Finite Group Databasebeta

← #10724007

Subgroups — #10724007

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4936C211yesmaximal, characteristic
2C3×#1434C211yesmaximal, characteristic
3#71534C211yesmaximal, characteristic
4#69417C212yesmaximal
5#69413C212yesmaximal
6#69411C212yesmaximal
7#69407C212yesmaximal
8#69406C212yesmaximal
9#69402C212yesmaximal
10#14477—31nomaximal, Sylow
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×Dic3⋊C4C2212yes—
15C22×C4×Dic3C2212yes—
16C2×C3×#95C2212yes—
17C2×C3×#95C2212yes—
18C2×C3×#95C2211yescharacteristic
19C2×C3×#95C2211yescharacteristic
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4936C418yescomplemented
33C2×#382—31no—
34#1443—32no—
35#1442—32no—
36#1441—32no—
37#1436—32no—
38#1436—32no—
39#1434—31no—
40#1434—32no—
41#1434D311yescharacteristic
42C23×C3×C4C2312yes—
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C2×C3×C42—24no—
52C2×C3×C42C2×C414yes—
53C2×C4⋊Dic3—42no—
54C2×C4⋊Dic3—42no—
55C2×C4⋊Dic3C2×C414yes—
56C2×C4⋊Dic3C2×C414yes—
57C2×Dic3⋊C4—24no—
58C2×Dic3⋊C4—42no—
59C2×Dic3⋊C4—42no—
60C2×C4×Dic3—24no—
61C2×C4×Dic3—42no—
62C2×C4×Dic3—42no—
63C3×#95—24no—
64C3×#95—24no—
65C3×#95—41no—
66C3×#95—41no—
67C3⋊#95—24no—
68C3⋊#95—24no—
69C3⋊#95—24no—
70C3⋊#95—24no—
71C3⋊#95—42no—
72C3⋊#95—41no—
73C3⋊#95—42no—
74C3⋊#95—42no—
75C3⋊#95—42no—
76C3⋊#95—41no—
77C3⋊#95C2×C418yes—
78C3⋊#95C2×C418yes—
79C22×C4⋊C4—31no—
80C22×C4⋊C4—31no—
81C22×C4⋊C4—32no—
82C22×C42—32no—
83C22×C42C2×D311yescharacteristic
84#382—38no—
85C2×#95—32no—
86C2×#95—32no—
87C2×#95—32no—
88C2×#95—32no—
89C2×#95—32no—
90C2×#95—32no—
91C2×#95—31no—
92C2×#95—32no—
93C2×#95—31no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—32no—
97C2×#95C2×D312yes—
98C2×#95C2×D312yes—
99C2×#95C2×D311yescharacteristic
100C2×#95C2×D311yescharacteristic
101C24×C3C2411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
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×C4C22×C412yes—
127C22×C3×C4C22×C412yes—
128C22×C3×C4C22×C412yes—
129C22×C3×C4#4114yes—
130C22×C3×C4#4114yes—
131C22×Dic3—22no—
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
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×Dic3C22×C414yes—
157C22×Dic3C22×C414yes—
158C22×Dic3#4114yes—
159C3×C42—28no—
160C4⋊Dic3—44no—
161C4⋊Dic3—44no—
162Dic3⋊C4—48no—
163C4×Dic3—48no—
164C23×C4—32no—
165C23×C4—32no—
166C23×C4—32no—
167C23×C4—32no—
168C23×C4C22×D312yes—
169C23×C4C22×D311yescharacteristic
170C23×C4C22×D311yescharacteristic
171C23×C4C22×D312yes—
172C23×C4C22×D311yescharacteristic
173C2×C4⋊C4—34no—
174C2×C4⋊C4—34no—
175C2×C4⋊C4—64no—
176C2×C4⋊C4—122no—
177C2×C4⋊C4—122no—
178C2×C4⋊C4—122no—
179C2×C4⋊C4—122no—
180C2×C42—24no—
181C2×C42—64no—
182C2×C42—122no—
183C2×C42—122no—
184C2×C42C4×D314yes—
185#95—24no—
186#95—24no—
187#95—38no—
188#95—38no—
189#95—41no—
190#95—41no—
191#95—64no—
192#95—64no—
193#95—64no—
194#95—64no—
195#95—121no—
196#95—121no—
197#95—122no—
198#95—122no—
199#95—122no—
200#95—122no—
201C23×C3C23×C411yescharacteristic
202C23×C3C2×#4112yes—
203C23×C3C2×#4112yes—
204C23×C3C2×#4112yes—
205C23×C3H(5,𝔽2)11yescharacteristic
206C23×C3H(5,𝔽2)12yes—
207C23×C3H(5,𝔽2)11yescharacteristic
208C23×C32−1+411yescharacteristic
209C23×C32−1+411yescharacteristic
210C23×C32−1+411yescharacteristic
211C23×C32−1+411yescharacteristic
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#11714yes—
