Finite Group Databasebeta

← #10848777

Subgroups — #10848777

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×#5719C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic, complemented
3#83261C211yesmaximal, characteristic, complemented
4C2×#5082C211yesmaximal, characteristic, complemented
5#72054C214yesmaximal, complemented
6#72047C214yesmaximal, complemented
7#71765C211yesmaximal, characteristic, complemented
8#69778C211yesmaximal, characteristic, complemented
9#69753C211yesmaximal, characteristic, complemented
10#25606—31nomaximal, Sylow
11C22×C3×SD16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×#946C2212yes—
15C2×#939—22no—
16C2×#938C2212yes—
17C2×#935—22no—
18C2×D4×Dic3C2212yes—
19#5719—24no—
20#5719—24no—
21C2×#872C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C3×#492—24no—
24C2×C3×C4×C8C2211yescharacteristic
25#5229C2212yes—
26C2×#835C2214yes—
27C2×#832C2214yes—
28C2×(C2×D3)⋊C8C2212yes—
29#5082—24no—
30#4992—22no—
31#4983—22no—
32#4936C2211yescharacteristic
33C2×#803C2211yescharacteristic
34#4551C2214yes—
35#4550C2212yes—
36C2×#540—31no—
37C2×#492D311yescharacteristic, complemented
38C2×#491—31no—
39#2621—31no—
40#2000—34no—
41#1991—31no—
42#1961—31no—
43#1877—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C22×C3×D4—22no—
47C22×C3×D4C2312yes—
48C22×#67—22no—
49C23×Dic3—22no—
50C22×C4×D3C2311yescharacteristic
51C2×C3×SD16—28no—
52C2×C3×SD16—28no—
53C2×C3×SD16—44no—
54C22×C3×C8C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×#127—22no—
58C3×#127D414yes—
59C2×C3×C4⋊C4—22no—
60C2×C3×C42C2311yescharacteristic
61C2×#217—22no—
62C2×#217—22no—
63#946—44no—
64#939—216no—
65#938—24no—
66#938—24no—
67#935—216no—
68D4×Dic3—24no—
69D4×Dic3—24no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4D412yes—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4D412yes—
77C2×C4×Dic3C2311yescharacteristic
78C22×C3⋊C8C2312yes—
79#872—22no—
80#872—22no—
81#870—22no—
82#870D414yes—
83C3×C4×C8—22no—
84#835—28no—
85#832—44no—
86C3⋊#95—22no—
87C3⋊#95—24no—
88(C2×D3)⋊C8—24no—
89#803—22no—
90C22×SD16C2×D314yes—
91#540—64no—
92#540—64no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#127C2×D311yescharacteristic
95C2×#125—31no—
96C2×#123—62no—
97C2×C22⋊Q8—32no—
98C2×#121—32no—
99C2×#121—62no—
100C2×C4×D4—32no—
101C2×#117—31no—
102#492—24no—
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#382—31no—
113#340—34no—
114#336—32no—
115C24×C3—22no—
116C23×D3C2×D411yescharacteristic
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×D4—28no—
122C2×C3×D4—22no—
123C2×C3×D4—24no—
124C2×C3×D4—24no—
125C2×C3×D4—84no—
126C2×C3×D4C2×D414yes—
127C22×C3×C4—22no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2411yescharacteristic
130C2×#67—28no—
131C2×#67—84no—
132C22×Dic3—22no—
133C22×Dic3—28no—
134C22×Dic3—22no—
135C22×Dic3—84no—
136C22×Dic3C2×D412yes—
137C22×Dic3C2×D411yescharacteristic
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—22no—
141C2×C4×D3—22no—
142C3×SD16—416no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8#4114yes—
146C3×C4⋊C4—44no—
147C3×C42C2×D412yes—
148C3×C42C2×D412yes—
149#217—28no—
150#217—28no—
151(C2×D3)⋊C4—24no—
152(C2×D3)⋊C4—44no—
153C4⋊Dic3—22no—
154C4⋊Dic3—24no—
155C4⋊Dic3C2×D414yes—
156Dic3⋊C4—28no—
157Dic3⋊C4—24no—
158Dic3⋊C4—44no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C22×Q8C22×D312yes—
165C22×D4—22no—
166C22×D4—62no—
167C22×D4C22×D312yes—
168C23×C4—31no—
169C23×C4—62no—
170C2×SD16—28no—
171C2×SD16—28no—
172C2×SD16—44no—
173C22×C8—32no—
174C22×C8C22×D312yes—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177#127—22no—
178#127#6714yes—
179#125—34no—
180#125—62no—
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—32no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—62no—
