Finite Group Databasebeta

← #10751704

Subgroups — #10751704

Order 768.

1706 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic, complemented
2C2×C3×#388C211yesmaximal, characteristic
3C2×#4947C212yesmaximal
4#71565C212yesmaximal, complemented
5#71546C212yesmaximal
6#71546C212yesmaximal
7#71534C211yesmaximal, characteristic, complemented
8#69424C212yesmaximal, complemented
9#69415C212yesmaximal, complemented
10#16602—31nomaximal, Sylow
11C23×Dic6C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C22×C4⋊Dic3C2212yes—
19C22×C4⋊Dic3C2212yes—
20C2×#869—28no—
21C2×#869—24no—
22C3×#388—24no—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31#4947—28no—
32C22×C4⋊Q8—31no—
33C2×#393—32no—
34C2×#388D311yescharacteristic
35#1446—32no—
36#1442—32no—
37#1441—32no—
38#1439—32no—
39#1439—32no—
40#1434—31no—
41C25×C3C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2314yes—
49C23×Dic3C2314yes—
50C22×Dic6—24no—
51C22×Dic6—24no—
52C22×Dic6—44no—
53C22×Dic6—44no—
54C22×Dic6D418yes—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4Q814yescomplemented
64C2×C3×C42—24no—
65C2×C3×C42D414yes—
66C2×#217—28no—
67C2×#217—44no—
68C2×#217—44no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—28no—
72C2×C4⋊Dic3—28no—
73#869—232no—
74C3×#95—22no—
75C3×#95—22no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—28no—
80C3⋊#95—44no—
81C3⋊#95—44no—
82C23×Q8—32no—
83C2×C4⋊Q8—68no—
84C2×C4⋊Q8—64no—
85C22×C4⋊C4—32no—
86C22×C4⋊C4—32no—
87C22×C4⋊C4C2×D311yescharacteristic
88C22×C22⋊C4—34no—
89C22×C22⋊C4C2×D311yescharacteristic
90C22×C22⋊C4C2×D311yescharacteristic
91C22×C22⋊C4C2×D311yescharacteristic
92C22×C42C2×D311yescharacteristic
93#393—68no—
94#388—24no—
95C2×#95—32no—
96C2×#95—34no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—34no—
100C2×#95—34no—
101C2×#95C2×D311yescharacteristic
102C2×#95C2×D311yescharacteristic
103C24×C3—22no—
104C24×C3—21no—
105C24×C3—21no—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3C2×Q812yes—
112C24×C3C2411yescharacteristic
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D414yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×Q812yes—
136C22×C3×C4C2×Q812yes—
137C22×C3×C4#4112yes—
138C22×C3×C4#4114yes—
139C22×Dic3—24no—
140C22×Dic3—24no—
141C22×Dic3—24no—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—24no—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3C2×D418yes—
153C2×Dic6—216no—
154C2×Dic6—416no—
155C2×Dic6—416no—
156C2×Dic6—48no—
157C2×Dic6—48no—
158C3×C4⋊C4—44no—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—44no—
161C3×C22⋊C4—44no—
162C3×C42—28no—
163#217—416no—
164C4⋊Dic3—216no—
165C4⋊Dic3—216no—
166C25C22×D311yescharacteristic
167C22×Q8—38no—
168C22×Q8—64no—
169C22×Q8—64no—
170C22×Q8—124no—
171C22×Q8—124no—
172C23×C4—34no—
173C23×C4—34no—
174C23×C4C22×D311yescharacteristic
175C23×C4C22×D311yescharacteristic
176C23×C4C22×D311yescharacteristic
177C23×C4C22×D311yescharacteristic
178C23×C4C22×D311yescharacteristic
179C23×C4C22×D311yescharacteristic
180C4⋊Q8—632no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—68no—
184C2×C4⋊C4—68no—
185C2×C4⋊C4—64no—
186C2×C4⋊C4—64no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—68no—
194C2×C22⋊C4—124no—
195C2×C22⋊C4—124no—
196C2×C22⋊C4Dic614yescomplemented
197C2×C42—24no—
198C2×C42#6714yes—
199#95—22no—
200#95—22no—
201#95—68no—
202#95—64no—
203#95—64no—
204#95—64no—
205#95—124no—
206#95—124no—
207C23×C3—22no—
208C23×C3—24no—
209C23×C3—24no—
210C23×C3—24no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—44no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—44no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—44no—
222C23×C3—42no—
223C23×C3—44no—
224C23×C3C22×D412yes—
225C23×C3C22×D411yescharacteristic
226C23×C3C22×Q811yescharacteristic
227C23×C3C2×#4111yescharacteristic
228C23×C3C2×#4112yes—
229C23×C3H(5,𝔽2)12yes—
230C23×C3H(5,𝔽2)12yes—
231C23×C32−1+412yes—
232C23×C32−1+412yes—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—28no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—28no—