241C2×Dic3—24no—
242C2×Dic3—24no—
243C2×Dic3—24no—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C24C23×D311yescharacteristic
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—34no—
295C22×C4—34no—
296C22×C4—34no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4C2×C4×D312yes—
323C22×C4C2×C4×D312yes—
324C22×C4C2×C4×D312yes—
325C22×C4#23514yes—
326C22×C4#23714yes—
327C4⋊C4—124no—
328C4⋊C4—124no—
329C4⋊C4—128no—
330C42—28no—
331C42—128no—
332C22×C3C2×#11711yescharacteristic
333C22×C3C4×#4111yescharacteristic
334C22×C3C4×#4111yescharacteristic
335C22×C3#51811yescharacteristic
336C22×C3#51911yescharacteristic
337C22×C3#52011yescharacteristic
338C22×C3#52011yescharacteristic
339C22×C3#53812yes—
340C22×C3#53812yes—
341C22×C3#53912yes—
342C22×C3#53912yes—
343C22×C3#54212yes—
344C22×C3#54212yes—
345C22×C3#55212yes—
346C22×C3#55212yes—
347C22×C3#55212yes—
348C22×C3#55312yes—
349C22×C3#55512yes—
350C22×C3#55612yes—
351C22×C3#56211yescharacteristic
352C22×C3#56311yescharacteristic
353C22×C3#56311yescharacteristic
354C22×C3#56411yescharacteristic
355C3×C4—24no—
356C3×C4—42no—
357C3×C4—42no—
358C3×C4—44no—
359C3×C4—82no—
360Dic3—44no—
361Dic3—44no—
362Dic3—44no—
363Dic3—82no—
364C23C22×C4×D311yescharacteristic
365C23C2×#23512yes—
366C23C2×#23712yes—
367C23#100411yescharacteristic
368C23#100711yescharacteristic
369C23D3×#4112yes—
370C23#100912yes—
371C23#100911yescharacteristic
372C23#101011yescharacteristic
373C23#101011yescharacteristic
374C23#101011yescharacteristic
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—64no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4#87714yes—
432C2×C3#231311yescharacteristic
433C2×C3#233711yescharacteristic
434C2×C3#233711yescharacteristic
435C2×C3#234611yescharacteristic
436C2×C3#234711yescharacteristic
437C2×C3#237111yescharacteristic
438C2×C3#238211yescharacteristic
439C2×C3#276512yes—
440C2×C3#276712yes—
441C2×C3#277512yes—
442C2×C3#278112yes—
443C22C4×#23511yescharacteristic
444C22C2×#87711yescharacteristic
445C22#550411yescharacteristic
446C22#551611yescharacteristic
447C22#552111yescharacteristic
448C22#552211yescharacteristic
449C22#552311yescharacteristic
450C22#552912yes—
451C22#553112yes—
452C22#554112yes—
453C22#556012yes—
454C22#556312yes—
455C22#558212yes—
456C22#561412yes—
457C22#564112yes—
458C22#566512yes—
459C22#566512yes—
460C22#567211yescharacteristic
461C22#568411yescharacteristic
462C22C3⋊#56411yescharacteristic
463C22#569911yescharacteristic
464C22#570412yes—
465C22#571012yes—
466C4—24no—
467C4—42no—
468C4—44no—
469C4—42no—
470C4—82no—
471C4—124no—
472C4—124no—
473C4—124no—
474C4—242no—
475C3#1447711yescharacteristic, complemented, Sylow
476C2#7339411yescharacteristic
477C2#7345911yescharacteristic
478C2#7363211yescharacteristic
479C2#7388211yescharacteristic
480C2#7405411yescharacteristic
481C2#7416111yescharacteristic
482C2#7426211yescharacteristic
483C2#7520812yes—
484C2#7522712yes—
485C2#7575112yes—
486C2#7595112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1434C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1434D3
hypocenterC3#14477
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
32. #4936C4semidirect
475. C3#14477semidirect

Maximal subgroups

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

subgroupindex
C2×#49362
C3×#14342
#715342
#694172
#694132
#694112
#694072
#694062
#694022
#144773