Finite Group Databasebeta

← #10769167

Subgroups — #10769167

Order 768.

1192 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85735C211yesmaximal, characteristic
2#82264C211yesmaximal, characteristic, complemented
3#79312C212yesmaximal, complemented
4C3×#1458C211yesmaximal, characteristic, complemented
5#70790C212yesmaximal, complemented
6#68133C214yesmaximal, complemented
7#68108C214yesmaximal, complemented
8#18301—31nomaximal, Sylow
9C22×#235—21no—
10C2×C3×#117C2211yescharacteristic
11#5526—22no—
12#5525—22no—
13#5524—24no—
14#5510—21no—
15#5510C2212yes—
16#5493—21no—
17#5493C2212yes—
18#5477—21no—
19#5477C2212yes—
20C2×#869C2211yescharacteristic, complemented
21C3×#431C2212yescomplemented
22#5212—21no—
23#4986—21no—
24#4979C2212yescomplemented
25#4949—21no—
26#4947C2212yes—
27#4936C2211yescharacteristic, complemented
28#4928—21no—
29#4841—24no—
30#4738C2218yescomplemented
31#4693—22no—
32#4681—22no—
33C3×#324C2212yescomplemented
34C3×#323C2212yes—
35#4470C2218yescomplemented
36#3449—31no—
37#3206—32no—
38#2628—31no—
39#1604—34no—
40#1601—34no—
41#1517—32no—
42#1458D311yescharacteristic, complemented
43C23×C3×C4D411yescharacteristic
44C22×#67—22no—
45C2×#235—22no—
46C2×#235—22no—
47C2×#235—22no—
48C2×#235—22no—
49C2×#235—44no—
50C2×#235—44no—
51C22×D12—21no—
52C22×C4×D3—22no—
53C22×Dic6—21no—
54C22×Dic6C2312yes—
55C3×#117—22no—
56C3×#117—22no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C22⋊C4D411yescharacteristic
60C2×C3×C22⋊C4C2311yescharacteristic
61C2×C3×C42D411yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×#217—22no—
64#930—22no—
65#930—22no—
66#930—22no—
67#930C2314yes—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×Dic3⋊C4—22no—
72C2×#204—28no—
73#897—24no—
74#897—24no—
75#895—24no—
76#895—24no—
77#886—24no—
78#886—24no—
79#882—24no—
80#882—24no—
81#875—22no—
82#875—22no—
83#874—21no—
84#874D412yes—
85C4×D12—22no—
86C4×D12—22no—
87C4×D12—22no—
88C4×D12D414yes—
89#869—21no—
90#869—22no—
91#869—22no—
92#869D412yes—
93C3×C4⋊C8—22no—
94C3×C22⋊C8C2314yes—
95C3×C4×C8—22no—
96C3⋊#95—22no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95C2312yes—
100#816—28no—
101C22×#41—61no—
102#537—32no—
103#537—61no—
104#536—62no—
105#535—62no—
106#534—32no—
107#534—61no—
108#532—32no—
109#532—61no—
110C2×C4⋊Q8—31no—
111#525—64no—
112C2×#117C2×D311yescharacteristic
113#492—62no—
114#487—62no—
115#461—64no—
116#448—38no—
117#431C2×D312yescomplemented
118#400—61no—
119#399—61no—
120#395—32no—
121#393—32no—
122#389—61no—
123#382—31no—
124#380—61no—
125#327—38no—
126#324C2×D312yescomplemented
127#323C2×D312yes—
128C24×C3C2×D411yescharacteristic
129C23×D3—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4SD1614yescomplemented
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C2×#67—24no—
141C2×#67—24no—
142C2×#67—44no—
143C2×#67—44no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147#235—44no—
148#235—48no—
149#235—48no—
150#235—44no—
151#235—44no—
152#235—44no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—44no—
159C2×D12C2×D414yes—
160C2×C4×D3—24no—
161C2×C4×D3—24no—
162C2×C4×D3—48no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—24no—
166C2×Dic6—44no—
167C2×Dic6—82no—
168C2×C3×C8—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C22⋊C4—22no—
172C3×C22⋊C4—22no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42C2×D412yes—
176C3×C42C2×D412yes—
177#217—82no—
178(C2×D3)⋊C4—24no—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—24no—
181(C2×D3)⋊C4—24no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—42no—
186C4⋊Dic3—41no—
187C4⋊Dic3—41no—
188C4⋊Dic3—42no—
189C4⋊Dic3C2×D414yes—
190Dic3⋊C4—82no—
191#204—88no—
192C2×#41—62no—
193C2×#41—62no—
194C2×#41—62no—
195C2×#41—62no—
196C2×#41—124no—
197C2×#41—124no—
198C22×Q8—32no—
199C22×Q8—61no—
200C22×D4—61no—
201C22×D4—62no—
202C23×C4—62no—
203C23×C4D1211yescharacteristic
204C2×SD16—68no—
205C4⋊Q8—32no—
206C4⋊Q8—61no—
207C4⋊Q8—62no—
208C4⋊Q8—62no—
209#127—32no—
210#127—61no—
211#124—62no—
212#124—62no—
213#123—64no—
214#123—64no—
215C22⋊Q8—64no—
216C22⋊Q8—64no—
217#121—34no—
218#121—64no—
219#121—64no—
220#121—62no—
221#121—62no—
222#121—62no—
223#120—64no—
224#120—64no—
225C4×D4—34no—
226C4×D4—62no—
227C4×D4—62no—
228C4×D4—62no—
229#117—22no—
230#117—22no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4D1211yescharacteristic
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—62no—
238C2×C22⋊C4D1211yescharacteristic
239C2×C22⋊C4C22×D311yescharacteristic
240C2×C42D1211yescharacteristic
241C2×C42C22×D311yescharacteristic
242C4⋊C8—22no—
243#103—68no—
