Finite Group Databasebeta

← #10732475

Subgroups — #10732475

Order 768.

1498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4976C214yesmaximal
3C2×#4928C211yesmaximal, characteristic
4#71568C212yesmaximal
5#71568C212yesmaximal
6C4×C3⋊#95C211yesmaximal, characteristic
7#69420C212yesmaximal
8#69420C212yesmaximal
9#15211—31nomaximal, Sylow
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2214yes—
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2214yes—
19C22×C4×Dic3C2212yes—
20C3×#389—24no—
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27#4976—216no—
28#4928—24no—
29C2×#389—31no—
30C2×#389—34no—
31C2×#389D311yescharacteristic
32#1444—32no—
33#1444—32no—
34#1444—32no—
35#1444—32no—
36C4×#95—31no—
37C23×C3×C4C2312yes—
38C23×C3×C4C2312yes—
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3C2312yes—
43C23×Dic3C2312yes—
44C23×Dic3C2314yes—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—24no—
47C2×C3×C4⋊C4—28no—
48C2×C3×C4⋊C4—22no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4—42no—
51C2×C3×C42—42no—
52C2×C3×C42D414yes—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3—24no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—42no—
57C2×Dic3⋊C4—28no—
58C2×Dic3⋊C4—28no—
59C2×Dic3⋊C4—28no—
60C2×Dic3⋊C4—24no—
61C2×Dic3⋊C4—28no—
62C2×Dic3⋊C4—28no—
63C2×Dic3⋊C4—44no—
64C2×Dic3⋊C4—44no—
65C2×C4×Dic3—42no—
66C2×C4×Dic3—42no—
67C2×C4×Dic3D418yes—
68C3×#95—24no—
69C3⋊#95—24no—
70C3⋊#95—28no—
71C3⋊#95—28no—
72C3⋊#95Q818yes—
73C3⋊#95Q818yes—
74C22×C4⋊C4—34no—
75C22×C4⋊C4—32no—
76C22×C4⋊C4—31no—
77C22×C4⋊C4—34no—
78C22×C4⋊C4—31no—
79C22×C4⋊C4C2×D311yescharacteristic
80C22×C4⋊C4C2×D312yes—
81C22×C4⋊C4C2×D311yescharacteristic
82C22×C42—32no—
83C22×C42C2×D311yescharacteristic
84#389—24no—
85#389—616no—
86#389—64no—
87C2×#95—32no—
88C2×#95—32no—
89C2×#95—34no—
90C2×#95—34no—
91C2×#95—32no—
92C2×#95C2×D312yes—
93C24×C3C2411yescharacteristic
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—21no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—24no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—24no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D414yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×Q814yes—
115C22×C3×C4C2×Q814yes—
116C22×C3×C4#4114yes—
117C22×Dic3—24no—
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—24no—
121C22×Dic3—24no—
122C22×Dic3—22no—
123C22×Dic3—24no—
124C22×Dic3—22no—
125C22×Dic3—24no—
126C22×Dic3—22no—
127C22×Dic3—22no—
128C22×Dic3—24no—
129C22×Dic3—22no—
130C22×Dic3C2×D418yes—
131C22×Dic3C2×Q818yes—
132C22×Dic3C2×Q818yes—
133C22×Dic3#4114yes—
134C3×C4⋊C4—28no—
135C3×C4⋊C4—44no—
136C3×C4⋊C4—48no—
137C3×C42—44no—
138C4⋊Dic3—28no—
139C4⋊Dic3—44no—
140Dic3⋊C4—232no—
141Dic3⋊C4—416no—
142Dic3⋊C4—48no—
143C4×Dic3—48no—
144C23×C4—34no—
145C23×C4—32no—
146C23×C4—32no—
147C23×C4C22×D312yes—
148C23×C4C22×D312yes—
149C23×C4C22×D311yescharacteristic
150C23×C4C22×D311yescharacteristic
151C23×C4C22×D311yescharacteristic
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—68no—
159C2×C4⋊C4—68no—
160C2×C4⋊C4—64no—
161C2×C4⋊C4—68no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—68no—
164C2×C4⋊C4—68no—
165C2×C4⋊C4—64no—
166C2×C4⋊C4—62no—
167C2×C4⋊C4—124no—
168C2×C4⋊C4—124no—
169C2×C4⋊C4—122no—
170C2×C42—38no—
171C2×C42—42no—
172C2×C42—122no—
173C2×C42—122no—
174C2×C42#6714yes—
175#95—24no—
176#95—38no—
177#95—38no—
178#95—68no—
179#95—68no—
180#95—64no—
181C23×C3C22×D412yes—
182C23×C3C22×D411yescharacteristic
183C23×C3C22×Q812yes—
184C23×C3C22×Q812yes—
185C23×C3C2×#4112yes—
186C23×C3C2×#4112yes—
187C23×C3H(5,𝔽2)12yes—
188C23×C3H(5,𝔽2)11yescharacteristic
189C23×C32−1+411yescharacteristic
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—28no—
