Finite Group Databasebeta

← #10723596

Subgroups — #10723596

Order 768.

1346 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×C3×#389C211yesmaximal, characteristic
2C2×#4984C212yesmaximal
3C2×#4980C212yesmaximal
4#71565C212yesmaximal
5#71565C212yesmaximal
6#71535C211yesmaximal, characteristic
7#71534C211yesmaximal, characteristic
8#69412C212yesmaximal
9#69407C212yesmaximal
10#14439—31nomaximal, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×C4⋊Dic3C2212yes—
17C22×C4⋊Dic3C2212yes—
18C22×C4×Dic3C2212yes—
19C3×#389C418yescomplemented
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⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33#4984—28no—
34#4980—28no—
35C2×#389—32no—
36C2×#389D311yescharacteristic
37C2×#387—32no—
38#1442—32no—
39#1441—32no—
40#1441—32no—
41#1436—32no—
42C4⋊#95—31no—
43#1434—31no—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—42no—
56C2×C3×C4⋊C4C2×C418yes—
57C2×C3×C4⋊C4C2×C414yes—
58C2×C3×C4⋊C4C2×C414yes—
59C2×C3×C4⋊C4D414yes—
60C2×C3×C4⋊C4Q814yes—
61C2×C3×C42—42no—
62C2×C3×C42C2×C414yes—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—24no—
65C2×C4⋊Dic3—24no—
66C2×C4⋊Dic3—24no—
67C2×C4⋊Dic3—24no—
68C2×C4⋊Dic3—24no—
69C2×C4⋊Dic3—42no—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3—42no—
72C2×C4×Dic3—24no—
73C2×C4×Dic3—24no—
74C2×C4×Dic3—42no—
75C3×#95C2×C418yes—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—22no—
81C3⋊#95—24no—
82C3⋊#95—24no—
83C3⋊#95—41no—
84C3⋊#95—42no—
85C3⋊#95—42no—
86C3⋊#95—41no—
87C3⋊#95D414yes—
88C22×C4⋊C4—32no—
89C22×C4⋊C4—32no—
90C22×C4⋊C4—32no—
91C22×C4⋊C4C2×D312yes—
92C22×C4⋊C4C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C42—32no—
95C22×C42C2×D311yescharacteristic
96#389—68no—
97#389Dic318yescomplemented
98#387—68no—
99C2×#95—31no—
100C2×#95—31no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—31no—
109C2×#95—31no—
110C2×#95—32no—
111C2×#95C2×D312yes—
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—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4C22×C412yes—
132C22×C3×C4C22×C414yes—
133C22×C3×C4C22×C414yes—
134C22×C3×C4C22×C412yes—
135C22×C3×C4C22×C412yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D414yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×Q812yes—
140C22×C3×C4C2×Q814yes—
141C22×C3×C4#4112yes—
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×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3C2×D414yes—
169C22×Dic3#4114yes—
170C3×C4⋊C4—216no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4C4⋊C4116yescomplemented
173C3×C42—44no—
174C4⋊Dic3—48no—
175C4⋊Dic3—48no—
176C4⋊Dic3—48no—
177C4×Dic3—48no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4C22×D312yes—
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D312yes—
185C23×C4C22×D311yescharacteristic
186C23×C4C22×D311yescharacteristic
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4—64no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—122no—
197C2×C4⋊C4—122no—
198C2×C4⋊C4—122no—
199C2×C4⋊C4Dic614yes—
200C2×C4⋊C4D1214yes—
201C2×C4⋊C4C2×Dic318yes—
202C2×C4⋊C4C2×Dic314yes—
203C2×C4⋊C4C2×Dic314yes—
204C2×C42—42no—
205C2×C42—64no—
206C2×C42—64no—
207C2×C42—122no—
208C2×C42C2×Dic314yes—
209#95—34no—
210#95—64no—
211#95—64no—
212#95—64no—
213#95—64no—
214#95—64no—
215#95—64no—
216#95—62no—
217#95—122no—
218#95—122no—
219#95—121no—
220#95—121no—
221#95C2×Dic318yes—
222C23×C3C23×C411yescharacteristic
223C23×C3C22×D411yescharacteristic
224C23×C3C22×D411yescharacteristic
225C23×C3C22×Q811yescharacteristic
226C23×C3C2×#4111yescharacteristic
227C23×C3C2×#4112yes—
228C23×C3H(5,𝔽2)12yes—
229C23×C3H(5,𝔽2)11yescharacteristic
230C23×C32−1+412yes—
231C23×C32−1+412yes—
232C23×C32−1+411yescharacteristic
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
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—24no—
248C2×C3×C4—24no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4C2×C4⋊C414yes—
260C2×C3×C4C2×C4⋊C418yes—
