Finite Group Databasebeta

← #10848782

Subgroups — #10848782

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5697C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic, complemented
3C2×#4693C211yesmaximal, characteristic, complemented
4#74193C211yesmaximal, characteristic, complemented
5C3×#1991C211yesmaximal, characteristic, complemented
6#69753C211yesmaximal, characteristic, complemented
7#69553C211yesmaximal, characteristic, complemented
8#68896C214yesmaximal, complemented
9#68887C214yesmaximal, complemented
10#25606—31nomaximal, Sylow
11C2×C3×#125C2211yescharacteristic
12C22×#204C2214yes—
13#5697—24no—
14#5697—24no—
15C2×#897C2212yes—
16C2×#895—22no—
17C2×#895C2212yes—
18C2×#894—22no—
19C2×#893C2212yes—
20C2×#874C2211yescharacteristic
21C2×#872C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×(C2×D3)⋊C8C2212yes—
26#5069—24no—
27#4973C2212yes—
28C2×#810C2214yes—
29C2×#809C2214yes—
30C2×#803C2211yescharacteristic
31#4693—24no—
32#4674—22no—
33#4661—22no—
34C3×#340C2214yes—
35#4550C2212yes—
36C2×#540—31no—
37C2×#492—31no—
38C2×#491—31no—
39#2621—31no—
40#2000—34no—
41#1991D311yescharacteristic, complemented
42#1961—31no—
43#1877—34no—
44#1750—31no—
45C22×D12—22no—
46C22×D12—22no—
47C22×D12C2312yes—
48C22×C4×D3—22no—
49C22×C4×D3C2311yescharacteristic
50C22×Dic6C2312yes—
51C22×C3×C8C2312yes—
52C3×#125—22no—
53C3×#125D414yes—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4C2312yes—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C42C2311yescharacteristic
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4D412yes—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3—22no—
64C2×Dic3⋊C4D412yes—
65C2×C4×Dic3C2311yescharacteristic
66C22×C3⋊C8C2312yes—
67C2×#204—28no—
68C2×#204—28no—
69C2×#204—44no—
70#897—44no—
71#895—216no—
72#895—24no—
73#895—24no—
74#894—216no—
75#893—24no—
76#893—24no—
77#874—22no—
78#874D414yes—
79#872—22no—
80#872—22no—
81#869—22no—
82#869—22no—
83C3×C4×C8—22no—
84C3×#95—22no—
85C3⋊#95—24no—
86(C2×D3)⋊C8—24no—
87#810—28no—
88#809—44no—
89#803—22no—
90C22×SD16—34no—
91#540—64no—
92#540—64no—
93C2×C4⋊Q8—31no—
94C2×#127—31no—
95C2×#125C2×D311yescharacteristic
96C2×#123—62no—
97C2×C22⋊Q8—32no—
98C2×#121—32no—
99C2×#121—62no—
100C2×C4×D4—32no—
101C2×#117—31no—
102#492—64no—
103#491—64no—
104C2×#103—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8C2×D311yescharacteristic
109#398—62no—
110#397—62no—
111#391—32no—
112#382C2×D311yescharacteristic
113#340C2×D314yes—
114#336—32no—
115C23×D3—22no—
116C23×D3C2×D411yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2411yescharacteristic
121C22×Dic3—22no—
122C22×Dic3C2×D412yes—
123C22×Dic3C2×D411yescharacteristic
124C2×D12—24no—
125C2×D12—24no—
126C2×D12—22no—
127C2×D12—28no—
128C2×D12—28no—
129C2×D12—84no—
130C2×D12—84no—
131C2×D12C2×D414yes—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C2×C4×D3—28no—
137C2×C4×D3—84no—
138C2×Dic6—22no—
139C2×Dic6—22no—
140C2×Dic6—24no—
141C2×Dic6—24no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8#4114yes—
145C3×C4⋊C4—28no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4C2×D414yes—
149C3×C42C2×D412yes—
150C3×C42C2×D412yes—
151(C2×D3)⋊C4—28no—
152(C2×D3)⋊C4—28no—
153(C2×D3)⋊C4—24no—
154(C2×D3)⋊C4—44no—
155C4⋊Dic3—44no—
156C4⋊Dic3—44no—
157Dic3⋊C4—24no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163#204—416no—
164C22×Q8—32no—
165C22×D4—32no—
166C22×D4—62no—
167C22×D4—62no—
168C23×C4—31no—
169C23×C4—62no—
170C2×SD16—68no—
171C2×SD16—68no—
172C2×SD16—124no—
173C22×C8—32no—
174C22×C8C22×D312yes—
175C4⋊Q8—62no—
176C4⋊Q8—62no—
177#127—34no—
178#127—62no—
179#125—22no—
180#125D1214yes—
181#123—616no—
182C22⋊Q8—124no—
183#121—64no—
184#121—616no—
185#121—64no—
186C4×D4—64no—
187C4×D4—64no—
188#117—62no—
