Finite Group Databasebeta

← #10729222

Subgroups — #10729222

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic
2C2×#4976C212yesmaximal
3C2×#4933C211yesmaximal, characteristic
4#71568C212yesmaximal
5#71568C211yesmaximal, characteristic
6#71568C211yesmaximal, characteristic
7#71567C211yesmaximal, characteristic
8#71567C211yesmaximal, characteristic
9#71566C212yesmaximal
10#71564C212yesmaximal
11C4×C3⋊#95C211yesmaximal, characteristic
12#14929—31nomaximal, Sylow
13C2×C3×#125—28no—
14C2×C3×#125—24no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38#4976—28no—
39#4933—24no—
40C22×#125D311yescharacteristic
41C2×#389—32no—
42C2×#387—31no—
43#1444—32no—
44#1444—31no—
45#1444—31no—
46#1442—31no—
47#1442—31no—
48#1441—32no—
49#1436—32no—
50C4×#95—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C3×#125—232no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4D418yes—
73C2×C3×C42—24no—
74C2×C3×C42D414yes—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—42no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—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—24no—
98C3⋊#95—24no—
99C3⋊#95—42no—
100C3⋊#95Q814yes—
101C3⋊#95Q814yes—
102C2×#125—28no—
103C2×#125—24no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—32no—
108C22×C4⋊C4C2×D312yes—
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C4⋊C4C2×D312yes—
112C22×C42—32no—
113C22×C42C2×D311yescharacteristic
114#389—68no—
115#387—64no—
116C2×#95—31no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C24×C3C2411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
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—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D414yes—
153C22×C3×C4C2×D414yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×Q812yes—
158C22×C3×C4C2×Q812yes—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3C2×Q814yes—
188C22×Dic3C2×Q814yes—
189C22×Dic3#4112yes—
190C22×Dic3#4112yes—
191C22×Dic3#4114yes—
192C22×Dic3#4114yes—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—216no—
196C3×C4⋊C4—216no—
197C3×C42—28no—
198Dic3⋊C4—216no—
199Dic3⋊C4—28no—
200Dic3⋊C4—48no—
201Dic3⋊C4—44no—
202C4×Dic3—48no—
203C23×C4—32no—
204C23×C4—31no—
205C23×C4—31no—
206C23×C4—32no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4C22×D312yes—
210C23×C4C22×D312yes—
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D311yescharacteristic
213C23×C4C22×D311yescharacteristic
214#125—232no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—122no—
235C2×C4⋊C4#6718yes—
236C2×C42—24no—
237C2×C42—64no—
238C2×C42—64no—
239C2×C42—122no—
240C2×C42#6714yes—
241#95—34no—
242#95—34no—
243#95—64no—
244#95—64no—
245#95—64no—
246#95—64no—
247#95—62no—
248#95—62no—
249#95—62no—
250#95—62no—
251#95—64no—
252#95—64no—
253#95—122no—
254C23×C3C22×D411yescharacteristic
255C23×C3C22×D412yes—
256C23×C3C22×Q811yescharacteristic
257C23×C3C22×Q811yescharacteristic
258C23×C3C2×#4111yescharacteristic
259C23×C3C2×#4112yes—
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4112yes—
262C23×C3H(5,𝔽2)11yescharacteristic
263C23×C32−1+412yes—
264C23×C32−1+411yescharacteristic
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
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—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4#12014yes—
293C2×C3×C4C4⋊Q814yes—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—42no—
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—
327C2×Dic3#12518yescomplemented
328C24C23×D311yescharacteristic
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
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—34no—
