Finite Group Databasebeta

← #10831746

Subgroups — #10831746

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×C3×#477C211yesmaximal, characteristic, complemented
3#82481C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#70638C212yesmaximal
7#70617C212yesmaximal, complemented
8#70480C211yesmaximal, characteristic
9#70473C211yesmaximal, characteristic, complemented
10#70276C212yesmaximal
11#70231C212yesmaximal, complemented
12#24342—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946—21no—
17C22×#215C2211yescharacteristic
18#5727—24no—
19#5727—24no—
20C2×#896—22no—
21C2×#893C2212yes—
22C2×#888C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#477—24no—
26C2×C3×#107C2212yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29#5229C2212yes—
30C2×#835C2211yescharacteristic
31C2×#821C2212yes—
32#5026—24no—
33#4986—21no—
34#4979—21no—
35#4972C2212yes—
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4936C2211yescharacteristic
39#4933—22no—
40C2×#803C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×D4×Q8—31no—
47C2×#477D311yescharacteristic, complemented
48C2×#440—31no—
49#2581—31no—
50#2057—32no—
51#2024—32no—
52#1993—31no—
53#1985—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C2×D3×Q8—28no—
60C2×D3×Q8—48no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C4×D3—22no—
64C22×Dic6—22no—
65C22×Dic6D411yescharacteristic
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77#946—28no—
78C2×#215—22no—
79C2×#215—22no—
80C2×#215Q814yescomplemented
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3D411yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4D412yes—
89C2×C4×Dic3C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92#896—216no—
93#893—24no—
94#893—24no—
95#888—24no—
96#888—24no—
97C4×D12—22no—
98C4×D12D414yes—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C3×#107—24no—
102C3×C4⋊C8—22no—
103C3×#103—24no—
104#835—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110#821—24no—
111#809—22no—
112#807—22no—
113#803—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×Q8C2×D311yescharacteristic
124C2×C4×D4—32no—
125C2×C4×D4—31no—
126#477—24no—
127#440—64no—
128C2×#107—31no—
129C2×#107C2×D312yes—
130C2×C4⋊C8C2×D311yescharacteristic
131C2×#103—31no—
132C2×#103C2×D312yes—
133C2×#102—32no—
134C2×#102—31no—
135C2×#97—31no—
136#400—61no—
137#395—61no—
138#391—32no—
139#389—32no—
140#387—62no—
141#382—31no—
142#340—32no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—32no—
147C23×D3—21no—
148C2×C3×Q8—24no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—44no—
155C2×C3×Q8C2×Q812yescomplemented
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3C2×D412yes—
165C22×Dic3C2×D412yes—
166D3×Q8—432no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12C2×D412yes—
171C2×D12C2×Q812yes—
172C2×D12#4112yes—
173C2×C4×D3—28no—
174C2×C4×D3—24no—
175C2×C4×D3—44no—
176C2×C4×D3—44no—
177C2×C4×D3—44no—
178C2×Dic6—22no—
179C2×Dic6—24no—
180C2×Dic6—28no—
181C2×Dic6—44no—
182C2×Dic6—44no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D414yes—
192C3×C42—42no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195#215—28no—
196(C2×D3)⋊C4—28no—
197(C2×D3)⋊C4—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3C2×D412yes—
203Dic3⋊C4—28no—
204Dic3⋊C4—24no—
205Dic3⋊C4—44no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8C2×Q812yescomplemented
214C22×Q8—21no—
215C22×Q8—31no—
216C22×Q8—62no—
217C22×Q8—68no—
218C22×Q8—128no—
219C22×Q8C22×D311yescharacteristic
220C22×D4—31no—
221C23×C4—62no—
222C23×C4—61no—
223C2×SD16—34no—
224C2×SD16—62no—
225C2×SD16—62no—
226C22×C8—31no—
227C22×C8—31no—
228C22×C8C22×D312yes—
229C4⋊Q8—22no—
230C4⋊Q8—64no—
231C4⋊Q8—64no—
232C4⋊Q8#6714yes—
233C22⋊Q8—616no—
234C22⋊Q8—68no—
235C4×Q8—22no—
236C4×Q8—22no—
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—21no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C42—21no—
256C2×C42—32no—
257C2×C42C22×D311yescharacteristic
258#107—24no—
259#107—62no—
260C4⋊C8—22no—
261#103—24no—
262#103—62no—
263#102—62no—
264#102—64no—
265#97—62no—
266#95—62no—
