Finite Group Databasebeta

← #10828789

Subgroups — #10828789

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5514C211yesmaximal, characteristic
2C22×#808C211yesmaximal, characteristic, complemented
3C2×#4736C211yesmaximal, characteristic, complemented
4#74027C211yesmaximal, characteristic, complemented
5C3×#1824C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic, complemented
7#69515C211yesmaximal, characteristic, complemented
8#68895C212yesmaximal, complemented
9#68877C212yesmaximal, complemented
10#68829C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#24091—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×#886—21no—
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C2×#821C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#818C2212yes—
34#4987C2212yes—
35#4984C2212yes—
36C2×#809C2212yes—
37C2×#808—24no—
38C2×#808C2212yes—
39C2×#808C2212yes—
40#4943—21no—
41#4736—24no—
42#4674—21no—
43#4661—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#480—31no—
50C22×#106—31no—
51#2486—31no—
52#2059—32no—
53#2034—32no—
54#1893—31no—
55#1869—31no—
56#1864—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—
113#886—44no—
114C3×C22⋊C8—22no—
115C3×#95—21no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#821—24no—
119(C2×D3)⋊C8—24no—
120#818—24no—
121#809—24no—
122#808—24no—
123#808—24no—
124C23×C8—31no—
125#553—64no—
126#553—64no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#123—61no—
130C2×#123—61no—
131C2×#121—31no—
132C2×#121—31no—
133C2×#121—61no—
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137C2×#117—31no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140#480—64no—
141C2×#107—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8—32no—
148C2×C22⋊C8C2×D311yescharacteristic
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—32no—
154#380C2×D311yescharacteristic
155#340—32no—
156#340C2×D312yes—
157#340C2×D312yes—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C23×D3—21no—
161C23×D3C2×D411yescharacteristic
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—82no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—22no—
174C2×#67—24no—
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—42no—
178C2×#67—42no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—42no—
182C22×Dic3—42no—
183C22×Dic3—42no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—21no—
190C2×D12—24no—
191C2×D12—82no—
192C2×D12C2×D412yes—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—24no—
198C2×C4×D3—82no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—42no—
210#217—22no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—44no—
217C4⋊Dic3—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—44no—
222C4⋊Dic3—42no—
223C4⋊Dic3C2×D412yes—
224Dic3⋊C4—24no—
225Dic3⋊C4—22no—
226C4×Dic3—24no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C2×C3⋊C8—28no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8#4114yes—
233C22×D4—31no—
234C22×D4—31no—
235C22×D4—61no—
236C22×D4—61no—
237C23×C4—21no—
238C23×C4—31no—
239C23×C4—61no—
240C23×C4—61no—
241C23×C4C22×D311yescharacteristic
242C22×C8—32no—
243C22×C8—64no—
244C22×C8—62no—
245C22×C8C22×D312yes—
246#126—68no—
247#125—62no—
248#125—62no—
249#123—68no—
250#123—124no—
251#121—34no—
252#121—62no—
253#121—68no—
254#121—62no—
255#121—62no—
256C4×D4—62no—
257C4×D4—68no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261#117—62no—
262#117—62no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4D1214yes—
275C2×C4⋊C4C22×D312yes—
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—62no—
286C2×C42—31no—
287C2×C42—61no—
288#107—64no—
289#106—64no—
290#106—64no—
291#102—64no—
292#102—64no—
293C22⋊C8—22no—
294C22⋊C8—64no—
295#95—21no—
296#95—62no—
297#95—62no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3#12012yes—
301C23×C3#13512yes—
302C23×C3AGL(1,ℤ/8)12yes—
303C23×C3C22×D411yescharacteristic
304C22×D3—21no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—22no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—42no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—84no—
