Finite Group Databasebeta

← #10817441

Subgroups — #10817441

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83229C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3C2×#4788C212yesmaximal
4#74360C211yesmaximal, characteristic
5C3×#1956C211yesmaximal, characteristic, complemented
6#70276C212yesmaximal
7#70254C212yesmaximal, complemented
8#70237C212yesmaximal, complemented
9C4×#821C211yesmaximal, characteristic
10C8×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#22996—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C22×#215C2212yes—
16C2×#893C2212yes—
17C2×D3×C4⋊C4—21no—
18C2×C4×D12C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22C3×#384C2211yescharacteristic
23#5229C2212yes—
24C2×#835C2212yes—
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×C8×Dic3C2212yes—
28#5083C418yescomplemented
29#4989—22no—
30#4987—21no—
31#4984—21no—
32#4980—22no—
33Dic3×C4⋊C4—21no—
34#4972C2212yes—
35C2×#808C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4788—28no—
40#4551C2211yescharacteristic
41#4551C2211yescharacteristic
42#4550C2212yes—
43#4529C2212yes—
44C2×#461—31no—
45C2×C4×SD16—32no—
46#2365—31no—
47D4×C4⋊C4—31no—
48#1956D311yescharacteristic, complemented
49#1948—31no—
50#1933—32no—
51#1877—32no—
52#1865—32no—
53C8×C4⋊C4—31no—
54C4×#102—31no—
55C22×C3×Q8C2312yes—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8C2×C414yescomplemented
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69Q8×Dic3—24no—
70Q8×Dic3—42no—
71C2×#215—24no—
72C2×#215—24no—
73C2×#215C2×C418yes—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3C2312yes—
86C22×C3⋊C8C2312yes—
87#893—24no—
88#893—42no—
89D3×C4⋊C4—44no—
90C4×D12—22no—
91C4×D12C2×C414yes—
92C3×#103—24no—
93C3×#103—42no—
94C3×C4×C8—22no—
95#835C2×C418yescomplemented
96C3⋊#95—21no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95D412yes—
101#821—22no—
102#821D414yes—
103C8×Dic3—24no—
104#808—24no—
105C4⋊(C3⋊C8)C2×C414yescomplemented
106C22×SD16—32no—
107C2×C4⋊Q8C2×D311yescharacteristic
108C2×C4×Q8—32no—
109C2×C4×D4—32no—
110C2×C4×D4—31no—
111C22×C4⋊C4—61no—
112#461—38no—
113C4×SD16—68no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#103—32no—
117C2×#103C2×D312yes—
118C2×#103C2×D312yes—
119C2×#102—31no—
120C2×#102—31no—
121C2×C4×C8—32no—
122C2×C4×C8C2×D311yescharacteristic
123#391—32no—
124#389—32no—
125#389—61no—
126#387—62no—
127#385—62no—
128#384C2×D311yescharacteristic
129#380—61no—
130C4×C4⋊C4—31no—
131C4×C4⋊C4—31no—
132C4×C4⋊C4—61no—
133#340—31no—
134#340—31no—
135#340—32no—
136#336—32no—
137C23×D3—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8C22×C414yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3C2×D412yes—
153C22×Dic3C2×D412yes—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12C22×C412yes—
158C2×D12C2×D412yes—
159C2×D12#4112yes—
160C2×C4×D3—24no—
161C2×C4×D3—42no—
162C2×C4×D3—42no—
163C2×C4×D3—44no—
164C2×C4×D3—44no—
165C2×C4×D3—82no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8C2×D412yes—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—81no—
178C3×C42C22×C412yes—
179C3×C42#4112yes—
180#215—216no—
181(C2×D3)⋊C4—24no—
182(C2×D3)⋊C4—44no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—44no—
189C4⋊Dic3SD1614yescomplemented
190C4⋊Dic3C22×C412yes—
191C4⋊Dic3C2×D412yes—
192Dic3⋊C4—44no—
193C4×Dic3—22no—
194C4×Dic3—22no—
195C4×Dic3—44no—
196C4×Dic3—42no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8C22×C414yes—
200C22×Q8C22×D312yes—
201C22×D4—31no—
202C23×C4—62no—
203C23×C4—61no—
204C2×SD16—38no—
205C2×SD16—64no—
206C2×SD16—64no—
207C22×C8—32no—
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C4⋊Q8—22no—
211C4⋊Q8C4×D314yescomplemented
212C4×Q8—64no—
213C4×Q8—122no—
214C4×D4—34no—
215C4×D4—64no—
216C4×D4—62no—
217C4×D4—122no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—124no—
230C2×C4⋊C4C22×D312yes—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—62no—
233C2×C42—32no—
234C2×C42—62no—
235C2×C42—61no—
236C2×C42C22×D311yescharacteristic
237#106—64no—
238C4⋊C8—34no—
239#103—24no—
240#103—38no—
241#103—42no—
242#102—34no—
243#102—62no—
244C4×C8—22no—
245C4×C8—64no—
246#95—32no—
247#95—62no—
248#95—62no—
249#95—62no—
250#95—61no—
251C23×C3C22×D411yescharacteristic
252C22×D3—22no—
253C22×D3—22no—
254C22×D3—22no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×Q8—42no—
262C3×Q8—42no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—24no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4C4×D412yes—
