Finite Group Databasebeta

← #10732266

Subgroups — #10732266

Order 768.

1374 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
1C2×#4975C211yesmaximal, characteristic
2C2×#4646C212yesmaximal
3C2×#4646C211yesmaximal, characteristic
4C3×#1445C211yesmaximal, characteristic
5#71569C211yesmaximal, characteristic
6#69423C211yesmaximal, characteristic
7#69422C211yesmaximal, characteristic
8#69414C212yesmaximal
9#69413C212yesmaximal
10Dic3⋊#95C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15185—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36#4975—24no—
37#4646—28no—
38#4646—24no—
39C2×#391—31no—
40C2×#391—32no—
41C2×#384—31no—
42#1445—31no—
43#1445—31no—
44#1445—31no—
45#1445D311yescharacteristic
46#1442—32no—
47#1442—32no—
48C4⋊#95—32no—
49C4×#95—31no—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×Dic3C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—24no—
64C22×Dic6—24no—
65C22×Dic6D414yes—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—42no—
79C2×Dic3⋊C4—42no—
80C2×C4×Dic3—24no—
81C2×C4×Dic3—24no—
82C2×C4×Dic3—24no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3D414yes—
85C3×#95—24no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—42no—
89C3⋊#95—24no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—24no—
93C3⋊#95—22no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—42no—
97C3⋊#95—42no—
98C3⋊#95—42no—
99C3⋊#95Q818yescomplemented
100C23×Q8—31no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—32no—
103C22×C4⋊C4—31no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4C2×D311yescharacteristic
106C22×C42—31no—
107C22×C42—32no—
108#391—64no—
109#391—68no—
110#384—64no—
111C2×#95—32no—
112C2×#95—31no—
113C2×#95—31no—
114C2×#95—31no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—31no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95C2×D311yescharacteristic
123C2×#95C2×D312yes—
124C2×#95C2×D311yescharacteristic
125C2×#95C2×D312yes—
126C24×C3C2411yescharacteristic
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—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×Q814yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3C2×D412yes—
182C22×Dic3C2×D412yes—
183C22×Dic3C2×D412yes—
184C22×Dic3C2×D412yes—
185C22×Dic3C2×Q814yes—
186C22×Dic3C2×Q814yes—
187C22×Dic3#4112yes—
188C22×Dic3#4114yes—
189C22×Dic3#4114yes—
190C2×Dic6—28no—
191C2×Dic6—216no—
192C2×Dic6—44no—
193C2×Dic6—44no—
194C2×Dic6—48no—
195C3×C4⋊C4—44no—
196C4⋊Dic3—44no—
197Dic3⋊C4—48no—
198Dic3⋊C4—48no—
199Dic3⋊C4—44no—
200C4×Dic3—28no—
201C4×Dic3—48no—
202C22×Q8—34no—
203C22×Q8—62no—
204C22×Q8—64no—
205C22×Q8—64no—
206C22×Q8—62no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—31no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—31no—
213C23×C4C22×D312yes—
214C23×C4C22×D311yescharacteristic
215C23×C4C22×D311yescharacteristic
216C23×C4C22×D312yes—
217C23×C4C22×D311yescharacteristic
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—64no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—122no—
231C2×C4⋊C4—122no—
232C2×C42—34no—
233C2×C42—64no—
234C2×C42—64no—
235C2×C42—64no—
236C2×C42—64no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—38no—
241#95—42no—
242#95—64no—
243#95—62no—
244#95—62no—
245#95—64no—
246#95—62no—
247#95—64no—
248#95—62no—
249#95—122no—
250#95—122no—
251#95—122no—
252C23×C3C22×D411yescharacteristic
253C23×C3C22×D411yescharacteristic
254C23×C3C22×Q812yes—
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4111yescharacteristic
257C23×C3C2×#4111yescharacteristic
258C23×C3C2×#4112yes—
259C23×C3C2×#4112yes—
260C23×C3H(5,𝔽2)11yescharacteristic
261C23×C3H(5,𝔽2)11yescharacteristic
262C23×C32−1+412yes—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—42no—
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×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4#12414yes—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—22no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
