Finite Group Databasebeta

← #10728728

Subgroups — #10728728

Order 768.

1590 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×#4986C212yesmaximal
3C2×#4928C211yesmaximal, characteristic
4#72134C212yesmaximal
5#71567C212yesmaximal
6#71565C212yesmaximal
7#71536C214yesmaximal
8#71534C211yesmaximal, characteristic
9#14884—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×C3×C4×Q8—28no—
12C2×C3×C4×Q8—24no—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28#4986—28no—
29#4928—24no—
30C22×C4×Q8D311yescharacteristic
31C2×#400—32no—
32C2×#389—31no—
33#1446—32no—
34#1442—32no—
35#1441—32no—
36#1436—34no—
37#1434—31no—
38C22×C3×Q8—24no—
39C22×C3×Q8—22no—
40C22×C3×Q8—24no—
41C22×C3×Q8—22no—
42C22×C3×Q8Q814yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2314yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C3×C4×Q8—232no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4D418yes—
57C2×C3×C42—24no—
58C2×C3×C42—24no—
59C2×C3×C42—24no—
60C2×C3×C42—24no—
61C2×C3×C42D414yes—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3—24no—
66C2×Dic3⋊C4—24no—
67C2×Dic3⋊C4—28no—
68C3⋊#95—24no—
69C3⋊#95—24no—
70C3⋊#95—28no—
71C3⋊#95—28no—
72C3⋊#95—28no—
73C3⋊#95—24no—
74C3⋊#95—24no—
75C3⋊#95—28no—
76C23×Q8C2×D311yescharacteristic
77C2×C4×Q8—28no—
78C2×C4×Q8—24no—
79C22×C4⋊C4—31no—
80C22×C4⋊C4—32no—
81C22×C4⋊C4—31no—
82C22×C4⋊C4C2×D311yescharacteristic
83C22×C4⋊C4C2×D312yes—
84C22×C42C2×D312yes—
85C22×C42C2×D311yescharacteristic
86#400—68no—
87#389—64no—
88C2×#95—32no—
89C2×#95—34no—
90C2×#95—34no—
91C2×#95—32no—
92C2×#95—32no—
93C2×#95—34no—
94C2×#95—34no—
95C2×#95—32no—
96C24×C3C2411yescharacteristic
97C2×C3×Q8—28no—
98C2×C3×Q8—28no—
99C2×C3×Q8—216no—
100C2×C3×Q8—28no—
101C2×C3×Q8—216no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
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—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4C2×D414yes—
117C22×C3×C4C2×D414yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×D414yes—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×Q814yes—
123C22×C3×C4C2×Q812yes—
124C22×C3×C4#4114yes—
125C22×C3×C4#4112yes—
126C22×C3×C4#4114yes—
127C22×Dic3—22no—
128C22×Dic3—24no—
129C22×Dic3—24no—
130C22×Dic3—22no—
131C22×Dic3—24no—
132C22×Dic3—24no—
133C22×Dic3—22no—
134C22×Dic3—24no—
135C22×Dic3—24no—
136C22×Dic3—24no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—24no—
140C22×Dic3—24no—
141C22×Dic3—24no—
142C22×Dic3#4114yes—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—216no—
145C3×C42—216no—
146C3×C42—28no—
147C4⋊Dic3—28no—
148C4⋊Dic3—44no—
149Dic3⋊C4—48no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8Dic614yes—
155C23×C4—32no—
156C23×C4—32no—
157C23×C4—34no—
158C23×C4C22×D312yes—
159C23×C4C22×D312yes—
160C23×C4C22×D311yescharacteristic
161C23×C4C22×D311yescharacteristic
162C23×C4C22×D311yescharacteristic
163C4×Q8—232no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—64no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—68no—
171C2×C4⋊C4—64no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—64no—
174C2×C4⋊C4#6718yes—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42#6714yes—
180#95—68no—
181#95—68no—
182#95—64no—
183#95—64no—
184#95—64no—
185#95—64no—
186#95—68no—
187#95—68no—
188C23×C3C22×D411yescharacteristic
189C23×C3C22×D412yes—
190C23×C3C22×Q811yescharacteristic
191C23×C3C2×#4112yes—
192C23×C3C2×#4111yescharacteristic
193C23×C3C2×#4112yes—
194C23×C3C2×#4112yes—
195C23×C32−1+412yes—
196C23×C32−1+412yes—
197C3×Q8—232no—
198C2×C3×C4—22no—
199C2×C3×C4—24no—
200C2×C3×C4—22no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—24no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—28no—
213C2×C3×C4—28no—
214C2×C3×C4—24no—
215C2×C3×C4—28no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—44no—