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—64no—
208#95—62no—
209C23×C3—24no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—44no—
213C23×C3—44no—
214C23×C3C22×D411yescharacteristic
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—24no—
219C3×Q8—44no—
220C3×Q8—44no—
221C3×D4—28no—
222C3×D4—44no—
223C3×D4—88no—
224C2×C3×C4—24no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4#12014yes—
229C2×C3×C4C22×D411yescharacteristic
230C2×C3×C4C22×D411yescharacteristic
231C2×C3×C4C2×#4112yes—
232C2×C3×C4H(5,𝔽2)12yes—
233#67—88no—
234C2×Dic3—22no—
235C2×Dic3—24no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—24no—
240C2×Dic3—44no—
241C2×Dic3—42no—
242C2×Dic3—44no—
243C2×Dic3—88no—
244C4×D3—42no—
245C4×D3—42no—
246C3×C8—24no—
247C3⋊C8—42no—
248C24—22no—
249C24—31no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×D4—28no—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—68no—
259C2×D4—84no—
260C2×D4—244no—
261C2×D4C2×#6714yes—
262C22×C4—22no—
263C22×C4—32no—
264C22×C4—31no—
265C22×C4—62no—
266C22×C4—68no—
267C22×C4—61no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4—62no—
272C22×C4—244no—
273C22×C4D3×D412yes—
274C22×C4C23×D311yescharacteristic
275SD16—416no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8#23714yes—
282C4⋊C4—34no—
283C4⋊C4—44no—
284C4⋊C4—64no—
285C4⋊C4—68no—
286C4⋊C4—62no—
287C4⋊C4—64no—
288C4⋊C4—124no—
289C22⋊C4—68no—
290C22⋊C4—68no—
291C22⋊C4—64no—
292C22⋊C4—124no—
293C42—61no—
294C42—61no—
295C42D3×D412yes—
296C42C2×#6712yes—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—84no—
301C22×C3—84no—
302C22×C3C2×#12011yescharacteristic
303C22×C3#53211yescharacteristic
304C22×C3#53511yescharacteristic
305C22×C3#57714yes—
306C2×D3—42no—
307C2×D3—41no—
308C2×D3—42no—
309C2×D3—42no—
310C3×C4—24no—
311C3×C4—82no—
312C3×C4#54612yes—
313C3×C4#55012yes—
314Dic3—41no—
315Dic3—42no—
316Dic3—44no—
317Dic3—82no—
318C23—24no—
319C23—42no—
320C23—42no—
321C23—44no—
322C23—44no—
323C23—61no—
324C23—61no—
325C23—64no—
326C23—61no—
327C23C2×D3×D411yescharacteristic
328Q8—44no—
329Q8—44no—
330D4—28no—
331D4—44no—
332D4—88no—
333D4—248no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—64no—
341C2×C4—62no—
342C2×C4—61no—
343C2×C4—64no—
344C2×C4—124no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—124no—
349C2×C4—248no—
350C2×C4#93714yes—
351C2×C4C2×D3×D411yescharacteristic
352C2×C4C2×#23712yes—
353C2×C4#100412yes—
354C2×C4C22×#6711yescharacteristic
355C8—24no—
356C8—122no—
357C2×C3—84no—
358C2×C3#243111yescharacteristic
359C2×C3#300912yes—
360C2×C3#317412yes—
361C2×C3#322112yes—
362D3—81no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—84no—
367C22—84no—
368C22—122no—
369C22—122no—
370C22—122no—
371C22—121no—
372C22#571511yescharacteristic
373C22#571611yescharacteristic
374C22#576214yes—
375C22C2×#93711yescharacteristic
376C4—24no—
377C4—82no—
378C4—122no—
379C4—124no—
380C4—121no—
381C4—242no—
382C4#571812yes—
383C4#580612yes—
384C3#2560611yescharacteristic, complemented, Sylow
385C2—84no—
386C2—241no—
387C2#8175312yes—
388C2#8176812yes—
389C2#8318711yescharacteristic
390C2#8385912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#492C2
socleC23×C3C22×D4
prime radicalC2×#5719C2
hypercenterC2×#492D3
hypocenterC3#25606
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
2. C2×C3×#492C2semidirect
3. #83261C2semidirect
4. C2×#5082C2semidirect
5. #72054C2semidirect
6. #72047C2semidirect
7. #71765C2semidirect
8. #69778C2semidirect
9. #69753C2semidirect
37. C2×#492D3semidirect
384. C3#25606semidirect

Maximal subgroups

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

subgroupindex
C2×#57192
C2×C3×#4922
#832612
C2×#50822
#720542
#720472
#717652
#697782
#697532
#256063