Finite Group Databasebeta

← #10848682

Subgroups — #10848682

Order 768.

1438 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4C2×#5079C211yesmaximal, characteristic, complemented
5#72054C214yesmaximal, complemented
6#72047C214yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8#69777C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
10#25597—31nomaximal, Sylow
11C22×#237—21no—
12C22×C3×SD16C2214yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×#946C2212yes—
16C2×#938—21no—
17C2×#938C2212yes—
18C2×#936—21no—
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2212yes—
21#5716—24no—
22#5716—24no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C3×#492—24no—
26C2×C3×C4×C8C2211yescharacteristic
27#5229C2212yes—
28C2×#835C2214yes—
29C2×#832C2214yes—
30C2×(C2×D3)⋊C8C2212yes—
31#5079—24no—
32#4987—21no—
33#4987—21no—
34#4972—21no—
35#4972—21no—
36#4935C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4551C2214yes—
39#4550C2212yes—
40C2×#532—31no—
41C2×#492D311yescharacteristic, complemented
42C2×#487—31no—
43#2857—31no—
44#2000—34no—
45#1990—31no—
46#1957—31no—
47#1877—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8C2312yes—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C22×#67—22no—
54C23×Dic3—22no—
55C2×#237—28no—
56C2×#237—44no—
57C2×#237—44no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×Dic6—21no—
62C2×C3×SD16—28no—
63C2×C3×SD16—28no—
64C2×C3×SD16—44no—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×#127—22no—
69C3×#127D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75#946—44no—
76#938—28no—
77#938—24no—
78#938—24no—
79#936—28no—
80D4×Dic3—28no—
81D4×Dic3—24no—
82D4×Dic3—24no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C42×D3—22no—
95C42×D3—22no—
96#869—22no—
97#869D414yes—
98C3×C4×C8—22no—
99#835—28no—
100#832—44no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103(C2×D3)⋊C8—24no—
104C4×C3⋊C8—22no—
105C22×#41—61no—
106C22×SD16C2×D314yes—
107#532—64no—
108#532—64no—
109C2×C4⋊Q8—31no—
110C2×C4⋊Q8C2×D311yescharacteristic
111C2×#127C2×D311yescharacteristic
112C2×#124—61no—
113C2×C22⋊Q8—32no—
114C2×#121—32no—
115C2×#121—61no—
116C2×C4×D4—32no—
117C2×C4×D4—61no—
118C22×C42—31no—
119#492—24no—
120#487—64no—
121C2×#103—34no—
122C2×#102—34no—
123C2×C22⋊C8—32no—
124C2×C4×C8—31no—
125C2×C4×C8C2×D311yescharacteristic
126#391—32no—
127#389—61no—
128#389—61no—
129#384—31no—
130#380—61no—
131#380—61no—
132#340—34no—
133#336—32no—
134C24×C3—22no—
135C23×D3C2×D411yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—22no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4C2×D414yes—
150C22×C3×C4—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C2×#67—28no—
155C2×#67—44no—
156C2×#67—44no—
157C22×Dic3—28no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—44no—
161C22×Dic3—44no—
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165#237—432no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3SD1614yescomplemented
176C2×Dic6—24no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C3×SD16—416no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8#4114yescomplemented
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C42C2×D412yes—
186C3×C42C2×D412yes—
187#217—28no—
188#217—28no—
189(C2×D3)⋊C4—44no—
190C4⋊Dic3—24no—
191C4⋊Dic3—22no—
192C4⋊Dic3—24no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4⋊Dic3C2×D414yes—
196Dic3⋊C4—44no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—21no—
200C4×Dic3—21no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×#41—68no—
205C2×#41—124no—
206C2×#41—124no—
207C22×Q8—61no—
208C22×Q8C22×D312yes—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—62no—
212C22×D4C22×D312yes—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—62no—
217C2×SD16—28no—
218C2×SD16—28no—
219C2×SD16—44no—
220C22×C8—32no—
221C22×C8C22×D312yes—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—34no—