272C2×C3×C4C4×D412yes—
273C2×C3×C4C2×SD1612yes—
274C2×C3×C4#13512yes—
275C2×C3×C4C23×C411yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4H(5,𝔽2)11yescharacteristic
280C2×C3×C42−1+411yescharacteristic
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3C2×SD1614yes—
296D12—22no—
297D12—22no—
298D12—22no—
299D12C4×D414yes—
300C4×D3—48no—
301C4×D3—88no—
302C3×C8—22no—
303C3×C8—41no—
304C3⋊C8—24no—
305C24—61no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8C2×C4×D314yes—
312C2×D4—32no—
313C2×D4—32no—
314C2×D4—32no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—64no—
325C22×C4—61no—
326C22×C4—124no—
327C22×C4—124no—
328C22×C4—122no—
329C22×C4—122no—
330C22×C4—242no—
331C22×C4D3×D411yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4C23×D311yescharacteristic
334SD16—616no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—34no—
341C2×C8—62no—
342C2×C8—62no—
343C2×C8D3×D412yes—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—34no—
349C4⋊C4—32no—
350C4⋊C4—32no—
351C4⋊C4—42no—
352C4⋊C4—62no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C4⋊C4—81no—
358C4⋊C4—124no—
359C4⋊C4—124no—
360C22⋊C4—64no—
361C22⋊C4—124no—
362C42—62no—
363C42—62no—
364C42—124no—
365C42—122no—
366C42C2×C4×D312yes—
367C42#23712yes—
368C22×C3C2×C4×D411yescharacteristic
369C22×C3D4211yescharacteristic
370C22×C3#54811yescharacteristic
371C22×C3C22×SD1611yescharacteristic
372C22×C3C2×#13511yescharacteristic
373C22×C3#57711yescharacteristic
374C22×C3#57811yescharacteristic
375C2×D3—24no—
376C2×D3—42no—
377C2×D3—42no—
378C2×D3—42no—
379C2×D3—42no—
380C2×D3—42no—
381C2×D3—42no—
382C3×C4—22no—
383C3×C4—44no—
384C3×C4—82no—
385C3×C4C4×SD1614yescomplemented
386C3×C4C2×C4×D411yescharacteristic
387C3×C4#52011yescharacteristic
388C3×C4#54711yescharacteristic
389C3×C4#55311yescharacteristic
390Dic3—24no—
391Dic3—42no—
392Dic3—42no—
393Dic3—42no—
394Dic3—81no—
395C23—62no—
396C23—62no—
397C23—62no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23—121no—
402C23C2×D3×D411yescharacteristic
403Q8—24no—
404Q8—24no—
405Q8—42no—
406Q8—42no—
407D4—34no—
408D4—62no—
409D4—62no—
410D4—62no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—34no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—128no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—248no—
436C2×C4#67⋊C412yes—
437C2×C4#67⋊C412yes—
438C2×C4D3×SD1612yes—
439C2×C4#91912yes—
440C2×C4C22×C4×D311yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×#23711yescharacteristic
443C2×C4#100411yescharacteristic
444C2×C4D3×#4111yescharacteristic
445C2×C4#101011yescharacteristic
446C8—22no—
447C8—41no—
448C8—64no—
449C2×C3#236111yescharacteristic
450C2×C3C2×C4×SD1611yescharacteristic
451C2×C3#294711yescharacteristic
452C2×C3D4×SD1611yescharacteristic
453C2×C3#328311yescharacteristic
454C2×C3#335911yescharacteristic
455C2×C3#336011yescharacteristic
456D3—44no—
457C22—64no—
458C22—122no—
459C22—122no—
460C22—122no—
461C22—122no—
462C22—122no—
463C22—122no—
464C22C2×#67⋊C411yescharacteristic
465C22#567711yescharacteristic
466C22#571711yescharacteristic
467C22C2×D3×SD1611yescharacteristic
468C22#576211yescharacteristic
469C22C2×#91911yescharacteristic
470C22#576611yescharacteristic
471C4—22no—
472C4—44no—
473C4—64no—
474C4—82no—
475C4—122no—
476C4—122no—
477C4—122no—
478C4—241no—
479C4#478814yescomplemented
480C4C4×D3×D411yescharacteristic
481C4#554611yescharacteristic
482C4#561111yescharacteristic
483C4#561411yescharacteristic
484C3#2299611yescharacteristic, complemented, Sylow
485C2—124no—
486C2#7643311yescharacteristic
487C2C2×#478811yescharacteristic
488C2#7877411yescharacteristic
489C2#8162011yescharacteristic
490C2#8162111yescharacteristic
491C2#8177111yescharacteristic
492C2#8180611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1956D3
hypocenterC3#22996
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. #83229C2semidirect
5. C3×#1956C2semidirect
7. #70254C2semidirect
8. #70237C2semidirect
10. C8×C4⋊Dic3C2semidirect
11. #69568C2semidirect
28. #5083C4semidirect
48. #1956D3semidirect
62. C3×C4⋊Q8C2×C4semidirect
95. #835C2×C4semidirect
105. C4⋊(C3⋊C8)C2×C4semidirect
189. C4⋊Dic3SD16semidirect
211. C4⋊Q8C4×D3semidirect
385. C3×C4C4×SD16semidirect
479. C4#4788semidirect
484. C3#22996semidirect

Maximal subgroups

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

subgroupindex
#832292
C2×#50832
C2×#47882
#743602
C3×#19562
#702762
#702542
#702372
C4×#8212
C8×C4⋊Dic32
#695682
#229963