Finite Group Databasebeta

← #10828857

Subgroups — #10828857

Order 768.

1138 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3#82373C211yesmaximal, characteristic
4C22×#808C211yesmaximal, characteristic, complemented
5#71783C212yesmaximal, complemented
6#71696C211yesmaximal, characteristic, complemented
7#71694C212yesmaximal, complemented
8#71629C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic
10#71556C212yesmaximal, complemented
11#71554C211yesmaximal, characteristic, complemented
12#24097—31nomaximal, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C2×#935—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5212C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2211yescharacteristic
35#5043—24no—
36#4981—21no—
37C2×#808—24no—
38C2×#808C2212yes—
39C2×#808C2212yes—
40C2×#808C2212yes—
41#4953—21no—
42#4943—21no—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#554D311yescharacteristic
49C2×#483—31no—
50C22×#106—31no—
51#2486—31no—
52#2058—32no—
53#2035—32no—
54#1892—31no—
55#1869—31no—
56#1865—32no—
57#1853—32no—
58#1824—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4D411yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×#125—28no—
70C3×C22⋊Q8—28no—
71C3×C22⋊Q8—28no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8D414yes—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×D4—28no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D414yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—22no—
103#935—44no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—24no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#835—24no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118C3⋊#95—21no—
119#830—22no—
120#830—24no—
121#808—24no—
122#808—24no—
123#808—24no—
124C23×C8—31no—
125#554—24no—
126#554—24no—
127C2×C4⋊Q8C2×D311yescharacteristic
128C2×#125—21no—
129C2×#123—61no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C2×D311yescharacteristic
133C2×C22⋊Q8C2×D311yescharacteristic
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×D4—21no—
136C2×C4×D4C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D311yescharacteristic
140#483—64no—
141C2×#106—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#103—32no—
146C2×#103—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
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#336—32no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—24no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—82no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4SD1614yes—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—82no—
183C22×C3×C4—82no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—42no—
194C22×Dic3—42no—
195C22×Dic3—42no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C42—24no—
214C3×C42—21no—
215C3×C42—21no—
216#217—42no—
217#217—44no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—42no—
223Dic3⋊C4—44no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8#4114yes—
231C22×Q8—21no—
232C22×Q8C22×D311yescharacteristic
233C22×D4#6711yescharacteristic
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—61no—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D311yescharacteristic
239C22×C8—32no—
240C22×C8—32no—
241C22×C8—64no—
242C22×C8—62no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245#125—28no—
246#123—124no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8#6714yes—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×D4—28no—
256C4×D4—22no—
257C4×D4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4#6711yescharacteristic
274C2×C4⋊C4#6714yes—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4#6711yescharacteristic
283C2×C22⋊C4#6711yescharacteristic
284C2×C42—21no—
285C2×C42C22×D311yescharacteristic
286#106—64no—
287#106—64no—
288#106—64no—
289#103—64no—
290#103—64no—
291C22⋊C8—64no—
292C22⋊C8—62no—
293#95—62no—
294#95—61no—
295#95—62no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3#12012yes—
303C23×C3C2×SD1612yes—
304C23×C3C2×SD1612yes—
305C23×C3#13712yes—
306C23×C3C22×D411yescharacteristic
307C3×Q8—24no—
308C3×Q8—41no—
309C3×Q8—41no—
310C3×Q8—84no—
311C3×D4—28no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—84no—
334C2×C3×C4—84no—
335C2×C3×C4#12012yes—
336C2×C3×C4C2×SD1612yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4112yes—
340C2×C3×C42−1+412yes—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—84no—
348C3⋊C8—24no—
349C3⋊C8—42no—
350C24C2×#6711yescharacteristic
351C24C2×#6711yescharacteristic
352C2×Q8—24no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—82no—
357C2×Q8C2×#6712yes—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4#21514yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—122no—
378C22×C4—122no—
379C22×C4—122no—
380C22×C4C2×#6711yescharacteristic
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6712yes—
385C22×C4C2×#6712yes—
386C22×C4C23×D311yescharacteristic
387C2×C8—34no—
388C2×C8—68no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—124no—
409C4⋊C4—122no—
410C4⋊C4C2×#6712yes—
411C4⋊C4C2×#6712yes—
412C4⋊C4C2×#6712yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—122no—
418C22⋊C4—124no—
419C42—24no—
420C42—21no—
421C42—21no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—42no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3#44814yes—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54712yes—
434C22×C3C22×SD1611yescharacteristic
435C22×C3C2×#13711yescharacteristic
436C22×C3#57712yes—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—41no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4#52511yescharacteristic
446C3×C4#53611yescharacteristic
447C3×C4#54712yes—
448Dic3—82no—
449Dic3—82no—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—21no—
456C23C2×#21512yes—
457C23C2×#21512yes—
458C23#94212yes—
459C23#95312yes—
460C23C22×#6711yescharacteristic
461Q8—24no—
462Q8—41no—
463Q8—41no—
464Q8—84no—
465D4—28no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—244no—
496C2×C4C2×#21512yes—
497C2×C4#95312yes—
498C2×C4D3×#4112yes—
499C2×C4#101012yes—
500C2×C4C22×#6711yescharacteristic
501C2×C4C22×#6711yescharacteristic
502C8—64no—
503C8—122no—
504C2×C3—24no—
505C2×C3—42no—
506C2×C3#240611yescharacteristic
507C2×C3C2×#44811yescharacteristic
508C2×C3#300911yescharacteristic
509C2×C3#333512yes—
510C2×C3#334412yes—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—24no—
515C22—22no—
516C22—24no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22#522714yes—
521C22C22×#21511yescharacteristic
522C22C2×#94211yescharacteristic
523C22#582412yes—
524C22#583612yes—
525C22C2×#95311yescharacteristic
526C4—22no—
527C4—22no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—242no—
535C4—242no—
536C4#582412yes—
537C4#582511yescharacteristic
538C4#582811yescharacteristic
539C3#2409711yescharacteristic, complemented, Sylow
540C2—24no—
541C2—42no—
542C2#8362212yes—
543C2#8362412yes—
544C2C2×#522711yescharacteristic
545C2#8409011yescharacteristic
546C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#554D3
hypocenterC3#24097
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
4. C22×#808C2semidirect
5. #71783C2semidirect
6. #71696C2semidirect
7. #71694C2semidirect
8. #71629C2semidirect
10. #71556C2semidirect
11. #71554C2semidirect
539. C3#24097semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×#50432
#823732
C22×#8082
#717832
#716962
#716942
#716292
#716102
#715562
#715542
#240973