Finite Group Databasebeta

← #10828557

Subgroups — #10828557

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×#5535C211yesmaximal, characteristic
2C3×#2316C211yesmaximal, characteristic, complemented
3C2×#5003C212yesmaximal, complemented
4#71584C211yesmaximal, characteristic, complemented
5#71560C212yesmaximal, complemented
6#69627C212yesmaximal, complemented
7#69626C212yesmaximal, complemented
8#69625C212yesmaximal, complemented
9#69606C212yesmaximal, complemented
10#24070—31nomaximal, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#930C2212yes—
15#5535—28no—
16C2×#894—22no—
17C2×#884—22no—
18C2×#874C2211yescharacteristic
19C2×C4×D12C2212yes—
20C2×#870C2211yescharacteristic
21C3×#389—22no—
22C3×#389C2212yes—
23C3×#389C2212yes—
24C3×#380—22no—
25C2×#830C2212yes—
26#5003—28no—
27C2×#809C2212yes—
28C2×#809C2212yes—
29C2×#809C2212yes—
30C2×#809C2212yes—
31C2×#808C2212yes—
32C2×#807C2212yes—
33#4529C2212yes—
34#4529C2212yes—
35#4524C2212yes—
36#4524C2211yescharacteristic
37#4524C2211yescharacteristic
38C2×#540—31no—
39C2×#482—32no—
40#2316D311yescharacteristic, complemented
41#2059—32no—
42#2041—32no—
43#1874—32no—
44#1864—32no—
45#1862—32no—
46#1818—31no—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50C22×D12—22no—
51C22×D12C2312yes—
52C22×C4×D3—22no—
53C3×#117—24no—
54C3×#117—24no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4D412yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4D412yes—
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67#930—24no—
68#930D418yes—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4—22no—
73C22×C3⋊C8C2312yes—
74C22×C3⋊C8C2312yes—
75#894—216no—
76#884—216no—
77#874—24no—
78C4×D12—24no—
79C4×D12—24no—
80#870—24no—
81C3×#95—22no—
82C3×#95—22no—
83#830—24no—
84#809—24no—
85#809—24no—
86#809—24no—
87#809—24no—
88#808—24no—
89#807—24no—
90#540—68no—
91C2×#127—31no—
92C2×#125—31no—
93C2×#123—62no—
94C2×#121—32no—
95C2×#121—62no—
96C2×C4×D4—32no—
97C2×#117C2×D311yescharacteristic
98C2×#117C2×D311yescharacteristic
99C22×C4⋊C4C2×D311yescharacteristic
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—22no—
109#389C2×D312yes—
110#389C2×D312yes—
111#380—22no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336—31no—
117C24×C3C2×D411yescharacteristic
118C23×D3—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—82no—
126C22×C3×C4C2×D414yes—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2411yescharacteristic
131C2×#67—28no—
132C2×#67—82no—
133C2×#67—82no—
134C22×Dic3—22no—
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12—28no—
141C2×D12—82no—
142C2×D12—82no—
143C2×D12C2×D414yes—
144C2×C4×D3—28no—
145C2×C4×D3—82no—
146C2×C4×D3—82no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—82no—
153C3×C4⋊C4—82no—
154C3×C4⋊C4#4114yes—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—42no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—42no—
160(C2×D3)⋊C4—28no—
161(C2×D3)⋊C4—28no—
162C4⋊Dic3—22no—
163C4⋊Dic3C2×D414yes—
164Dic3⋊C4—28no—
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—32no—
172C22×D4—62no—
173C22×D4—62no—
174C23×C4—21no—
175C23×C4—62no—
176C23×C4C22×D311yescharacteristic
177C22×C8—32no—
178C22×C8—32no—
179#127—64no—
180#125—64no—
181#123—616no—
182#121—38no—
183#121—616no—
184#121—64no—
185C4×D4—64no—
186C4×D4—64no—
187#117—24no—
188#117—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4#6712yes—
196C2×C4⋊C4C22×D312yes—
197C2×C4⋊C4C22×D312yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4#6712yes—
202C2×C42—21no—
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—22no—
