Finite Group Databasebeta

← #10751739

Subgroups — #10751739

Order 768.

1382 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#870C211yesmaximal, characteristic
2C2×#4981C212yesmaximal
3C2×#4933C211yesmaximal, characteristic
4C3×C4⋊#95C211yesmaximal, characteristic
5#71566C212yesmaximal
6#69419C211yesmaximal, characteristic
7#69413C212yesmaximal
8#69412C211yesmaximal, characteristic
9#69411C212yesmaximal
10#69410C211yesmaximal, characteristic
11#69410C211yesmaximal, characteristic
12#16605—31nomaximal, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C2×#870—24no—
23C2×#870—28no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4981—28no—
39#4933—24no—
40C22×#125—31no—
41C2×#398—32no—
42C2×#387—31no—
43#1444—31no—
44#1442—31no—
45#1442—32no—
46#1441—31no—
47#1441—31no—
48#1441—32no—
49#1441—32no—
50C4⋊#95D311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C42—24no—
66C2×C3×C42D414yes—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—24no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—42no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4D418yes—
84C2×Dic3⋊C4Q814yes—
85#870—232no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—42no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—42no—
98C3⋊#95—42no—
99C3⋊#95—42no—
100C2×#125—68no—
101C2×#125—64no—
102C22×C4⋊C4—32no—
103C22×C4⋊C4—31no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4C2×D312yes—
110C22×C42C2×D311yescharacteristic
111#398—68no—
112#387—64no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—31no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95C2×D311yescharacteristic
125C2×#95C2×D311yescharacteristic
126C2×#95C2×D312yes—
127C24×C3C2411yescharacteristic
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D414yes—
154C22×C3×C4C2×Q812yes—
155C22×C3×C4#4114yes—
156C22×C3×C4#4112yes—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D414yes—
191C22×Dic3C2×Q812yes—
192C22×Dic3C2×Q812yes—
193C3×C4⋊C4—48no—
194C3×C42—28no—
195C4⋊Dic3—28no—
196C4⋊Dic3—28no—
197C4⋊Dic3—48no—
198Dic3⋊C4—216no—
199Dic3⋊C4—216no—
200Dic3⋊C4—28no—
201Dic3⋊C4—44no—
202C23×C4—32no—
203C23×C4—32no—
204C23×C4—31no—
205C23×C4—31no—
206C23×C4—31no—
207C23×C4—31no—
208C23×C4C22×D312yes—
209C23×C4C22×D311yescharacteristic
210C23×C4C22×D312yes—
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D311yescharacteristic
213#125—632no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—38no—
217C2×C4⋊C4—34no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—64no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—64no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—122no—
235C2×C42—24no—
236C2×C42D1214yes—
237#95—22no—
238#95—22no—
239#95—42no—
240#95—64no—
241#95—64no—
242#95—64no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—64no—
248#95—122no—
249#95—122no—
250#95—122no—
251C23×C3C22×D411yescharacteristic
252C23×C3C22×D412yes—
253C23×C3C22×Q811yescharacteristic
254C23×C3C2×#4112yes—
255C23×C3C2×#4111yescharacteristic
256C23×C3H(5,𝔽2)11yescharacteristic
257C23×C3H(5,𝔽2)11yescharacteristic
258C23×C32−1+411yescharacteristic
259C23×C32−1+411yescharacteristic
260C23×C32−1+412yes—
261C23×C32−1+412yes—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4#12014yes—
290C2×C3×C4C22⋊Q814yes—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C24C23×D311yescharacteristic
