Finite Group Databasebeta

← #10822757

Subgroups — #10822757

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5510C211yesmaximal, characteristic
2C22×C3×#107C211yesmaximal, characteristic, complemented
3C2×#5113C211yesmaximal, characteristic, complemented
4C2×#5110C211yesmaximal, characteristic, complemented
5#82186C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#70635C214yesmaximal, complemented
9#70634C214yesmaximal, complemented
10#23488—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930—22no—
17C2×#926C2211yescharacteristic
18C22×#208C2211yescharacteristic
19#5510—24no—
20#5510—24no—
21C2×#897—21no—
22C2×#897C2212yes—
23C2×#895—21no—
24C2×#895C2212yes—
25C2×C3×#107—24no—
26C2×C3×#107C2214yes—
27#5213—22no—
28#5113—24no—
29C2×#821C2212yes—
30#5110—24no—
31C2×(C2×D3)⋊C8C2214yes—
32C2×#816C2212yes—
33#4988—21no—
34#4987C2214yes—
35#4973—21no—
36C2×#810C2212yes—
37#4966—24no—
38C2×#809C2212yes—
39#4529C2214yes—
40C2×#537—31no—
41C22×#107D311yescharacteristic, complemented
42C2×#417—31no—
43C2×#416—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2033—34no—
48#1736—31no—
49C23×C3×C4—21no—
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C22×#67—21no—
53C22×#67D412yes—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235D414yes—
66C22×D12—22no—
67C22×D12D411yescharacteristic
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3C2312yes—
71C22×Dic6D411yescharacteristic
72C22×Dic6C2311yescharacteristic
73C22×C3×C8—22no—
74C22×C3×C8—22no—
75C22×C3×C8D414yescomplemented
76C22×C3×C8C2312yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4D414yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×#217—21no—
86#930—216no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—24no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4×Dic3—21no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208—22no—
98C22×C3⋊C8C2312yes—
99#897—28no—
100#897—24no—
101#897—24no—
102#895—28no—
103#895—24no—
104#895—24no—
105C3×#107—28no—
106C3⋊#95—24no—
107#821—24no—
108(C2×D3)⋊C8—28no—
109#816—24no—
110#810—24no—
111#809—24no—
112C22×#41—31no—
113C22×#41—31no—
114C22×M16—31no—
115C23×C8C2×D311yescharacteristic
116#537—64no—
117#537—64no—
118C2×C22⋊Q8—32no—
119C2×C22⋊Q8—61no—
120C2×#121—32no—
121C2×#121—61no—
122C2×#121—62no—
123C2×#117—31no—
124C22×C4⋊C4C2×D311yescharacteristic
125C22×C4⋊C4C2×D311yescharacteristic
126C2×#107—24no—
127C2×#107C2×D314yes—
128#417—64no—
129#416—64no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134#408—64no—
135C2×C22⋊C8—34no—
136#391—61no—
137#390—61no—
138#386—62no—
139#380—34no—
140#340—34no—
141C24×C3C2×D411yescharacteristic
142C23×D3—21no—
143C23×D3C2×D412yes—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—42no—
148C22×C3×C4—82no—
149C22×C3×C4—82no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4112yes—
157C22×C3×C4C2411yescharacteristic
158C2×#67—24no—
159C2×#67—24no—
160C2×#67—24no—
161C2×#67—24no—
162C2×#67—42no—
163C2×#67—42no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3C2×D412yes—
167#235—28no—
168#235—28no—
169#235—44no—
170#235—44no—
171#235—44no—
172#235—44no—
173#235—44no—
174#235—44no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—21no—
183C2×D12—28no—
184C2×D12—84no—
185C2×D12C2×D412yes—
186C2×C4×D3—24no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—42no—
194C2×C4×D3—42no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—21no—
203C2×Dic6C2×D412yes—
204C2×C3×C8—28no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8C2×D414yes—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4C2×D414yes—
214#217—42no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—84no—
218C4⋊Dic3—28no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4×Dic3—42no—
222#208—44no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×#41—34no—
227C2×#41—62no—
228C2×#41—62no—
229C2×#41—62no—
230C2×#41—62no—
231C2×#41—62no—
232C2×#41—62no—
233C2×#41—62no—
234C2×#41—62no—
235C2×#41—62no—
236C2×#41—62no—
237C2×#41—62no—
238C22×Q8—31no—
239C22×Q8—31no—
240C22×D4—32no—
241C22×D4—31no—
242C22×D4—31no—
243C22×D4—62no—
244C22×D4—61no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—32no—
248C23×C4—61no—
249C23×C4C22×D311yescharacteristic
250C2×M16—62no—
251C2×M16—62no—
252C2×M16—62no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—32no—
256C22×C8D1214yescomplemented
257C22×C8C22×D312yes—
258C22⋊Q8—64no—
259C22⋊Q8—68no—
260C22⋊Q8—64no—
261#121—64no—
262#121—68no—
263#121—616no—
264#121—64no—
265#117—62no—
266#117—62no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4#6714yes—
275C2×C4⋊C4C22×D312yes—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—64no—
281C2×C42—61no—
282#107—28no—
283#103—64no—
284#103—64no—
285#102—64no—
286#102—64no—
287C22⋊C8—68no—
288#95—64no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3#12012yes—
292C23×C3#12112yes—
293C23×C3#12312yes—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—42no—