213#95—22no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3#12012yes—
219C23×C3C22×D411yescharacteristic
220C22×D3—24no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—42no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—84no—
244C2×C3×C4#12012yes—
245C2×C3×C4C22×D412yes—
246C2×C3×C4C2×#4112yes—
247C2×C3×C42−1+411yescharacteristic
248C2×C3×C42−1+411yescharacteristic
249#67—88no—
250C2×Dic3—24no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253D12—24no—
254D12—24no—
255D12—42no—
256D12—42no—
257D12—88no—
258C4×D3—88no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C24—62no—
262C24C2×#6711yescharacteristic
263C2×D4—34no—
264C2×D4—68no—
265C2×D4—68no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—242no—
272C2×D4—242no—
273C2×D4—242no—
274C2×D4—242no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—68no—
282C22×C4—62no—
283C22×C4—82no—
284C22×C4—242no—
285C22×C4—242no—
286C22×C4D3×D414yes—
287C22×C4C2×#6712yes—
288C22×C4C2×#6712yes—
289C22×C4C2×#6711yescharacteristic
290C22×C4C23×D311yescharacteristic
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—62no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—34no—
303C4⋊C4—68no—
304C4⋊C4—62no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4#23514yes—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C22⋊C4—68no—
311C22⋊C4—68no—
312C42—22no—
313C42—22no—
314C42—42no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3C2×#12011yescharacteristic
321C22×C3#52511yescharacteristic
322C22×C3#53611yescharacteristic
323C22×C3#57612yes—
324C22×C3#57712yes—
325C2×D3—44no—
326C2×D3—44no—
327C2×D3—44no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—82no—
337C3×C4#54712yes—
338C3×C4#54712yes—
339Dic3—44no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—64no—
345C23—122no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23#93712yes—
352C23C22×#6711yescharacteristic
353D4—64no—
354D4—64no—
355D4—122no—
356D4—122no—
357D4—248no—
358D4—248no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—64no—
376C2×C4—84no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—248no—
380C2×C4#93712yes—
381C2×C4C2×#23512yes—
382C2×C4C2×D3×D412yes—
383C2×C4#100711yescharacteristic
384C2×C4#100711yescharacteristic
385C8—122no—
386C8—122no—
387C2×C3—42no—
388C2×C3#240611yescharacteristic
389C2×C3#301212yes—
390C2×C3#312211yescharacteristic
391C2×C3#312611yescharacteristic
392C2×C3#324112yes—
393D3—84no—
394C22—22no—
395C22—24no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—124no—
400C22—124no—
401C22—124no—
402C22—242no—
403C22—242no—
404C22—242no—
405C22—242no—
406C22#579111yescharacteristic
407C22C2×#93711yescharacteristic
408C22#581611yescharacteristic
409C22#583512yes—
410C22#583712yes—
411C4—22no—
412C4—44no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—124no—
417C4#550712yes—
418C4#550712yes—
419C3#2407011yescharacteristic, complemented, Sylow
420C2—42no—
421C2—244no—
422C2#8219211yescharacteristic
423C2#8254211yescharacteristic
424C2#8255311yescharacteristic
425C2#8342412yes—
426C2#8347712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2316C2
socleC23×C3C22×D4
prime radicalC2×#5535C2
hypercenter#2316D3
hypocenterC3#24070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C3×#2316C2semidirect
3. C2×#5003C2semidirect
4. #71584C2semidirect
5. #71560C2semidirect
6. #69627C2semidirect
7. #69626C2semidirect
8. #69625C2semidirect
9. #69606C2semidirect
40. #2316D3semidirect
419. C3#24070semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C3×#23162
C2×#50032
#715842
#715602
#696272
#696262
#696252
#696062
#240703