Finite Group Databasebeta

← #10728688

Subgroups — #10728688

Order 768.

1718 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic6C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C2×#4661C212yesmaximal
4C2×#4656C212yesmaximal
5C3×#1434C211yesmaximal, characteristic
6#69410C212yesmaximal
7#69410C212yesmaximal
8#69407C214yesmaximal
9#14880—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3C2214yes—
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2212yes—
19C2×C4×Dic6—24no—
20C2×C4×Dic6—28no—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28#4928—24no—
29#4661—28no—
30#4656—28no—
31C22×C4×Q8—31no—
32C2×#398—32no—
33C2×#395—32no—
34C2×#389—31no—
35#1441—32no—
36#1441—32no—
37#1436—34no—
38#1434D311yescharacteristic
39C23×C3×C4C2314yes—
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2312yes—
44C23×Dic3C2312yes—
45C23×Dic3C2312yes—
46C23×Dic3C2312yes—
47C22×Dic6—22no—
48C22×Dic6—22no—
49C22×Dic6—24no—
50C22×Dic6—24no—
51C22×Dic6D414yes—
52C2×C3×C42—24no—
53C2×C3×C42D414yes—
54C2×C4⋊Dic3—22no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—24no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3—28no—
61C2×C4⋊Dic3—28no—
62C2×C4⋊Dic3—28no—
63C2×Dic3⋊C4—24no—
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4—24no—
66C2×Dic3⋊C4—24no—
67C2×Dic3⋊C4—24no—
68C2×Dic3⋊C4D418yes—
69C2×C4×Dic3—24no—
70C2×C4×Dic3—24no—
71C2×C4×Dic3—24no—
72C4×Dic6—232no—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95Q818yescomplemented
76C3⋊#95—24no—
77C3⋊#95—28no—
78C3⋊#95—28no—
79C3⋊#95—28no—
80C23×Q8—31no—
81C2×C4×Q8—68no—
82C2×C4×Q8—64no—
83C22×C4⋊C4—32no—
84C22×C4⋊C4—34no—
85C22×C4⋊C4—32no—
86C22×C4⋊C4—31no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—31no—
89C22×C42—32no—
90C22×C42C2×D311yescharacteristic
91#398—68no—
92#395—68no—
93#389—64no—
94C2×#95—34no—
95C2×#95—34no—
96C2×#95—32no—
97C2×#95—34no—
98C2×#95C2×D312yes—
99C2×#95C2×D312yes—
100C2×#95C2×D312yes—
101C24×C3C2411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—24no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×D414yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×Q818yes—
120C22×C3×C4C2×Q814yes—
121C22×C3×C4#4112yes—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—22no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—24no—
131C22×Dic3—22no—
132C22×Dic3—22no—
133C22×Dic3—24no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—24no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—24no—
141C22×Dic3C2×D414yes—
142C22×Dic3C2×D414yes—
143C22×Dic3C2×D414yes—
144C22×Dic3#4114yes—
145C22×Dic3#4114yes—
146C2×Dic6—28no—
147C2×Dic6—216no—
148C2×Dic6—28no—
149C2×Dic6—28no—
150C2×Dic6—216no—
151C3×C42—28no—
152C4⋊Dic3—28no—
153C4⋊Dic3—28no—
154C4⋊Dic3—44no—
155C4⋊Dic3—416no—
156Dic3⋊C4—216no—
157Dic3⋊C4—48no—
158C4×Dic3—216no—
159C22×Q8—34no—
160C22×Q8—64no—
161C22×Q8—64no—
162C22×Q8—62no—
163C22×Q8—62no—
164C23×C4—32no—
165C23×C4—32no—
166C23×C4—32no—
167C23×C4—32no—
168C23×C4C22×D314yes—
169C23×C4C22×D311yescharacteristic
170C23×C4C22×D311yescharacteristic
171C23×C4C22×D311yescharacteristic
172C4×Q8—632no—
173C2×C4⋊C4—38no—
174C2×C4⋊C4—64no—
175C2×C4⋊C4—64no—
176C2×C4⋊C4—64no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—68no—
181C2×C4⋊C4—68no—
182C2×C4⋊C4—68no—
183C2×C4⋊C4—64no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—64no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4—64no—
188C2×C42—24no—
189C2×C42—64no—
190C2×C42—64no—
191C2×C42—64no—
192C2×C42D1214yes—
193#95—24no—
194#95—24no—
195#95—68no—
196#95—68no—
197#95—68no—
198#95—64no—
199#95Dic618yescomplemented
200C23×C3C22×D411yescharacteristic
201C23×C3C22×D412yes—
202C23×C3C22×D411yescharacteristic
203C23×C3C22×Q812yes—
204C23×C3C2×#4111yescharacteristic
205C23×C3C2×#4112yes—
206C23×C3C2×#4112yes—
207C23×C32−1+412yes—
208C23×C32−1+412yes—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—24no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—28no—
