Finite Group Databasebeta

← #10823314

Subgroups — #10823314

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2C2×#5121C212yesmaximal
3#82234C211yesmaximal, characteristic
4C3×#1742C211yesmaximal, characteristic
5#71982C212yesmaximal
6#68979C214yesmaximal, complemented
7#68978C214yesmaximal, complemented
8#23562—31nomaximal, Sylow
9C22×C3×M16C2212yes—
10C2×C3×#117C2211yescharacteristic
11C2×#926C2212yes—
12C2×#926C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14#5519—22no—
15#5519—22no—
16#5517—24no—
17#5500C2212yes—
18#5500C2212yes—
19#5483—22no—
20#5470—22no—
21#5121—24no—
22#5121C2218yes—
23C2×#818—22no—
24C2×#817—22no—
25#4977C2212yes—
26#4975C2212yes—
27#4946—22no—
28#4928—22no—
29C3×#343C2212yes—
30C3×#343C2212yes—
31#4555C2218yes—
32#3458—31no—
33C2×#428—32no—
34#2317—31no—
35#2061—34no—
36#2060—34no—
37#1742D311yescharacteristic
38#1733—32no—
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C2×C3×M16—22no—
42C2×C3×M16—24no—
43C2×C3×M16Q818yescomplemented
44C2×C3×M16C2314yes—
45C22×C3×C8—22no—
46C3×#117—22no—
47C3×#117—22no—
48C2×C3×C4⋊C4C2311yescharacteristic
49C2×C3×C4⋊C4C2311yescharacteristic
50C2×C3×C22⋊C4D412yes—
51C2×C3×C42D412yes—
52C2×#217—21no—
53C2×#217—22no—
54#926—22no—
55#926—22no—
56#926—22no—
57#926D414yes—
58#926Q814yescomplemented
59#926C2314yes—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3D412yes—
65C2×C4⋊Dic3D412yes—
66C2×C4⋊Dic3C2312yes—
67#924—24no—
68#924—24no—
69C2×Dic3⋊C4—24no—
70C2×C4×Dic3—22no—
71C2×C4×Dic3C2312yes—
72#890—24no—
73#890—24no—
74#888—24no—
75#888—24no—
76#878—28no—
77#878—28no—
78#870—24no—
79#869—22no—
80#869—22no—
81C4×Dic6—24no—
82C4×Dic6—24no—
83C3⋊#95—24no—
84#818—28no—
85#817—28no—
86C22×M16C2×D312yes—
87#544—32no—
88#544—32no—
89#544—62no—
90#544—62no—
91#543—62no—
92#533—64no—
93#533—62no—
94C2×#117—31no—
95C2×#117—32no—
96C2×#117C2×D311yescharacteristic
97C22×C4⋊C4—31no—
98#428—38no—
99#428—64no—
100C2×#107—62no—
101C2×#106—62no—
102#389—62no—
103#385—62no—
104#384—32no—
105#382—32no—
106#343C2×D312yes—
107#343C2×D312yes—
108#339—38no—
109C24×C3C2×D411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×Q814yes—
116C22×C3×C4C2×Q812yes—
117C22×C3×C4#4112yes—
118C22×C3×C4C2411yescharacteristic
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—22no—
122C22×Dic3—22no—
123C22×Dic3—42no—
124C22×Dic3—82no—
125C2×Dic6—24no—
126C2×Dic6—24no—
127C3×M16—28no—
128C3×M16—24no—
129C3×M16C2×Q818yes—
130C2×C3×C8—28no—
131C2×C3×C8—24no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C22⋊C4—24no—
135C3×C42—22no—
136C3×C42—22no—
137#217—24no—
138#217—24no—
139#217—24no—
140C4⋊Dic3—24no—
141C4⋊Dic3—24no—
142C4⋊Dic3—22no—
143C4⋊Dic3—22no—
144C4⋊Dic3—24no—
145C4⋊Dic3—24no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3C2×D414yes—
149C4⋊Dic3C2×D414yes—
150Dic3⋊C4—28no—
151Dic3⋊C4—28no—
152Dic3⋊C4—84no—
153C4×Dic3—24no—
154C4×Dic3—24no—
155C4×Dic3—22no—
156C4×Dic3C2×Q814yes—
157C23×C4—61no—
158C23×C4C22×D311yescharacteristic
159C2×M16—22no—
160C2×M16—24no—
161C2×M16Dic618yescomplemented
162C2×M16C22×D314yes—
163C22×C8—22no—
164C4⋊Q8—62no—
165C4⋊Q8—62no—
166C4⋊Q8—64no—
167C4⋊Q8—64no—
168#125—64no—
169#125—64no—
170#125—64no—
171C22⋊Q8—68no—
172C22⋊Q8—68no—
173C22⋊Q8—64no—
174C22⋊Q8—64no—
175C4×Q8—64no—
176C4×Q8—64no—
177#117—22no—
178#117—22no—
179#117—34no—
180#117—34no—
181#117—34no—
182#117—62no—
183#117—62no—
184#117—62no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C22⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4D1212yes—
198C2×C42—32no—
199C2×C42—62no—
200C2×C42D1212yes—
201#107—68no—
202#106—68no—
203#95—64no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3#12012yes—
