Finite Group Databasebeta

← #10731915

Subgroups — #10731915

Order 768.

1566 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#384C211yesmaximal, characteristic
2C2×#4983C214yesmaximal
3C2×#4975C212yesmaximal
4C2×#4928C211yesmaximal, characteristic
5#71568C214yesmaximal
6#71563C212yesmaximal
7C4×C3⋊#95C211yesmaximal, characteristic
8#15145—31nomaximal, Sylow
9C22×C3×C4⋊C4C2212yes—
10C22×C3×C4⋊C4C2212yes—
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C42C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2212yes—
16C22×Dic3⋊C4C2218yes—
17C22×C4×Dic3C2212yes—
18C3×#384—24no—
19C2×C3⋊#95C2218yes—
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23#4983—216no—
24#4975—28no—
25#4928—24no—
26C2×#398—34no—
27C2×#389—31no—
28C2×#384—32no—
29C2×#384D311yescharacteristic
30#1444—34no—
31C4⋊#95—32no—
32C4×#95—31no—
33C23×C3×C4C2314yes—
34C23×C3×C4C2311yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3C2312yes—
38C23×Dic3C2312yes—
39C23×Dic3C2314yes—
40C2×C3×C4⋊C4—24no—
41C2×C3×C4⋊C4—28no—
42C2×C3×C4⋊C4—24no—
43C2×C3×C4⋊C4—24no—
44C2×C3×C4⋊C4—24no—
45C2×C3×C4⋊C4—24no—
46C2×C3×C4⋊C4—44no—
47C2×C3×C42—24no—
48C2×C3×C42D414yes—
49C2×C4⋊Dic3—22no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3—24no—
53C2×C4⋊Dic3D414yes—
54C2×Dic3⋊C4—24no—
55C2×Dic3⋊C4—28no—
56C2×Dic3⋊C4—216no—
57C2×Dic3⋊C4—216no—
58C2×Dic3⋊C4—48no—
59C2×C4×Dic3—24no—
60C2×C4×Dic3—24no—
61C2×C4×Dic3Q818yes—
62C3⋊#95—24no—
63C3⋊#95—216no—
64C3⋊#95D418yes—
65C3⋊#95Q818yes—
66C22×C4⋊C4—31no—
67C22×C4⋊C4—32no—
68C22×C4⋊C4—31no—
69C22×C4⋊C4—38no—
70C22×C4⋊C4C2×D312yes—
71C22×C4⋊C4C2×D312yes—
72C22×C4⋊C4C2×D312yes—
73C22×C42—32no—
74C22×C42C2×D311yescharacteristic
75#398—616no—
76#389—64no—
77#384—24no—
78#384—68no—
79C2×#95—32no—
80C2×#95—32no—
81C2×#95—38no—
82C2×#95—32no—
83C24×C3C2411yescharacteristic
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—28no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—28no—
93C22×C3×C4C2×D414yes—
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×Q814yes—
99C22×C3×C4C2×Q814yes—
100C22×C3×C4#4114yes—
101C22×C3×C4#4112yes—
102C22×Dic3—24no—
103C22×Dic3—24no—
104C22×Dic3—24no—
105C22×Dic3—24no—
106C22×Dic3—22no—
107C22×Dic3—24no—
108C22×Dic3—22no—
109C22×Dic3—24no—
110C22×Dic3—24no—
111C22×Dic3—22no—
112C22×Dic3—24no—
113C22×Dic3—24no—
114C22×Dic3C2×D414yes—
115C22×Dic3C2×D418yes—
116C22×Dic3C2×Q818yes—
117C22×Dic3C2×Q818yes—
118C3×C4⋊C4—48no—
119C3×C4⋊C4—48no—
120C3×C4⋊C4—48no—
121C3×C42—28no—
122C4⋊Dic3—28no—
123C4⋊Dic3—44no—
124Dic3⋊C4—432no—
125Dic3⋊C4—48no—
126C4×Dic3—216no—
127C23×C4—34no—
128C23×C4—32no—
129C23×C4—32no—
130C23×C4C22×D314yes—
131C23×C4C22×D311yescharacteristic
132C23×C4C22×D311yescharacteristic
133C23×C4C22×D311yescharacteristic
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—34no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—616no—
145C2×C4⋊C4—616no—
146C2×C4⋊C4—64no—
147C2×C4⋊C4—68no—
148C2×C4⋊C4—64no—
149C2×C4⋊C4—62no—
150C2×C4⋊C4—128no—
151C2×C42—24no—
152C2×C42—38no—
153C2×C42—64no—
154C2×C42—64no—
155C2×C42#6714yes—
156#95—38no—
157#95—38no—
158#95—616no—
159#95—64no—
160C23×C3C22×D411yescharacteristic
161C23×C3C22×D412yes—
162C23×C3C22×D411yescharacteristic
163C23×C3C22×Q812yes—
164C23×C3C22×Q812yes—
165C23×C3C2×#4112yes—
166C23×C3C2×#4111yescharacteristic
167C23×C3H(5,𝔽2)12yes—
168C23×C32−1+412yes—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—28no—
174C2×C3×C4—28no—
175C2×C3×C4—44no—
176C2×C3×C4—44no—
177C2×C3×C4—44no—
178C2×C3×C4—48no—
179C2×C3×C4—48no—
180C2×C3×C4#12114yes—
181C2×C3×C4C22⋊Q818yes—
182C2×C3×C4#12314yes—
183C2×C3×C4#12518yes—
184C2×C3×C4#12714yes—
185C2×C3×C4C4⋊Q814yes—