189#117—62no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4C22×D312yes—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C42—31no—
201C2×C42C22×D311yescharacteristic
202#103—68no—
203#102—124no—
204C22⋊C8—64no—
205#97—62no—
206C4×C8—22no—
207#95—22no—
208#95—64no—
209C23×C3C22×D411yescharacteristic
210C22×D3—21no—
211C22×D3—24no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—24no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—44no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—44no—
224C2×C3×C4#12014yes—
225C2×C3×C4C22×D411yescharacteristic
226C2×C3×C4C22×D411yescharacteristic
227C2×C3×C4C2×#4112yes—
228C2×C3×C4H(5,𝔽2)12yes—
229C2×Dic3—22no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—24no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237D12—28no—
238D12—44no—
239D12—88no—
240D12—88no—
241C4×D3—42no—
242C4×D3—42no—
243C4×D3—88no—
244Dic6—44no—
245Dic6—44no—
246C3×C8—24no—
247C3⋊C8—42no—
248C24—31no—
249C24—62no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—64no—
253C2×Q8—64no—
254C2×D4—34no—
255C2×D4—68no—
256C2×D4—68no—
257C2×D4—62no—
258C2×D4—64no—
259C2×D4—64no—
260C2×D4—244no—
261C2×D4—244no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—32no—
265C22×C4—31no—
266C22×C4—68no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—244no—
273C22×C4D3×D412yes—
274C22×C4C23×D311yescharacteristic
275SD16—1216no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8#23914yes—
282C4⋊C4—28no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—64no—
286C4⋊C4—124no—
287C4⋊C4—124no—
288C4⋊C4C2×D1214yes—
289C22⋊C4—68no—
290C22⋊C4—68no—
291C22⋊C4—64no—
292C22⋊C4—124no—
293C42—61no—
294C42—61no—
295C42C2×D1212yes—
296C42D3×D412yes—
297C22×C3C2×#12011yescharacteristic
298C22×C3#53211yescharacteristic
299C22×C3#53511yescharacteristic
300C22×C3#57714yes—
301C2×D3—42no—
302C2×D3—41no—
303C2×D3—44no—
304C2×D3—42no—
305C2×D3—44no—
306C2×D3—42no—
307C2×D3—44no—
308C2×D3—84no—
309C2×D3—84no—
310C3×C4—24no—
311C3×C4—44no—
312C3×C4—82no—
313C3×C4#54612yes—
314C3×C4#55012yes—
315Dic3—41no—
316Dic3—42no—
317Dic3—82no—
318C23—61no—
319C23—61no—
320C23—64no—
321C23—61no—
322C23—64no—
323C23—122no—
324C23—122no—
325C23—124no—
326C23—124no—
327C23C2×D3×D411yescharacteristic
328Q8—124no—
329Q8—124no—
330D4—68no—
331D4—124no—
332D4—248no—
333D4—248no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—62no—
342C2×C4—61no—
343C2×C4—64no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—248no—
350C2×C4#88214yes—
351C2×C4C22×D1211yescharacteristic
352C2×C4C2×D3×D411yescharacteristic
353C2×C4C2×#23912yes—
354C2×C4#100912yes—
355C8—24no—
356C8—122no—
357C2×C3#243111yescharacteristic
358C2×C3#300912yes—
359C2×C3#317412yes—
360C2×C3#322112yes—
361D3—81no—
362D3—84no—
363C22—121no—
364C22—124no—
365C22—122no—
366C22—124no—
367C22—122no—
368C22—124no—
369C22—122no—
370C22—244no—
371C22—244no—
372C22C2×#88211yescharacteristic
373C22#567411yescharacteristic
374C22#572511yescharacteristic
375C22#577014yes—
376C4—24no—
377C4—44no—
378C4—82no—
379C4—121no—
380C4—122no—
381C4—242no—
382C4#557712yes—
383C4#563012yes—
384C3#2560611yescharacteristic, complemented, Sylow
385C2—241no—
386C2—244no—
387C2#7662611yescharacteristic
388C2#7883512yes—
389C2#8164512yes—
390C2#8190812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1991C2
socleC23×C3C22×D4
prime radicalC2×#5697C2
hypercenter#1991D3
hypocenterC3#25606
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×#5069C2semidirect
3. C2×#4693C2semidirect
4. #74193C2semidirect
5. C3×#1991C2semidirect
6. #69753C2semidirect
7. #69553C2semidirect
8. #68896C2semidirect
9. #68887C2semidirect
41. #1991D3semidirect
384. C3#25606semidirect

Maximal subgroups

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

subgroupindex
C2×#56972
C2×#50692
C2×#46932
#741932
C3×#19912
#697532
#695532
#688962
#688872
#256063