Finite Group Databasebeta

← #10732162

Subgroups — #10732162

Order 768.

1530 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4929C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C2×#4661C212yesmaximal
4C2×#4656C212yesmaximal
5C3×C4×#95C211yesmaximal, characteristic
6#71535C214yesmaximal
7#69419C212yesmaximal
8#69419C212yesmaximal
9#15173—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C22×C3×C42C2212yes—
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2214yes—
16C22×C4⋊Dic3C2214yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26#4929—24no—
27#4928—24no—
28#4661—28no—
29#4656—28no—
30C2×#398—32no—
31C2×#395—32no—
32C2×#391—31no—
33C2×#389—31no—
34#1444—32no—
35#1444—32no—
36C4⋊#95—34no—
37C4×#95D311yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2314yes—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3C2312yes—
43C23×Dic3C2312yes—
44C23×Dic3C2312yes—
45C23×Dic3C2312yes—
46C22×Dic6—22no—
47C22×Dic6—24no—
48C22×Dic6—24no—
49C22×Dic6—42no—
50C2×C3×C42—42no—
51C2×C3×C42—42no—
52C2×C3×C42—42no—
53C2×C3×C42D414yes—
54C2×C3×C42Q818yescomplemented
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—28no—
58C2×C4⋊Dic3—28no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3—28no—
63C2×C4⋊Dic3—28no—
64C2×C4⋊Dic3—44no—
65C2×C4⋊Dic3—44no—
66C2×C4⋊Dic3—42no—
67C2×Dic3⋊C4—24no—
68C2×Dic3⋊C4—24no—
69C2×Dic3⋊C4—24no—
70C3×#95D418yes—
71C3×#95Q818yescomplemented
72C3⋊#95—28no—
73C3⋊#95—28no—
74C3⋊#95—24no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C23×Q8—31no—
78C22×C4⋊C4—32no—
79C22×C4⋊C4—31no—
80C22×C4⋊C4—31no—
81C22×C4⋊C4—34no—
82C22×C4⋊C4—34no—
83C22×C4⋊C4—31no—
84C22×C42C2×D312yes—
85C22×C42C2×D311yescharacteristic
86#398—68no—
87#395—68no—
88#391—64no—
89#389—64no—
90C2×#95—34no—
91C2×#95—34no—
92C2×#95—32no—
93C2×#95—32no—
94C2×#95—32no—
95C2×#95C2×D312yes—
96C2×#95C2×D312yes—
97C24×C3C2411yescharacteristic
98C22×C3×C4—22no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D414yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D418yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×Q814yes—
115C22×C3×C4C2×Q818yes—
116C22×C3×C4C2×Q818yes—
117C22×C3×C4C2×Q814yes—
118C22×Dic3—22no—
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—24no—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—24no—
127C22×Dic3—22no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—22no—
131C22×Dic3—24no—
132C22×Dic3—22no—
133C22×Dic3—24no—
134C22×Dic3—24no—
135C22×Dic3—24no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3#4114yes—
139C22×Dic3#4114yes—
140C2×Dic6—28no—
141C2×Dic6—416no—
142C2×Dic6—44no—
143C2×Dic6—44no—
144C3×C42—44no—
145C3×C42—48no—
146C4⋊Dic3—28no—
147C4⋊Dic3—416no—
148C4⋊Dic3—416no—
149C4⋊Dic3—44no—
150C4⋊Dic3—44no—
151Dic3⋊C4—48no—
152C22×Q8—64no—
153C22×Q8—62no—
154C22×Q8—64no—
155C22×Q8—122no—
156C23×C4—32no—
157C23×C4—32no—
158C23×C4—32no—
159C23×C4—32no—
160C23×C4C22×D311yescharacteristic
161C23×C4C22×D314yes—
162C23×C4C22×D311yescharacteristic
163C23×C4C22×D311yescharacteristic
164C2×C4⋊C4—64no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4—68no—
167C2×C4⋊C4—68no—
168C2×C4⋊C4—64no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—64no—
172C2×C4⋊C4—68no—
173C2×C4⋊C4—64no—
174C2×C4⋊C4—68no—
175C2×C4⋊C4—64no—
176C2×C4⋊C4—124no—
177C2×C4⋊C4—124no—
178C2×C4⋊C4—122no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42Dic618yescomplemented
183C2×C42D1214yes—
184#95—64no—
185#95—64no—
186#95—64no—
187#95—68no—
188#95—68no—
189#95Dic618yescomplemented
190#95D1218yes—
191C23×C3C22×D411yescharacteristic
192C23×C3C22×D412yes—
193C23×C3C22×Q812yes—
194C23×C3C22×Q812yes—
195C23×C3C2×#4112yes—
196C23×C3C2×#4112yes—
197C23×C3H(5,𝔽2)12yes—
198C23×C3H(5,𝔽2)11yescharacteristic
199C23×C32−1+411yescharacteristic
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—28no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—24no—
208C2×C3×C4—28no—
