Finite Group Databasebeta

← #11069289

Subgroups — #11069289

Order 768.

2890 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5727C211yesmaximal, characteristic, complemented
2C2×#5726C211yesmaximal, characteristic
3C2×#5724C212yesmaximal, complemented
4C3×#2868C211yesmaximal, characteristic, complemented
5#83261C212yesmaximal, complemented
6#83255C212yesmaximal, complemented
7#83240C211yesmaximal, characteristic, complemented
8#83239C211yesmaximal, characteristic, complemented
9#83232C212yesmaximal, complemented
10#74371C218yesmaximal, complemented
11#74362C214yesmaximal, complemented
12#74326C214yesmaximal, complemented
13C4×#875C211yesmaximal, characteristic, complemented
14#73299C211yesmaximal, characteristic, complemented
15#39360—31nomaximal, Sylow
16C22×#239C2212yes—
17C22×D3×Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4⋊Q8C2212yes—
21C2×C3×C4⋊Q8C2212yes—
22C2×#947C2212yes—
23C2×#946C2212yes—
24C2×#946C2212yes—
25C2×#944C2212yes—
26#5727—28no—
27#5726—28no—
28#5724—216no—
29C2×#896C2218yes—
30C2×#896C2214yes—
31C2×#895C2212yes—
32C2×#894C2218yes—
33C2×#894C2214yes—
34C2×#893C2212yes—
35C2×D3×C4⋊C4C2212yes—
36C2×#888C2212yes—
37C2×#875C2211yescharacteristic
38C2×C4×D12C2211yescharacteristic
39C2×C4×D12C2211yescharacteristic
40C2×#872C2212yes—
41C2×C4×Dic6C2211yescharacteristic
42C3×#384C2214yes—
43C3×#384C2214yes—
44C3×C43C2211yescharacteristic
45#5229C2212yes—
46#5229C2212yes—
47#4992C2214yes—
48#4992C2218yes—
49#4991C2218yes—
50#4989C2218yes—
51#4989C2218yes—
52#4987C2214yes—
53#4983C2214yes—
54#4983C2218yes—
55#4979C2218yes—
56#4972C2214yes—
57#4940C2214yes—
58#4940C2214yes—
59#4939C2211yescharacteristic
60#4939C2212yes—
61C4×(C2×D3)⋊C4C2212yes—
62C4×(C2×D3)⋊C4C2214yes—
63#4936C2211yescharacteristic
64C4×C4⋊Dic3C2211yescharacteristic
65#4929C2211yescharacteristic
66#4928C2212yes—
67C2×D4×Q8—31no—
68C2×#547—31no—
69C2×#536—32no—
70#2868D311yescharacteristic, complemented
71#2680—38no—
72#2621—32no—
73#2547—31no—
74#2546—31no—
75#2440—34no—
76#2434—32no—
77#2432—32no—
78#2422—34no—
79C4×#124—31no—
80#2293—31no—
81C22×C3×Q8C2312yes—
82C22×C3×Q8C2312yes—
83C2×#239—28no—
84C2×#239—28no—
85C2×#239—44no—
86C2×D3×Q8—28no—
87C2×D3×Q8—28no—
88C2×D3×Q8—44no—
89C22×D12C2314yes—
90C22×D12C2311yescharacteristic
91C22×D12C2311yescharacteristic
92C22×C4×D3C2312yes—
93C22×C4×D3C2314yes—
94C22×C4×D3C2314yes—
95C22×C4×D3C2312yes—
96C22×Dic6C2314yes—
97C22×Dic6C2311yescharacteristic
98C3×C4⋊Q8—28no—
99C3×C4⋊Q8—24no—
100C3×C4⋊Q8—24no—
101C3×C4⋊Q8—44no—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2318yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2318yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113#947—28no—
114#946—28no—
115#946—28no—
116#944—28no—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2318yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2311yescharacteristic
