Finite Group Databasebeta

← #10831748

Subgroups — #10831748

Order 768.

1248 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5727C211yesmaximal, characteristic
2C2×#5091C211yesmaximal, characteristic, complemented
3C2×#4710C211yesmaximal, characteristic
4#74191C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69838C211yesmaximal, characteristic, complemented
7#69556C211yesmaximal, characteristic
8#68910C212yesmaximal, complemented
9#68896C212yesmaximal
10#68894C212yesmaximal
11#68826C212yesmaximal, complemented
12#24342—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946—21no—
16C22×#204C2211yescharacteristic
17#5727—24no—
18#5727—24no—
19C2×#896—22no—
20C2×#893C2212yes—
21C2×#888C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#5091—24no—
31#4984C2212yes—
32#4973C2212yes—
33C2×#810C2212yes—
34C2×#809C2212yes—
35C2×#807C2212yes—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4710—24no—
38#4659—21no—
39#4656—21no—
40#4653—22no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×D4×Q8—31no—
47C2×#477—31no—
48C2×#440—31no—
49#2581—31no—
50#2057—32no—
51#2024—32no—
52#1993D311yescharacteristic, complemented
53#1985—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C2×D3×Q8—28no—
59C2×D3×Q8—48no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3—22no—
63C22×Dic6—22no—
64C22×Dic6D411yescharacteristic
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74#946—28no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4D412yes—
84C2×C4×Dic3—21no—
85C2×C4×Dic3C2312yes—
86C22×C3⋊C8C2312yes—
87C2×#204—22no—
88C2×#204—22no—
89C2×#204Q814yescomplemented
90#896—216no—
91#893—24no—
92#893—24no—
93#888—24no—
94#888—24no—
95C4×D12—22no—
96C4×D12D414yes—
97C4×Dic6—22no—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C3×#97—22no—
102C3×#95—21no—
103C3×#95—21no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#821—22no—
108#818—22no—
109#816—22no—
110#810—24no—
111#809—24no—
112#807—24no—
113C4⋊(C3⋊C8)—22no—
114C23×Q8—61no—
115C22×SD16—31no—
116D4×Q8—64no—
117D4×Q8—64no—
118C2×C4⋊Q8—32no—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—61no—
122C2×C4×Q8—31no—
123C2×C4×Q8—31no—
124C2×C4×D4—32no—
125C2×C4×D4—31no—
126#477—64no—
127#440—64no—
128C2×#107—31no—
129C2×#107—32no—
130C2×C4⋊C8—31no—
131C2×#103—31no—
132C2×#103—32no—
133C2×#102—31no—
134C2×#102—32no—
135C2×#97C2×D311yescharacteristic
136#400—61no—
137#395—61no—
138#391—32no—
139#389—32no—
140#387—62no—
141#382C2×D311yescharacteristic
142#340—32no—
143#340C2×D312yes—
144#340C2×D312yes—
145#336—32no—
146#336—32no—
147C23×D3—21no—
148C2×C3×Q8—24no—
149C2×C3×Q8—44no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3C2×D412yes—
159C22×Dic3C2×D412yes—
160D3×Q8—432no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12C2×D412yes—
165C2×D12C2×Q812yes—
166C2×D12#4112yes—
167C2×C4×D3—28no—
168C2×C4×D3—24no—
169C2×C4×D3—44no—
170C2×C4×D3—44no—
171C2×C4×D3—44no—
172C2×Dic6—22no—
173C2×Dic6—24no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—28no—
179C2×Dic6—21no—
180C2×Dic6—44no—
181C2×Dic6—44no—
182C2×Dic6C2×Q812yescomplemented
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8C2×Q812yescomplemented
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4C2×D414yes—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196(C2×D3)⋊C4—28no—
197(C2×D3)⋊C4—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—44no—
203C4⋊Dic3C2×D412yes—
204Dic3⋊C4—28no—
205Dic3⋊C4—24no—
206Dic3⋊C4—42no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—42no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213#204—28no—
214C22×Q8—21no—
215C22×Q8—31no—
216C22×Q8—31no—
217C22×Q8—62no—
218C22×Q8—68no—
219C22×Q8—128no—
220C22×D4—31no—
221C23×C4—62no—
222C23×C4—61no—
223C2×SD16—34no—
224C2×SD16—62no—
225C2×SD16—62no—
226C22×C8—32no—
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229C4⋊Q8—22no—
230C4⋊Q8—64no—
231C4⋊Q8—64no—
232C4⋊Q8D1214yes—
233C22⋊Q8—616no—
234C22⋊Q8—68no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×Q8—62no—
238C4×Q8—62no—
239C4×D4—34no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—62no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4C22×D312yes—
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C42—32no—
256C2×C42—61no—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#107—64no—