244C22⋊C8C22×D314yes—
245C4×C8—22no—
246#95—32no—
247#95—62no—
248#95—61no—
249#95—62no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3C2×SD1612yes—
255C23×C3C22×D411yescharacteristic
256C22×D3—24no—
257C22×D3—44no—
258C22×D3—82no—
259C22×D3—82no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4#12012yes—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4C2×SD1612yes—
274C2×C3×C4C2×SD1612yes—
275C2×C3×C4AGL(1,ℤ/8)12yes—
276C2×C3×C4C22×D412yes—
277#67—48no—
278#67—44no—
279#67—44no—
280C2×Dic3—24no—
281C2×Dic3—44no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288D12—44no—
289D12—44no—
290D12—44no—
291D12—82no—
292D12—82no—
293D12—82no—
294C4×D3—48no—
295C4×D3—48no—
296Dic6—44no—
297Dic6—44no—
298Dic6—48no—
299C3×C8—44no—
300C24—62no—
301C24C2×D1211yescharacteristic
302#41—124no—
303#41—128no—
304#41—128no—
305#41—124no—
306#41—124no—
307#41—124no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—64no—
311C2×Q8—124no—
312C2×Q8—242no—
313C2×D4—34no—
314C2×D4—62no—
315C2×D4—62no—
316C2×D4—62no—
317C2×D4—64no—
318C2×D4—64no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—124no—
322C2×D4—124no—
323C2×D4—124no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—64no—
328C22×C4—62no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—128no—
333C22×C4#20414yescomplemented
334C22×C4C2×D1211yescharacteristic
335C22×C4C2×D1211yescharacteristic
336C22×C4C2×D1211yescharacteristic
337C22×C4C2×D1211yescharacteristic
338C22×C4C2×D1211yescharacteristic
339C22×C4C23×D311yescharacteristic
340SD16—248no—
341C2×C8—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—34no—
345C4⋊C4—62no—
346C4⋊C4—62no—
347C4⋊C4—62no—
348C4⋊C4—122no—
349C4⋊C4—121no—
350C4⋊C4—121no—
351C4⋊C4—122no—
352C4⋊C4—242no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—64no—
356C22⋊C4—64no—
357C22⋊C4—64no—
358C22⋊C4—64no—
359C22⋊C4—242no—
360C42—21no—
361C42—21no—
362C42D3×D412yes—
363C42D3×D412yes—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—22no—
367C22×C3—42no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—81no—
371C22×C3C2×#12011yescharacteristic
372C22×C3C22×SD1611yescharacteristic
373C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
374C2×D3—44no—
375C2×D3—44no—
376C2×D3—44no—
377C2×D3—88no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—22no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4#45012yes—
384C3×C4#45012yes—
385Dic3—44no—
386Dic3—82no—
387Dic3—82no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—64no—
393C23—124no—
394C23—242no—
395C23—242no—
396C23C2×#20412yes—
397C23C22×D1211yescharacteristic
398Q8—124no—
399Q8—128no—
400Q8—124no—
401D4—124no—
402D4—124no—
403D4—128no—
404D4—124no—
405D4—124no—
406D4—124no—
407D4—242no—
408D4—242no—
409D4—242no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—64no—
421C2×C4—128no—
422C2×C4—124no—
423C2×C4—128no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4#88212yes—
431C2×C4#88211yescharacteristic
432C2×C4#88211yescharacteristic
433C2×C4C2×#20412yes—
434C2×C4C2×#20412yes—
435C2×C4#90812yes—
436C2×C4C2×D3×D412yes—
437C8—44no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—42no—
441C2×C3C2×#45011yescharacteristic
442C2×C3#302211yescharacteristic
443C2×C3#302711yescharacteristic
444D3—84no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—81no—
452C22—124no—
453C22—124no—
454C22—124no—
455C22—248no—
456C22C2×#88211yescharacteristic
457C22C22×#20411yescharacteristic
458C22C2×#90811yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—124no—
465C4—242no—
466C4—242no—
467C4#473112yes—
468C4#473112yes—
469C3#1830111yescharacteristic, complemented, Sylow
470C2—21no—
471C2—21no—
472C2—42no—
473C2—244no—
474C2C2×#473111yescharacteristic
475C2#7869111yescharacteristic
476C2#7871811yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1458C2
socleC22×C3C2×#120
prime radical#85735C2
hypercenter#1458D3
hypocenterC3#18301
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #82264C2semidirect
3. #79312C2semidirect
4. C3×#1458C2semidirect
5. #70790C2semidirect
6. #68133C2semidirect
7. #68108C2semidirect
20. C2×#869C22semidirect
21. C3×#431C22semidirect
24. #4979C22semidirect
27. #4936C22semidirect
30. #4738C22semidirect
33. C3×#324C22semidirect
35. #4470C22semidirect
42. #1458D3semidirect
117. #431C2×D3semidirect
126. #324C2×D3semidirect
133. C22×C3×C4SD16semidirect
333. C22×C4#204semidirect
469. C3#18301semidirect

Maximal subgroups

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

subgroupindex
#857352
#822642
#793122
C3×#14582
#707902
#681332
#681082
#183013