Finite Group Databasebeta

← #10732498

Subgroups — #10732498

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#387C211yesmaximal, characteristic
2C2×C4×C4⋊Dic3C211yesmaximal, characteristic
3#71566C212yesmaximal
4#71564C212yesmaximal
5#71534C211yesmaximal, characteristic
6#69423C211yesmaximal, characteristic
7#69423C211yesmaximal, characteristic
8#69420C211yesmaximal, characteristic
9#69418C212yesmaximal
10#69415C211yesmaximal, characteristic
11#69406C212yesmaximal
12#15213—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2212yes—
20C3×#387—24no—
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C4×C4⋊Dic3—24no—
38C2×#387D311yescharacteristic
39C2×C4×C4⋊C4—31no—
40#1445—31no—
41#1445—31no—
42#1444—31no—
43#1443—32no—
44#1442—31no—
45#1441—32no—
46#1436—32no—
47#1436—32no—
48#1434—31no—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C42—24no—
66C2×C3×C42D414yes—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3Q814yes—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—42no—
75C2×C4×Dic3—24no—
76C2×C4×Dic3—42no—
77C2×C4×Dic3—42no—
78C3×#95—22no—
79C3×#95—22no—
80C3×#95—22no—
81C3×#95—22no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—42no—
92C3⋊#95—42no—
93C3⋊#95—42no—
94C22×C4⋊C4—31no—
95C22×C4⋊C4—31no—
96C22×C4⋊C4—32no—
97C22×C4⋊C4C2×D311yescharacteristic
98C22×C4⋊C4C2×D311yescharacteristic
99C22×C42—32no—
100C22×C42C2×D311yescharacteristic
101#387—24no—
102C4×C4⋊C4—64no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—31no—
106C2×#95—31no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95C2×D311yescharacteristic
116C2×#95C2×D311yescharacteristic
117C2×#95C2×D311yescharacteristic
118C2×#95C2×D311yescharacteristic
119C24×C3C2411yescharacteristic
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×Q812yes—
151C22×C3×C4#4112yes—
152C22×C3×C4#4112yes—
153C22×C3×C4#4114yes—
154C22×C3×C4#4112yes—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3C2×Q814yes—
181C22×Dic3#4114yes—
182C22×Dic3#4114yes—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—44no—
185C3×C42—28no—
186C4⋊Dic3—28no—
187C4⋊Dic3—28no—
188Dic3⋊C4—48no—
189C4×Dic3—48no—
190C23×C4—32no—
191C23×C4—32no—
192C23×C4—32no—
193C23×C4—32no—
194C23×C4C22×D311yescharacteristic
195C23×C4C22×D311yescharacteristic
196C23×C4C22×D311yescharacteristic
197C23×C4C22×D311yescharacteristic
198C23×C4C22×D311yescharacteristic
199C23×C4C22×D311yescharacteristic
200C23×C4C22×D311yescharacteristic
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—34no—
207C2×C4⋊C4—64no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—62no—
211C2×C4⋊C4—64no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—122no—
214C2×C42—24no—
215C2×C42—64no—
216C2×C42—122no—
217C2×C42—122no—
218C2×C42#6714yes—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—64no—
224#95—64no—
225#95—64no—
226#95—64no—
227#95—64no—
228#95—64no—
229#95—62no—
230#95—62no—
231#95—64no—
232#95—122no—
233#95—122no—
234#95—122no—
235C23×C3C22×D411yescharacteristic
236C23×C3C22×Q811yescharacteristic
237C23×C3C2×#4111yescharacteristic
238C23×C3C2×#4111yescharacteristic
239C23×C3C2×#4112yes—
240C23×C3C2×#4111yescharacteristic
241C23×C3C2×#4112yes—
242C23×C3C2×#4112yes—
243C23×C3H(5,𝔽2)12yes—
244C23×C32−1+411yescharacteristic
245C23×C32−1+411yescharacteristic
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4C22⋊Q814yes—
280C2×C3×C4#12314yes—
281C2×Dic3—24no—
