Finite Group Databasebeta

← #10751761

Subgroups — #10751761

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#870C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic
3C2×#4983C212yesmaximal
4#71568C214yesmaximal
5#71566C212yesmaximal
6#71534C211yesmaximal, characteristic
7#69420C212yesmaximal
8#69415C212yesmaximal
9#16607—31nomaximal, Sylow
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2214yes—
19C2×#870—24no—
20C2×#870—28no—
21C3×#389—24no—
22C2×C3×#95C2212yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29#4983—28no—
30C22×#125—31no—
31C2×#398—32no—
32C2×#389D311yescharacteristic
33#1444—32no—
34#1444—34no—
35#1442—32no—
36#1441—32no—
37#1434—31no—
38C23×C3×C4C2312yes—
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2312yes—
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2312yes—
44C23×Dic3C2312yes—
45C23×Dic3C2314yes—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—24no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—44no—
52C2×C3×C42—24no—
53C2×C3×C42D414yes—
54C2×C4⋊Dic3—22no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—24no—
58C2×Dic3⋊C4—24no—
59C2×Dic3⋊C4—24no—
60C2×Dic3⋊C4—24no—
61C2×Dic3⋊C4—24no—
62C2×Dic3⋊C4—24no—
63C2×Dic3⋊C4—28no—
64C2×Dic3⋊C4—44no—
65C2×Dic3⋊C4—44no—
66C2×Dic3⋊C4D418yes—
67#870—232no—
68C3×#95—24no—
69C3⋊#95—24no—
70C3⋊#95—28no—
71C3⋊#95—28no—
72C3⋊#95—44no—
73C3⋊#95—42no—
74C3⋊#95Q818yes—
75C2×#125—68no—
76C2×#125—64no—
77C22×C4⋊C4—32no—
78C22×C4⋊C4—32no—
79C22×C4⋊C4—31no—
80C22×C4⋊C4—31no—
81C22×C4⋊C4—34no—
82C22×C4⋊C4C2×D311yescharacteristic
83C22×C4⋊C4C2×D311yescharacteristic
84C22×C4⋊C4C2×D312yes—
85C22×C42C2×D311yescharacteristic
86#398—68no—
87#389—24no—
88C2×#95—32no—
89C2×#95—32no—
90C2×#95—34no—
91C2×#95—34no—
92C2×#95—32no—
93C2×#95—34no—
94C2×#95C2×D312yes—
95C24×C3C2411yescharacteristic
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—24no—
108C22×C3×C4—24no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—22no—
113C22×C3×C4C2×D414yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×Q814yes—
118C22×C3×C4#4114yes—
119C22×Dic3—24no—
120C22×Dic3—22no—
121C22×Dic3—22no—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—24no—
125C22×Dic3—24no—
126C22×Dic3—22no—
127C22×Dic3—24no—
128C22×Dic3—24no—
129C22×Dic3—22no—
130C22×Dic3—22no—
131C22×Dic3—24no—
132C22×Dic3—24no—
133C22×Dic3—22no—
134C22×Dic3—24no—
135C22×Dic3C2×D414yes—
136C22×Dic3C2×D414yes—
137C22×Dic3C2×Q818yes—
138C3×C4⋊C4—28no—
139C3×C4⋊C4—48no—
140C3×C4⋊C4—44no—
141C3×C42—28no—
142C4⋊Dic3—28no—
143C4⋊Dic3—28no—
144Dic3⋊C4—216no—
145Dic3⋊C4—216no—
146Dic3⋊C4—416no—
147C23×C4—32no—
148C23×C4—32no—
149C23×C4—34no—
150C23×C4C22×D312yes—
151C23×C4C22×D312yes—
152C23×C4C22×D311yescharacteristic
153C23×C4C22×D311yescharacteristic
154C23×C4C22×D311yescharacteristic
155#125—632no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—38no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—64no—
164C2×C4⋊C4—64no—
165C2×C4⋊C4—64no—
166C2×C4⋊C4—64no—
167C2×C4⋊C4—64no—
168C2×C4⋊C4—64no—
169C2×C4⋊C4—64no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4—68no—
173C2×C4⋊C4—124no—
174C2×C4⋊C4—124no—
175C2×C42—24no—
176C2×C42#6714yes—
177#95—24no—
178#95—38no—
179#95—68no—
180#95—68no—
181#95—64no—
182#95—124no—
183#95—122no—
184C23×C3C22×D412yes—
185C23×C3C22×D411yescharacteristic
186C23×C3C22×Q812yes—
187C23×C3C2×#4112yes—
188C23×C3H(5,𝔽2)12yes—
189C23×C3H(5,𝔽2)11yescharacteristic
190C23×C32−1+412yes—
191C23×C32−1+412yes—
192C23×C32−1+411yescharacteristic
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—28no—
198C2×C3×C4—28no—
199C2×C3×C4—22no—