207C23×C3C22⋊Q812yes—
208C23×C3C4⋊Q812yes—
209C23×C3C22×D411yescharacteristic
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4C22⋊Q814yes—
217C2×C3×C4C22⋊Q814yes—
218C2×C3×C4C22⋊Q812yes—
219C2×C3×C4C4⋊Q812yes—
220C2×C3×C4C4⋊Q814yes—
221C2×C3×C4C4⋊Q814yes—
222C2×C3×C4C22×D412yes—
223C2×C3×C4C22×Q812yes—
224C2×C3×C4C22×Q811yescharacteristic
225C2×C3×C4C2×#4111yescharacteristic
226C2×Dic3—24no—
227C2×Dic3—24no—
228C2×Dic3—24no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—44no—
232C2×Dic3—42no—
233C2×Dic3—82no—
234C2×Dic3—82no—
235C2×Dic3—84no—
236Dic6—84no—
237Dic6—84no—
238C3×C8—28no—
239C24C2×D1211yescharacteristic
240C2×Q8—64no—
241C2×Q8—64no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—122no—
247C22×C4—242no—
248C22×C4C2×Dic614yes—
249C22×C4C2×D1212yes—
250C22×C4C2×D1212yes—
251C22×C4C2×D1211yescharacteristic
252C22×C4D3×D412yes—
253C22×C4D3×D412yes—
254C22×C4#23712yes—
255C22×C4D3×Q812yes—
256C22×C4C23×D311yescharacteristic
257M16—28no—
258M16—24no—
259M16C2×Dic618yes—
260C2×C8—28no—
261C2×C8—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—34no—
265C4⋊C4—34no—
266C4⋊C4—68no—
267C4⋊C4—68no—
268C4⋊C4—64no—
269C4⋊C4—64no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—64no—
273C4⋊C4—64no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—244no—
277C22⋊C4—24no—
278C22⋊C4—64no—
279C22⋊C4—64no—
280C22⋊C4—64no—
281C42—22no—
282C42—22no—
283C42—34no—
284C42—64no—
285C42—64no—
286C42—62no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3#39514yes—
294C22×C3C2×#12011yescharacteristic
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3C2×C4⋊Q811yescharacteristic
297C3×C4—42no—
298C3×C4—42no—
299C3×C4#39512yes—
300C3×C4#39512yes—
301C3×C4C2×C22⋊Q812yes—
302C3×C4C2×C4⋊Q812yes—
303Dic3—44no—
304Dic3—44no—
305Dic3—44no—
306Dic3—82no—
307C23—41no—
308C23—41no—
309C23#86912yes—
310C23#88212yes—
311C23#89712yes—
312C23C22×D1211yescharacteristic
313Q8—244no—
314Q8—244no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—64no—
320C2×C4—64no—
321C2×C4—64no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—124no—
325C2×C4—122no—
326C2×C4—242no—
327C2×C4—242no—
328C2×C4—244no—
329C2×C4#86912yes—
330C2×C4#87814yes—
331C2×C4#87814yes—
332C2×C4#88211yescharacteristic
333C2×C4#88211yescharacteristic
334C2×C4#88814yes—
335C2×C4#88814yes—
336C2×C4#89712yes—
337C2×C4C22×Dic612yes—
338C2×C4C2×D3×D412yes—
339C2×C4C2×#23711yescharacteristic
340C2×C4C2×D3×Q811yescharacteristic
341C8—28no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—24no—
345C2×C3C2×#39511yescharacteristic
346C2×C3#302111yescharacteristic
347C2×C3#302211yescharacteristic
348C22—21no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—42no—
353C22—42no—
354C22#465614yes—
355C22C2×#86911yescharacteristic
356C22C2×#88211yescharacteristic
357C22C2×#89711yescharacteristic
358C4—42no—
359C4—42no—
360C4—124no—
361C4—124no—
362C4—124no—
363C4—242no—
364C4#465612yes—
365C4#465612yes—
366C4C2×#87812yes—
367C4C2×#88812yes—
368C3#2356211yescharacteristic, complemented, Sylow
369C2—21no—
370C2—24no—
371C2—21no—
372C2C2×#465611yescharacteristic
373C2#7871711yescharacteristic
374C2#7871811yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1742C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenter#1742D3
hypocenterC3#23562
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
6. #68979C2semidirect
7. #68978C2semidirect
43. C2×C3×M16Q8semidirect
58. #926Q8semidirect
161. C2×M16Dic6semidirect
368. C3#23562semidirect

Maximal subgroups

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

subgroupindex
#857252
C2×#51212
#822342
C3×#17422
#719822
#689792
#689782
#235623