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2318yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2312yes—
137#896—232no—
138#896—216no—
139#895—28no—
140#894—232no—
141#894—216no—
142#893—28no—
143D3×C4⋊C4—28no—
144#888—28no—
145#875D418yescomplemented
146C4×D12—24no—
147C4×D12D418yes—
148#872—28no—
149C4×Dic6—24no—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2318yes—
155C3⋊#95C2318yes—
156C22×#41—32no—
157C23×Q8—32no—
158D4×Q8—68no—
159#547—68no—
160#536—616no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8—32no—
163C2×C4⋊Q8C2×D312yes—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×C4⋊Q8C2×D312yes—
167C2×#124—32no—
168C2×#124—31no—
169C2×#123—38no—
170C2×#123—34no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—38no—
174C2×C22⋊Q8—34no—
175C2×#121—32no—
176C2×C4×Q8—31no—
177C2×C4×D4—32no—
178C2×C4×D4—31no—
179C2×C4×D4—31no—
180C2×#117—32no—
181C22×C4⋊C4—32no—
182#398—38no—
183#398—34no—
184#397—38no—
185#397—34no—
186#395—38no—
187#394—38no—
188#391—32no—
189#391—32no—
190#391—31no—
191#390—32no—
192#390—31no—
193#389—34no—
194#389—32no—
195#388—34no—
196#388—34no—
197#385—38no—
198#385—38no—
199#384C2×D314yes—
200#384C2×D314yes—
201#382—31no—
202#380—34no—
203C4×C4⋊C4—31no—
204C4×C22⋊C4—34no—
205C4×C22⋊C4—32no—
206C43C2×D311yescharacteristic
207C23×D3C2412yes—
208C23×D3C2412yes—
209C2×C3×Q8—24no—
210C2×C3×Q8—24no—
211C2×C3×Q8—24no—
212C2×C3×Q8—24no—
213C2×C3×Q8—24no—
214C2×C3×Q8—24no—
215C22×C3×C4C2414yes—
216C22×C3×C4C2414yes—
217C22×C3×C4C2411yescharacteristic
218C22×C3×C4C2412yes—
219C22×C3×C4C2412yes—
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C22×Dic3C2414yes—
225C22×Dic3C2412yes—
226C22×Dic3C2412yes—
227#239—432no—
228D3×Q8—432no—
229C2×D12—22no—
230C2×D12—22no—
231C2×D12—24no—
232C2×D12—28no—
233C2×D12—28no—
234C2×D12—44no—
235C2×D12C2×D414yes—
236C2×D12C2×D414yes—
237C2×C4×D3—24no—
238C2×C4×D3—24no—
239C2×C4×D3—28no—
240C2×C4×D3—28no—
241C2×C4×D3—24no—
242C2×C4×D3—28no—
243C2×C4×D3—28no—
244C2×C4×D3—28no—
245C2×C4×D3—44no—
246C2×C4×D3—44no—
247C2×C4×D3C2×D418yes—
248C2×Dic6—22no—
249C2×Dic6—22no—
250C2×Dic6—28no—
251C2×Dic6—28no—
252C2×Dic6—44no—
253C2×Dic6C2×D414yes—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—216no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—48no—
260C3×C42—22no—
261C3×C42—24no—
262C3×C42—24no—
263C3×C42C2×D414yes—
264C3×C42C2×D414yes—
265(C2×D3)⋊C4—28no—
266(C2×D3)⋊C4—28no—
267(C2×D3)⋊C4—216no—
268(C2×D3)⋊C4—28no—
269(C2×D3)⋊C4—24no—
270(C2×D3)⋊C4—28no—
271(C2×D3)⋊C4—28no—
272(C2×D3)⋊C4C2×D418yes—
273(C2×D3)⋊C4C2×D4116yes—
274C4⋊Dic3—22no—
275C4⋊Dic3—22no—
276C4⋊Dic3—24no—
277C4⋊Dic3C2×D414yes—
278Dic3⋊C4—24no—
279Dic3⋊C4—28no—
280Dic3⋊C4—28no—
281Dic3⋊C4—28no—
282Dic3⋊C4—216no—
283C4×Dic3—24no—
284C4×Dic3—24no—
285C2×#41—68no—
286C2×#41—68no—
287C2×#41—124no—
288C22×Q8—31no—
289C22×Q8—34no—
290C22×Q8—68no—
291C22×Q8—68no—
292C22×Q8—124no—
293C22×Q8C22×D312yes—
294C22×Q8C22×D312yes—
295C22×D4—31no—
296C22×D4—31no—
297C22×D4—34no—
