4—24no—
286(C2×D3)⋊C4—28no—
287(C2×D3)⋊C4—44no—
288C4⋊Dic3—22no—
289C4⋊Dic3—22no—
290C4⋊Dic3—28no—
291C4⋊Dic3—28no—
292C4⋊Dic3—28no—
293C4⋊Dic3—44no—
294C4⋊Dic3C2×D418yes—
295C4⋊Dic3C2×D418yes—
296Dic3⋊C4—24no—
297Dic3⋊C4—28no—
298Dic3⋊C4—28no—
299Dic3⋊C4—44no—
300C4×Dic3—24no—
301C4×Dic3—22no—
302C4×Dic3—22no—
303C4×Dic3—28no—
304C4×Dic3—44no—
305C22×Q8—32no—
306C22×Q8—31no—
307C22×Q8—34no—
308C22×Q8—68no—
309C22×Q8—68no—
310C22×Q8—124no—
311C22×Q8C22×D311yescharacteristic
312C22×Q8C22×D311yescharacteristic
313C22×D4—31no—
314C23×C4—32no—
315C23×C4—34no—
316C4⋊Q8—38no—
317C4⋊Q8—68no—
318C4⋊Q8—68no—
319C4⋊Q8—64no—
320#125—616no—
321#125—68no—
322C22⋊Q8—616no—
323C22⋊Q8—68no—
324C22⋊Q8—124no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—68no—
328C4×Q8—68no—
329C4×Q8—64no—
330C4×Q8—124no—
331C4×D4—68no—
332C4×D4—64no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4C22×D312yes—
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D312yes—
348C2×C4⋊C4C22×D312yes—
349C2×C4⋊C4C22×D311yescharacteristic
350C2×C4⋊C4C22×D314yes—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C42—34no—
360C2×C42—32no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—34no—
364C2×C42C22×D312yes—
365C2×C42C22×D312yes—
366C2×C42C22×D311yescharacteristic
367#95—32no—
368#95—32no—
369#95—34no—
370#95—32no—
371#95—32no—
372#95—32no—
373#95—32no—
374#95—34no—
375#95—34no—
376#95—34no—
377#95—34no—
378#95—34no—
379#95C22×D314yes—
380#95C22×D314yes—
381#95C22×D312yes—
382#95C22×D312yes—
383#95C22×D314yes—
384C23×C3C2511yescharacteristic
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—22no—
389C22×D3—22no—
390C22×D3—22no—
391C22×D32−1+414yes—
392C3×Q8—28no—
393C3×Q8—28no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—24no—
406C2×C3×C4—24no—
407C2×C3×C4—24no—
408C2×C3×C4C22×D412yes—
409C2×C3×C4C22×D412yes—
410C2×C3×C4C22×D412yes—
411C2×C3×C4C2×#4114yes—
412C2×C3×C4C2×#4112yes—
413C2×C3×C4H(5,𝔽2)12yes—
414C2×C3×C42−1+414yes—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—24no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3C22×D418yes—
432D12—28no—
433C4×D3—416no—
434C4×D3—48no—
435Dic6—216no—
436Dic6—416no—
437Dic6—44no—
438C24—32no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—38no—
444C2×Q8—68no—
445C2×Q8—62no—
446C2×Q8—64no—
447C2×Q8—62no—
448C2×Q8—68no—
449C2×Q8—64no—
450C2×Q8—62no—
451C2×Q8—124no—
452C2×Q8—1232no—
453C2×Q8#23914yes—
454C2×D4—64no—
455C2×D4—62no—
456C22×C4—34no—
457C22×C4—32no—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—32no—
462C22×C4—68no—
463C22×C4—64no—
464C22×C4—68no—
465C22×C4—64no—
466C22×C4—64no—
467C22×C4—124no—
468C22×C4C23×D312yes—
469C22×C4C23×D314yes—
470C22×C4C23×D312yes—
471C22×C4C23×D311yescharacteristic
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D312yes—
474C22×C4C23×D312yes—
475C22×C4C23×D311yescharacteristic
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—38no—
483C4⋊C4—38no—
484C4⋊C4—68no—
485C4⋊C4—68no—
486C4⋊C4—68no—
487C4⋊C4—68no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—64no—
491C4⋊C4—68no—
492C4⋊C4—124no—
493C4⋊C4—124no—
494C22⋊C4—68no—
495C22⋊C4—64no—
496C22⋊C4—68no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—124no—
502C42—24no—
503C42—24no—
504C42—68no—
505C42—64no—
506C42—62no—
507C42—62no—
508C42—124no—
509C42D3×D414yes—
510C22×C3C23×D411yescharacteristic
511C22×C3C22×#4111yescharacteristic
512C22×C3C2×H(5,𝔽2)11yescharacteristic
513C22×C3C2×2−1+412yes—
514C22×C3C2×2−1+412yes—
515C2×D3—24no—
516C2×D3—24no—
517C2×D3—24no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—42no—
522C3×C4—24no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—24no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—44no—
529C3×C4#54614yes—
530Dic3—28no—
531Dic3—42no—
532Dic3—42no—
533Dic3—42no—
534Dic3—42no—
535Dic3—42no—
536C23—34no—
537C23—62no—
538C23—62no—
539C23—62no—
540C23—62no—
541C23—62no—
542C23—62no—
543C23C24×D311yescharacteristic
544Q8—28no—
545Q8—28no—
546Q8—616no—
547Q8—1216no—
548Q8—124no—
549D4—68no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—38no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—1216no—
582C2×C4—128no—
583C2×C4C2×D3×D412yes—
584C2×C4C2×D3×D412yes—
585C2×C4C2×D3×D412yes—
586C2×C4C2×#23914yes—
587C2×C4C2×#23912yes—
588C2×C4#100912yes—
589C2×C4#101014yes—
590C2×C3C2×#54611yescharacteristic
591C2×C3#348611yescharacteristic
592C2×C3#348611yescharacteristic
593C2×C3#352312yes—
594C2×C3#355212yes—
595D3—44no—
596C22—64no—
597C22—64no—
598C22—64no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22C22×D3×D411yescharacteristic
604C22C22×#23911yescharacteristic
605C22C2×#100911yescharacteristic
606C22C2×#101012yes—
607C22D3×2−1+412yes—
608C4—22no—
609C4—24no—
610C4—22no—
611C4—24no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—68no—
616C4—122no—
617C4—122no—
618C4—122no—
619C4—122no—
620C4—122no—
621C4#563014yes—
622C3#4533611yescharacteristic, complemented, Sylow
623C2—124no—
624C2C2×#563011yescharacteristic
625C2#8602211yescharacteristic
626C2#8635511yescharacteristic
627C2#8701012yes—
628C2#8702412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2581C2
prime radicalC22×D12C23
hypercenter#2581D3
hypocenterC3#45336
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C2×#5570C2semidirect
3. C3×#2581C2semidirect
5. #82456C2semidirect
6. #78237C2semidirect
7. #77179C2semidirect
8. #76614C2semidirect
9. #76499C2semidirect
10. #76221C2semidirect
11. #76214C2semidirect
12. #75158C2semidirect
15. #74152C2semidirect
82. #2581D3semidirect
622. C3#45336semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D122
C2×#55702
C3×#25812
#824922
#824562
#782372
#771792
#766142
#764992
#762212
#762142
#751582
#749922
#742402
#741522
#453363