Finite Group Databasebeta

← #10716869

Subgroups — #10716869

Order 768.

2702 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C22×#926C211yesmaximal, characteristic, complemented
3C2×C4×#217C212yesmaximal, complemented
4C2×C4×#217C211yesmaximal, characteristic, complemented
5#72131C212yesmaximal
6#71565C212yesmaximal, complemented
7#71548C212yesmaximal, complemented
8#71546C212yesmaximal, complemented
9C4×C3⋊#95C212yesmaximal, complemented
10#13839—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C2×C3×C4×D4—24no—
15C2×C3×C4×D4—24no—
16C2×C3×C4×D4C418yescomplemented
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×#217C2214yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C2×#926—24no—
25C2×#926—28no—
26C22×C4⋊Dic3C2212yes—
27C22×C4×Dic3C2212yes—
28C22×C4×Dic3C2212yes—
29C4×#217—24no—
30C4×#217—28no—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2214yes—
34C2×C3⋊#95C2214yes—
35C22×C4×D4D311yescharacteristic
36C22×#117—31no—
37C2×C4×C22⋊C4—32no—
38C2×C4×C22⋊C4—31no—
39#1441—32no—
40#1440—32no—
41#1439—32no—
42#1438—32no—
43C4×#95—32no—
44C25×C3C2311yescharacteristic
45C25×C3C2311yescharacteristic
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—24no—
49C22×C3×D4—24no—
50C22×C3×D4C2×C414yes—
51C23×C3×C4—24no—
52C23×C3×C4—24no—
53C23×C3×C4—24no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—24no—
57C23×C3×C4C2×C414yes—
58C23×C3×C4C2×C414yes—
59C23×C3×C4D414yes—
60C23×C3×C4D414yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3C2314yes—
67C23×Dic3C2314yes—
68C3×C4×D4—232no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2×C414yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4C2×C414yes—
77C2×C3×C22⋊C4C2×C414yes—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42C2×C414yes—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—28no—
84C2×#217—28no—
85C2×#217—28no—
86C2×#217—28no—
87C2×#217—28no—
88#926—232no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—28no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3D418yescomplemented
97C3⋊#95—24no—
98C3⋊#95—28no—
99C3⋊#95—28no—
100C3⋊#95—28no—
101C23×D4C2×D311yescharacteristic
102C24×C4C2×D311yescharacteristic
103C24×C4C2×D311yescharacteristic
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4Dic318yescomplemented
107C2×#117—68no—
108C2×#117—64no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C22⋊C4—32no—
112C22×C22⋊C4—34no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4C2×D311yescharacteristic
115C22×C22⋊C4C2×D311yescharacteristic
116C22×C42—32no—
117C22×C42—32no—
118C22×C42C2×D311yescharacteristic
119C4×C22⋊C4—68no—
120C4×C22⋊C4—64no—
121C2×#95—34no—
122C2×#95—34no—
123C2×#95—32no—
124C2×#95—34no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—21no—
128C24×C3—21no—
129C24×C3—24no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3C22×C412yes—
137C24×C3C22×C412yes—
138C24×C3C2×D412yes—
139C24×C3C2×D412yes—
140C24×C3#4114yes—
141C24×C3C2411yescharacteristic
142C2×C3×D4—216no—
143C2×C3×D4—216no—
144C2×C3×D4—28no—
145C2×C3×D4—28no—
146C2×C3×D4—28no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—216no—
153C22×C3×C4—28no—
154C22×C3×C4—216no—
155C22×C3×C4—28no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—28no—
164C22×C3×C4—216no—
165C22×C3×C4—44no—
166C22×C3×C4—48no—
167C22×C3×C4—44no—
168C22×C3×C4C22⋊C4116yescomplemented
169C22×C3×C4C22×C412yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C22×C412yes—
172C22×C3×C4C22×C412yes—
173C22×C3×C4C22×C412yes—
