Finite Group Databasebeta

← #10828800

Subgroups — #10828800

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5514C211yesmaximal, characteristic
2C22×#807C211yesmaximal, characteristic, complemented
3C2×#4735C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic, complemented
5C3×#1824C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic, complemented
7#69515C211yesmaximal, characteristic, complemented
8#68894C212yesmaximal, complemented
9#68876C212yesmaximal, complemented
10#68828C212yesmaximal, complemented
11#68822C212yesmaximal, complemented
12#24092—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#930—21no—
15C2×#930C2211yescharacteristic
16C2×C4×#67—21no—
17C2×C4×#67C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20#5514—24no—
21#5514—24no—
22C2×#898—21no—
23C2×#895C2211yescharacteristic
24C2×#894—21no—
25C2×#893C2211yescharacteristic
26C2×#892C2211yescharacteristic
27C2×#890C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C2×#821C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2212yes—
33#4987C2212yes—
34#4984C2212yes—
35C2×#809C2212yes—
36C2×#807—24no—
37C2×#807C2212yes—
38C2×#807C2212yes—
39#4958—21no—
40#4950—21no—
41#4735—24no—
42#4677—21no—
43#4659—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#481—31no—
50C22×#107—31no—
51#2487—31no—
52#2059—32no—
53#2033—32no—
54#1893—31no—
55#1869—31no—
56#1862—32no—
57#1854—32no—
58#1824D311yescharacteristic, complemented
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67D411yescharacteristic
63C23×Dic3—21no—
64C22×D12—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8C2312yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D414yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×#217D411yescharacteristic
77#930—28no—
78#930—22no—
79#930D414yes—
80C4×#67—28no—
81C4×#67—22no—
82C4×#67—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4D411yescharacteristic
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4D411yescharacteristic
98C2×C4×Dic3—21no—
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8—22no—
101C22×C3⋊C8—24no—
102C22×C3⋊C8C2312yes—
103#898—28no—
104#895—22no—
105#895—22no—
106#894—28no—
107#893—22no—
108#893—22no—
109#892—22no—
110#892—22no—
111#890—22no—
112#890—22no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117#821—24no—
118(C2×D3)⋊C8—24no—
119#817—24no—
120#809—24no—
121#807—24no—
122#807—24no—
123C23×C8—31no—
124#553—64no—
125#553—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#123—61no—
129C2×#121—31no—
130C2×#121—31no—
131C2×#121—61no—
132C2×C4×D4—31no—
133C2×C4×D4—31no—
134C2×C4×D4—61no—
135C2×#117—31no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#481—64no—
139C2×#107—32no—
140C2×#107—32no—
141C2×#107—64no—
142C2×#106—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—32no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#399—61no—
149#389—32no—
150#386—61no—
151#385—61no—
152#380—32no—
153#380C2×D311yescharacteristic
154#340—32no—
155#340C2×D312yes—
156#340C2×D312yes—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C23×D3C2×D411yescharacteristic
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—82no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C2×#67—22no—
173C2×#67—24no—
174C2×#67—22no—
175C2×#67—22no—
176C2×#67—42no—
177C2×#67—42no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—42no—
181C22×Dic3—82no—
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—21no—
188C2×D12—24no—
189C2×D12—82no—
190C2×D12C2×D412yes—
191C2×C4×D3—21no—
192C2×C4×D3—21no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—24no—
196C2×C4×D3—82no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D412yes—
206C3×C4⋊C4C2×D412yes—
207C3×C22⋊C4—42no—
208#217—22no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—22no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—82no—
215C4⋊Dic3—24no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220C4⋊Dic3—82no—
221C4⋊Dic3C2×D412yes—
222Dic3⋊C4—24no—
223Dic3⋊C4—22no—
224C4×Dic3—24no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C2×C3⋊C8—28no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8#4114yes—
231C22×D4—31no—
232C22×D4—31no—
233C22×D4—61no—
234C22×D4—61no—
235C23×C4—21no—
236C23×C4—31no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C22×C8—32no—
241C22×C8—64no—
242C22×C8—62no—
243C22×C8C22×D312yes—
244#126—68no—
245#125—62no—
246#125—62no—
247#123—68no—
248#121—34no—
249#121—62no—
250#121—68no—
251#121—62no—
252#121—62no—
253C4×D4—62no—
254C4×D4—68no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—62no—
258#117—62no—
259#117—62no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4D1214yes—
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—31no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—62no—
