Finite Group Databasebeta

← #10720413

Subgroups — #10720413

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4929C211yesmaximal, characteristic
2C3×C4×#95C211yesmaximal, characteristic
3#71536C212yesmaximal
4#71534C212yesmaximal
5C4×C3⋊#95C211yesmaximal, characteristic
6#69424C212yesmaximal
7#69411C212yesmaximal
8#69406C212yesmaximal
9#69401C212yesmaximal
10#14162—31nomaximal, Sylow
11C23×Dic6C2211yescharacteristic
12C22×C3×C42C2212yes—
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2212yes—
16C22×C4×Dic3C2212yes—
17C2×C3×#95C2211yescharacteristic
18C2×C3×#95C2212yes—
19C2×C3×#95C2211yescharacteristic
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32#4929C418yescomplemented
33C2×#391—31no—
34#1446—32no—
35#1441—32no—
36#1436—32no—
37#1436—32no—
38#1434—32no—
39#1434—32no—
40C4×#95—31no—
41C4×#95D311yescharacteristic
42C23×C3×C4C2312yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C22×Dic6—24no—
52C22×Dic6—41no—
53C22×Dic6—41no—
54C22×Dic6—42no—
55C22×Dic6C2×C414yes—
56C2×C3×C42—24no—
57C2×C3×C42—24no—
58C2×C3×C42—42no—
59C2×C3×C42—42no—
60C2×C3×C42C2×C414yes—
61C2×C4⋊Dic3—41no—
62C2×C4⋊Dic3—41no—
63C2×C4⋊Dic3C2×C414yes—
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4—24no—
66C2×Dic3⋊C4—42no—
67C2×C4×Dic3—24no—
68C2×C4×Dic3—24no—
69C2×C4×Dic3—42no—
70C3×#95—24no—
71C3×#95—22no—
72C3×#95Q814yescomplemented
73C3⋊#95—24no—
74C3⋊#95—24no—
75C3⋊#95—22no—
76C3⋊#95—24no—
77C3⋊#95—42no—
78C3⋊#95—42no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95—42no—
82C3⋊#95C2×C418yes—
83C3⋊#95C2×C418yes—
84C3⋊#95Q814yescomplemented
85C23×Q8—31no—
86C22×C4⋊C4—31no—
87C22×C4⋊C4—32no—
88C22×C42—32no—
89C22×C42C2×D312yes—
90C22×C42C2×D311yescharacteristic
91#391—38no—
92C2×#95—31no—
93C2×#95—32no—
94C2×#95—32no—
95C2×#95—31no—
96C2×#95—32no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95C2×D311yescharacteristic
105C2×#95C2×D312yes—
106C2×#95C2×D311yescharacteristic
107C24×C3C2411yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
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—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4C22×C412yes—
128C22×C3×C4C22×C412yes—
129C22×C3×C4C22×C412yes—
130C22×C3×C4C2×Q814yes—
131C22×C3×C4C2×Q812yes—
132C22×C3×C4C2×Q812yes—
133C22×C3×C4#4114yes—
134C22×C3×C4#4114yes—
135C22×C3×C4#4112yes—
136C22×C3×C4#4112yes—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3C22×C414yes—
162C22×Dic3C22×C414yes—
163C22×Dic3C2×Q814yes—
164C22×Dic3#4114yes—
165C2×Dic6—28no—
166C2×Dic6—44no—
167C2×Dic6—44no—
168C2×Dic6—48no—
169C2×Dic6—48no—
170C3×C42—44no—
171C3×C42—48no—
172C4⋊Dic3—44no—
173Dic3⋊C4—48no—
174C4×Dic3—48no—
175C22×Q8—34no—
176C22×Q8—64no—
177C22×Q8—122no—
178C22×Q8—121no—
179C22×Q8—121no—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4—32no—
183C23×C4—32no—
184C23×C4C22×D312yes—
185C23×C4C22×D312yes—
186C23×C4C22×D311yescharacteristic
187C23×C4C22×D311yescharacteristic
188C23×C4C22×D311yescharacteristic
189C2×C4⋊C4—34no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4—64no—
192C2×C4⋊C4—122no—
193C2×C4⋊C4—121no—
194C2×C4⋊C4—121no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—64no—
200C2×C42—64no—
201C2×C42—122no—
202C2×C42C4×D314yes—
203#95—22no—
204#95—24no—
205#95—38no—
206#95—38no—
207#95—34no—
208#95—64no—
209#95—64no—
210#95—64no—
211#95—62no—
212#95—122no—
213#95—122no—
214#95—122no—
215#95—122no—
216#95—122no—
217#95Dic614yescomplemented
218C23×C3C23×C411yescharacteristic
219C23×C3C22×Q811yescharacteristic
220C23×C3C22×Q811yescharacteristic
221C23×C3C2×#4112yes—
222C23×C3C2×#4111yescharacteristic
223C23×C3C2×#4112yes—
224C23×C3C2×#4111yescharacteristic
225C23×C3C2×#4112yes—
226C23×C3H(5,𝔽2)12yes—
227C23×C3H(5,𝔽2)11yescharacteristic
228C23×C3H(5,𝔽2)11yescharacteristic