209C2×C3×C4—24no—
210C2×C3×C4—28no—
211C2×C3×C4—28no—
212C2×C3×C4—28no—
213C2×C3×C4—28no—
214C2×C3×C4—28no—
215C2×C3×C4—28no—
216C2×C3×C4#12714yes—
217C2×C3×C4C4⋊Q814yes—
218C2×C3×C4C4⋊Q814yes—
219C2×C3×C4C4⋊Q8116yes—
220C2×Dic3—24no—
221C2×Dic3—24no—
222C2×Dic3—24no—
223C2×Dic3—28no—
224C2×Dic3—24no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—44no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—44no—
241C2×Dic3—44no—
242Dic6—416no—
243C24C23×D311yescharacteristic
244C2×Q8—68no—
245C2×Q8—1216no—
246C2×Q8—124no—
247C2×Q8—124no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—34no—
260C22×C4—34no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—64no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—64no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—64no—
275C22×C4—62no—
276C22×C4—64no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—64no—
280C22×C4—64no—
281C22×C4C2×Dic618yes—
282C22×C4C2×Dic614yes—
283C22×C4C2×Dic614yes—
284C22×C4C2×Dic618yes—
285C22×C4C2×D1212yes—
286C22×C4C2×D1218yes—
287C22×C4C2×D1212yes—
288C22×C4C2×D1214yes—
289C22×C4C2×D1212yes—
290C4⋊C4—68no—
291C4⋊C4—1216no—
292C4⋊C4—1216no—
293C4⋊C4—124no—
294C4⋊C4—128no—
295C4⋊C4—124no—
296C42—48no—
297C42—44no—
298C22×C3#52511yescharacteristic
299C22×C3#52511yescharacteristic
300C22×C3C2×#12711yescharacteristic
301C22×C3C2×C4⋊Q811yescharacteristic
302C22×C3C2×C4⋊Q811yescharacteristic
303C22×C3C2×C4⋊Q814yes—
304C22×C3#53314yes—
305C22×C3#53411yescharacteristic
306C22×C3#53711yescharacteristic
307C22×C3#54312yes—
308C22×C3#54412yes—
309C22×C3#54614yes—
310C22×C3#54612yes—
311C22×C3#54812yes—
312C22×C3#55414yes—
313C22×C3#55412yes—
314C22×C3#55712yes—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—216no—
319Dic3—44no—
320Dic3—44no—
321Dic3—82no—
322Dic3—82no—
323C23C22×Dic612yes—
324C23C22×Dic612yes—
325C23C22×D1211yescharacteristic
326C23C22×D1212yes—
327C23#100411yescharacteristic
328C23#100412yes—
329C23#100711yescharacteristic
330C23D3×#4112yes—
331C23D3×#4112yes—
332Q8—1216no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—64no—
350C2×C4—64no—
351C2×C4—64no—
352C2×C4—68no—
353C2×C4—64no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—124no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—124no—
368C2×C4—124no—
369C2×C4—124no—
370C2×C4—124no—
371C2×C4#869116yes—
372C2×C4#86914yes—
373C2×C4#86914yes—
374C2×C4#87414yes—
375C2×C3#239911yescharacteristic
376C2×C3#253911yescharacteristic
377C2×C3#254011yescharacteristic
378C2×C3#254914yes—
379C2×C3#255014yes—
380C2×C3#258311yescharacteristic
381C2×C3#258411yescharacteristic
382C2×C3#265911yescharacteristic
383C2×C3#286811yescharacteristic
384C22C2×#86914yes—
385C22C2×#86911yescharacteristic
386C22C2×#86911yescharacteristic
387C22C2×#87411yescharacteristic
388C22#548312yes—
389C22#549311yescharacteristic
390C22#550012yes—
391C22#551011yescharacteristic
392C22#551714yes—
393C22#552411yescharacteristic
394C22#552411yescharacteristic
395C22#553912yes—
396C22#553914yes—
397C22#555412yes—
398C22#555414yes—
399C22#556912yes—
400C22#557612yes—
401C4—216no—
402C4—24no—
403C4—24no—
404C4—24no—
405C4—124no—
406C4—124no—
407C4—242no—
408C4—242no—
409C3#1517311yescharacteristic, complemented, Sylow
410C2#7331111yescharacteristic
411C2#7336111yescharacteristic
412C2#7339711yescharacteristic
413C2#7353414yes—
414C2#7354711yescharacteristic
415C2#7359711yescharacteristic
416C2#7371414yes—
417C2#7372711yescharacteristic
418C2#7379811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×C4×#95C2
hypercenterC4×#95D3
hypocenterC3#15173
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
54. C2×C3×C42Q8semidirect
71. C3×#95Q8semidirect
182. C2×C42Dic6semidirect
189. #95Dic6semidirect
409. C3#15173semidirect

Maximal subgroups

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

subgroupindex
C2×#49292
C2×#49282
C2×#46612
C2×#46562
C3×C4×#952
#715352
#694192
#151733