267#95—62no—
268#95—62no—
269#95—61no—
270#95—61no—
271C23×C3C22×D411yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—42no—
276C22×D3—42no—
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—48no—
280C3×Q8—41no—
281C3×Q8—41no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12012yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×Q811yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299C2×C3×C42−1+411yescharacteristic
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3AGL(1,ℤ/8)14yes—
310D12—22no—
311D12—22no—
312D12—22no—
313D12C22⋊Q814yes—
314C4×D3—416no—
315C4×D3—48no—
316Dic6—28no—
317Dic6—416no—
318C3×C8—42no—
319C3⋊C8—22no—
320C3⋊C8—41no—
321C24—61no—
322C2×Q8—24no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—44no—
329C2×Q8—64no—
330C2×Q8—68no—
331C2×Q8—62no—
332C2×Q8—124no—
333C2×Q8—124no—
334C2×Q8—1232no—
335C2×Q8D3×Q812yescomplemented
336C2×D4—32no—
337C2×D4—32no—
338C2×D4—32no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—61no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—32no—
346C22×C4—32no—
347C22×C4—68no—
348C22×C4—61no—
349C22×C4—64no—
350C22×C4—62no—
351C22×C4—124no—
352C22×C4—124no—
353C22×C4—124no—
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4C23×D311yescharacteristic
357SD16—68no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—32no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—61no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—32no—
372C4⋊C4—42no—
373C4⋊C4—68no—
374C4⋊C4—64no—
375C4⋊C4—62no—
376C4⋊C4—61no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—124no—
380C4⋊C4C2×#6714yes—
381C22⋊C4—68no—
382C22⋊C4—64no—
383C42—42no—
384C42—62no—
385C42—62no—
386C42D3×D412yes—
387C42C2×#6712yes—
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—42no—
401C3×C4—44no—
402C3×C4—81no—
403C3×C4C2×C22⋊Q811yescharacteristic
404C3×C4#53611yescharacteristic
405C3×C4#54811yescharacteristic
406C3×C4D4×Q811yescharacteristic
407Dic3—24no—
408Dic3—42no—
409Dic3—42no—
410Dic3—82no—
411C23—62no—
412C23—62no—
413C23—62no—
414C23—122no—
415C23—122no—
416C23C2×D3×D411yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—48no—
420Q8—41no—
421Q8—41no—
422Q8—68no—
423Q8—1216no—
424D4—34no—
425D4—62no—
426D4—62no—
427D4—62no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—34no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—64no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—128no—
445C2×C4—1216no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4#93712yes—
451C2×C4#93712yes—
452C2×C4C2×D3×D411yescharacteristic
453C2×C4C2×D3×Q811yescharacteristic
454C2×C4#100711yescharacteristic
455C2×C4D3×#4111yescharacteristic
456C2×C4#101011yescharacteristic
457C2×C4C22×#6711yescharacteristic
458C8—42no—
459C8—62no—
460C8—121no—
461C2×C3#240911yescharacteristic
462C2×C3#300311yescharacteristic
463C2×C3#301111yescharacteristic
464C2×C3#336712yes—
465C2×C3#338612yes—
466D3—44no—
467C22—64no—
468C22—122no—
469C22—124no—
470C22—124no—
471C22—122no—
472C22#563211yescharacteristic
473C22#563311yescharacteristic
474C22#576612yes—
475C22D3×AGL(1,ℤ/8)12yes—
476C22C2×#93711yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—44no—
481C4—64no—
482C4—81no—
483C4—122no—
484C4—122no—
485C4—242no—
486C4D3×C22⋊Q811yescharacteristic
487C4#562811yescharacteristic
488C4Q8×#6711yescharacteristic
489C4#583111yescharacteristic
490C3#2434211yescharacteristic, complemented, Sylow
491C2—124no—
492C2#8110012yes—
493C2#8110312yes—
494C2#8275111yescharacteristic
495C2#8388411yescharacteristic
496C2#8394411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#477D3
hypocenterC3#24342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
3. #82481C2semidirect
5. #71788C2semidirect
7. #70617C2semidirect
9. #70473C2semidirect
11. #70231C2semidirect
47. C2×#477D3semidirect
80. C2×#215Q8semidirect
155. C2×C3×Q8C2×Q8semidirect
213. C2×C3⋊C8C2×Q8semidirect
335. C2×Q8D3×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×C3×#4772
#824812
C2×#50262
#717882
#706382
#706172
#704802
#704732
#702762
#702312
#243423