Finite Group Databasebeta

← #10828797

Subgroups — #10828797

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×#5618C211yesmaximal, characteristic
2C22×#818C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic, complemented
5C3×#1893C211yesmaximal, characteristic, complemented
6#71975C211yesmaximal, characteristic, complemented
7#70175C212yesmaximal, complemented
8#70168C212yesmaximal, complemented
9#69502C211yesmaximal, characteristic, complemented
10#68877C212yesmaximal, complemented
11#68829C212yesmaximal, complemented
12#24092—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×#939—21no—
16C2×#938C2211yescharacteristic
17C2×D4×Dic3—21no—
18C2×D4×Dic3C2211yescharacteristic
19C2×C4×#67C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22#5618—24no—
23#5618—24no—
24C2×#892C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#886—21no—
27C2×#879—21no—
28C2×C3×#102C2212yes—
29C3×#380C2211yescharacteristic
30C2×#832C2212yes—
31C2×#830C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33C2×#818—24no—
34C2×#818C2212yes—
35C2×#818C2212yes—
36#5034—24no—
37#4987C2212yes—
38#4984C2212yes—
39C2×#808C2212yes—
40#4958—21no—
41#4950—21no—
42#4677—21no—
43#4659—21no—
44C3×#340C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#481—31no—
50C22×#107—31no—
51#2487—31no—
52#2059—32no—
53#2033—32no—
54#1893D311yescharacteristic, complemented
55#1869—31no—
56#1862—32no—
57#1854—32no—
58#1824—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67D411yescharacteristic
64C23×Dic3—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8—22no—
69C22×C3×C8—24no—
70C22×C3×C8C2312yes—
71C3×#121—22no—
72C3×#121D414yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×#217—21no—
78C2×#217—21no—
79C2×#217D411yescharacteristic
80#939—28no—
81#938—22no—
82#938—22no—
83D4×Dic3—28no—
84D4×Dic3—22no—
85D4×Dic3—22no—
86C4×#67—22no—
87C4×#67—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3D414yes—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107#892—22no—
108#892—22no—
109#890—22no—
110#890—22no—
111#886—28no—
112#879—28no—
113C3×#102—24no—
114C3×#95—21no—
115#832—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119(C2×D3)⋊C8—24no—
120#818—24no—
121#818—24no—
122#808—24no—
123C23×C8C2×D311yescharacteristic
124#553—64no—
125#553—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#123—61no—
129C2×#121—31no—
130C2×#121—61no—
131C2×#121C2×D311yescharacteristic
132C2×C4×D4—31no—
133C2×C4×D4—31no—
134C2×C4×D4—61no—
135C2×#117—31no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—31no—
138#481—64no—
139C2×#107—32no—
140C2×#107—32no—
141C2×#107—64no—
142C2×#106—32no—
143C2×#102—32no—
144C2×#102C2×D312yes—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—32no—
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#340—32no—
156#340C2×D312yes—
157#336—32no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C23×D3C2×D411yescharacteristic
161C2×C3×D4—24no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—22no—
178C2×#67—22no—
179C2×#67—24no—
180C2×#67—82no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3—42no—
187C22×Dic3—82no—
188C22×Dic3—82no—
189C22×Dic3—82no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C3×C8—28no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8#4114yes—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—42no—
207#217—24no—
208#217—24no—
209#217—22no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—22no—
212(C2×D3)⋊C4—82no—
213C4⋊Dic3—24no—
214C4⋊Dic3—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220C4⋊Dic3—82no—
221C4⋊Dic3C2×D412yes—
222C4⋊Dic3C2×D412yes—
223Dic3⋊C4—24no—
224Dic3⋊C4—22no—
225C4×Dic3—24no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C22×D4—21no—
232C22×D4—31no—
233C22×D4—61no—
234C22×D4C22×D311yescharacteristic
235C23×C4—31no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C22×C8—22no—
241C22×C8—24no—
242C22×C8—32no—
243C22×C8C22×D312yes—
244#126—68no—
245#125—62no—
246#125—62no—
247#123—68no—
248#121—22no—
249#121—62no—
250#121—68no—
251#121—62no—
252#121D1214yes—
253C4×D4—62no—
254C4×D4—68no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—62no—
258#117—62no—
259#117—62no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4—31no—
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—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4C22×D312yes—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—31no—
