Finite Group Databasebeta

← #10732462

Subgroups — #10732462

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3#72134C212yesmaximal
4#71569C212yesmaximal
5C4×C3⋊#95C211yesmaximal, characteristic
6#69415C212yesmaximal
7#69415C212yesmaximal
8#69405C214yesmaximal
9#15209—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2212yes—
16C22×C4×Dic3C2212yes—
17C3×#391—24no—
18C2×C3×#95C2212yes—
19C2×C3×#95C2212yes—
20C2×C3⋊#95C2214yes—
21C2×C3⋊#95C2214yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2212yes—
27#4928—24no—
28C2×#391D311yescharacteristic
29C2×#389—31no—
30#1446—32no—
31#1445—32no—
32#1442—32no—
33#1442—32no—
34#1436—34no—
35C4×#95—31no—
36C22×C3×Q8—22no—
37C22×C3×Q8—24no—
38C22×C3×Q8—24no—
39C22×C3×Q8—42no—
40C23×C3×C4C2312yes—
41C23×C3×C4C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3C2312yes—
46C23×Dic3C2314yes—
47C23×Dic3C2312yes—
48C2×C3×C4⋊C4—22no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C42—42no—
51C2×C3×C42D414yes—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3—24no—
54C2×C4⋊Dic3—22no—
55C2×C4⋊Dic3—42no—
56C2×Dic3⋊C4—24no—
57C2×Dic3⋊C4—28no—
58C2×C4×Dic3—24no—
59C2×C4×Dic3—42no—
60C2×C4×Dic3—42no—
61C3×#95—24no—
62C3×#95—24no—
63C3⋊#95—24no—
64C3⋊#95—28no—
65C3⋊#95—24no—
66C3⋊#95—28no—
67C3⋊#95—44no—
68C3⋊#95—44no—
69C3⋊#95Q818yescomplemented
70C23×Q8C2×D311yescharacteristic
71C22×C4⋊C4—32no—
72C22×C4⋊C4—31no—
73C22×C4⋊C4—31no—
74C22×C4⋊C4C2×D311yescharacteristic
75C22×C42—32no—
76C22×C42C2×D311yescharacteristic
77#391—24no—
78#389—64no—
79C2×#95—32no—
80C2×#95—32no—
81C2×#95—34no—
82C2×#95—34no—
83C2×#95—32no—
84C2×#95—34no—
85C2×#95—34no—
86C2×#95C2×D312yes—
87C2×#95C2×D312yes—
88C24×C3C2411yescharacteristic
89C2×C3×Q8—28no—
90C2×C3×Q8—416no—
91C2×C3×Q8—44no—
92C2×C3×Q8—44no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×Q814yes—
113C22×C3×C4#4114yes—
114C22×C3×C4#4114yes—
115C22×C3×C4#4114yes—
116C22×Dic3—22no—
117C22×Dic3—24no—
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—24no—
121C22×Dic3—24no—
122C22×Dic3—24no—
123C22×Dic3—22no—
124C22×Dic3—24no—
125C22×Dic3—24no—
126C22×Dic3—22no—
127C22×Dic3—24no—
128C22×Dic3—22no—
129C22×Dic3—24no—
130C22×Dic3—22no—
131C22×Dic3C2×Q818yes—
132C22×Dic3#4114yes—
133C3×C4⋊C4—44no—
134C3×C42—44no—
135C4⋊Dic3—28no—
136C4⋊Dic3—44no—
137Dic3⋊C4—48no—
138C4×Dic3—48no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—42no—
143C23×C4—32no—
144C23×C4—34no—
145C23×C4—32no—
146C23×C4C22×D312yes—
147C23×C4C22×D312yes—
148C23×C4C22×D311yescharacteristic
149C23×C4C22×D311yescharacteristic
150C23×C4C22×D311yescharacteristic
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4—62no—
155C2×C4⋊C4—68no—
156C2×C4⋊C4—64no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4—122no—
159C2×C42—42no—
160C2×C42—64no—
161C2×C42—122no—
162C2×C42—122no—
163C2×C42#6714yes—
164#95—24no—
165#95—24no—
166#95—38no—
167#95—68no—
168#95—68no—
169#95—64no—
170#95—64no—
171#95—124no—
172#95—124no—
173C23×C3C22×D411yescharacteristic
174C23×C3C22×Q812yes—
175C23×C3C2×#4112yes—
176C23×C3C2×#4112yes—
177C23×C3C2×#4112yes—
178C23×C3C2×#4112yes—
179C23×C3H(5,𝔽2)12yes—
180C23×C3H(5,𝔽2)11yescharacteristic
181C23×C32−1+411yescharacteristic
182C3×Q8—416no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4—24no—
194C2×C3×C4—28no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—44no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4#12414yes—
204C2×C3×C4#12414yes—
205C2×C3×C4C4⋊Q814yes—
206C2×Dic3—24no—
207C2×Dic3—28no—
208C2×Dic3—28no—
209C2×Dic3—28no—