282C2×Dic3—24no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C24C23×D311yescharacteristic
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—34no—
338C22×C4—34no—
339C22×C4—34no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4#23512yes—
366C22×C4#23512yes—
367C22×C4#23714yes—
368C22×C4D3×Q812yes—
369C22×C4#23912yes—
370C22×C4C2×#6712yes—
371C22×C4C2×#6712yes—
372C22×C4C2×#6712yes—
373C4⋊C4—28no—
374C4⋊C4—44no—
375C4⋊C4—68no—
376C4⋊C4—68no—
377C4⋊C4—128no—
378C42—28no—
379C42—128no—
380C22×C3C2×C22⋊Q811yescharacteristic
381C22×C3C2×#12311yescharacteristic
382C22×C3#52511yescharacteristic
383C22×C3#52511yescharacteristic
384C22×C3#52912yes—
385C22×C3#52912yes—
386C22×C3#52912yes—
387C22×C3#53211yescharacteristic
388C22×C3#53312yes—
389C22×C3#53511yescharacteristic
390C22×C3#53611yescharacteristic
391C22×C3#53912yes—
392C22×C3#53912yes—
393C22×C3#55112yes—
394C22×C3#55212yes—
395C22×C3#55312yes—
396C22×C3#55312yes—
397C22×C3#55412yes—
398C22×C3#55612yes—
399C22×C3#55611yescharacteristic
400C22×C3#55611yescharacteristic
401C22×C3#55711yescharacteristic
402C22×C3#55711yescharacteristic
403C3×C4—24no—
404C3×C4—24no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—81no—
409C3×C4—81no—
410Dic3—44no—
411Dic3—44no—
412Dic3—44no—
413Dic3—82no—
414C23C2×#23511yescharacteristic
415C23C2×#23511yescharacteristic
416C23C2×#23712yes—
417C23C2×D3×Q811yescharacteristic
418C23C2×#23911yescharacteristic
419C23D3×#4112yes—
420C23D3×#4112yes—
421C23#100912yes—
422C23#101011yescharacteristic
423C23#101011yescharacteristic
424C23C22×#6711yescharacteristic
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—64no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—64no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4#93514yes—
487C2×C4#94614yes—
488C2×C3#239511yescharacteristic
489C2×C3#256611yescharacteristic
490C2×C3#257712yes—
491C2×C3#258111yescharacteristic
492C2×C3#258812yes—
493C2×C3#259012yes—
494C2×C3#259412yes—
495C2×C3#259811yescharacteristic
496C2×C3#260411yescharacteristic
497C2×C3#262311yescharacteristic
498C2×C3#285511yescharacteristic
499C22#552712yes—
500C22#553012yes—
501C22#553812yes—
502C22#555712yes—
503C22#556711yescharacteristic
504C22#556711yescharacteristic
505C22#558411yescharacteristic
506C22#558411yescharacteristic
507C22#562212yes—
508C22#562412yes—
509C22#566512yes—
510C22#566512yes—
511C22#569012yes—
512C22#569512yes—
513C22#570212yes—
514C22#570812yes—
515C22#579111yescharacteristic
516C22C2×#93511yescharacteristic
517C22C2×#94611yescharacteristic
518C22#582511yescharacteristic
519C22#582811yescharacteristic
520C22#583011yescharacteristic
521C22#583211yescharacteristic
522C4—24no—
523C4—24no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—81no—
528C4—81no—
529C4—124no—
530C4—124no—
531C4—124no—
532C4—242no—
533C3#1521311yescharacteristic, complemented, Sylow
534C2#7508712yes—
535C2#7510612yes—
536C2#7511312yes—
537C2#7511612yes—
538C2#8228411yescharacteristic
539C2#8234611yescharacteristic
540C2#8248011yescharacteristic
541C2#8281011yescharacteristic
542C2#8286611yescharacteristic
543C2#8304011yescharacteristic
544C2#8308811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#387C2
hypercenterC2×#387D3
hypocenterC3#15213
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
533. C3#15213semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3872
C2×C4×C4⋊Dic32
#715662
#715642
#715342
#694232
#694202
#694182
#694152
#694062
#152133