260C4⋊C8—62no—
261#103—62no—
262#103—64no—
263#102—64no—
264#102—62no—
265#97—22no—
266#95—21no—
267#95—21no—
268#95—62no—
269#95—62no—
270#95—62no—
271C23×C3C22×D411yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—42no—
276C22×D3—42no—
277C3×Q8—48no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4#12012yes—
287C2×C3×C4#12012yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3AGL(1,ℤ/8)14yes—
306D12—22no—
307D12—22no—
308D12—22no—
309D12C22⋊Q814yes—
310C4×D3—416no—
311C4×D3—48no—
312Dic6—28no—
313Dic6—22no—
314Dic6—22no—
315Dic6—416no—
316Dic6—41no—
317Dic6—41no—
318C3×C8—22no—
319C3×C8—41no—
320C3⋊C8—42no—
321C24—61no—
322C2×Q8—24no—
323C2×Q8—32no—
324C2×Q8—44no—
325C2×Q8—64no—
326C2×Q8—68no—
327C2×Q8—62no—
328C2×Q8—61no—
329C2×Q8—61no—
330C2×Q8—61no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—124no—
334C2×Q8—124no—
335C2×Q8—1232no—
336C2×D4—32no—
337C2×D4—32no—
338C2×D4—32no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—61no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—68no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—64no—
350C22×C4—61no—
351C22×C4—124no—
352C22×C4—124no—
353C22×C4—124no—
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4C23×D311yescharacteristic
357SD16—68no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8D3×Q812yescomplemented
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—32no—
372C4⋊C4—68no—
373C4⋊C4—64no—
374C4⋊C4—62no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—124no—
379C4⋊C4—122no—
380C4⋊C4C2×D1214yes—
381C22⋊C4—68no—
382C22⋊C4—64no—
383C42—62no—
384C42—62no—
385C42—122no—
386C42C2×D1212yes—
387C42D3×D412yes—
388C22×C3C2×#12011yescharacteristic
389C22×C3#54811yescharacteristic
390C22×C3D4×Q811yescharacteristic
391C22×C3C2×AGL(1,ℤ/8)12yes—
392C22×C3#57812yes—
393C2×D3—24no—
394C2×D3—44no—
395C2×D3—44no—
396C2×D3—42no—
397C2×D3—42no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—44no—
401C3×C4—82no—
402C3×C4C2×C22⋊Q811yescharacteristic
403C3×C4#53611yescharacteristic
404C3×C4#54811yescharacteristic
405C3×C4D4×Q811yescharacteristic
406Dic3—24no—
407Dic3—42no—
408Dic3—42no—
409Dic3—42no—
410Dic3—81no—
411C23—62no—
412C23—62no—
413C23—62no—
414C23—122no—
415C23—122no—
416C23C2×D3×D411yescharacteristic
417Q8—48no—
418Q8—68no—
419Q8—62no—
420Q8—62no—
421Q8—1216no—
422Q8—121no—
423Q8—121no—
424D4—34no—
425D4—62no—
426D4—62no—
427D4—62no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—34no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—64no—
443C2×C4—128no—
444C2×C4—1216no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—122no—
450C2×C4#88212yes—
451C2×C4#88212yes—
452C2×C4C22×D1211yescharacteristic
453C2×C4C2×D3×D411yescharacteristic
454C2×C4C2×D3×Q811yescharacteristic
455C2×C4#100711yescharacteristic
456C2×C4D3×#4111yescharacteristic
457C2×C4#101011yescharacteristic
458C8—22no—
459C8—41no—
460C8—122no—
461C2×C3#240911yescharacteristic
462C2×C3#300311yescharacteristic
463C2×C3#301111yescharacteristic
464C2×C3#336712yes—
465C2×C3#338612yes—
466D3—44no—
467C22—64no—
468C22—124no—
469C22—124no—
470C22—122no—
471C22—122no—
472C22C2×#88211yescharacteristic
473C22#567711yescharacteristic
474C22#572711yescharacteristic
475C22D3×AGL(1,ℤ/8)12yes—
476C22#577912yes—
477C4—22no—
478C4—22no—
479C4—44no—
480C4—64no—
481C4—82no—
482C4—122no—
483C4—122no—
484C4—122no—
485C4—241no—
486C4Q8×D1211yescharacteristic
487C4#557611yescharacteristic
488C4D3×C22⋊Q811yescharacteristic
489C4#562811yescharacteristic
490C3#2434211yescharacteristic, complemented, Sylow
491C2—124no—
492C2#7661911yescharacteristic
493C2#7892611yescharacteristic
494C2#7896111yescharacteristic
495C2#8167012yes—
496C2#8196612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1993D3
hypocenterC3#24342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#5091C2semidirect
4. #74191C2semidirect
5. C3×#1993C2semidirect
6. #69838C2semidirect
8. #68910C2semidirect
11. #68826C2semidirect
52. #1993D3semidirect
89. C2×#204Q8semidirect
182. C2×Dic6C2×Q8semidirect
188. C2×C3×C8C2×Q8semidirect
366. C2×C8D3×Q8semidirect
490. C3#24342semidirect

Maximal subgroups

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

subgroupindex
C2×#57272
C2×#50912
C2×#47102
#741912
C3×#19932
#698382
#695562
#689102
#688962
#688942
#688262
#243423