199C2×C3×C4—28no—
200C2×C3×C4—28no—
201C2×C3×C4—24no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—44no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—44no—
210C2×C3×C4#12714yes—
211C2×C3×C4C4⋊Q814yes—
212C2×C3×C4C4⋊Q814yes—
213C2×Dic3—28no—
214C2×Dic3—28no—
215C2×Dic3—28no—
216C2×Dic3—28no—
217C2×Dic3—28no—
218C2×Dic3—28no—
219C2×Dic3—28no—
220C2×Dic3—24no—
221C2×Dic3—42no—
222C2×Dic3—44no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—44no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—44no—
229C2×Dic3C4⋊Q8116yes—
230C24C23×D311yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—38no—
248C22×C4—38no—
249C22×C4—38no—
250C22×C4—34no—
251C22×C4—64no—
252C22×C4—64no—
253C22×C4—64no—
254C22×C4—64no—
255C22×C4—64no—
256C22×C4—64no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—64no—
260C22×C4—64no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4#23514yes—
265C22×C4D3×D414yes—
266C22×C4D3×Q814yes—
267C22×C4D3×Q814yes—
268C22×C4C2×#6712yes—
269C22×C4C2×#6712yes—
270C22×C4C2×#6712yes—
271C4⋊C4—28no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—632no—
275C4⋊C4—68no—
276C4⋊C4—124no—
277C4⋊C4—1216no—
278C4⋊C4—128no—
279C42—44no—
280C42—128no—
281C22×C3#52511yescharacteristic
282C22×C3#52514yes—
283C22×C3#52511yescharacteristic
284C22×C3C2×#12711yescharacteristic
285C22×C3C2×C4⋊Q814yes—
286C22×C3C2×C4⋊Q811yescharacteristic
287C22×C3C2×C4⋊Q811yescharacteristic
288C22×C3#53411yescharacteristic
289C22×C3#53412yes—
290C22×C3#53711yescharacteristic
291C22×C3#53712yes—
292C22×C3#55414yes—
293C22×C3#55412yes—
294C22×C3#55414yes—
295C22×C3#55412yes—
296C22×C3#55712yes—
297C22×C3#55712yes—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—44no—
302C3×C4—82no—
303Dic3—216no—
304Dic3—44no—
305Dic3—82no—
306C23C2×#23512yes—
307C23C2×D3×D412yes—
308C23#100412yes—
309C23C2×D3×Q812yes—
310C23C2×D3×Q812yes—
311C23D3×#4112yes—
312C23#100911yescharacteristic
313C23#101011yescharacteristic
314C23C22×#6711yescharacteristic
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—316no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—68no—
337C2×C4—68no—
338C2×C4—68no—
339C2×C4—68no—
340C2×C4—68no—
341C2×C4—68no—
342C2×C4—68no—
343C2×C4—64no—
344C2×C4—122no—
345C2×C4—124no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—124no—
351C2×C4—124no—
352C2×C4#94014yes—
353C2×C4#94414yes—
354C2×C4#94414yes—
355C2×C3#239711yescharacteristic
356C2×C3#258311yescharacteristic
357C2×C3#258311yescharacteristic
358C2×C3#258314yes—
359C2×C3#258414yes—
360C2×C3#258411yescharacteristic
361C2×C3#258411yescharacteristic
362C2×C3#265611yescharacteristic
363C2×C3#286811yescharacteristic
364C22#551114yes—
365C22#554714yes—
366C22#556712yes—
367C22#557912yes—
368C22#559712yes—
369C22#560512yes—
370C22#563614yes—
371C22#568512yes—
372C22#569012yes—
373C22D3×C4⋊Q814yes—
374C22#579111yescharacteristic
375C22C2×#94011yescharacteristic
376C22#580411yescharacteristic
377C22C2×#94411yescharacteristic
378C22C2×#94411yescharacteristic
379C22#582811yescharacteristic
380C22#583311yescharacteristic
381C4—24no—
382C4—24no—
383C4—24no—
384C4—44no—
385C4—616no—
386C4—82no—
387C4—124no—
388C4—242no—
389C3#1521111yescharacteristic, complemented, Sylow
390C2#7608014yes—
391C2#7624714yes—
392C2#8218111yescharacteristic
393C2#8234711yescharacteristic
394C2#8248311yescharacteristic
395C2#8268011yescharacteristic
396C2#8278511yescharacteristic
397C2#8304111yescharacteristic
398C2#8325311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#15211
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
389. C3#15211semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49762
C2×#49282
#715682
C4×C3⋊#952
#694202
#152113