Finite Group Databasebeta

← #10811997

Subgroups — #10811997

Order 768.

1138 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C3×#2355C211yesmaximal, characteristic, complemented
3C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4#69588C211yesmaximal, characteristic, complemented
5#69580C211yesmaximal, characteristic
6#69556C212yesmaximal, complemented
7#69551C212yesmaximal, complemented
8#69488C212yesmaximal, complemented
9#69487C211yesmaximal, characteristic
10C4×#810C211yesmaximal, characteristic, complemented
11C4×#809C212yesmaximal, complemented
12#22543—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15#5585—24no—
16#5585C418yescomplemented
17C2×#894—22no—
18C2×#893—21no—
19C2×#875C2212yes—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2212yes—
22C2×C4×Dic6C2211yescharacteristic
23C3×#391—21no—
24C3×#387—22no—
25C3×#384C2211yescharacteristic
26C3×#382C2212yes—
27C3×C4×C4⋊C4—21no—
28C3×C4×C4⋊C4C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30Q8×C3⋊C8—24no—
31C2×#810C2211yescharacteristic
32C2×#810C2211yescharacteristic
33C2×#810C2212yes—
34C2×#809C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2212yes—
40C2×C4×C3⋊C8C2212yes—
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2211yescharacteristic
43#4529C2212yes—
44#4529C2211yescharacteristic
45C2×#559—31no—
46C2×C8×Q8—31no—
47#2355D311yescharacteristic, complemented
48#1983—32no—
49#1982—31no—
50#1958—32no—
51#1957—31no—
52#1804—32no—
53#1803—31no—
54C4×#103—31no—
55C4×#102—32no—
56C22×C3×Q8D411yescharacteristic
57C22×D12—21no—
58C22×D12—21no—
59C22×D12C2312yes—
60C22×C4×D3—22no—
61C22×Dic6C2311yescharacteristic
62C3×C4×Q8C2×C414yes—
63C3×C4×Q8D414yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4D412yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C42—21no—
72C2×C3×C42—22no—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2312yes—
75#947—28no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×C4×Dic3—21no—
82C22×C3⋊C8C2312yes—
83C22×C3⋊C8C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85#894—216no—
86#893—28no—
87#875—24no—
88#875C2×C418yes—
89#874—22no—
90#874C2×C414yes—
91C4×D12—24no—
92C4×D12C2×C418yes—
93C4×Dic6—22no—
94C4×Dic6C2×C414yes—
95C3×#95—22no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—22no—
99#810—22no—
100#810—22no—
101#810—42no—
102#809—24no—
103#809—24no—
104#809—42no—
105#809—42no—
106C4⋊(C3⋊C8)—22no—
107C4⋊(C3⋊C8)—24no—
108C4×C3⋊C8—22no—
109C4×C3⋊C8—24no—
110#559—38no—
111#559—64no—
112C2×#127—31no—
113C2×#124—32no—
114C2×#124—61no—
115C2×#123—62no—
116C2×C4×Q8—31no—
117C2×C4×Q8C2×D311yescharacteristic
118C2×C4×D4—32no—
119C2×C4×D4—61no—
120C8×Q8—64no—
121C2×C4⋊C8—31no—
122C2×C4⋊C8—32no—
123C2×#103—31no—
124C2×#103—31no—
125C2×#103—32no—
126C2×#102—32no—
127C2×#102—32no—
128C2×#102—32no—
129C2×#102—32no—
130C2×C4×C8—31no—
131C2×C4×C8—32no—
132#391—21no—
133#387—22no—
134#384C2×D311yescharacteristic
135#382C2×D312yes—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×D311yescharacteristic
138C4×C4⋊C4C2×D312yes—
139#340—32no—
140#340—31no—
141#340—31no—
