Finite Group Databasebeta

← #10732464

Subgroups — #10732464

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×#4929C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C3×C4×#95C211yesmaximal, characteristic
4#71536C214yesmaximal
5#69424C212yesmaximal
6#69423C212yesmaximal
7#69412C212yesmaximal
8#69412C212yesmaximal
9#15209—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C22×C3×C42C2212yes—
12C22×C3×C42C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C2×C3×#95C2212yes—
18C2×C3×#95C2212yes—
19C2×C3⋊#95C2212yes—
20C2×C3⋊#95C2214yes—
21C2×C3⋊#95C2214yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2214yes—
26#4929—24no—
27#4928—24no—
28C2×#391—31no—
29C2×#389—31no—
30#1446—32no—
31#1445—32no—
32#1442—32no—
33#1442—32no—
34#1436—34no—
35C4×#95D311yescharacteristic
36C23×C3×C4C2314yes—
37C23×C3×C4C2311yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3C2312yes—
41C23×Dic3C2312yes—
42C23×Dic3C2312yes—
43C23×Dic3C2312yes—
44C22×Dic6—22no—
45C22×Dic6—24no—
46C22×Dic6—24no—
47C22×Dic6—42no—
48C2×C3×C42—24no—
49C2×C3×C42—42no—
50C2×C3×C42—42no—
51C2×C3×C42—42no—
52C2×C3×C42D414yes—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3—24no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3—24no—
58C2×C4⋊Dic3—42no—
59C2×Dic3⋊C4—24no—
60C2×Dic3⋊C4—28no—
61C3×#95—24no—
62C3×#95Q818yescomplemented
63C3⋊#95—24no—
64C3⋊#95—24no—
65C3⋊#95—24no—
66C3⋊#95—28no—
67C3⋊#95—28no—
68C3⋊#95—44no—
69C3⋊#95—44no—
70C23×Q8—31no—
71C22×C4⋊C4—32no—
72C22×C4⋊C4—31no—
73C22×C4⋊C4—31no—
74C22×C4⋊C4—31no—
75C22×C42C2×D312yes—
76C22×C42C2×D311yescharacteristic
77#391—64no—
78#389—64no—
79C2×#95—32no—
80C2×#95—34no—
81C2×#95—34no—
82C2×#95—32no—
83C2×#95—32no—
84C2×#95—34no—
85C2×#95—34no—
86C2×#95C2×D312yes—
87C2×#95C2×D312yes—
88C24×C3C2411yescharacteristic
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—22no—
95C22×C3×C4—21no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×Q818yes—
106C22×C3×C4C2×Q814yes—
107C22×C3×C4#4114yes—
108C22×C3×C4#4114yes—
109C22×Dic3—22no—
110C22×Dic3—22no—
111C22×Dic3—24no—
112C22×Dic3—24no—
113C22×Dic3—22no—
114C22×Dic3—24no—
115C22×Dic3—22no—
116C22×Dic3—24no—
117C22×Dic3—22no—
118C22×Dic3—22no—
119C22×Dic3—24no—
120C22×Dic3—24no—
121C22×Dic3—24no—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—24no—
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3#4114yes—
128C22×Dic3#4114yes—
129C2×Dic6—28no—
130C2×Dic6—416no—
131C2×Dic6—44no—
132C2×Dic6—44no—
133C3×C42—44no—
134C3×C42—48no—
135C4⋊Dic3—28no—
136C4⋊Dic3—44no—
137C4⋊Dic3—44no—
138Dic3⋊C4—48no—
139C22×Q8—64no—
140C22×Q8—62no—
141C22×Q8—64no—
142C22×Q8—122no—
143C23×C4—32no—
144C23×C4—32no—
145C23×C4—32no—
146C23×C4—32no—
147C23×C4C22×D314yes—
148C23×C4C22×D311yescharacteristic
149C23×C4C22×D311yescharacteristic
150C23×C4C22×D311yescharacteristic
151C2×C4⋊C4—64no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4—62no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—68no—
157C2×C4⋊C4—64no—
158C2×C4⋊C4—122no—
159C2×C42—24no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42D1214yes—
164#95—24no—
165#95—68no—
166#95—68no—
167#95—64no—
168#95—64no—
169#95—64no—
170#95—124no—
171#95—124no—
172#95Dic618yescomplemented
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