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—34no—
351C22×C4—34no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4C2×D1212yes—
387C22×C4C2×D1212yes—
388C22×C4C2×D1212yes—
389C22×C4#23514yes—
390C22×C4D3×D414yes—
391C22×C4#23712yes—
392C22×C4D3×Q812yes—
393C4⋊C4—48no—
394C4⋊C4—616no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—616no—
398C4⋊C4—68no—
399C4⋊C4—128no—
400C4⋊C4—124no—
401C42—28no—
402C22×C3C2×#12011yescharacteristic
403C22×C3C2×C22⋊Q811yescharacteristic
404C22×C3#53211yescharacteristic
405C22×C3#53611yescharacteristic
406C22×C3#53611yescharacteristic
407C22×C3#53612yes—
408C22×C3#53711yescharacteristic
409C22×C3#53711yescharacteristic
410C22×C3#53712yes—
411C22×C3#53912yes—
412C22×C3#54011yescharacteristic
413C22×C3#54111yescharacteristic
414C22×C3#54112yes—
415C22×C3#54411yescharacteristic
416C22×C3#54411yescharacteristic
417C22×C3#54612yes—
418C22×C3#54712yes—
419C22×C3#54812yes—
420C22×C3D4×Q812yes—
421C22×C3#55212yes—
422C22×C3#55712yes—
423C22×C3#56312yes—
424C22×C3#56412yes—
425C3×C4—24no—
426C3×C4—24no—
427C3×C4—42no—
428C3×C4—44no—
429C3×C4—82no—
430Dic3—44no—
431Dic3—42no—
432Dic3—42no—
433Dic3—44no—
434Dic3—81no—
435Dic3—81no—
436C23C22×D1211yescharacteristic
437C23C2×#23512yes—
438C23C2×D3×D412yes—
439C23C2×#23711yescharacteristic
440C23#100411yescharacteristic
441C23#100411yescharacteristic
442C23C2×D3×Q811yescharacteristic
443C23#100711yescharacteristic
444C23#100711yescharacteristic
445C23#101012yes—
446C23#101012yes—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—64no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4#88214yes—
511C2×C4#89714yes—
512C2×C3#240911yescharacteristic
513C2×C3#243011yescharacteristic
514C2×C3#256511yescharacteristic
515C2×C3#261611yescharacteristic
516C2×C3#262911yescharacteristic
517C2×C3#266411yescharacteristic
518C2×C3#270712yes—
519C2×C3#271312yes—
520C2×C3#271912yes—
521C2×C3#272612yes—
522C2×C3#282211yescharacteristic
523C22#547711yescharacteristic
524C22C2×#88211yescharacteristic
525C22#549012yes—
526C22#550712yes—
527C22C2×#89711yescharacteristic
528C22#551011yescharacteristic
529C22#551011yescharacteristic
530C22#552611yescharacteristic
531C22#552611yescharacteristic
532C22#554112yes—
533C22#556412yes—
534C22#556712yes—
535C22#558312yes—
536C22#559812yes—
537C22#560812yes—
538C22#562812yes—
539C22#563212yes—
540C22#567212yes—
541C22C3⋊#56412yes—
542C22#571911yescharacteristic
543C22#572911yescharacteristic
544C22#572911yescharacteristic
545C22#573111yescharacteristic
546C4—24no—
547C4—24no—
548C4—44no—
549C4—42no—
550C4—82no—
551C4—122no—
552C4—122no—
553C4—124no—
554C4—124no—
555C4—241no—
556C4—241no—
557C3#1660511yescharacteristic, complemented, Sylow
558C2#7337611yescharacteristic
559C2#7371911yescharacteristic
560C2#7374711yescharacteristic
561C2#7395011yescharacteristic
562C2#7401711yescharacteristic
563C2#7421711yescharacteristic
564C2#7434411yescharacteristic
565C2#7611712yes—
566C2#7612612yes—
567C2#7639112yes—
568C2#7656712yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×C4⋊#95C2
hypercenterC4⋊#95D3
hypocenterC3#16605
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
557. C3#16605semidirect

Maximal subgroups

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

subgroupindex
C22×#8702
C2×#49812
C2×#49332
C3×C4⋊#952
#715662
#694192
#694132
#694122
#694112
#694102
#166053