142C23×D3—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8C22⋊C414yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—21no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—24no—
161C2×D12—22no—
162C2×D12—24no—
163C2×D12—42no—
164C2×D12—42no—
165C2×D12—81no—
166C2×D12—81no—
167C2×D12C22×C414yes—
168C2×C4×D3—28no—
169C2×C4×D3—82no—
170C2×C4×D3—82no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6C22×C412yes—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—81no—
184C3×C4⋊C4C22⋊C414yes—
185C3×C4⋊C4C22⋊C418yes—
186C3×C4⋊C4#4112yes—
187C3×C42—42no—
188C3×C42—82no—
189C3×C42C22×C412yes—
190C3×C42C22×C414yes—
191C3×C42C2×D412yes—
192C3×C42C2×D414yes—
193(C2×D3)⋊C4—28no—
194(C2×D3)⋊C4—28no—
195(C2×D3)⋊C4—28no—
196C4⋊Dic3—21no—
197C4⋊Dic3C22×C412yes—
198Dic3⋊C4—24no—
199C4×Dic3—24no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8#4112yes—
209C22×Q8—31no—
210C22×Q8#6711yescharacteristic
211C22×D4—32no—
212C22×D4—61no—
213C22×D4—61no—
214C23×C4—62no—
215C22×C8—31no—
216C22×C8—32no—
217C22×C8—31no—
218#127—34no—
219#127—62no—
220#124—38no—
221#124—68no—
222#124—64no—
223#123—616no—
224C4×Q8—34no—
225C4×Q8—62no—
226C4×Q8C4×D314yes—
227C4×Q8D1214yes—
228C4×D4—38no—
229C4×D4—68no—
230C4×D4—64no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4#6711yescharacteristic
236C2×C4⋊C4#6712yes—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D312yes—
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—61no—
246C2×C42C22×D312yes—
247C2×C42C22×D311yescharacteristic
248C4⋊C8—64no—
249C4⋊C8—62no—
250#103—62no—
251#103—62no—
252#103—122no—
253#102—64no—
254#102—64no—
255#102—122no—
256#102—122no—
257C4×C8—64no—
258C4×C8—62no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—22no—
263C23×C3C22×D411yescharacteristic
264C22×D3—24no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C3×Q8—28no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4C2×C22⋊C412yes—
288C2×C3×C4C2×C22⋊C414yes—
289C2×C3×C4C2×C22⋊C414yes—
290C2×C3×C4C2×C22⋊C412yes—
291C2×C3×C4#13512yes—
292C2×C3×C4#13512yes—
293C2×C3×C4C23×C411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305D12—24no—
306D12—24no—
307D12—42no—
308D12—42no—
309D12—48no—
310D12—84no—
311C4×D3—88no—
312Dic6—22no—
313Dic6—22no—
314Dic6—41no—
315Dic6—41no—
316C3⋊C8—22no—
317C3⋊C8—42no—
318C3⋊C8—41no—
319C24—62no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—32no—
323C2×Q8—61no—
324C2×Q8—61no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8(C2×D3)⋊C414yes—
329C2×D4—34no—
330C2×D4—64no—
331C2×D4—64no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—241no—
340C2×D4—241no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—68no—
347C22×C4—242no—
348C22×C4—242no—
349C22×C4C2×#6712yes—
350C22×C4C2×#6712yes—
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C22×C4C23×D311yescharacteristic
354C2×C8—32no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—62no—
358C2×C8—61no—
359C2×C8—62no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—62no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—32no—
369C4⋊C4—42no—
370C4⋊C4—64no—
371C4⋊C4—61no—
372C4⋊C4—81no—
373C4⋊C4(C2×D3)⋊C414yes—
374C4⋊C4(C2×D3)⋊C418yes—
375C4⋊C4#23912yes—
376C22⋊C4—68no—
377C22⋊C4—68no—
378C22⋊C4—68no—
379C42—42no—
