Finite Group Databasebeta

← #10828792

Subgroups — #10828792

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×#5565C211yesmaximal, characteristic
2C22×#817C211yesmaximal, characteristic, complemented
3#82373C211yesmaximal, characteristic, complemented
4C2×#4736C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69935C212yesmaximal, complemented
8#69929C212yesmaximal, complemented
9#69925C211yesmaximal, characteristic, complemented
10#68671C212yesmaximal, complemented
11#68634C212yesmaximal, complemented
12#24091—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#939—21no—
16C2×#935—21no—
17C2×#930C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22#5565—24no—
23#5565—24no—
24C2×#898—21no—
25C2×#883—21no—
26C2×#67⋊C4—21no—
27C2×C4×D12C2211yescharacteristic
28C2×#870C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2212yes—
31#5212C2211yescharacteristic
32#5212C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#818C2212yes—
36C2×#817—24no—
37C2×#817C2212yes—
38C2×#817C2212yes—
39#4981—21no—
40#4953—21no—
41#4943—21no—
42#4928C2212yes—
43#4736—24no—
44C3×#336C2212yes—
45#4551C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#553—31no—
49C2×#480—31no—
50C22×#106—31no—
51#2486—31no—
52#2059—32no—
53#2034—32no—
54#1893—31no—
55#1869D311yescharacteristic, complemented
56#1864—32no—
57#1854—32no—
58#1824—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C23×Dic3—21no—
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C22×C3×C8—22no—
69C22×C3×C8—24no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D411yescharacteristic
76C2×C3×C22⋊C4D411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217—21no—
79C2×#217—21no—
80C2×#217—22no—
81#939—28no—
82#935—44no—
83#930—22no—
84#930—22no—
85#930—22no—
86#930D414yes—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3D414yes—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—22no—
106C2×C4×Dic3—21no—
107#898—28no—
108#883—28no—
109#67⋊C4—28no—
110C4×D12—22no—
111C4×D12—22no—
112#870—22no—
113#870—22no—
114C3×C22⋊C8—22no—
115C3×C22⋊C8—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118C3⋊#95—21no—
119#821—24no—
120#821—24no—
121#818—24no—
122#817—24no—
123#817—24no—
124C23×C8C2×D311yescharacteristic
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—61no—
136C2×C4×D4C2×D311yescharacteristic
137C2×#117—31no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—31no—
140#480—64no—
141C2×#107—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8C2×D311yescharacteristic
148C2×C22⋊C8C2×D312yes—
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—31no—
154#380—32no—
155#340—32no—
156#340—32no—
157#340—32no—
158#336C2×D312yes—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C23×D3—21no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—24no—
179C2×#67—24no—
180C2×#67—42no—
181C2×#67—42no—
182C2×#67—82no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—24no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—42no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C22×Dic3—82no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—21no—
196C2×D12C2×D412yes—
197C2×C4×D3—24no—
198C2×C4×D3—82no—
199C2×C3×C8—28no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8#4114yes—
206C3×C4⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C42—21no—
210C3×C42—21no—
211#217—24no—
212#217—44no—
213#217—42no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217C4⋊Dic3—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—42no—
226C4⋊Dic3C2×D412yes—
227C4⋊Dic3C2×D412yes—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—24no—
231Dic3⋊C4—44no—
232C4×Dic3—24no—
233C22×D4—31no—
234C22×D4—61no—
235C22×D4—61no—
236C22×D4D1211yescharacteristic
237C23×C4—61no—
238C23×C4—61no—
239C23×C4—61no—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C22×C8—22no—
243C22×C8—24no—
244C22×C8C22×D312yes—
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—62no—
254#121—68no—
255#121—62no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—68no—
261#117—62no—
262#117—62no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4D1211yescharacteristic
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4D1211yescharacteristic
285C2×C22⋊C4D1211yescharacteristic
286C2×C42—61no—
287C2×C42C22×D311yescharacteristic
288#107—64no—