242C2×C3×C4—24no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4#12014yes—
257C2×C3×C4C22⋊Q814yes—
258C2×C3×C4#12414yes—
259C2×Dic3—28no—
260C2×Dic3—28no—
261C2×Dic3—28no—
262C2×Dic3—44no—
263C2×Dic3—44no—
264C2×Dic3—44no—
265C2×Dic3—44no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—44no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273Dic6—432no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24C2×Dic612yes—
283C24C23×D311yescharacteristic
284C2×Q8—616no—
285C2×Q8—1216no—
286C2×Q8—1216no—
287C2×Q8—128no—
288C2×Q8—128no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—38no—
308C22×C4—64no—
309C22×C4—64no—
310C22×C4—64no—
311C22×C4—64no—
312C22×C4—64no—
313C22×C4—64no—
314C22×C4—64no—
315C22×C4—64no—
316C22×C4—64no—
317C22×C4—64no—
318C22×C4—64no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4C2×Dic612yes—
322C22×C4C2×Dic612yes—
323C22×C4#23512yes—
324C22×C4D3×D414yes—
325C22×C4#23714yes—
326C22×C4C2×#6712yes—
327C22×C4C2×#6712yes—
328C22×C4C2×#6712yes—
329C4⋊C4—44no—
330C4⋊C4—616no—
331C4⋊C4—616no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—1216no—
336C42—28no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—48no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—84no—
346C22×C3C2×#12011yescharacteristic
347C22×C3C2×C22⋊Q811yescharacteristic
348C22×C3C2×#12411yescharacteristic
349C22×C3#53414yes—
350C22×C3#53511yescharacteristic
351C22×C3#53511yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3#53611yescharacteristic
354C22×C3#54214yes—
355C22×C3#54212yes—
356C22×C3#54412yes—
357C22×C3#54812yes—
358C22×C3#54814yes—
359C22×C3D4×Q812yes—
360C22×C3#55412yes—
361C22×C3#55512yes—
362C22×C3#56314yes—
363C3×C4—24no—
364C3×C4—24no—
365C3×C4—24no—
366C3×C4—42no—
367C3×C4—81no—
368C3×C4—81no—
369Dic3—48no—
370Dic3—84no—
371C23—24no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—44no—
388C23C22×Dic611yescharacteristic
389C23C2×#23511yescharacteristic
390C23C2×D3×D412yes—
391C23C2×#23712yes—
392C23#100912yes—
393C23#100912yes—
394C23#101012yes—
395C23#101012yes—
396C23C22×#6711yescharacteristic
397Q8—1232no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—68no—
422C2×C4—68no—
423C2×C4—68no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4—124no—
431C2×C4—124no—
432C2×C4—124no—
433C2×C4—124no—
434C2×C4—124no—
435C2×C4#92414yes—
436C2×C4#93614yes—
437C2×C4#93714yes—
438C2×C3—82no—
439C2×C3#240911yescharacteristic
440C2×C3#243611yescharacteristic
441C2×C3#256311yescharacteristic
442C2×C3#261311yescharacteristic
443C2×C3#262811yescharacteristic
444C2×C3#264011yescharacteristic
445C2×C3#272914yes—
446C2×C3#273114yes—
447C2×C3#282111yescharacteristic
448C22—42no—
449C22—42no—
450C22—48no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22—84no—
455C22—81no—
456C22—81no—
457C22#551912yes—
458C22#553412yes—
459C22D4×Dic612yes—
460C22#554312yes—
461C22#555212yes—
462C22#556312yes—
463C22#559814yes—
464C22#565314yes—
465C22#568214yes—
466C22#571214yes—
467C22C2×#92411yescharacteristic
468C22C2×#93611yescharacteristic
469C22C2×#93711yescharacteristic
470C22#582511yescharacteristic
471C22#582511yescharacteristic
472C22#583011yescharacteristic
473C22#583011yescharacteristic
474C4—24no—
475C4—24no—
476C4—24no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—128no—
481C4—244no—
482C3#1660211yescharacteristic, complemented, Sylow
483C2—82no—
484C2#7570614yes—
485C2#7571014yes—
486C2#8226411yescharacteristic
487C2#8234911yescharacteristic
488C2#8238411yescharacteristic
489C2#8265411yescharacteristic
490C2#8290311yescharacteristic
491C2#8297811yescharacteristic
492C2#8310311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#388C2
prime radicalC25×C3C23
hypercenterC2×#388D3
hypocenterC3#16602
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. C22×#869C2semidirect
4. #71565C2semidirect
7. #71534C2semidirect
8. #69424C2semidirect
9. #69415C2semidirect
63. C2×C3×C22⋊C4Q8semidirect
196. C2×C22⋊C4Dic6semidirect
482. C3#16602semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
C2×C3×#3882
C2×#49472
#715652
#715462
#715342
#694242
#694152
#166023