261C2×C3×C4C4×D414yes—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—42no—
269C2×Dic3—42no—
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—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C24C23×D311yescharacteristic
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—34no—
310C22×C4—34no—
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×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×C4C2×Dic612yes—
338C22×C4C2×Dic614yes—
339C22×C4C2×D1212yes—
340C22×C4C2×D1214yes—
341C22×C4D3×D412yes—
342C22×C4#23712yes—
343C22×C4C22×Dic312yes—
344C22×C4C22×Dic314yes—
345C22×C4C22×Dic314yes—
346C22×C4C22×Dic312yes—
347C22×C4C22×Dic312yes—
348C4⋊C4—216no—
349C4⋊C4—44no—
350C4⋊C4—128no—
351C4⋊C4—128no—
352C4⋊C4—128no—
353C4⋊C4C4⋊Dic3116yescomplemented
354C42—44no—
355C42—128no—
356C22×C3C22×C4⋊C411yescharacteristic
357C22×C3C2×C4×D411yescharacteristic
358C22×C3C4×#4111yescharacteristic
359C22×C3#51811yescharacteristic
360C22×C3#51911yescharacteristic
361C22×C3#51911yescharacteristic
362C22×C3#52011yescharacteristic
363C22×C3#53612yes—
364C22×C3#53712yes—
365C22×C3#53912yes—
366C22×C3#54112yes—
367C22×C3#54112yes—
368C22×C3#54412yes—
369C22×C3D4211yescharacteristic
370C22×C3#54612yes—
371C22×C3#54711yescharacteristic
372C22×C3#54812yes—
373C22×C3D4×Q811yescharacteristic
374C22×C3#55312yes—
375C22×C3#55411yescharacteristic
376C22×C3#55712yes—
377C22×C3#56212yes—
378C22×C3#56312yes—
379C3×C4—24no—
380C3×C4—28no—
381C3×C4—24no—
382C3×C4—44no—
383C3×C4—42no—
384Dic3—44no—
385Dic3—44no—
386Dic3—82no—
387Dic3—82no—
388C23C22×Dic611yescharacteristic
389C23C22×D1211yescharacteristic
390C23C2×D3×D411yescharacteristic
391C23C2×#23711yescharacteristic
392C23#100412yes—
393C23#100712yes—
394C23#100712yes—
395C23D3×#4112yes—
396C23#100911yescharacteristic
397C23#101011yescharacteristic
398C23C23×Dic311yescharacteristic
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
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—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
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×C4C2×C4⋊Dic318yes—
454C2×C4C2×C4⋊Dic314yes—
455C2×C4D4×Dic314yes—
456C2×C3#231611yescharacteristic
457C2×C3#232711yescharacteristic
458C2×C3#233611yescharacteristic
459C2×C3D4×C4⋊C411yescharacteristic
460C2×C3#235111yescharacteristic
461C2×C3#236411yescharacteristic
462C2×C3#237811yescharacteristic
463C2×C3#268112yes—
464C2×C3#271312yes—
465C2×C3#272512yes—
466C2×C3#280212yes—
467C22#550012yes—
468C22#551012yes—
469C22D4×Dic611yescharacteristic
470C22#554311yescharacteristic
471C22D4×D1211yescharacteristic
472C22#555511yescharacteristic
473C22#556912yes—
474C22#557612yes—
475C22#560112yes—
476C22#561012yes—
477C22#564612yes—
478C22#564712yes—
479C22#569212yes—
480C22#569612yes—
481C22#572412yes—
482C22#573112yes—
483C22C22×C4⋊Dic311yescharacteristic
484C22C2×D4×Dic311yescharacteristic
485C22#579811yescharacteristic
486C22#581211yescharacteristic
487C22#581211yescharacteristic
488C22Dic3×#4111yescharacteristic
489C22#582711yescharacteristic
490C4—28no—
491C4—24no—
492C4—24no—
493C4—42no—
494C4—44no—
495C4—124no—
496C4—124no—
497C4—242no—
498C4—242no—
499C3#1443911yescharacteristic, complemented, Sylow
500C2#7611512yes—
501C2#7612312yes—
502C2#7617712yes—
503C2#7625912yes—
504C2#8215611yescharacteristic
505C2D4×C4⋊Dic311yescharacteristic
506C2#8246011yescharacteristic
507C2#8263811yescharacteristic
508C2#8275011yescharacteristic
509C2#8302411yescharacteristic
510C2#8322711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#389C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#389D3
hypocenterC3#14439
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
19. C3×#389C4semidirect
97. #389Dic3semidirect
172. C3×C4⋊C4C4⋊C4semidirect
353. C4⋊C4C4⋊Dic3semidirect
499. C3#14439semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49842
C2×#49802
#715652
#715352
#715342
#694122
#694072
#144393