278C2×C22⋊C4—31no—
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—24no—
289#102—64no—
290C22⋊C8—62no—
291C22⋊C8—64no—
292#95—21no—
293#95—62no—
294#95—62no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3#12012yes—
303C23×C3#13512yes—
304C23×C3#13712yes—
305C23×C3C22×D411yescharacteristic
306C22×D3—21no—
307C22×D3—22no—
308C22×D3—21no—
309C22×D3—21no—
310C22×D3—22no—
311C3×D4—24no—
312C3×D4—48no—
313C3×D4—41no—
314C3×D4—41no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4#12012yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4112yes—
329C2×C3×C42−1+412yes—
330#67—44no—
331#67—84no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—84no—
347C2×Dic3—84no—
348C2×Dic3—84no—
349C4×D3—41no—
350C4×D3—41no—
351C4×D3—42no—
352C3×C8—24no—
353C3⋊C8—42no—
354C24—21no—
355C24—31no—
356C24D3×D411yescharacteristic
357C2×D4—24no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—64no—
367C2×D4—242no—
368C2×D4C2×D1212yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—122no—
386C22×C4—242no—
387C22×C4—242no—
388C22×C4—242no—
389C22×C4C2×D1212yes—
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D412yes—
392C22×C4C23×D311yescharacteristic
393C2×C8—28no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8#23714yes—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—32no—
404C4⋊C4—32no—
405C4⋊C4—64no—
406C4⋊C4—64no—
407C4⋊C4—64no—
408C4⋊C4—62no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—122no—
414C4⋊C4—242no—
415C4⋊C4C2×D1212yes—
416C22⋊C4—24no—
417C22⋊C4—42no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—62no—
422C22⋊C4—62no—
423C22⋊C4—242no—
424C42—64no—
425C42—61no—
426C42—61no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3#45214yes—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54712yes—
439C22×C3C2×#13511yescharacteristic
440C22×C3C2×#13711yescharacteristic
441C22×C3#57612yes—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—41no—
445C2×D3—41no—
446C2×D3—41no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4#52511yescharacteristic
451C3×C4#53611yescharacteristic
452C3×C4#54712yes—
453Dic3—41no—
454Dic3—41no—
455Dic3—41no—
456Dic3—42no—
457Dic3—42no—
458Dic3—82no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—61no—
467C23—61no—
468C23—61no—
469C23—62no—
470C23—62no—
471C23#88212yes—
472C23#91212yes—
473C23#91512yes—
474C23C2×D3×D411yescharacteristic
475D4—24no—
476D4—48no—
477D4—41no—
478D4—41no—
479D4—124no—
480D4—244no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—61no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—62no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—244no—
509C2×C4—244no—
510C2×C4—244no—
511C2×C4#88212yes—
512C2×C4C22×D1211yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23712yes—
515C2×C4#101012yes—
516C8—24no—
517C8—122no—
518C2×C3—24no—
519C2×C3—82no—
520C2×C3#240611yescharacteristic
521C2×C3C2×#45211yescharacteristic
522C2×C3#300811yescharacteristic
523C2×C3#333912yes—
524C2×C3#334612yes—
525D3—81no—
526C22—24no—
527C22—24no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—82no—
535C22—122no—
536C22—122no—
537C22—121no—
538C22—121no—
539C22—121no—
540C22#478314yes—
541C22C2×#88211yescharacteristic
542C22#561112yes—
543C22C2×#91211yescharacteristic
544C22C2×#91511yescharacteristic
545C22#576912yes—
546C4—22no—
547C4—42no—
548C4—82no—
549C4—121no—
550C4—121no—
551C4—121no—
552C4—122no—
553C4—122no—
554C4—242no—
555C4#555512yes—
556C4#558611yescharacteristic
557C4#564811yescharacteristic
558C3#2409211yescharacteristic, complemented, Sylow
559C2—24no—
560C2—82no—
561C2—241no—
562C2#7468711yescharacteristic
563C2C2×#478311yescharacteristic
564C2#7883411yescharacteristic
565C2#8065212yes—
566C2#8070912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radicalC2×#5618C2
hypercenter#1893D3
hypocenterC3#24092
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#818C2semidirect
3. C2×#5034C2semidirect
4. #74028C2semidirect
5. C3×#1893C2semidirect
6. #71975C2semidirect
7. #70175C2semidirect
8. #70168C2semidirect
9. #69502C2semidirect
10. #68877C2semidirect
11. #68829C2semidirect
54. #1893D3semidirect
558. C3#24092semidirect

Maximal subgroups

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

subgroupindex
C2×#56182
C22×#8182
C2×#50342
#740282
C3×#18932
#719752
#701752
#701682
#695022
#688772
#688292
#240923