174C22×C3×C4C2×D414yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4#4114yes—
180C22×C3×C4#4114yes—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3C2×D418yes—
194C22×Dic3#4118yes—
195C3×C4⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C42—28no—
199#217—216no—
200#217—232no—
201#217—216no—
202C4⋊Dic3—216no—
203C4×Dic3—216no—
204C4×Dic3—48no—
205C25C22×D311yescharacteristic
206C25C22×D311yescharacteristic
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4C2×Dic314yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—34no—
219C23×C4—34no—
220C23×C4C2×Dic314yes—
221C23×C4C2×Dic314yes—
222C23×C4#6714yes—
223C23×C4#6714yes—
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D311yescharacteristic
229C4×D4—232no—
230#117—632no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—68no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4C2×Dic314yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—68no—
241C2×C22⋊C4—68no—
242C2×C22⋊C4—68no—
243C2×C22⋊C4—68no—
244C2×C22⋊C4—68no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4C2×Dic314yes—
248C2×C22⋊C4C2×Dic314yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—38no—
252C2×C42—64no—
253C2×C42—64no—
254C2×C42—64no—
255C2×C42—64no—
256C2×C42—64no—
257C2×C42C2×Dic314yes—
258#95—68no—
259#95—68no—
260#95—68no—
261#95—64no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3—24no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—24no—
276C23×C3—24no—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—28no—
280C23×C3—24no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3—44no—
285C23×C3—42no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—42no—
292C23×C3C2×C22⋊C414yes—
293C23×C3#11714yescomplemented
294C23×C3C23×C411yescharacteristic
295C23×C3C22×D412yes—
296C23×C3C22×D411yescharacteristic
297C23×C3C22×D411yescharacteristic
298C23×C3C2×#4112yes—
299C23×C3C2×#4112yes—
300C23×C3C2×#4112yes—
301C23×C3C2×#4114yes—
302C3×D4—232no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—232no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—416no—
322C2×C3×C4C2×C22⋊C414yes—
323C2×C3×C4C2×C22⋊C414yes—
324C2×C3×C4#11714yescomplemented
325C2×C3×C4C4×D418yes—
326C2×C3×C4#12014yes—
327C2×C3×C4#12118yes—
328C2×C3×C4#12314yes—
329C2×Dic3—28no—
330C2×Dic3—28no—
331C2×Dic3—28no—
332C2×Dic3—28no—
333C2×Dic3—28no—
334C2×Dic3—28no—
335C2×Dic3—44no—
336C2×Dic3—44no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3—44no—
340C2×Dic3—44no—
341C2×Dic3—44no—
342C2×Dic3—44no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—21no—
353C24—21no—
354C24#23514yes—
355C24C22×Dic312yes—
356C24C22×Dic312yes—
357C24C2×#6712yes—
358C24C2×#6712yes—
359C24C23×D311yescharacteristic
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—216no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—38no—
384C22×C4—38no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—64no—
392C22×C4—64no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4#217116yescomplemented
401C22×C4#23514yes—
402C22×C4D3×D414yes—
403C22×C4#23714yes—
404C22×C4C22×Dic312yes—
405C22×C4C22×Dic312yes—
406C22×C4C22×Dic312yes—
407C22×C4C22×Dic312yes—
408C22×C4C22×Dic312yes—
409C22×C4C2×#6712yes—
410C22×C4C2×#6712yes—
411C22×C4C2×#6712yes—
412C22×C4C2×#6712yes—
413C4⋊C4—28no—
414C4⋊C4—616no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—616no—
418C22⋊C4—632no—
419C22⋊C4—616no—
420C42—28no—
421C42—616no—
422C42—128no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—28no—
