Finite Group Databasebeta

← #10848780

Subgroups — #10848780

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×#5535C211yesmaximal, characteristic
2#82262C211yesmaximal, characteristic, complemented
3C2×#4703C211yesmaximal, characteristic, complemented
4C2×#4693C211yesmaximal, characteristic, complemented
5C3×#1750C211yesmaximal, characteristic, complemented
6#69962C214yesmaximal, complemented
7#69823C211yesmaximal, characteristic, complemented
8#69778C211yesmaximal, characteristic, complemented
9#68672C214yesmaximal, complemented
10#25606—31nomaximal, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×#930C2212yes—
13C2×#924C2212yes—
14C22×#204C2214yes—
15#5535—24no—
16#5535—24no—
17C2×#894—22no—
18C2×#884—22no—
19C2×#874C2211yescharacteristic
20C2×C4×D12C2212yes—
21C2×#870C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C3×C22⋊C8C2212yes—
24C2×C3×#97C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C2×#821C2214yes—
27C2×#816C2214yes—
28#4981—22no—
29#4953—22no—
30#4936C2211yescharacteristic
31#4929C2212yes—
32#4703—24no—
33#4693—24no—
34C3×#336C2212yes—
35#4551C2214yes—
36C2×#540—31no—
37C2×#492—31no—
38C2×#491—31no—
39#2621—31no—
40#2000—34no—
41#1991—31no—
42#1961—31no—
43#1877—34no—
44#1750D311yescharacteristic, complemented
45C23×C3×C4C2311yescharacteristic
46C22×#67—22no—
47C22×D12—22no—
48C22×D12C2312yes—
49C22×C4×D3—22no—
50C22×Dic6C2312yes—
51C22×C3×C8C2312yes—
52C22×C3×C8C2312yes—
53C3×#117—22no—
54C3×#117—22no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4D412yes—
57C2×C3×C42C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60#930—24no—
61#930—24no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3C2312yes—
66C2×C4⋊Dic3C2312yes—
67#924—44no—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—22no—
70C2×#204—28no—
71C2×#204—28no—
72C2×#204—44no—
73#894—216no—
74#884—216no—
75#874—22no—
76#874D414yes—
77C4×D12—24no—
78C4×D12—24no—
79#870—22no—
80#870D414yes—
81#869—22no—
82#869—22no—
83C3×C22⋊C8—24no—
84C3×#97—22no—
85C3×C4×C8—22no—
86C3⋊#95—24no—
87C3⋊#95—22no—
88#821—44no—
89#816—28no—
90C22×SD16—34no—
91#540—64no—
92#540—64no—
93C2×C4⋊Q8—31no—
94C2×#127—31no—
95C2×#125—31no—
96C2×#123—62no—
97C2×C22⋊Q8—32no—
98C2×#121—32no—
99C2×#121—62no—
100C2×C4×D4—32no—
101C2×#117C2×D311yescharacteristic
102#492—64no—
103#491—64no—
104C2×#103—34no—
105C2×#102—34no—
106C2×C22⋊C8C2×D312yes—
107C2×#97C2×D311yescharacteristic
108C2×C4×C8C2×D311yescharacteristic
109#398—62no—
110#397—62no—
111#391—32no—
112#382—31no—
113#340—34no—
114#336C2×D312yes—
115C24×C3C2×D411yescharacteristic
116C23×D3—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2411yescharacteristic
125C2×#67—28no—
126C2×#67—84no—
127C22×Dic3—22no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C2×D12—24no—
131C2×D12—24no—
132C2×D12—22no—
133C2×D12—28no—
134C2×D12—84no—
135C2×D12C2×D414yes—
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—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8#4114yes—
148C3×C4⋊C4—24no—
149C3×C22⋊C4—24no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42C2×D412yes—
153C3×C42C2×D412yes—
154#217—44no—
155(C2×D3)⋊C4—28no—
156(C2×D3)⋊C4—28no—
157C4⋊Dic3—22no—
158C4⋊Dic3—24no—
159C4⋊Dic3—44no—
160C4⋊Dic3C2×D414yes—
161Dic3⋊C4—28no—
162Dic3⋊C4—44no—
163#204—416no—
164C22×Q8—32no—
165C22×D4—32no—
166C22×D4—62no—
167C22×D4—62no—
168C23×C4—62no—
169C23×C4C22×D311yescharacteristic
170C2×SD16—68no—
171C2×SD16—68no—
172C2×SD16—124no—
173C22×C8C22×D312yes—
174C22×C8C22×D312yes—
175C4⋊Q8—62no—
176C4⋊Q8—62no—
177#127—34no—
178#127—62no—
179#125—34no—
180#125—62no—
181#123—616no—
182C22⋊Q8—124no—
183#121—616no—
184#121—64no—
185#121—64no—
186C4×D4—64no—
187C4×D4—64no—
188#117—22no—
189#117—22no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4D1212yes—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4D1212yes—
