Finite Group Databasebeta

← #10732440

Subgroups — #10732440

Order 768.

1210 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4984C211yesmaximal, characteristic
2C2×#4647C211yesmaximal, characteristic
3C3×#1442C211yesmaximal, characteristic
4#71567C211yesmaximal, characteristic
5#69415C211yesmaximal, characteristic
6#69412C211yesmaximal, characteristic
7#69411C212yesmaximal
8#69411C212yesmaximal
9#69407C212yesmaximal
10#69406C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15207—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2211yescharacteristic
20C22×C4×Dic3C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3×#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4984—24no—
37#4647—24no—
38C2×#389—31no—
39C2×#389—31no—
40#1442—31no—
41#1442—31no—
42#1442—31no—
43#1442D311yescharacteristic
44#1441—32no—
45#1441—32no—
46#1436—32no—
47#1436—32no—
48C4×#95—31no—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2311yescharacteristic
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—24no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3—22no—
64C2×C4⋊Dic3—24no—
65C2×C4⋊Dic3—24no—
66C2×C4⋊Dic3—24no—
67C2×C4⋊Dic3—24no—
68C2×C4⋊Dic3—42no—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—42no—
76C2×C4×Dic3—24no—
77C2×C4×Dic3—42no—
78C2×C4×Dic3—42no—
79C2×C4×Dic3—42no—
80C2×C4×Dic3D414yes—
81C3×#95—22no—
82C3×#95—24no—
83C3×#95—42no—
84C3×#95Q814yes—
85C3⋊#95—22no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—42no—
92C3⋊#95—42no—
93C3⋊#95—42no—
94C3⋊#95Q814yes—
95C22×C4⋊C4—31no—
96C22×C4⋊C4—31no—
97C22×C4⋊C4—31no—
98C22×C4⋊C4—32no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4C2×D311yescharacteristic
102C22×C42—31no—
103C22×C42—32no—
104#389—64no—
105#389—64no—
106C2×#95—31no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—31no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95C2×D312yes—
117C2×#95C2×D311yescharacteristic
118C2×#95C2×D312yes—
119C2×#95C2×D311yescharacteristic
120C24×C3C2411yescharacteristic
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×Q814yes—
146C22×C3×C4C2×Q812yes—
147C22×C3×C4C2×Q812yes—
148C22×C3×C4#4114yes—
149C22×C3×C4#4112yes—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3C2×D412yes—
179C22×Dic3C2×D412yes—
180C22×Dic3C2×Q814yes—
181C22×Dic3#4112yes—
182C22×Dic3#4114yes—
183C22×Dic3#4114yes—
184C3×C4⋊C4—28no—
185C4⋊Dic3—44no—
186C4⋊Dic3—44no—
187C4⋊Dic3—48no—
188Dic3⋊C4—28no—
189Dic3⋊C4—44no—
190Dic3⋊C4—44no—
191C4×Dic3—44no—
192C4×Dic3—48no—
193C23×C4—31no—
194C23×C4—31no—
195C23×C4—31no—
196C23×C4—32no—
197C23×C4—31no—
198C23×C4—32no—
199C23×C4C22×D311yescharacteristic
200C23×C4C22×D312yes—
201C23×C4C22×D312yes—
202C23×C4C22×D311yescharacteristic
203C23×C4C22×D311yescharacteristic
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4—62no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—62no—
211C2×C4⋊C4—62no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—64no—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4—122no—
219C2×C4⋊C4—122no—
220C2×C42—34no—
221C2×C42—64no—
222C2×C42—122no—
223C2×C42—122no—
224C2×C42—122no—
225#95—22no—
226#95—24no—
227#95—34no—
228#95—42no—
229#95—64no—
230#95—64no—
231#95—64no—
232#95—62no—
233#95—64no—
234#95—64no—
235#95—122no—
236#95—122no—
237#95—122no—
238#95Dic614yes—
239C23×C3C22×D411yescharacteristic
240C23×C3C22×Q811yescharacteristic
241C23×C3C22×Q811yescharacteristic
242C23×C3C2×#4112yes—
243C23×C3C2×#4111yescharacteristic
244C23×C3C2×#4111yescharacteristic
245C23×C3C2×#4112yes—
246C23×C3C2×#4112yes—
247C23×C3H(5,𝔽2)11yescharacteristic
248C23×C3H(5,𝔽2)12yes—
249C23×C32−1+411yescharacteristic