427C22×C3—24no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—48no—
431C22×C3—48no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—42no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—42no—
439C22×C3C22×C22⋊C411yescharacteristic
440C22×C3C2×#11711yescharacteristic
441C22×C3C2×#11711yescharacteristic
442C22×C3C2×C4×D412yes—
443C22×C3C4×#4112yes—
444C22×C3C2×#12011yescharacteristic
445C22×C3C2×#12112yes—
446C22×C3C2×#12311yescharacteristic
447C22×C3#52512yes—
448C22×C3#52512yes—
449C22×C3#52514yes—
450C22×C3#52512yes—
451C22×C3#52914yes—
452C22×C3#52914yes—
453C22×C3#53214yes—
454C22×C3#53212yes—
455C3×C4—24no—
456C3×C4—24no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4#38618yes—
460Dic3—48no—
461Dic3—48no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—44no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23#92614yescomplemented
493C23C2×#21714yes—
494C23C2×#23512yes—
495C23C2×#23512yes—
496C23C2×D3×D412yes—
497C23C2×#23712yes—
498C23D3×#4114yes—
499C23C23×Dic311yescharacteristic
500C23C22×#6711yescharacteristic
501C23C22×#6711yescharacteristic
502D4—232no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—232no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—416no—
522C2×C4—68no—
523C2×C4—68no—
524C2×C4—68no—
525C2×C4—68no—
526C2×C4—68no—
527C2×C4—68no—
528C2×C4—124no—
529C2×C4—124no—
530C2×C4—124no—
531C2×C4—124no—
532C2×C4—124no—
533C2×C4—124no—
534C2×C4—124no—
535C2×C4—124no—
536C2×C4#92614yescomplemented
537C2×C4D4×Dic318yes—
538C2×C4#93514yes—
539C2×C4#93714yes—
540C2×C4#93918yes—
541C2×C4C2×#21714yes—
542C2×C4C2×#21714yes—
543C2×C3—44no—
544C2×C3—44no—
545C2×C3#228011yescharacteristic
546C2×C3C2×#38611yescharacteristic
547C2×C3#229012yes—
548C2×C3#229312yes—
549C2×C3#229711yescharacteristic
550C2×C3#238812yes—
551C2×C3#239214yes—
552C2×C3#239512yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—28no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—48no—
564C22—48no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22#553814yes—
570C22#554714yes—
571C22#554814yes—
572C22#555714yes—
573C22C2×#92611yescharacteristic
574C22C2×#92611yescharacteristic
575C22#579112yes—
576C22#579112yes—
577C22C2×D4×Dic312yes—
578C22C2×#93511yescharacteristic
579C22C2×#93711yescharacteristic
580C22C2×#93912yes—
581C22Dic3×#4112yes—
582C22#582812yes—
583C22#583212yes—
584C22C22×#21711yescharacteristic
585C4—24no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—128no—
590C4—128no—
591C4#522018yes—
592C3#1383911yescharacteristic, complemented, Sylow
593C2—44no—
594C2—44no—
595C2#8231812yes—
596C2#8231912yes—
597C2#8234514yes—
598C2#8234612yes—
599C2#8240012yes—
600C2#8402711yescharacteristic
601C2C2×#522011yescharacteristic
602C2#8408611yescharacteristic
subgroupG/H
centerC23×C4C22×D3
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×C4×D4C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC22×C4×D4D3
hypocenterC3#13839
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×#926C2semidirect
3. C2×C4×#217C2semidirect
4. C2×C4×#217C2semidirect
6. #71565C2semidirect
7. #71548C2semidirect
8. #71546C2semidirect
9. C4×C3⋊#95C2semidirect
16. C2×C3×C4×D4C4semidirect
96. C2×C4×Dic3D4semidirect
106. C2×C4×D4Dic3semidirect
168. C22×C3×C4C22⋊C4semidirect
293. C23×C3#117semidirect
324. C2×C3×C4#117semidirect
400. C22×C4#217semidirect
492. C23#926semidirect
536. C2×C4#926semidirect
592. C3#13839semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C22×#9262
C2×C4×#2172
#721312
#715652
#715482
#715462
C4×C3⋊#952
#138393