Finite Group Databasebeta

← #10828552

Subgroups — #10828552

Order 768.

1130 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×#5035C212yesmaximal, complemented
3#82232C211yesmaximal, characteristic, complemented
4#71762C212yesmaximal, complemented
5#71739C212yesmaximal, complemented
6#71677C212yesmaximal, complemented
7#71607C212yesmaximal, complemented
8#71606C212yesmaximal, complemented
9#71584C211yesmaximal, characteristic, complemented
10#24070—31nomaximal, Sylow
11C3×#540—28no—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#125C2211yescharacteristic
14C2×C3×#123—22no—
15C2×C3×#121—22no—
16C2×C3×#121C2212yes—
17C2×C3×C4×D4C2212yes—
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#832C2212yes—
22C2×#832C2212yes—
23C2×#832C2212yes—
24C2×#832C2212yes—
25C2×#830C2212yes—
26#5035—28no—
27#4972—22no—
28C2×#808C2212yes—
29C2×#807C2212yes—
30#4942—22no—
31#4928C2212yes—
32#4928C2212yes—
33#4529C2212yes—
34#4529C2212yes—
35#4524C2212yes—
36#4524C2211yescharacteristic
37#4524C2211yescharacteristic
38C2×#540D311yescharacteristic
39C2×#482—32no—
40#2316—31no—
41#2059—32no—
42#2041—32no—
43#1874—32no—
44#1864—32no—
45#1862—32no—
46#1818—31no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4C2312yes—
50C23×C3×C4—22no—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3—21no—
53C3×#127—24no—
54C3×#125—24no—
55C3×#123—216no—
56C3×#121—216no—
57C3×#121—24no—
58C3×#121D418yes—
59C3×C4×D4—24no—
60C3×C4×D4—24no—
61C3×#117—24no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4D412yes—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71#926—24no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81#832—24no—
82#832—24no—
83#832—24no—
84#832—24no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#830—24no—
88#808—24no—
89#807—24no—
90#540—28no—
91C2×#127C2×D311yescharacteristic
92C2×#125C2×D311yescharacteristic
93C2×#123—22no—
94C2×#121—22no—
95C2×#121C2×D312yes—
96C2×C4×D4C2×D312yes—
97C2×#117—31no—
98C2×#117C2×D311yescharacteristic
99C22×C4⋊C4—31no—
100#482—68no—
101C2×#107—32no—
102C2×#106—32no—
103C2×#102—32no—
104C2×#102—32no—
105C2×#102—32no—
106C2×#102—32no—
107C2×C22⋊C8—32no—
108#389—32no—
109#389—32no—
110#389—62no—
111#380—62no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336—31no—
117C24×C3—22no—
118C24×C3C2×D411yescharacteristic
119C2×C3×D4—28no—
120C2×C3×D4—28no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—82no—
127C2×C3×D4—82no—
128C2×C3×D4—82no—
129C2×C3×D4—82no—
130C2×C3×D4C2×D414yes—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—28no—
136C22×C3×C4—82no—
137C22×C3×C4—82no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D414yes—
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—22no—
144C22×Dic3—24no—
145C22×Dic3—22no—
146C22×Dic3—82no—
147C3×C4⋊C4—28no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4C2×D414yes—
151C3×C22⋊C4—28no—
152C3×C22⋊C4—24no—
153C3×C22⋊C4—28no—
154C3×C42—22no—
155C3×C42—22no—
156#217—42no—
157C4⋊Dic3—24no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3#4114yes—
162Dic3⋊C4—82no—
163Dic3⋊C4—82no—
164C4×Dic3—42no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4C22×D312yes—
174C23×C4—22no—
175C23×C4—61no—
176C23×C4C22×D311yescharacteristic
177C22×C8—32no—
178C22×C8—32no—
179#127—24no—
180#125—24no—
181#123—216no—
182#121—216no—
183#121—24no—
184#121#6718yes—
185C4×D4—24no—
186C4×D4—24no—
187#117—24no—
188#117—64no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4#6712yes—
197C2×C4⋊C4C22×D312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4#6712yes—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204C2×C42C22×D311yescharacteristic
205#107—64no—
206#106—64no—
207#102—64no—
208#102—64no—
209#102—64no—
210#102—64no—
211C22⋊C8—64no—
212#95—62no—
213#95—62no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—24no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3#12012yes—
226C23×C3C22×D411yescharacteristic
227C3×D4—24no—
228C3×D4—24no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—88no—
232C3×D4—88no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—88no—
247C2×C3×C4#12012yes—
248C2×C3×C4C22×D412yes—
249C2×C3×C4C2×#4112yes—
250C2×C3×C42−1+411yescharacteristic
251C2×C3×C42−1+411yescharacteristic
252C2×Dic3—24no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—84no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C24—22no—
262C24C2×#6711yescharacteristic
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4C2×#6714yes—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—62no—
281C22×C4—64no—
282C22×C4—62no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—242no—
286C22×C4C2×#6712yes—
287C22×C4C2×#6712yes—
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6714yes—
290C22×C4C23×D311yescharacteristic
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—62no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—34no—
301C4⋊C4—64no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—242no—
306C4⋊C4—242no—
307C4⋊C4C2×#6714yes—
308C22⋊C4—28no—
309C22⋊C4—24no—
310C22⋊C4—28no—
311C22⋊C4—122no—
312C42—22no—
313C42—22no—
314C42—122no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—44no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3C2×#12011yescharacteristic
328C22×C3#52511yescharacteristic
329C22×C3#53611yescharacteristic
330C22×C3#57612yes—
331C22×C3#57712yes—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4—44no—
336C3×C4#54712yes—
337C3×C4#54712yes—
338Dic3—44no—
339Dic3—82no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—24no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23#95312yes—
352C23C22×#6711yescharacteristic
353D4—24no—
354D4—24no—
355D4—42no—
356D4—42no—
357D4—88no—
358D4—88no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—64no—
373C2×C4—88no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—244no—
380C2×C4#95312yes—
381C2×C4D3×#4112yes—
382C2×C4#101011yescharacteristic
383C2×C4#101011yescharacteristic
384C2×C4C22×#6712yes—
385C8—122no—
386C8—122no—
387C2×C3—42no—
388C2×C3—84no—
389C2×C3#240611yescharacteristic
390C2×C3#301212yes—
391C2×C3#312211yescharacteristic
392C2×C3#312611yescharacteristic
393C2×C3#324112yes—
394C22—22no—
395C22—24no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22#582511yescharacteristic
407C22#582811yescharacteristic
408C22#583512yes—
409C22#583612yes—
410C22C2×#95311yescharacteristic
411C4—22no—
412C4—42no—
413C4—42no—
414C4—44no—
415C4—124no—
416C4—242no—
417C4#582412yes—
418C4#582412yes—
419C3#2407011yescharacteristic, complemented, Sylow
420C2—42no—
421C2—84no—
422C2#8343012yes—
423C2#8343511yescharacteristic
424C2#8343911yescharacteristic
425C2#8409011yescharacteristic
426C2#8410212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#540C2
socleC23×C3C22×D4
hypercenterC2×#540D3
hypocenterC3#24070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
2. C2×#5035C2semidirect
3. #82232C2semidirect
4. #71762C2semidirect
5. #71739C2semidirect
6. #71677C2semidirect
7. #71607C2semidirect
8. #71606C2semidirect
9. #71584C2semidirect
419. C3#24070semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×#50352
#822322
#717622
#717392
#716772
#716072
#716062
#715842
#240703