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4#11714yes—
256C2×C3×C4C4×Q814yes—
257C2×C3×C4C4×Q814yes—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286Dic6—416no—
287C24C23×D311yescharacteristic
288C2×Q8—68no—
289C2×Q8—124no—
290C2×Q8—124no—
291C2×Q8—128no—
292C2×Q8—128no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—34no—
313C22×C4—34no—
314C22×C4—34no—
315C22×C4—34no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4C2×Dic614yes—
341C22×C4C2×Dic612yes—
342C22×C4C2×C4×D312yes—
343C22×C4C2×C4×D312yes—
344C22×C4C2×C4×D312yes—
345C22×C4#23514yes—
346C22×C4#23512yes—
347C22×C4#23514yes—
348C22×C4#23712yes—
349C22×C4D3×Q812yes—
350C4⋊C4—124no—
351C4⋊C4—128no—
352C42—44no—
353C42—48no—
354C42—128no—
355C22×C3C2×#11711yescharacteristic
356C22×C3C2×C4×Q811yescharacteristic
357C22×C3C2×C4×Q811yescharacteristic
358C22×C3C4×#4111yescharacteristic
359C22×C3C4×#4111yescharacteristic
360C22×C3#51811yescharacteristic
361C22×C3#51811yescharacteristic
362C22×C3#52912yes—
363C22×C3#52912yes—
364C22×C3#53312yes—
365C22×C3#53312yes—
366C22×C3#53812yes—
367C22×C3#53812yes—
368C22×C3#55112yes—
369C22×C3#55112yes—
370C22×C3#55112yes—
371C22×C3#55412yes—
372C22×C3#55412yes—
373C22×C3#55411yescharacteristic
374C22×C3#55411yescharacteristic
375C22×C3#55512yes—
376C22×C3#55511yescharacteristic
377C22×C3Q8211yescharacteristic
378C3×C4—24no—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—44no—
382C3×C4—44no—
383Dic3—44no—
384Dic3—44no—
385Dic3—44no—
386Dic3—44no—
387C23C22×Dic611yescharacteristic
388C23C22×C4×D311yescharacteristic
389C23C2×#23511yescharacteristic
390C23C2×#23512yes—
391C23C2×#23512yes—
392C23C2×#23711yescharacteristic
393C23#100412yes—
394C23C2×D3×Q811yescharacteristic
395C23D3×#4112yes—
396C23#100911yescharacteristic
397C23#100911yescharacteristic
398Q8—1216no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—64no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4C4×Dic614yes—
454C2×C4#87214yes—
455C2×C4#88714yes—
456C2×C3#229511yescharacteristic
457C2×C3#229511yescharacteristic
458C2×C3#231211yescharacteristic
459C2×C3#236811yescharacteristic
460C2×C3#236811yescharacteristic
461C2×C3#236911yescharacteristic
462C2×C3#237211yescharacteristic
463C2×C3#257212yes—
464C2×C3#257412yes—
465C2×C3#257412yes—
466C2×C3#258612yes—
467C22C2×C4×Dic611yescharacteristic
468C22#547012yes—
469C22C2×#87211yescharacteristic
470C22C4×#23511yescharacteristic
471C22#547912yes—
472C22#548611yescharacteristic
473C22#549512yes—
474C22C2×#88711yescharacteristic
475C22#551112yes—
476C22#552111yescharacteristic
477C22#552211yescharacteristic
478C22#552712yes—
479C22#553212yes—
480C22#553912yes—
481C22#554311yescharacteristic
482C22#555612yes—
483C22#555712yes—
484C22#556012yes—
485C22#556012yes—
486C22#556311yescharacteristic
487C22Q8×Dic611yescharacteristic
488C22#557812yes—
489C22#557911yescharacteristic
490C4—24no—
491C4—24no—
492C4—24no—
493C4—44no—
494C4—44no—
495C4—124no—
496C4—124no—
497C4—124no—
498C4—124no—
499C3#1416211yescharacteristic, complemented, Sylow
500C2#7330011yescharacteristic
501C2#7333911yescharacteristic
502C2#7339511yescharacteristic
503C2#7347311yescharacteristic
504C2#7354012yes—
505C2#7354212yes—
506C2#7356711yescharacteristic
507C2#7361811yescharacteristic
508C2#7365611yescharacteristic
509C2#7372612yes—
510C2#7373012yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4×#95C2
socle, prime radical, cosocleC24×C3C24
hypercenterC4×#95D3
hypocenterC3#14162
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
32. #4929C4semidirect
72. C3×#95Q8semidirect
84. C3⋊#95Q8semidirect
217. #95Dic6semidirect
499. C3#14162semidirect

Maximal subgroups

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

subgroupindex
C2×#49292
C3×C4×#952
#715362
#715342
C4×C3⋊#952
#694242
#694112
#694062
#694012
#141623