326C2×C3×C4#12012yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4112yes—
330C2×C3×C42−1+412yes—
331#67—44no—
332#67—48no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—84no—
347D12—24no—
348D12—41no—
349D12—41no—
350D12—84no—
351C4×D3—41no—
352C4×D3—41no—
353C4×D3—42no—
354C4×D3—84no—
355C3×C8—42no—
356C3⋊C8—24no—
357C24—31no—
358C24—61no—
359C24D3×D411yescharacteristic
360C2×D4—32no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4—242no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—64no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—82no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4—242no—
393C22×C4C2×D1212yes—
394C22×C4D3×D411yescharacteristic
395C22×C4D3×D412yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—34no—
401C2×C8—68no—
402C2×C8—62no—
403C2×C8—62no—
404C4⋊C4—24no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—32no—
410C4⋊C4—64no—
411C4⋊C4—64no—
412C4⋊C4—62no—
413C4⋊C4—61no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—124no—
417C4⋊C4—122no—
418C4⋊C4C2×D1212yes—
419C4⋊C4C2×D1212yes—
420C22⋊C4—42no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—64no—
427C22⋊C4—124no—
428C42—64no—
429C42—61no—
430C42—61no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3#44914yes—
436C22×C3C2×#12011yescharacteristic
437C22×C3#54712yes—
438C22×C3C2×#13511yescharacteristic
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C22×C3#57712yes—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—41no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—41no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—82no—
455C3×C4#52511yescharacteristic
456C3×C4#53611yescharacteristic
457C3×C4#54712yes—
458Dic3—41no—
459Dic3—41no—
460Dic3—42no—
461Dic3—41no—
462Dic3—82no—
463C23—22no—
464C23—22no—
465C23—62no—
466C23—62no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—62no—
471C23—122no—
472C23—121no—
473C23—121no—
474C23—122no—
475C23#88212yes—
476C23#91412yes—
477C23#91912yes—
478C23C2×D3×D411yescharacteristic
479D4—64no—
480D4—124no—
481D4—128no—
482D4—121no—
483D4—121no—
484D4—244no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—62no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—62no—
504C2×C4—84no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—244no—
514C2×C4—244no—
515C2×C4#88212yes—
516C2×C4C22×D1211yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4D3×#4112yes—
519C2×C4#101012yes—
520C8—42no—
521C8—64no—
522C2×C3—24no—
523C2×C3#240611yescharacteristic
524C2×C3C2×#44911yescharacteristic
525C2×C3#300911yescharacteristic
526C2×C3#333712yes—
527C2×C3#334512yes—
528D3—81no—
529D3—82no—
530C22—24no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—122no—
535C22—121no—
536C22—122no—
537C22—122no—
538C22—121no—
539C22—122no—
540C22—121no—
541C22—122no—
542C22—242no—
543C22—242no—
544C22#481014yes—
545C22C2×#88211yescharacteristic
546C22#565512yes—
547C22C2×#91411yescharacteristic
548C22C2×#91911yescharacteristic
549C22#577512yes—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—121no—
555C4—121no—
556C4—121no—
557C4—122no—
558C4—242no—
559C4#552411yescharacteristic
560C4#552611yescharacteristic
561C4#565512yes—
562C3#2409111yescharacteristic, complemented, Sylow
563C2—24no—
564C2—241no—
565C2—242no—
566C2#7406311yescharacteristic
567C2C2×#481011yescharacteristic
568C2#7892911yescharacteristic
569C2#8132112yes—
570C2#8146412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1824C2
socleC23×C3C22×D4
prime radicalC2×#5514C2
hypercenter#1824D3
hypocenterC3#24091
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×#808C2semidirect
3. C2×#4736C2semidirect
4. #74027C2semidirect
5. C3×#1824C2semidirect
6. #69952C2semidirect
7. #69515C2semidirect
8. #68895C2semidirect
9. #68877C2semidirect
10. #68829C2semidirect
11. #68823C2semidirect
58. #1824D3semidirect
562. C3#24091semidirect

Maximal subgroups

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

subgroupindex
C2×#55142
C22×#8082
C2×#47362
#740272
C3×#18242
#699522
#695152
#688952
#688772
#688292
#688232
#240913