Finite Group Databasebeta

← #10831752

Subgroups — #10831752

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×Q8×D12C211yesmaximal, characteristic
2C3×#2581C211yesmaximal, characteristic, complemented
3C2×#5026C211yesmaximal, characteristic
4C2×#5021C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69687C212yesmaximal
7#69668C212yesmaximal, complemented
8#69625C212yesmaximal
9#69604C212yesmaximal, complemented
10#69556C211yesmaximal, characteristic
11#69555C211yesmaximal, characteristic, complemented
12#24342—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C22×#215C2211yescharacteristic
17Q8×D12—24no—
18Q8×D12—24no—
19C2×#897—22no—
20C2×#897—21no—
21C2×C4×D12C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2212yes—
24C2×#869C2211yescharacteristic
25C3×#400—21no—
26C3×#395—21no—
27C3×#391C2212yes—
28C3×#389C2212yes—
29C3×#387—22no—
30C3×#382C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33#5026—24no—
34#5021—24no—
35C2×#809C2211yescharacteristic
36C2×#809C2212yes—
37C2×#807C2211yescharacteristic
38C2×#807C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×D4×Q8—31no—
47C2×#477—31no—
48C2×#440—31no—
49#2581D311yescharacteristic, complemented
50#2057—32no—
51#2024—32no—
52#1993—31no—
53#1985—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C2×D3×Q8—28no—
60C2×D3×Q8—48no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—21no—
64C22×Dic6—22no—
65C22×Dic6—21no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C42—21no—
77C2×C3×C42C2312yes—
78C2×C3×C42C2311yescharacteristic
79C2×#215—22no—
80C2×#215—22no—
81C2×#215Q814yescomplemented
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91#897—28no—
92#897—216no—
93C4×D12—24no—
94C4×D12—22no—
95C4×D12—24no—
96C4×D12D414yes—
97#869—22no—
98#869—24no—
99#869—24no—
100#869D414yes—
101C3×#95—21no—
102C3×#95—21no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—22no—
106#835—24no—
107#835—22no—
108#809—22no—
109#809—24no—
110#807—24no—
111#807—22no—
112C4⋊(C3⋊C8)—22no—
113#803—22no—
114C23×Q8—61no—
115C22×SD16—31no—
116D4×Q8—64no—
117D4×Q8—64no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8—32no—
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—61no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×D4—31no—
125C2×C4×D4—32no—
126#477—64no—
127#440—64no—
128C2×#107—32no—
129C2×#107—31no—
130C2×C4⋊C8—31no—
131C2×#103—32no—
132C2×#103—31no—
133C2×#102—32no—
134C2×#102—31no—
135C2×#97—31no—
136#400—21no—
137#395—21no—
138#391C2×D312yes—
139#389C2×D312yes—
140#387—22no—
141#382C2×D311yescharacteristic
142#340—32no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—32no—
147C23×D3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8C2×Q812yescomplemented
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—22no—
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—28no—
179C2×Dic6—24no—
180C2×Dic6—44no—
181C2×Dic6—44no—
182C2×Dic6—44no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—44no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—42no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#215—28no—
197(C2×D3)⋊C4—28no—
198(C2×D3)⋊C4—24no—
199C4⋊Dic3—28no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203C4⋊Dic3C2×D412yes—
204C4⋊Dic3C2×D414yes—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8C2×Q812yescomplemented
214C22×Q8—68no—
215C22×Q8—61no—
216C22×Q8—62no—
217C22×Q8—128no—
218C22×Q8#6711yescharacteristic
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×C8—32no—
229C4⋊Q8—34no—
230C4⋊Q8—64no—
231C4⋊Q8—64no—
232C4⋊Q8—62no—
233C22⋊Q8—68no—
234C22⋊Q8—616no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—34no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—62no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4#6712yes—
250C2×C4⋊C4#6711yescharacteristic
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C42—21no—
256C2×C42C22×D312yes—
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—62no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271C23×C3C22×D411yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—42no—
276C22×D3—42no—
277C3×Q8—28no—
278C3×Q8—22no—
279C3×Q8—22no—
280C3×Q8—41no—
281C3×Q8—41no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4#12012yes—
297C2×C3×C4#12012yes—
298C2×C3×C4AGL(1,ℤ/8)14yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×Q811yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C42−1+411yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×Dic3—22no—
306C2×Dic3—24no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310D12—22no—
311D12—22no—
312D12—22no—
313D12C22⋊Q814yes—
314C4×D3—48no—
315C4×D3—416no—
316Dic6—48no—
317Dic6—416no—
318C3⋊C8—22no—
319C3⋊C8—41no—
320C3⋊C8—42no—
321C24—61no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—68no—
330C2×Q8—64no—
331C2×Q8—124no—
332C2×Q8—124no—
333C2×Q8—1232no—
334C2×Q8—124no—
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—68no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—64no—
349C22×C4—124no—
350C22×C4—124no—
351C22×C4—124no—
352C22×C4C2×#6712yes—
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6712yes—
356C22×C4C23×D311yescharacteristic
357SD16—68no—
358C2×C8—32no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—61no—
365C2×C8—62no—
366C2×C8—61no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—32no—
374C4⋊C4—34no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—68no—
378C4⋊C4—64no—
379C4⋊C4—61no—
380C4⋊C4—62no—
381C22⋊C4—68no—
382C22⋊C4—64no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42D3×D412yes—
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—24no—
400C3×C4—22no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—82no—
404C3×C4—81no—
405C3×C4C2×C22⋊Q811yescharacteristic
406C3×C4#53611yescharacteristic
407C3×C4#54811yescharacteristic
408C3×C4D4×Q811yescharacteristic
409Dic3—44no—
410Dic3—42no—
411C23—62no—
412C23—62no—
413C23—62no—
414C23—122no—
415C23—122no—
416C23C22×#6711yescharacteristic
417Q8—28no—
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422Q8—128no—
423Q8—1216no—
424D4—34no—
425D4—62no—
426D4—62no—
427D4—62no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—64no—
443C2×C4—62no—
444C2×C4—128no—
445C2×C4—1216no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4#93712yes—
450C2×C4#93712yes—
451C2×C4#94914yes—
452C2×C4C2×D3×D411yescharacteristic
453C2×C4C2×D3×D411yescharacteristic
454C2×C4C2×D3×Q811yescharacteristic
455C2×C4#100711yescharacteristic
456C2×C4D3×#4111yescharacteristic
457C2×C4#101011yescharacteristic
458C8—62no—
459C8—122no—
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—
472C22C2×#93711yescharacteristic
473C22Q8×#6711yescharacteristic
474C22C2×#94912yes—
475C22#583111yescharacteristic
476C22#583812yes—
477C4—22no—
478C4—22no—
479C4—24no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—81no—
484C4—124no—
485C4—122no—
486C4D3×C22⋊Q811yescharacteristic
487C4#562811yescharacteristic
488C4#563211yescharacteristic
489C4#563311yescharacteristic
490C3#2434211yescharacteristic, complemented, Sylow
491C2—124no—
492C2#8275111yescharacteristic
493C2#8292711yescharacteristic
494C2#8341111yescharacteristic
495C2#8366812yes—
496C2#8369412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2581C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2581D3
hypocenterC3#24342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C3×#2581C2semidirect
4. C2×#5021C2semidirect
5. #71635C2semidirect
7. #69668C2semidirect
9. #69604C2semidirect
11. #69555C2semidirect
49. #2581D3semidirect
81. 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×Q8×D122
C3×#25812
C2×#50262
C2×#50212
#716352
#696872
#696682
#696252
#696042
#695562
#695552
#243423