219C2×C3×C4#12014yes—
220C2×C3×C4C22⋊Q814yes—
221C2×C3×C4C22⋊Q818yes—
222C2×C3×C4#12414yes—
223C2×Dic3—24no—
224C2×Dic3—28no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—44no—
228C2×Dic3—42no—
229C2×Dic3—44no—
230C2×Dic3—42no—
231C2×Dic3—44no—
232C2×Dic3—42no—
233C2×Dic3—44no—
234C2×Dic3—44no—
235C2×Dic3—48no—
236C2×Dic3—44no—
237C2×Dic3—44no—
238C2×Dic3—44no—
239C24C23×D311yescharacteristic
240C2×Q8—28no—
241C2×Q8—216no—
242C2×Q8—28no—
243C2×Q8—216no—
244C2×Q8—28no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—34no—
260C22×C4—64no—
261C22×C4—64no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—64no—
265C22×C4—64no—
266C22×C4—64no—
267C22×C4—64no—
268C22×C4—64no—
269C22×C4—64no—
270C22×C4—64no—
271C22×C4—64no—
272C22×C4—64no—
273C22×C4—64no—
274C22×C4—62no—
275C22×C4C2×Dic612yes—
276C22×C4C2×Dic614yes—
277C22×C4#23512yes—
278C22×C4#23514yes—
279C22×C4#23914yes—
280C22×C4C2×#6714yes—
281C22×C4C2×#6714yes—
282C22×C4C2×#6712yes—
283C22×C4C2×#6712yes—
284C22×C4C2×#6714yes—
285C22×C4C2×#6712yes—
286C4⋊C4—28no—
287C4⋊C4—216no—
288C4⋊C4—68no—
289C4⋊C4—128no—
290C4⋊C4—124no—
291C42—216no—
292C42—28no—
293C22×C3C2×#12011yescharacteristic
294C22×C3C2×C22⋊Q811yescharacteristic
295C22×C3C2×C22⋊Q812yes—
296C22×C3#52511yescharacteristic
297C22×C3#52512yes—
298C22×C3#52511yescharacteristic
299C22×C3C2×#12411yescharacteristic
300C22×C3#52914yes—
301C22×C3#53611yescharacteristic
302C22×C3#53611yescharacteristic
303C22×C3#54114yes—
304C22×C3#54714yes—
305C22×C3#54814yes—
306C22×C3#55612yes—
307C22×C3#55612yes—
308C22×C3#55712yes—
309C22×C3#55712yes—
310C3×C4—28no—
311C3×C4—24no—
312C3×C4—24no—
313C3×C4—24no—
314C3×C4—44no—
315Dic3—44no—
316Dic3—82no—
317Dic3—84no—
318C23C22×Dic611yescharacteristic
319C23C2×#23511yescharacteristic
320C23C2×#23512yes—
321C23C2×#23912yes—
322C23#100712yes—
323C23D3×#4112yes—
324C23#101012yes—
325C23C22×#6711yescharacteristic
326C23C22×#6712yes—
327Q8—232no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—28no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—64no—
350C2×C4—68no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—124no—
354C2×C4—124no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—124no—
358C2×C4—128no—
359C2×C4—124no—
360C2×C4—124no—
361C2×C4—124no—
362C2×C4—124no—
363C2×C4—124no—
364C2×C4—122no—
365C2×C4#92418yes—
366C2×C4#92414yes—
367C2×C4#94714yes—
368C2×C4#95314yes—
369C2×C3#239811yescharacteristic
370C2×C3#240611yescharacteristic
371C2×C3#243611yescharacteristic
372C2×C3#245314yes—
373C2×C3#246914yes—
374C2×C3#256811yescharacteristic
375C2×C3#258111yescharacteristic
376C2×C3#263411yescharacteristic
377C2×C3#284811yescharacteristic
378C22#556812yes—
379C22#556912yes—
380C22#557012yes—
381C22#558014yes—
382C22#558314yes—
383C22#558412yes—
384C22C2×#92412yes—
385C22C2×#92411yescharacteristic
386C22#579112yes—
387C22#579111yescharacteristic
388C22C2×#94711yescharacteristic
389C22#581611yescharacteristic
390C22#581714yes—
391C22#582414yes—
392C22#582511yescharacteristic
393C22#582811yescharacteristic
394C22C2×#95311yescharacteristic
395C4—24no—
396C4—28no—
397C4—24no—
398C4—24no—
399C4—44no—
400C4—124no—
401C4—244no—
402C4—242no—
403C3#1488411yescharacteristic, complemented, Sylow
404C2#8247114yes—
405C2#8247314yes—
406C2#8247911yescharacteristic
407C2#8248111yescharacteristic
408C2#8248611yescharacteristic
409C2#8249211yescharacteristic
410C2#8403111yescharacteristic
411C2#8406611yescharacteristic
412C2#8409011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC22×C3×C4×Q8C2
hypercenterC22×C4×Q8D3
hypocenterC3#14884
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
403. C3#14884semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×#49862
C2×#49282
#721342
#715672
#715652
#715362
#715342
#148843