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4C4⋊Q814yes—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3#12414yes—
312C2×Dic3#12414yes—
313C24C23×D311yescharacteristic
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—34no—
340C22×C4—34no—
341C22×C4—34no—
342C22×C4—32no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4C2×Dic614yes—
372C22×C4C2×Dic612yes—
373C22×C4D3×D412yes—
374C22×C4#23714yes—
375C22×C4D3×Q812yes—
376C22×C4#23912yes—
377C4⋊C4—28no—
378C4⋊C4—68no—
379C4⋊C4—124no—
380C4⋊C4—124no—
381C4⋊C4—128no—
382C4⋊C4—124no—
383C4⋊C4—124no—
384C42—124no—
385C42—128no—
386C22×C3#52511yescharacteristic
387C22×C3#52511yescharacteristic
388C22×C3C2×#12411yescharacteristic
389C22×C3C2×#12411yescharacteristic
390C22×C3#52912yes—
391C22×C3#52912yes—
392C22×C3C2×C4⋊Q811yescharacteristic
393C22×C3#53312yes—
394C22×C3#53312yes—
395C22×C3#53411yescharacteristic
396C22×C3#53711yescharacteristic
397C22×C3#53812yes—
398C22×C3#53912yes—
399C22×C3#55212yes—
400C22×C3#55312yes—
401C22×C3#55312yes—
402C22×C3#55312yes—
403C22×C3#55412yes—
404C22×C3#55411yescharacteristic
405C22×C3#55411yescharacteristic
406C22×C3#55412yes—
407C22×C3#55711yescharacteristic
408C22×C3#55711yescharacteristic
409C3×C4—24no—
410C3×C4—44no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4—81no—
414Dic3—24no—
415Dic3—24no—
416Dic3—44no—
417Dic3—42no—
418Dic3—82no—
419Dic3—81no—
420C23C22×Dic611yescharacteristic
421C23C2×D3×D411yescharacteristic
422C23C2×#23712yes—
423C23#100411yescharacteristic
424C23C2×D3×Q811yescharacteristic
425C23C2×#23911yescharacteristic
426C23D3×#4112yes—
427C23D3×#4112yes—
428C23D3×#4111yescharacteristic
429C23#100912yes—
430C23#101011yescharacteristic
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—34no—
444C2×C4—34no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—64no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—64no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—121no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4#88814yes—
494C2×C3#239811yescharacteristic
495C2×C3#256211yescharacteristic
496C2×C3#258311yescharacteristic
497C2×C3#258411yescharacteristic
498C2×C3#258712yes—
499C2×C3#258712yes—
500C2×C3#259012yes—
501C2×C3#259012yes—
502C2×C3#259111yescharacteristic
503C2×C3#263911yescharacteristic
504C2×C3#286211yescharacteristic
505C22C2×#88811yescharacteristic
506C22#551712yes—
507C22#553911yescharacteristic
508C22#554312yes—
509C22#557011yescharacteristic
510C22#558611yescharacteristic
511C22#560112yes—
512C22#560811yescharacteristic
513C22#562412yes—
514C22#562711yescharacteristic
515C22#563512yes—
516C22#565012yes—
517C22#565311yescharacteristic
518C22#565712yes—
519C22D3×#12411yescharacteristic
520C22D3×#12411yescharacteristic
521C22#568012yes—
522C22#568511yescharacteristic
523C22#568812yes—
524C22#569512yes—
525C22#570712yes—
526C22#570912yes—
527C22#573011yescharacteristic
528C4—24no—
529C4—44no—
530C4—44no—
531C4—42no—
532C4—64no—
533C4—64no—
534C4—81no—
535C4—124no—
536C4—122no—
537C4—242no—
538C4—241no—
539C3#1520711yescharacteristic, complemented, Sylow
540C2#7422111yescharacteristic
541C2#7448811yescharacteristic
542C2#7510912yes—
543C2#7561612yes—
544C2#7607911yescharacteristic
545C2#7677911yescharacteristic
546C2#7695511yescharacteristic
547C2#7768711yescharacteristic
548C2#7780212yes—
549C2#7819712yes—
550C2#7835811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1442C2
hypercenter#1442D3
hypocenterC3#15207
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
539. C3#15207semidirect

Maximal subgroups

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

subgroupindex
C2×#49842
C2×#46472
C3×#14422
#715672
#694152
#694122
#694112
#694072
#694062
Dic3×#952
#152073