Finite Group Databasebeta

← #10750426

Subgroups — #10750426

Order 768.

1638 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic
2C2×#4933C211yesmaximal, characteristic
3C2×#4661C212yesmaximal
4C2×#4656C212yesmaximal
5C3×#1434C211yesmaximal, characteristic
6#69424C212yesmaximal
7#69423C212yesmaximal
8#69419C212yesmaximal
9#69412C212yesmaximal
10#16507—31nomaximal, Sylow
11C23×Dic6C2212yes—
12C22×C3×C42C2211yescharacteristic
13C22×C4⋊Dic3C2212yes—
14C22×C4⋊Dic3C2214yes—
15C22×C4⋊Dic3C2212yes—
16C22×Dic3⋊C4C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C2×#869—24no—
19C2×#869—28no—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2214yes—
30#4933—24no—
31#4661—28no—
32#4656—28no—
33C22×C4⋊Q8—31no—
34C2×#398—32no—
35C2×#395—32no—
36C2×#387—31no—
37#1446—32no—
38#1445—32no—
39#1444—32no—
40#1442—32no—
41#1434D311yescharacteristic
42C23×C3×C4C2314yes—
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3C2311yescharacteristic
47C23×Dic3C2311yescharacteristic
48C23×Dic3C2314yes—
49C23×Dic3C2311yescharacteristic
50C23×Dic3C2311yescharacteristic
51C22×Dic6—24no—
52C22×Dic6—28no—
53C22×Dic6—24no—
54C22×Dic6—24no—
55C22×Dic6—44no—
56C2×C3×C42—24no—
57C2×C3×C42D414yes—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3—28no—
62C2×C4⋊Dic3—28no—
63C2×C4⋊Dic3—28no—
64C2×C4⋊Dic3—44no—
65C2×C4⋊Dic3D418yes—
66C2×Dic3⋊C4—22no—
67C2×Dic3⋊C4—22no—
68C2×Dic3⋊C4—24no—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4D414yes—
71#869—232no—
72C3×#95—24no—
73C3×#95—24no—
74C3×#95Q818yescomplemented
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—28no—
80C3⋊#95—28no—
81C3⋊#95—44no—
82C23×Q8—32no—
83C2×C4⋊Q8—68no—
84C2×C4⋊Q8—64no—
85C22×C4⋊C4—31no—
86C22×C4⋊C4—32no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—32no—
89C22×C4⋊C4—34no—
90C22×C42C2×D311yescharacteristic
91#398—68no—
92#395—68no—
93#387—64no—
94C2×#95—34no—
95C2×#95—34no—
96C2×#95—31no—
97C2×#95—31no—
98C2×#95—31no—
99C2×#95—31no—
100C2×#95—34no—
101C2×#95C2×D312yes—
102C2×#95C2×D312yes—
103C2×#95C2×D312yes—
104C24×C3C2411yescharacteristic
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D414yes—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×Q818yes—
123C22×C3×C4C2×Q814yes—
124C22×C3×C4#4112yes—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—24no—
128C22×Dic3—22no—
129C22×Dic3—24no—
130C22×Dic3—24no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—24no—
134C22×Dic3—24no—
135C22×Dic3—22no—
136C22×Dic3—24no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3C2×D412yes—
148C22×Dic3C2×D418yes—
149C22×Dic3C2×D412yes—
150C2×Dic6—216no—
151C2×Dic6—416no—
152C2×Dic6—416no—
153C2×Dic6—48no—
154C2×Dic6—48no—
155C3×C42—28no—
156C4⋊Dic3—216no—
157C4⋊Dic3—216no—
158C4⋊Dic3—416no—
159Dic3⋊C4—28no—
160Dic3⋊C4—44no—
161C22×Q8—68no—
162C22×Q8—64no—
163C22×Q8—64no—
164C22×Q8—64no—
165C22×Q8—124no—
166C23×C4—31no—
167C23×C4—34no—
168C23×C4—31no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4C22×D314yes—
172C23×C4C22×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C23×C4C22×D311yescharacteristic
175C4⋊Q8—632no—
176C2×C4⋊C4—38no—
177C2×C4⋊C4—34no—
178C2×C4⋊C4—68no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—68no—
185C2×C4⋊C4—68no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—64no—
188C2×C4⋊C4—124no—
189C2×C42—24no—
190C2×C42D1214yes—
191#95—24no—
192#95—24no—
193#95—68no—
194#95—68no—
195#95—62no—
196#95—62no—
197#95—62no—
198#95—62no—
199#95—124no—
200#95Dic618yescomplemented
201C23×C3C22×D411yescharacteristic
202C23×C3C22×D412yes—
203C23×C3C22×D411yescharacteristic
204C23×C3C22×Q812yes—
205C23×C3C2×#4111yescharacteristic
206C23×C3H(5,𝔽2)12yes—
207C23×C32−1+412yes—
208C23×C32−1+412yes—
209C23×C32−1+412yes—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—28no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—28no—