298C23×C4—32no—
299C23×C4—32no—
300C23×C4—34no—
301C23×C4—34no—
302C4⋊Q8—28no—
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305C4⋊Q8—44no—
306C4⋊Q8—68no—
307C4⋊Q8—68no—
308#124—38no—
309#124—68no—
310#123—632no—
311#123—616no—
312C22⋊Q8—632no—
313C22⋊Q8—616no—
314C22⋊Q8—68no—
315C22⋊Q8—68no—
316#121—68no—
317C4×Q8—64no—
318C4×D4—38no—
319C4×D4—68no—
320C4×D4—64no—
321#117—68no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—38no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—68no—
332C2×C4⋊C4C22×D312yes—
333C2×C4⋊C4C22×D312yes—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D318yes—
336C2×C4⋊C4C22×D312yes—
337C2×C4⋊C4C22×D318yes—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—38no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C42—32no—
348C2×C42—32no—
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D312yes—
351C2×C42C22×D312yes—
352C2×C42C22×D311yescharacteristic
353C2×C42C22×D311yescharacteristic
354#95—34no—
355#95—34no—
356#95—34no—
357#95—32no—
358#95—38no—
359#95—38no—
360C23×C3C2511yescharacteristic
361C22×D3—22no—
362C22×D3—22no—
363C22×D3—22no—
364C22×D3—22no—
365C22×D3—24no—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3—24no—
369C22×D3C22×D414yes—
370C22×D3C22×D414yes—
371C3×Q8—48no—
372C3×Q8—48no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—24no—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C22×D414yes—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4H(5,𝔽2)12yes—
390C2×C3×C42−1+414yes—
391C2×C3×C42−1+414yes—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3C22×D414yes—
404C2×Dic3C22×D414yes—
405D12—28no—
406D12—28no—
407D12—416no—
408C4×D3—216no—
409C4×D3—416no—
410C4×D3—416no—
411C4×D3—48no—
412Dic6—28no—
413Dic6—416no—
414C24—32no—
415C24—32no—
416#41—1232no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—34no—
424C2×Q8—68no—
425C2×Q8—68no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8—124no—
429C2×Q8—1232no—
430C2×D4—34no—
431C2×D4—34no—
432C2×D4—68no—
433C2×D4—68no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—64no—
437C2×D4—124no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—38no—
441C22×C4—32no—
442C22×C4—34no—
443C22×C4—32no—
444C22×C4—68no—
445C22×C4—68no—
446C22×C4—68no—
447C22×C4—68no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—68no—
452C22×C4—124no—
453C22×C4—124no—
454C22×C4C23×D314yes—
455C22×C4C23×D314yes—
456C22×C4C23×D312yes—
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D312yes—
459C22×C4C23×D311yescharacteristic
460C22×C4C23×D311yescharacteristic
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—216no—
465C4⋊C4—24no—
466C4⋊C4—34no—
467C4⋊C4—48no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—616no—
471C4⋊C4—68no—
472C4⋊C4—62no—
473C4⋊C4—64no—
474C4⋊C4—62no—
475C4⋊C4—64no—
476C22⋊C4—38no—
477C22⋊C4—316no—
478C22⋊C4—68no—
479C22⋊C4—68no—
480C22⋊C4—616no—
481C22⋊C4—68no—
482C22⋊C4—64no—
483C22⋊C4—68no—
484C22⋊C4—68no—