210C2×Dic3—28no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—44no—
214C2×Dic3—44no—
215C2×Dic3—44no—
216C2×Dic3—44no—
217C2×Dic3—44no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—44no—
222C2×Dic3—44no—
223C24C23×D311yescharacteristic
224C2×Q8—28no—
225C2×Q8—416no—
226C2×Q8—44no—
227C2×Q8—44no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—38no—
245C22×C4—34no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—64no—
249C22×C4—64no—
250C22×C4—64no—
251C22×C4—64no—
252C22×C4—64no—
253C22×C4—62no—
254C22×C4—64no—
255C22×C4—62no—
256C22×C4—64no—
257C22×C4—64no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4#23514yes—
262C22×C4#23714yes—
263C22×C4D3×Q814yes—
264C22×C4#23914yes—
265C22×C4C2×#6712yes—
266C22×C4C2×#6712yes—
267C22×C4C2×#6712yes—
268C4⋊C4—44no—
269C4⋊C4—68no—
270C4⋊C4—124no—
271C4⋊C4—128no—
272C42—44no—
273C42—128no—
274C22×C3#52511yescharacteristic
275C22×C3#52511yescharacteristic
276C22×C3C2×#12411yescharacteristic
277C22×C3C2×#12411yescharacteristic
278C22×C3#52914yes—
279C22×C3C2×C4⋊Q811yescharacteristic
280C22×C3#53314yes—
281C22×C3#53411yescharacteristic
282C22×C3#53711yescharacteristic
283C22×C3#53812yes—
284C22×C3#53912yes—
285C22×C3#55114yes—
286C22×C3#55414yes—
287C22×C3#55412yes—
288C22×C3#55612yes—
289C22×C3#55712yes—
290C22×C3#55912yes—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—24no—
294C3×C4—44no—
295C3×C4—82no—
296Dic3—48no—
297Dic3—44no—
298Dic3—82no—
299C23C2×#23512yes—
300C23C2×#23712yes—
301C23#100412yes—
302C23C2×D3×Q812yes—
303C23C2×#23912yes—
304C23D3×#4112yes—
305C23#100911yescharacteristic
306C23#101011yescharacteristic
307C23C22×#6711yescharacteristic
308Q8—416no—
309C2×C4—28no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—24no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—68no—
330C2×C4—68no—
331C2×C4—64no—
332C2×C4—68no—
333C2×C4—68no—
334C2×C4—124no—
335C2×C4—124no—
336C2×C4—124no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—124no—
340C2×C4—124no—
341C2×C4—124no—
342C2×C4—124no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4#93614yes—
347C2×C4#94414yes—
348C2×C4#94714yes—
349C2×C3#239811yescharacteristic
350C2×C3#256111yescharacteristic
351C2×C3#258311yescharacteristic
352C2×C3#258411yescharacteristic
353C2×C3#258814yes—
354C2×C3#258914yes—
355C2×C3#259211yescharacteristic
356C2×C3#263911yescharacteristic
357C2×C3#286211yescharacteristic
358C22#549514yes—
359C22#556712yes—
360C22#557912yes—
361C22#558014yes—
362C22#563614yes—
363C22#564412yes—
364C22#564512yes—
365C22#567112yes—
366C22#568112yes—
367C22#572814yes—
368C22#579111yescharacteristic
369C22C2×#93611yescharacteristic
370C22#580411yescharacteristic
371C22C2×#94411yescharacteristic
372C22C2×#94711yescharacteristic
373C22#582811yescharacteristic
374C22#583311yescharacteristic
375C4—24no—
376C4—24no—
377C4—24no—
378C4—44no—
379C4—82no—
380C4—128no—
381C4—124no—
382C4—242no—
383C3#1520911yescharacteristic, complemented, Sylow
384C2#7655214yes—
385C2#7655414yes—
386C2#8211611yescharacteristic
387C2#8248311yescharacteristic
388C2#8249211yescharacteristic
389C2#8278511yescharacteristic
390C2#8282411yescharacteristic
391C2#8297611yescharacteristic
392C2#8325211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#391C2
hypercenterC2×#391D3
hypocenterC3#15209
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
69. C3⋊#95Q8semidirect
383. C3#15209semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×#49282
#721342
#715692
C4×C3⋊#952
#694152
#694052
#152093