186C2×C3×C4C4⋊Q814yes—
187C2×Dic3—28no—
188C2×Dic3—28no—
189C2×Dic3—28no—
190C2×Dic3—28no—
191C2×Dic3—28no—
192C2×Dic3—28no—
193C2×Dic3—24no—
194C2×Dic3—28no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—44no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—48no—
202C2×Dic3C4⋊Q8116yes—
203C24C23×D311yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—38no—
214C22×C4—38no—
215C22×C4—38no—
216C22×C4—34no—
217C22×C4—64no—
218C22×C4—64no—
219C22×C4—64no—
220C22×C4—64no—
221C22×C4—64no—
222C22×C4—62no—
223C22×C4—64no—
224C22×C4—62no—
225C22×C4—64no—
226C22×C4—64no—
227C22×C4—64no—
228C22×C4—62no—
229C22×C4#23514yes—
230C22×C4D3×D414yes—
231C22×C4D3×D412yes—
232C22×C4#23712yes—
233C22×C4D3×Q814yes—
234C22×C4D3×Q814yes—
235C22×C4C2×#6712yes—
236C22×C4C2×#6712yes—
237C22×C4C2×#6712yes—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—48no—
241C4⋊C4—68no—
242C4⋊C4—1232no—
243C4⋊C4—124no—
244C4⋊C4—128no—
245C42—28no—
246C42—616no—
247C22×C3C2×#12111yescharacteristic
248C22×C3C2×C22⋊Q812yes—
249C22×C3C2×#12311yescharacteristic
250C22×C3C2×#12512yes—
251C22×C3C2×#12711yescharacteristic
252C22×C3C2×C4⋊Q814yes—
253C22×C3C2×C4⋊Q811yescharacteristic
254C22×C3C2×C4⋊Q811yescharacteristic
255C22×C3#53511yescharacteristic
256C22×C3#53611yescharacteristic
257C22×C3#54414yes—
258C22×C3D4212yes—
259C22×C3#54714yes—
260C22×C3D4×Q812yes—
261C22×C3#55012yes—
262C22×C3#55414yes—
263C22×C3#55712yes—
264C3×C4—24no—
265C3×C4—24no—
266C3×C4—84no—
267C3×C4#39818yescomplemented
268Dic3—216no—
269Dic3—44no—
270Dic3—82no—
271C23C2×#23512yes—
272C23C2×D3×D411yescharacteristic
273C23C2×D3×D412yes—
274C23C2×#23711yescharacteristic
275C23#100412yes—
276C23C2×D3×Q812yes—
277C23C2×D3×Q812yes—
278C23#100712yes—
279C23C22×#6711yescharacteristic
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—316no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—68no—
293C2×C4—68no—
294C2×C4—68no—
295C2×C4—68no—
296C2×C4—68no—
297C2×C4—68no—
298C2×C4—68no—
299C2×C4—64no—
300C2×C4—124no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—128no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4#88918yes—
308C2×C4#89618yes—
309C2×C4#92914yes—
310C2×C4#93814yes—
311C2×C4#94014yes—
312C2×C4#94414yes—
313C2×C4#94414yes—
314C2×C3C2×#39811yescharacteristic
315C2×C3#244014yes—
316C2×C3#250111yescharacteristic
317C2×C3#250211yescharacteristic
318C2×C3#254111yescharacteristic
319C2×C3#254611yescharacteristic
320C2×C3#255214yes—
321C2×C3#266011yescharacteristic
322C2×C3#286811yescharacteristic
323C22C2×#88912yes—
324C22#550714yes—
325C22C2×#89612yes—
326C22#551114yes—
327C22#571712yes—
328C22#571812yes—
329C22#572112yes—
330C22D3×C4⋊Q814yes—
331C22#572712yes—
332C22#572914yes—
333C22C2×#92911yescharacteristic
334C22C2×#93811yescharacteristic
335C22C2×#94011yescharacteristic
336C22#580511yescharacteristic
337C22C2×#94411yescharacteristic
338C22C2×#94411yescharacteristic
339C22#581611yescharacteristic
340C4—24no—
341C4—24no—
342C4—616no—
343C4—84no—
344C4—124no—
345C4—242no—
346C4#498318yescomplemented
347C3#1514511yescharacteristic, complemented, Sylow
348C2#7432614yes—
349C2#7433714yes—
350C2C2×#498311yescharacteristic
351C2#8217711yescharacteristic
352C2#8217811yescharacteristic
353C2#8318811yescharacteristic
354C2#8323911yescharacteristic
355C2#8324611yescharacteristic
356C2#8325311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#384C2
hypercenterC2×#384D3
hypocenterC3#15145
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
267. C3×C4#398semidirect
346. C4#4983semidirect
347. C3#15145semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3842
C2×#49832
C2×#49752
C2×#49282
#715682
#715632
C4×C3⋊#952
#151453