305C2×C3×C4—24no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—44no—
309C2×C3×C4—84no—
310C2×C3×C4—84no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12114yes—
313C2×C3×C4#12112yes—
314C2×C3×C4#12114yes—
315C2×C3×C4#12314yes—
316C2×C3×C4#12312yes—
317C2×C3×C4#12414yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324#67—48no—
325#67—48no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—22no—
336D12—22no—
337D12—41no—
338D12—41no—
339D12—44no—
340D12—88no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—42no—
344C4×D3—42no—
345C4×D3—48no—
346Dic6—22no—
347Dic6—22no—
348Dic6—41no—
349Dic6—41no—
350Dic6—44no—
351C3×C8—24no—
352C3⋊C8—42no—
353C24—32no—
354C24—61no—
355C24D3×D411yescharacteristic
356#41—68no—
357#41—68no—
358#41—124no—
359#41—124no—
360#41—124no—
361#41—124no—
362#41—124no—
363#41—124no—
364C2×Q8—32no—
365C2×Q8—62no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—61no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×D4—32no—
374C2×D4—68no—
375C2×D4—64no—
376C2×D4—64no—
377C2×D4—64no—
378C2×D4—64no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—122no—
388C2×D4—122no—
389C2×D4—244no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—32no—
394C22×C4—42no—
395C22×C4—61no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4C2×D1212yes—
409C22×C4#23512yes—
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D412yes—
412C22×C4#23912yes—
413C22×C4#23912yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416M16—124no—
417C2×C8—28no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8C2×D1214yes—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—42no—
429C4⋊C4—68no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4C2×#6714yes—
433C22⋊C4—68no—
434C22⋊C4—68no—
435C22⋊C4—122no—
436C22⋊C4—244no—
437C42—122no—
438C22×C3—24no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3#39414yes—
443C22×C3C2×#12011yescharacteristic
444C22×C3C2×#12111yescharacteristic
445C22×C3C2×#12311yescharacteristic
446C22×C3#57612yes—
447C22×C3#57812yes—
448C2×D3—44no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C2×D3—44no—
456C2×D3—82no—
457C2×D3—82no—
458C3×C4—44no—
459C3×C4—82no—
460C3×C4#39414yes—
461C3×C4C2×#12111yescharacteristic
462C3×C4C2×#12111yescharacteristic
463C3×C4C2×#12311yescharacteristic
464C3×C4C2×#12411yescharacteristic
465Dic3—42no—
466Dic3—42no—
467Dic3—82no—
468C23—22no—
469C23—22no—
470C23—64no—
471C23—62no—
472C23—62no—
473C23—64no—
474C23—62no—
475C23—62no—
476C23—122no—
477C23—121no—
478C23—121no—
479C23—122no—
480C23#89412yes—
481C23#89512yes—
482C23#93712yes—
483C23C2×D3×D411yescharacteristic
484Q8—62no—
485Q8—62no—
486Q8—124no—
487Q8—121no—
488Q8—121no—
489D4—62no—
490D4—62no—
491D4—128no—
492D4—128no—
493D4—124no—
494D4—121no—
495D4—121no—
496D4—248no—
497C2×C4—24no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—128no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4#89412yes—
518C2×C4#89414yes—
519C2×C4#89512yes—
520C2×C4#89514yes—
521C2×C4#93014yes—
522C2×C4#93712yes—
523C2×C4#94714yes—
524C2×C4C22×D1211yescharacteristic
525C2×C4C2×#23511yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×#23911yescharacteristic
528C2×C4C2×#23911yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—24no—
531C8—122no—
532C2×C3—24no—
533C2×C3C2×#39411yescharacteristic
534C2×C3#300811yescharacteristic
535C2×C3#301111yescharacteristic
536C2×C3#306112yes—
537C2×C3#309812yes—
538D3—82no—
539D3—82no—
540C22—24no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—122no—
545C22—122no—
546C22—124no—
547C22—124no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—242no—
553C22—242no—
554C22#499114yes—
555C22C2×#89411yescharacteristic
556C22C2×#89511yescharacteristic
557C22#575712yes—
558C22#576612yes—
559C22C2×#93711yescharacteristic
560C4—44no—
561C4—82no—
562C4—122no—
563C4—122no—
564C4—242no—
565C4#499114yes—
566C4C2×#89411yescharacteristic
567C4C2×#89511yescharacteristic
568C4C2×#93011yescharacteristic
569C4C2×#94711yescharacteristic
570C3#2348811yescharacteristic, complemented, Sylow
571C2—24no—
572C2—242no—
573C2—242no—
574C2#7914112yes—
575C2#7915012yes—
576C2C2×#499111yescharacteristic
577C2#8383311yescharacteristic
578C2#8388411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#107C2
socleC23×C3C22×D4
prime radicalC2×#5510C2
hypercenterC22×#107D3
hypocenterC3#23488
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C22×C3×#107C2semidirect
3. C2×#5113C2semidirect
4. C2×#5110C2semidirect
5. #82186C2semidirect
6. C2×#4966C2semidirect
7. #71558C2semidirect
8. #70635C2semidirect
9. #70634C2semidirect
41. C22×#107D3semidirect
75. C22×C3×C8D4semidirect
256. C22×C8D12semidirect
570. C3#23488semidirect

Maximal subgroups

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

subgroupindex
C2×#55102
C22×C3×#1072
C2×#51132
C2×#51102
#821862
C2×#49662
#715582
#706352
#706342
#234883