218C2×C3×C4—28no—
219C2×C3×C4—28no—
220C2×C3×C4—28no—
221C2×C3×C4—28no—
222C2×C3×C4—44no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—44no—
226C2×C3×C4#12014yes—
227C2×C3×C4#12114yes—
228C2×C3×C4C4⋊Q814yes—
229C2×Dic3—24no—
230C2×Dic3—24no—
231C2×Dic3—24no—
232C2×Dic3—24no—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—24no—
237C2×Dic3—28no—
238C2×Dic3—24no—
239C2×Dic3—24no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—44no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—44no—
250C2×Dic3—42no—
251C2×Dic3—44no—
252C2×Dic3#12118yes—
253Dic6—232no—
254C24C23×D311yescharacteristic
255C2×Q8—616no—
256C2×Q8—68no—
257C2×Q8—68no—
258C2×Q8—616no—
259C2×Q8—68no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—34no—
273C22×C4—34no—
274C22×C4—34no—
275C22×C4—34no—
276C22×C4—34no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—64no—
287C22×C4—62no—
288C22×C4—64no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—64no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—64no—
296C22×C4C2×Dic618yes—
297C22×C4C2×Dic614yes—
298C22×C4C2×D1212yes—
299C22×C4C2×D1212yes—
300C22×C4C2×D1212yes—
301C22×C4D3×D414yes—
302C22×C4D3×D412yes—
303C22×C4#23912yes—
304C4⋊C4—68no—
305C4⋊C4—68no—
306C4⋊C4—616no—
307C4⋊C4—1216no—
308C4⋊C4—124no—
309C4⋊C4—128no—
310C42—28no—
311C42—616no—
312C22×C3C2×#12011yescharacteristic
313C22×C3C2×#12111yescharacteristic
314C22×C3C2×#12112yes—
315C22×C3#52511yescharacteristic
316C22×C3#52511yescharacteristic
317C22×C3#52512yes—
318C22×C3C2×C4⋊Q811yescharacteristic
319C22×C3#53314yes—
320C22×C3#53611yescharacteristic
321C22×C3#53611yescharacteristic
322C22×C3#54114yes—
323C22×C3#54714yes—
324C22×C3#54812yes—
325C22×C3D4×Q812yes—
326C22×C3D4×Q814yes—
327C22×C3#55212yes—
328C22×C3#55712yes—
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—48no—
333Dic3—28no—
334Dic3—44no—
335Dic3—44no—
336Dic3—82no—
337C23C22×Dic612yes—
338C23C22×D1211yescharacteristic
339C23C2×D3×D412yes—
340C23C2×D3×D411yescharacteristic
341C23C2×#23911yescharacteristic
342C23D3×#4112yes—
343C23D3×#4112yes—
344C23#101012yes—
345C23#101012yes—
346Q8—632no—
347C2×C4—28no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—38no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—64no—
366C2×C4—64no—
367C2×C4—64no—
368C2×C4—64no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—68no—
372C2×C4—64no—
373C2×C4—64no—
374C2×C4—64no—
375C2×C4—64no—
376C2×C4—124no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4#86914yes—
389C2×C4#88214yes—
390C2×C4#89514yes—
391C2×C3#239911yescharacteristic
392C2×C3#240611yescharacteristic
393C2×C3#242914yes—
394C2×C3#243411yescharacteristic
395C2×C3#246214yes—
396C2×C3#253611yescharacteristic
397C2×C3#254611yescharacteristic
398C2×C3#263411yescharacteristic
399C2×C3#284711yescharacteristic
400C22C2×#86911yescharacteristic
401C22C2×#88211yescharacteristic
402C22C2×#89511yescharacteristic
403C22#551714yes—
404C22#552411yescharacteristic
405C22#552411yescharacteristic
406C22#552611yescharacteristic
407C22#552611yescharacteristic
408C22D4×Dic614yes—
409C22D4×Dic612yes—
410C22#557012yes—
411C22D3×#12112yes—
412C22#562712yes—
413C22#565514yes—
414C22#567712yes—
415C22#569412yes—
416C22#570114yes—
417C4—24no—
418C4—24no—
419C4—24no—
420C4—48no—
421C4—68no—
422C4—124no—
423C4—124no—
424C4—242no—
425C3#1488011yescharacteristic, complemented, Sylow
426C2#7339711yescharacteristic
427C2#7349011yescharacteristic
428C2#7371111yescharacteristic
429C2#7404811yescharacteristic
430C2#7406311yescharacteristic
431C2#7424011yescharacteristic
432C2#7427811yescharacteristic
433C2#7499814yes—
434C2#7550014yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1434C2
hypercenter#1434D3
hypocenterC3#14880
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
75. C3×#95Q8semidirect
199. #95Dic6semidirect
425. C3#14880semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic62
C2×#49282
C2×#46612
C2×#46562
C3×#14342
#694102
#694072
#148803