353C22×C4—34no—
354C22×C4—32no—
355C22×C4—32no—
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—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4D3×Q812yes—
385C22×C4D3×Q812yes—
386C22×C4C2×#6712yes—
387C22×C4C2×#6714yes—
388C22×C4C2×#6714yes—
389C22×C4C2×#6712yes—
390C22×C4C2×#6712yes—
391C22×C4C2×#6714yes—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—216no—
395C4⋊C4—216no—
396C4⋊C4—616no—
397C4⋊C4—68no—
398C4⋊C4—128no—
399C4⋊C4—124no—
400C42—28no—
401C42—128no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#52511yescharacteristic
404C22×C3#52512yes—
405C22×C3C2×#12512yes—
406C22×C3C2×C4⋊Q811yescharacteristic
407C22×C3#53211yescharacteristic
408C22×C3#53211yescharacteristic
409C22×C3#53212yes—
410C22×C3#53312yes—
411C22×C3#53611yescharacteristic
412C22×C3#53711yescharacteristic
413C22×C3#53912yes—
414C22×C3#54112yes—
415C22×C3#54612yes—
416C22×C3#54712yes—
417C22×C3D4×Q812yes—
418C22×C3D4×Q812yes—
419C22×C3#55212yes—
420C22×C3#55212yes—
421C22×C3#55411yescharacteristic
422C22×C3#55411yescharacteristic
423C22×C3#55711yescharacteristic
424C22×C3#55711yescharacteristic
425C3×C4—24no—
426C3×C4—24no—
427C3×C4—42no—
428C3×C4—44no—
429C3×C4—44no—
430Dic3—28no—
431Dic3—42no—
432Dic3—42no—
433Dic3—44no—
434Dic3—81no—
435Dic3—81no—
436C23C2×D3×Q811yescharacteristic
437C23C2×D3×Q811yescharacteristic
438C23#100712yes—
439C23D3×#4111yescharacteristic
440C23D3×#4111yescharacteristic
441C23D3×#4112yes—
442C23D3×#4112yes—
443C23#100911yescharacteristic
444C23#101011yescharacteristic
445C23C22×#6711yescharacteristic
446C23C22×#6712yes—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—38no—
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—42no—
475C2×C4—64no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—62no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—121no—
495C2×C4—122no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4#94414yes—
509C2×C4#95314yes—
510C2×C3#238811yescharacteristic
511C2×C3#243011yescharacteristic
512C2×C3#243411yescharacteristic
513C2×C3#246212yes—
514C2×C3#247312yes—
515C2×C3#250612yes—
516C2×C3#255612yes—
517C2×C3#258411yescharacteristic
518C2×C3#261111yescharacteristic
519C2×C3#266411yescharacteristic
520C2×C3#285911yescharacteristic
521C22#568511yescharacteristic
522C22#568511yescharacteristic
523C22D3×#12512yes—
524C22#568812yes—
525C22#569011yescharacteristic
526C22#569011yescharacteristic
527C22#569212yes—
528C22#569312yes—
529C22#569312yes—
530C22#569612yes—
531C22C2×#94411yescharacteristic
532C22Q8×#6712yes—
533C22Q8×#6712yes—
534C22#581611yescharacteristic
535C22#582412yes—
536C22#582812yes—
537C22#582811yescharacteristic
538C22#582912yes—
539C22#583212yes—
540C22#583211yescharacteristic
541C22#583211yescharacteristic
542C22#583311yescharacteristic
543C22C2×#95311yescharacteristic
544C4—24no—
545C4—24no—
546C4—44no—
547C4—42no—
548C4—44no—
549C4—68no—
550C4—124no—
551C4—122no—
552C4—122no—
553C4—241no—
554C4—241no—
555C3#1492911yescharacteristic, complemented, Sylow
556C2#8303112yes—
557C2#8303212yes—
558C2#8303412yes—
559C2#8303812yes—
560C2#8304111yescharacteristic
561C2#8304511yescharacteristic
562C2#8305611yescharacteristic
563C2#8306411yescharacteristic
564C2#8406511yescharacteristic
565C2#8409111yescharacteristic
566C2#8409511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC22×C3×#125C2
hypercenterC22×#125D3
hypocenterC3#14929
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
327. C2×Dic3#125semidirect
555. C3#14929semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×#49762
C2×#49332
#715682
#715672
#715662
#715642
C4×C3⋊#952
#149293