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—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3#12114yes—
321C2×Dic3#12314yes—
322C2×Dic3#12318yes—
323C2×Dic3#12414yes—
324C2×Dic3#12618yes—
325C2×Dic3C4⋊Q814yes—
326Dic6—416no—
327C24C23×D311yescharacteristic
328C2×Q8—616no—
329C2×Q8—68no—
330C2×Q8—124no—
331C2×Q8—124no—
332C2×Q8—128no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—32no—
358C22×C4—32no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—34no—
364C22×C4—32no—
365C22×C4—34no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4#23512yes—
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C22×C4#23712yes—
395C22×C4D3×Q814yes—
396C4⋊C4—44no—
397C4⋊C4—124no—
398C4⋊C4—128no—
399C4⋊C4—128no—
400C4⋊C4—124no—
401C42—68no—
402C42—128no—
403C22×C3C2×#12111yescharacteristic
404C22×C3C2×#12312yes—
405C22×C3C2×#12311yescharacteristic
406C22×C3C2×#12411yescharacteristic
407C22×C3C2×#12411yescharacteristic
408C22×C3C2×#12612yes—
409C22×C3C2×C4⋊Q811yescharacteristic
410C22×C3#53312yes—
411C22×C3#53312yes—
412C22×C3#53511yescharacteristic
413C22×C3#53611yescharacteristic
414C22×C3#54212yes—
415C22×C3#54212yes—
416C22×C3D4×Q812yes—
417C22×C3#55011yescharacteristic
418C22×C3#55011yescharacteristic
419C22×C3#55212yes—
420C22×C3#55212yes—
421C22×C3#55412yes—
422C22×C3#55412yes—
423C22×C3#55712yes—
424C22×C3#55911yescharacteristic
425C22×C3#55911yescharacteristic
426C3×C4—24no—
427C3×C4—42no—
428C3×C4—44no—
429C3×C4—81no—
430C3×C4—82no—
431Dic3—24no—
432Dic3—42no—
433Dic3—44no—
434Dic3—81no—
435Dic3—82no—
436Dic3#39718yes—
437C23C2×#23511yescharacteristic
438C23C2×D3×D411yescharacteristic
439C23C2×D3×D411yescharacteristic
440C23C2×#23711yescharacteristic
441C23#100411yescharacteristic
442C23C2×D3×Q812yes—
443C23D3×#4112yes—
444C23D3×#4112yes—
445C23D3×#4111yescharacteristic
446C23#100911yescharacteristic
447C23#101012yes—
448Q8—1216no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—34no—
459C2×C4—38no—
460C2×C4—34no—
461C2×C4—38no—
462C2×C4—34no—
463C2×C4—34no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—64no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—62no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—121no—
511C2×C4#88514yes—
512C2×C3C2×#39711yescharacteristic
513C2×C3#242312yes—
514C2×C3#242312yes—
515C2×C3#250211yescharacteristic
516C2×C3#250311yescharacteristic
517C2×C3#251112yes—
518C2×C3#251112yes—
519C2×C3#254611yescharacteristic
520C2×C3#254811yescharacteristic
521C2×C3#264011yescharacteristic
522C2×C3#286211yescharacteristic
523C22C2×#88511yescharacteristic
524C22#553112yes—
525C22#555311yescharacteristic
526C22#555911yescharacteristic
527C22#557812yes—
528C22D3×#12111yescharacteristic
529C22#560711yescharacteristic
530C22#562212yes—
531C22#563212yes—
532C22#563612yes—
533C22#564811yescharacteristic
534C22#565012yes—
535C22D3×#12311yescharacteristic
536C22D3×#12312yes—
537C22#566212yes—
538C22D3×#12411yescharacteristic
539C22#568111yescharacteristic
540C22#568512yes—
541C22#569012yes—
542C22D3×#12612yes—
543C22#571012yes—
544C22#571811yescharacteristic
545C22D3×C4⋊Q811yescharacteristic
546C4—24no—
547C4—38no—
548C4—42no—
549C4—44no—
550C4—64no—
551C4—82no—
552C4—81no—
553C4—122no—
554C4—124no—
555C4—242no—
556C4—241no—
557C3#1518511yescharacteristic, complemented, Sylow
558C2#7407911yescharacteristic
559C2#7498212yes—
560C2#7503012yes—
561C2#7635311yescharacteristic
562C2#7660711yescharacteristic
563C2#7754711yescharacteristic
564C2D3×#39711yescharacteristic
565C2#7776511yescharacteristic
566C2#7778511yescharacteristic
567C2#7802012yes—
568C2#7815712yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1445C2
hypercenter#1445D3
hypocenterC3#15185
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
99. C3⋊#95Q8semidirect
557. C3#15185semidirect

Maximal subgroups

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

subgroupindex
C2×#49752
C2×#46462
C3×#14452
#715692
#694232
#694222
#694142
#694132
Dic3⋊#952
Dic3×#952
#151853