485C42—22no—
486C42—24no—
487C42—24no—
488C42—64no—
489C42—64no—
490C42D3×D414yes—
491C42D3×D414yes—
492C22×C3C23×D411yescharacteristic
493C22×C3C23×D411yescharacteristic
494C22×C3C2×H(5,𝔽2)11yescharacteristic
495C22×C3C2×2−1+412yes—
496C22×C3C2×2−1+412yes—
497C2×D3—24no—
498C2×D3—24no—
499C2×D3—24no—
500C2×D3—24no—
501C2×D3—28no—
502C2×D3—42no—
503C2×D3—44no—
504C2×D3—42no—
505C2×D3—44no—
506C2×D3—44no—
507C3×C4—22no—
508C3×C4—24no—
509C3×C4—24no—
510C3×C4—44no—
511C3×C4—44no—
512C3×C4#53614yescomplemented
513C3×C4D4214yes—
514Dic3—24no—
515Dic3—24no—
516Dic3—42no—
517Dic3—42no—
518Dic3—44no—
519C23—34no—
520C23—34no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23C24×D311yescharacteristic
530Q8—48no—
531Q8—48no—
532Q8—68no—
533Q8—1216no—
534D4—68no—
535D4—68no—
536D4—1216no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—34no—
549C2×C4—34no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—616no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—1216no—
563C2×C4—1216no—
564C2×C4—128no—
565C2×C4C2×D3×D412yes—
566C2×C4C2×D3×D414yes—
567C2×C4C2×D3×D412yes—
568C2×C4C2×D3×D412yes—
569C2×C4C2×D3×D412yes—
570C2×C4#100412yes—
571C2×C4#100714yes—
572C2×C4#100714yes—
573C2×C3C2×#53611yescharacteristic
574C2×C3C2×D4211yescharacteristic
575C2×C3#349911yescharacteristic
576C2×C3#349912yes—
577C2×C3#355212yes—
578D3—44no—
579D3—44no—
580C22—64no—
581C22—64no—
582C22—64no—
583C22—68no—
584C22—64no—
585C22—122no—
586C22—124no—
587C22—122no—
588C22—124no—
589C22—124no—
590C22C22×D3×D411yescharacteristic
591C22C22×D3×D411yescharacteristic
592C22C2×#100411yescharacteristic
593C22C2×#100712yes—
594C22C2×#100712yes—
595C4—22no—
596C4—24no—
597C4—24no—
598C4—44no—
599C4—44no—
600C4—64no—
601C4—64no—
602C4—122no—
603C4—124no—
604C4—122no—
605C4#571714yes—
606C4#572414yescomplemented
607C3#3936011yescharacteristic, complemented, Sylow
608C2—124no—
609C2—124no—
610C2C2×#571711yescharacteristic
611C2C2×#572411yescharacteristic
612C2#8621411yescharacteristic
613C2#8621412yes—
614C2#8621812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2868C2
prime radicalC2×#5726C2
hypercenter#2868D3
hypocenterC3#39360
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2868

Complements

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

NH ≅ G/Ntype
1. C2×#5727C2semidirect
3. C2×#5724C2semidirect
4. C3×#2868C2semidirect
5. #83261C2semidirect
6. #83255C2semidirect
7. #83240C2semidirect
8. #83239C2semidirect
9. #83232C2semidirect
10. #74371C2semidirect
11. #74362C2semidirect
12. #74326C2semidirect
13. C4×#875C2semidirect
14. #73299C2semidirect
70. #2868D3semidirect
145. #875D4semidirect
512. C3×C4#536semidirect
606. C4#5724semidirect
607. C3#39360semidirect

Maximal subgroups

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

subgroupindex
C2×#57272
C2×#57262
C2×#57242
C3×#28682
#832612
#832552
#832402
#832392
#832322
#743712
#743622
#743262
C4×#8752
#732992
#393603