200C2×C42C22×D311yescharacteristic
201C2×C42C22×D311yescharacteristic
202#103—68no—
203#102—124no—
204C22⋊C8—24no—
205#97—22no—
206C4×C8—22no—
207#95—62no—
208#95—64no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—24no—
213C23×C3C22×D411yescharacteristic
214C22×D3—24no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—44no—
219C2×C3×C4—24no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—24no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4#12014yes—
228C2×C3×C4C22×D411yescharacteristic
229C2×C3×C4C22×D411yescharacteristic
230C2×C3×C4C2×#4112yes—
231C2×C3×C4H(5,𝔽2)12yes—
232#67—88no—
233C2×Dic3—24no—
234C2×Dic3—44no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—44no—
240D12—28no—
241D12—44no—
242D12—88no—
243C4×D3—88no—
244Dic6—44no—
245Dic6—44no—
246C3×C8—24no—
247C3×C8—42no—
248C24—62no—
249C24C2×D1211yescharacteristic
250C2×Q8—64no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—64no—
254C2×D4—34no—
255C2×D4—68no—
256C2×D4—68no—
257C2×D4—64no—
258C2×D4—62no—
259C2×D4—64no—
260C2×D4—244no—
261C2×D4—244no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—68no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—244no—
271C22×C4C2×D1212yes—
272C22×C4C2×D1211yescharacteristic
273C22×C4C2×D1212yes—
274C22×C4C23×D311yescharacteristic
275SD16—1216no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8#23514yes—
282C4⋊C4—24no—
283C4⋊C4—34no—
284C4⋊C4—68no—
285C4⋊C4—62no—
286C4⋊C4—64no—
287C4⋊C4—124no—
288C4⋊C4—124no—
289C22⋊C4—24no—
290C22⋊C4—68no—
291C22⋊C4—68no—
292C22⋊C4—124no—
293C42—21no—
294C42—21no—
295C42D3×D412yes—
296C42D3×D412yes—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3C2×#12011yescharacteristic
302C22×C3#53211yescharacteristic
303C22×C3#53511yescharacteristic
304C22×C3#57714yes—
305C2×D3—44no—
306C2×D3—44no—
307C2×D3—44no—
308C2×D3—84no—
309C2×D3—84no—
310C3×C4—24no—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4#54612yes—
314C3×C4#55012yes—
315Dic3—44no—
316Dic3—82no—
317Dic3—82no—
318C23—21no—
319C23—21no—
320C23—24no—
321C23—21no—
322C23—64no—
323C23—122no—
324C23—124no—
325C23—124no—
326C23—122no—
327C23C22×D1211yescharacteristic
328Q8—124no—
329Q8—124no—
330D4—68no—
331D4—124no—
332D4—248no—
333D4—248no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—64no—
343C2×C4—124no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—124no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—248no—
350C2×C4#88214yes—
351C2×C4C2×#23512yes—
352C2×C4C2×D3×D411yescharacteristic
353C2×C4C2×D3×D411yescharacteristic
354C2×C4#100912yes—
355C8—24no—
356C8—42no—
357C2×C3—81no—
358C2×C3#243111yescharacteristic
359C2×C3#300912yes—
360C2×C3#317412yes—
361C2×C3#322112yes—
362D3—84no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—42no—
367C22—124no—
368C22—124no—
369C22—124no—
370C22—244no—
371C22—244no—
372C22#547711yescharacteristic
373C22C2×#88211yescharacteristic
374C22#552511yescharacteristic
375C22#575114yes—
376C4—24no—
377C4—42no—
378C4—41no—
379C4—124no—
380C4—242no—
381C4—242no—
382C4#555012yes—
383C4#555312yes—
384C3#2560611yescharacteristic, complemented, Sylow
385C2—81no—
386C2—244no—
387C2#7358611yescharacteristic
388C2#7856712yes—
389C2#7867112yes—
390C2#7939712yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1750C2
socleC23×C3C22×D4
prime radicalC2×#5535C2
hypercenter#1750D3
hypocenterC3#25606
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. #82262C2semidirect
3. C2×#4703C2semidirect
4. C2×#4693C2semidirect
5. C3×#1750C2semidirect
6. #69962C2semidirect
7. #69823C2semidirect
8. #69778C2semidirect
9. #68672C2semidirect
44. #1750D3semidirect
384. C3#25606semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
#822622
C2×#47032
C2×#46932
C3×#17502
#699622
#698232
#697782
#686722
#256063