380C42—64no—
381C42—82no—
382C42C2×C4×D314yes—
383C42C2×C4×D312yes—
384C42C2×D1214yes—
385C42C2×D1212yes—
386C22×C3C22×C22⋊C411yescharacteristic
387C22×C3#54811yescharacteristic
388C22×C3#55011yescharacteristic
389C22×C3C2×#13511yescharacteristic
390C22×C3C2×#13511yescharacteristic
391C22×C3#57611yescharacteristic
392C22×C3#57711yescharacteristic
393C2×D3—44no—
394C2×D3—44no—
395C2×D3—44no—
396C2×D3—82no—
397C2×D3—82no—
398C2×D3—82no—
399C2×D3—82no—
400C3×C4—24no—
401C3×C4—24no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—81no—
405C3×C4—82no—
406C3×C4#41614yes—
407C3×C4C4×#4111yescharacteristic
408C3×C4#52011yescharacteristic
409C3×C4#54811yescharacteristic
410C3×C4#55011yescharacteristic
411Dic3—42no—
412Dic3—42no—
413C23—64no—
414C23—122no—
415C23—122no—
416C23—122no—
417C23—122no—
418C23—122no—
419C23—122no—
420C23C22×#6711yescharacteristic
421Q8—28no—
422Q8—62no—
423Q8—62no—
424Q8—121no—
425Q8—121no—
426D4—64no—
427D4—64no—
428D4—128no—
429D4—122no—
430D4—122no—
431D4—244no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—248no—
454C2×C4C2×(C2×D3)⋊C414yes—
455C2×C4C2×(C2×D3)⋊C414yes—
456C2×C4C2×(C2×D3)⋊C412yes—
457C2×C4C2×(C2×D3)⋊C412yes—
458C2×C4#95012yes—
459C2×C4#95012yes—
460C2×C4C22×C4×D311yescharacteristic
461C2×C4C22×D1211yescharacteristic
462C2×C4C2×#23911yescharacteristic
463C2×C4#100711yescharacteristic
464C2×C4D3×#4111yescharacteristic
465C2×C4#100911yescharacteristic
466C8—62no—
467C8—122no—
468C8—121no—
469C2×C3#234111yescharacteristic
470C2×C3C2×#41611yescharacteristic
471C2×C3#289711yescharacteristic
472C2×C3#336111yescharacteristic
473C2×C3#336511yescharacteristic
474C2×C3#339411yescharacteristic
475C2×C3#339611yescharacteristic
476D3—84no—
477C22—124no—
478C22—124no—
479C22—124no—
480C22—242no—
481C22—242no—
482C22—242no—
483C22—242no—
484C22C22×(C2×D3)⋊C411yescharacteristic
485C22#581711yescharacteristic
486C22C2×#95011yescharacteristic
487C22C2×#95011yescharacteristic
488C22#583411yescharacteristic
489C22#583511yescharacteristic
490C22#583711yescharacteristic
491C4—24no—
492C4—24no—
493C4—22no—
494C4—42no—
495C4—82no—
496C4—81no—
497C4—122no—
498C4—122no—
499C4#500214yes—
500C4C4×#23911yescharacteristic
501C4#557411yescharacteristic
502C4#557611yescharacteristic
503C4#557711yescharacteristic
504C3#2254311yescharacteristic, complemented, Sylow
505C2—244no—
506C2C2×#500211yescharacteristic
507C2#8245711yescharacteristic
508C2#8256311yescharacteristic
509C2#8364311yescharacteristic
510C2#8367911yescharacteristic
511C2#8369711yescharacteristic
512C2#8372011yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2355C2
socleC23×C3C22×D4
prime radicalC2×#874C22
cosocleC22×C3×C4C24
hypercenter#2355D3
hypocenterC3#22543
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C3×#2355C2semidirect
3. C2×Q8×C3⋊C8C2semidirect
4. #69588C2semidirect
6. #69556C2semidirect
7. #69551C2semidirect
8. #69488C2semidirect
10. C4×#810C2semidirect
11. C4×#809C2semidirect
16. #5585C4semidirect
47. #2355D3semidirect
504. C3#22543semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#23552
C2×Q8×C3⋊C82
#695882
#695802
#695562
#695512
#694882
#694872
C4×#8102
C4×#8092
#225433