289#106—64no—
290#106—64no—
291#102—64no—
292#102—64no—
293C22⋊C8—22no—
294C22⋊C8—24no—
295#95—62no—
296#95—61no—
297#95—62no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—22no—
305C23×C3#12012yes—
306C23×C3#13512yes—
307C23×C3AGL(1,ℤ/8)12yes—
308C23×C3C22×D411yescharacteristic
309C22×D3—22no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—42no—
314C3×D4—44no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4#12012yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4112yes—
333C2×C3×C42−1+412yes—
334#67—48no—
335#67—84no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—84no—
350C2×Dic3—84no—
351D12—24no—
352D12—41no—
353D12—41no—
354C4×D3—84no—
355C3×C8—24no—
356C3×C8—42no—
357C24—61no—
358C24C2×D1211yescharacteristic
359C24C2×D1211yescharacteristic
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—32no—
364C2×D4—64no—
365C2×D4—64no—
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—22no—
377C22×C4—22no—
378C22×C4—64no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—64no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—242no—
389C22×C4—242no—
390C22×C4C2×D1211yescharacteristic
391C22×C4C2×D1211yescharacteristic
392C22×C4C2×D1211yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4D3×D412yes—
395C22×C4D3×D412yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—28no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8#23714yes—
404C4⋊C4—22no—
405C4⋊C4—32no—
406C4⋊C4—32no—
407C4⋊C4—32no—
408C4⋊C4—64no—
409C4⋊C4—64no—
410C4⋊C4—64no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—124no—
419C4⋊C4—122no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C22⋊C4—64no—
426C22⋊C4—124no—
427C22⋊C4—122no—
428C42—21no—
429C42—21no—
430C42—64no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3#44914yes—
441C22×C3C2×#12011yescharacteristic
442C22×C3#54712yes—
443C22×C3C2×#13511yescharacteristic
444C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
445C22×C3#57712yes—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—22no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4#52511yescharacteristic
456C3×C4#53611yescharacteristic
457C3×C4#54712yes—
458Dic3—42no—
459Dic3—42no—
460Dic3—42no—
461Dic3—82no—
462Dic3—82no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—62no—
471C23—121no—
472C23—121no—
473C23—122no—
474C23—122no—
475C23#88212yes—
476C23#90412yes—
477C23#90812yes—
478C23C22×D1211yescharacteristic
479D4—44no—
480D4—64no—
481D4—128no—
482D4—121no—
483D4—121no—
484D4—244no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—244no—
513C2×C4—244no—
514C2×C4—244no—
515C2×C4#88212yes—
516C2×C4C2×D3×D411yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×#23712yes—
519C2×C4#101012yes—
520C8—24no—
521C8—42no—
522C2×C3—24no—
523C2×C3—81no—
524C2×C3#240611yescharacteristic
525C2×C3C2×#44911yescharacteristic
526C2×C3#300911yescharacteristic
527C2×C3#333712yes—
528C2×C3#334512yes—
529D3—82no—
530C22—24no—
531C22—22no—
532C22—24no—
533C22—22no—
534C22—41no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—122no—
540C22—122no—
541C22—122no—
542C22—242no—
543C22—242no—
544C22#473414yes—
545C22C2×#88211yescharacteristic
546C22#555512yes—
547C22C2×#90411yescharacteristic
548C22C2×#90811yescharacteristic
549C22#575112yes—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—122no—
555C4—122no—
556C4—122no—
557C4—242no—
558C4—242no—
559C4#558611yescharacteristic
560C4#561112yes—
561C4#564811yescharacteristic
562C3#2409111yescharacteristic, complemented, Sylow
563C2—24no—
564C2—81no—
565C2—242no—
566C2#7468711yescharacteristic
567C2C2×#473411yescharacteristic
568C2#7867111yescharacteristic
569C2#7941412yes—
570C2#7947112yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
prime radicalC2×#5565C2
hypercenter#1869D3
hypocenterC3#24091
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#817C2semidirect
3. #82373C2semidirect
4. C2×#4736C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69935C2semidirect
8. #69929C2semidirect
9. #69925C2semidirect
10. #68671C2semidirect
11. #68634C2semidirect
55. #1869D3semidirect
562. C3#24091semidirect

Maximal subgroups

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

subgroupindex
C2×#55652
C22×#8172
#823732
C2×#47362
C3×#18692
#719732
#699352
#699292
#699252
#686712
#686342
#240913