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—28no—
195C2×C3×C4—44no—
196C2×C3×C4—44no—
197C2×C3×C4—44no—
198C2×C3×C4—44no—
199C2×C3×C4#12414yes—
200C2×C3×C4#12414yes—
201C2×C3×C4C4⋊Q814yes—
202C2×Dic3—24no—
203C2×Dic3—24no—
204C2×Dic3—28no—
205C2×Dic3—28no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—44no—
210C2×Dic3—42no—
211C2×Dic3—44no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—44no—
219C2×Dic3—44no—
220C2×Dic3—44no—
221C2×Dic3—44no—
222Dic6—416no—
223C24C23×D311yescharacteristic
224C2×Q8—68no—
225C2×Q8—1216no—
226C2×Q8—124no—
227C2×Q8—124no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—34no—
242C22×C4—34no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—64no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—64no—
249C22×C4—62no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—64no—
254C22×C4—64no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—64no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—64no—
261C22×C4C2×Dic618yes—
262C22×C4C2×Dic614yes—
263C22×C4C2×D1212yes—
264C22×C4C2×D1212yes—
265C22×C4C2×D1212yes—
266C22×C4#23514yes—
267C22×C4#23514yes—
268C4⋊C4—68no—
269C4⋊C4—124no—
270C4⋊C4—128no—
271C4⋊C4—124no—
272C42—44no—
273C42—48no—
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—48no—
295Dic3—44no—
296Dic3—44no—
297Dic3—82no—
298Dic3—82no—
299C23C22×Dic612yes—
300C23C22×D1211yescharacteristic
301C23C2×#23512yes—
302C23C2×#23512yes—
303C23#100412yes—
304C23#100411yescharacteristic
305C23#100711yescharacteristic
306C23D3×#4112yes—
307C23D3×#4112yes—
308Q8—1216no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—64no—
327C2×C4—64no—
328C2×C4—68no—
329C2×C4—68no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—124no—
335C2×C4—124no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—124no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—124no—
343C2×C4—122no—
344C2×C4—124no—
345C2×C4—124no—
346C2×C4#86914yes—
347C2×C4#87514yes—
348C2×C4#87514yes—
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
358C22C2×#86911yescharacteristic
359C22#547014yes—
360C22C2×#87511yescharacteristic
361C22C2×#87511yescharacteristic
362C22#549311yescharacteristic
363C22#549412yes—
364C22#551011yescharacteristic
365C22#551512yes—
366C22#552411yescharacteristic
367C22#552411yescharacteristic
368C22#552814yes—
369C22#553914yes—
370C22#553912yes—
371C22#555614yes—
372C22#555912yes—
373C22#556812yes—
374C22#556912yes—
375C4—24no—
376C4—24no—
377C4—24no—
378C4—48no—
379C4—124no—
380C4—124no—
381C4—242no—
382C4—242no—
383C3#1520911yescharacteristic, complemented, Sylow
384C2#7330811yescharacteristic
385C2#7337911yescharacteristic
386C2#7341911yescharacteristic
387C2#7354711yescharacteristic
388C2#7354914yes—
389C2#7360511yescharacteristic
390C2#7372711yescharacteristic
391C2#7373214yes—
392C2#7373511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×C4×#95C2
hypercenterC4×#95D3
hypocenterC3#15209
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
62. C3×#95Q8semidirect
172. #95Dic6semidirect
383. C3#15209semidirect

Maximal subgroups

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

subgroupindex
C2×#49292
C2×#49282
C3×C4×#952
#715362
#694242
#694232
#694122
#152093