200C2×C3×C4—24no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—28no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—44no—
214C2×C3×C4#12014yes—
215C2×C3×C4C4⋊Q814yes—
216C2×Dic3—24no—
217C2×Dic3—24no—
218C2×Dic3—24no—
219C2×Dic3—24no—
220C2×Dic3—28no—
221C2×Dic3—28no—
222C2×Dic3—28no—
223C2×Dic3—28no—
224C2×Dic3—42no—
225C2×Dic3—44no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—44no—
232C2×Dic3—44no—
233C2×Dic3—44no—
234C2×Dic3—44no—
235C24C23×D311yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—34no—
254C22×C4—34no—
255C22×C4—38no—
256C22×C4—64no—
257C22×C4—64no—
258C22×C4—64no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—64no—
263C22×C4—62no—
264C22×C4—64no—
265C22×C4—64no—
266C22×C4—64no—
267C22×C4—62no—
268C22×C4—64no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4#23514yes—
273C22×C4D3×D414yes—
274C22×C4D3×Q814yes—
275C22×C4C2×#6712yes—
276C22×C4C2×#6712yes—
277C22×C4C2×#6712yes—
278C4⋊C4—28no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—616no—
282C4⋊C4—68no—
283C4⋊C4—68no—
284C4⋊C4—616no—
285C4⋊C4—1216no—
286C42—28no—
287C22×C3C2×#12011yescharacteristic
288C22×C3#52511yescharacteristic
289C22×C3C2×C4⋊Q811yescharacteristic
290C22×C3#53511yescharacteristic
291C22×C3#53512yes—
292C22×C3#53611yescharacteristic
293C22×C3#53611yescharacteristic
294C22×C3#53711yescharacteristic
295C22×C3#53712yes—
296C22×C3#53914yes—
297C22×C3#54414yes—
298C22×C3#54714yes—
299C22×C3D4×Q814yes—
300C22×C3#55412yes—
301C22×C3#55712yes—
302C22×C3#56312yes—
303C22×C3#56412yes—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—42no—
307C3×C4—44no—
308C3×C4—82no—
309Dic3—44no—
310Dic3—44no—
311Dic3—48no—
312C23C2×#23512yes—
313C23C2×D3×D412yes—
314C23#100412yes—
315C23C2×D3×Q812yes—
316C23#100712yes—
317C23#100911yescharacteristic
318C23#101012yes—
319C23#101011yescharacteristic
320C23C22×#6711yescharacteristic
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—68no—
343C2×C4—68no—
344C2×C4—68no—
345C2×C4—64no—
346C2×C4—64no—
347C2×C4—68no—
348C2×C4—64no—
349C2×C4—64no—
350C2×C4—124no—
351C2×C4—124no—
352C2×C4—124no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—124no—
357C2×C4—122no—
358C2×C4—124no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4#93714yes—
362C2×C4#94414yes—
363C2×C3#240611yescharacteristic
364C2×C3#243411yescharacteristic
365C2×C3#258411yescharacteristic
366C2×C3#261511yescharacteristic
367C2×C3#263311yescharacteristic
368C2×C3#266011yescharacteristic
369C2×C3#272014yes—
370C2×C3#272614yes—
371C2×C3#282211yescharacteristic
372C22#550714yes—
373C22#556414yes—
374C22#556712yes—
375C22#557912yes—
376C22#560712yes—
377C22#560812yes—
378C22#563214yes—
379C22C3⋊#56412yes—
380C22#569912yes—
381C22#572914yes—
382C22#579111yescharacteristic
383C22C2×#93711yescharacteristic
384C22#580511yescharacteristic
385C22C2×#94411yescharacteristic
386C22#581611yescharacteristic
387C22#582511yescharacteristic
388C22#583311yescharacteristic
389C4—24no—
390C4—24no—
391C4—42no—
392C4—44no—
393C4—82no—
394C4—124no—
395C4—124no—
396C4—128no—
397C3#1660711yescharacteristic, complemented, Sylow
398C2#7611914yes—
399C2#7612614yes—
400C2#8219211yescharacteristic
401C2#8239111yescharacteristic
402C2#8248311yescharacteristic
403C2#8270911yescharacteristic
404C2#8276011yescharacteristic
405C2#8305511yescharacteristic
406C2#8324611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#16607
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
397. C3#16607semidirect

Maximal subgroups

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

subgroupindex
C22×#8702
C2×C3×#3892
C2×#49832
#715682
#715662
#715342
#694202
#694152
#166073