220C2×C3×C4—28no—
221C2×C3×C4—28no—
222C2×C3×C4—28no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—44no—
226C2×C3×C4—44no—
227C2×C3×C4#12014yes—
228C2×C3×C4#12114yes—
229C2×C3×C4C4⋊Q814yes—
230C2×Dic3—28no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—28no—
236C2×Dic3—28no—
237C2×Dic3—41no—
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—44no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—44no—
255Dic6—432no—
256C24C23×D311yescharacteristic
257C2×Q8—616no—
258C2×Q8—1216no—
259C2×Q8—1216no—
260C2×Q8—128no—
261C2×Q8—128no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—32no—
275C22×C4—38no—
276C22×C4—32no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—64no—
280C22×C4—64no—
281C22×C4—62no—
282C22×C4—64no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—64no—
286C22×C4—64no—
287C22×C4—62no—
288C22×C4—64no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4C2×Dic618yes—
300C22×C4C2×Dic614yes—
301C22×C4C2×D1212yes—
302C22×C4C2×D1212yes—
303C22×C4C2×D1212yes—
304C22×C4D3×D414yes—
305C22×C4D3×D412yes—
306C22×C4#23912yes—
307C4⋊C4—616no—
308C4⋊C4—616no—
309C4⋊C4—68no—
310C4⋊C4—1216no—
311C4⋊C4—124no—
312C42—28no—
313C22×C3C2×#12011yescharacteristic
314C22×C3C2×#12111yescharacteristic
315C22×C3C2×C4⋊Q811yescharacteristic
316C22×C3#53511yescharacteristic
317C22×C3#53614yes—
318C22×C3#53612yes—
319C22×C3#53611yescharacteristic
320C22×C3#53611yescharacteristic
321C22×C3#53611yescharacteristic
322C22×C3#54112yes—
323C22×C3#54414yes—
324C22×C3D4212yes—
325C22×C3#54612yes—
326C22×C3D4×Q812yes—
327C22×C3D4×Q814yes—
328C22×C3#55712yes—
329C22×C3#56314yes—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—48no—
334Dic3—42no—
335Dic3—48no—
336Dic3—42no—
337Dic3—81no—
338Dic3—81no—
339C23C22×Dic612yes—
340C23C22×D1211yescharacteristic
341C23C2×D3×D412yes—
342C23C2×D3×D411yescharacteristic
343C23C2×#23911yescharacteristic
344C23#100912yes—
345C23#101012yes—
346C23#101012yes—
347C23#101012yes—
348Q8—1232no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—64no—
367C2×C4—68no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—64no—
371C2×C4—68no—
372C2×C4—68no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—124no—
376C2×C4—124no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—122no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4#86914yes—
392C2×C4#88214yes—
393C2×C4#89514yes—
394C2×C3#240811yescharacteristic
395C2×C3#243411yescharacteristic
396C2×C3#254611yescharacteristic
397C2×C3#261411yescharacteristic
398C2×C3#262111yescharacteristic
399C2×C3#266011yescharacteristic
400C2×C3#267714yes—
401C2×C3#271314yes—
402C2×C3#282111yescharacteristic
403C22C2×#86911yescharacteristic
404C22C2×#88211yescharacteristic
405C22C2×#89511yescharacteristic
406C22#551914yes—
407C22#552511yescharacteristic
408C22#552611yescharacteristic
409C22#552611yescharacteristic
410C22#552611yescharacteristic
411C22D4×Dic614yes—
412C22D4×Dic612yes—
413C22#557012yes—
414C22#560912yes—
415C22#563012yes—
416C22#564814yes—
417C22#567512yes—
418C22#569812yes—
419C22#571214yes—
420C4—24no—
421C4—24no—
422C4—24no—
423C4—48no—
424C4—128no—
425C4—122no—
426C4—122no—
427C4—241no—
428C4—241no—
429C3#1650711yescharacteristic, complemented, Sylow
430C2#7341011yescharacteristic
431C2#7349011yescharacteristic
432C2#7371111yescharacteristic
433C2#7401611yescharacteristic
434C2#7413511yescharacteristic
435C2#7419311yescharacteristic
436C2#7429111yescharacteristic
437C2#7515814yes—
438C2#7566714yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1434C2
hypercenter#1434D3
hypocenterC3#16507
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
74. C3×#95Q8semidirect
200. #95Dic6semidirect
429. C3#16507semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
C2×#49332
C2×#46612
C2×#46562
C3×#14342
#694242
#694232
#694192
#694122
#165073