283C2×C42—31no—
284C2×C42—61no—
285#107—64no—
286#107—64no—
287#106—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—22no—
291C22⋊C8—64no—
292#95—21no—
293#95—62no—
294#95—62no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3#12012yes—
298C23×C3#13512yes—
299C23×C3#13712yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—21no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—22no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—42no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—84no—
323C2×C3×C4#12012yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4112yes—
327C2×C3×C42−1+412yes—
328#67—44no—
329#67—48no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—22no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—84no—
344D12—24no—
345D12—41no—
346D12—41no—
347D12—84no—
348C4×D3—41no—
349C4×D3—41no—
350C4×D3—42no—
351C4×D3—84no—
352C3×C8—42no—
353C3⋊C8—24no—
354C24—31no—
355C24—61no—
356C24D3×D411yescharacteristic
357C2×D4—32no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—61no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—242no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—64no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—82no—
386C22×C4—122no—
387C22×C4—242no—
388C22×C4—242no—
389C22×C4C2×D1212yes—
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D412yes—
392C22×C4C23×D311yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—34no—
397C2×C8—68no—
398C2×C8—62no—
399C2×C8—62no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—32no—
406C4⋊C4—64no—
407C4⋊C4—64no—
408C4⋊C4—62no—
409C4⋊C4—61no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—122no—
413C4⋊C4—242no—
414C4⋊C4C2×D1212yes—
415C4⋊C4C2×D1212yes—
416C22⋊C4—42no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C22⋊C4—62no—
421C22⋊C4—62no—
422C22⋊C4—64no—
423C22⋊C4—242no—
424C42—64no—
425C42—61no—
426C42—61no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3#45214yes—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54712yes—
434C22×C3C2×#13511yescharacteristic
435C22×C3C2×#13711yescharacteristic
436C22×C3#57612yes—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—41no—
440C2×D3—42no—
441C2×D3—41no—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—41no—
445C2×D3—82no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4—82no—
451C3×C4#52511yescharacteristic
452C3×C4#53611yescharacteristic
453C3×C4#54712yes—
454Dic3—41no—
455Dic3—41no—
456Dic3—42no—
457Dic3—41no—
458Dic3—82no—
459C23—22no—
460C23—22no—
461C23—62no—
462C23—62no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—62no—
467C23—122no—
468C23—121no—
469C23—121no—
470C23—122no—
471C23#88212yes—
472C23#91612yes—
473C23#91812yes—
474C23C2×D3×D411yescharacteristic
475D4—64no—
476D4—124no—
477D4—128no—
478D4—121no—
479D4—121no—
480D4—244no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—62no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—84no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—244no—
510C2×C4—244no—
511C2×C4#88212yes—
512C2×C4C22×D1211yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4D3×#4112yes—
515C2×C4#101012yes—
516C8—42no—
517C8—64no—
518C2×C3—24no—
519C2×C3#240611yescharacteristic
520C2×C3C2×#45211yescharacteristic
521C2×C3#300811yescharacteristic
522C2×C3#333912yes—
523C2×C3#334612yes—
524D3—81no—
525D3—82no—
526C22—24no—
527C22—22no—
528C22—22no—
529C22—24no—
530C22—122no—
531C22—121no—
532C22—122no—
533C22—122no—
534C22—121no—
535C22—122no—
536C22—121no—
537C22—122no—
538C22—242no—
539C22—242no—
540C22#480814yes—
541C22C2×#88211yescharacteristic
542C22#565512yes—
543C22C2×#91611yescharacteristic
544C22C2×#91811yescharacteristic
545C22#577412yes—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—82no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4—122no—
554C4—242no—
555C4#552411yescharacteristic
556C4#552611yescharacteristic
557C4#565512yes—
558C3#2409211yescharacteristic, complemented, Sylow
559C2—24no—
560C2—241no—
561C2—242no—
562C2#7406311yescharacteristic
563C2C2×#480811yescharacteristic
564C2#7892811yescharacteristic
565C2#8133412yes—
566C2#8147812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1824C2
socleC23×C3C22×D4
prime radicalC2×#5514C2
hypercenter#1824D3
hypocenterC3#24092
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. C22×#807C2semidirect
3. C2×#4735C2semidirect
4. #74028C2semidirect
5. C3×#1824C2semidirect
6. #69952C2semidirect
7. #69515C2semidirect
8. #68894C2semidirect
9. #68876C2semidirect
10. #68828C2semidirect
11. #68822C2semidirect
58. #1824D3semidirect
558. C3#24092semidirect

Maximal subgroups

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

subgroupindex
C2×#55142
C22×#8072
C2×#47352
#740282
C3×#18242
#699522
#695152
#688942
#688762
#688282
#688222
#240923