225C4⋊Q8—62no—
226#127—22no—
227#127#6714yes—
228#124—68no—
229C22⋊Q8—124no—
230#121—64no—
231#121—68no—
232#121—64no—
233C4×D4—64no—
234C4×D4—68no—
235C4×D4—64no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—62no—
251C2×C42—62no—
252C2×C42C22×D311yescharacteristic
253#103—68no—
254#102—124no—
255C22⋊C8—64no—
256C4×C8—22no—
257C4×C8—62no—
258#95—62no—
259#95—62no—
260C23×C3—24no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3C22×D411yescharacteristic
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—21no—
274C22×D3C2×SD1612yes—
275C3×Q8—44no—
276C3×Q8—44no—
277C3×D4—28no—
278C3×D4—44no—
279C3×D4—48no—
280C3×D4—48no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C2×SD1612yes—
287C2×C3×C4#13512yes—
288C2×C3×C4AGL(1,ℤ/8)12yes—
289C2×C3×C4#13712yes—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4112yes—
293C2×C3×C4H(5,𝔽2)12yes—
294#67—416no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—24no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—416no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3C2×SD1612yes—
311C4×D3—28no—
312C4×D3—42no—
313C4×D3—42no—
314C4×D3—44no—
315Dic6—48no—
316C3×C8—24no—
317C3⋊C8—42no—
318C24—22no—
319C24—31no—
320#41—1232no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—64no—
326C2×Q8—122no—
327C2×Q8—122no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—68no—
338C2×D4—124no—
339C2×D4—124no—
340C2×D4C2×#6714yes—
341C22×C4—22no—
342C22×C4—31no—
343C22×C4—34no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—68no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—124no—
359C22×C4—124no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—416no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8#23714yescomplemented
370C4⋊C4—34no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—64no—
374C4⋊C4—62no—
375C4⋊C4—64no—
376C4⋊C4—124no—
377C4⋊C4—122no—
378C4⋊C4—122no—
379C22⋊C4—68no—
380C22⋊C4—68no—
381C22⋊C4—124no—
382C42—62no—
383C42—62no—
384C42—61no—
385C42—61no—
386C42D3×D412yes—
387C42C2×#6712yes—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3C2×#12011yescharacteristic
396C22×C3#53211yescharacteristic
397C22×C3#53511yescharacteristic
398C22×C3C22×SD1611yescharacteristic
399C22×C3C2×#13511yescharacteristic
400C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
401C22×C3C2×#13711yescharacteristic
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—41no—
406C2×D3—41no—
407C2×D3—41no—
408C2×D3—41no—
409C3×C4—82no—
410C3×C4#45014yes—
411C3×C4#45214yes—
412C3×C4#54612yes—
413C3×C4#55012yes—
414Dic3—22no—
415Dic3—41no—
416Dic3—44no—
417Dic3—41no—
418Dic3—82no—
419C23—24no—
420C23—32no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23C2×D3×D411yescharacteristic
434Q8—44no—
435Q8—44no—
436Q8—128no—
437D4—28no—
438D4—48no—
439D4—48no—
440D4—44no—
441D4—1216no—
442C2×C4—32no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—68no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—64no—
456C2×C4—1216no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—124no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4D3×SD1612yes—
466C2×C4#91412yes—
467C2×C4#91512yes—
468C2×C4#91612yes—
469C2×C4#93712yes—
470C2×C4#93712yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×#23712yes—
473C2×C4#100412yes—
474C2×C4C22×#6711yescharacteristic
475C8—24no—
476C8—122no—
477C2×C3—84no—
478C2×C3#243111yescharacteristic
479C2×C3C2×#45011yescharacteristic
480C2×C3C2×#45211yescharacteristic
481C2×C3#316711yescharacteristic
482C2×C3#318211yescharacteristic
483C2×C3#320611yescharacteristic
484C2×C3#320711yescharacteristic
485D3—42no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—62no—
490C22—62no—
491C22—62no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22#571511yescharacteristic
501C22#571611yescharacteristic
502C22C2×D3×SD1611yescharacteristic
503C22C2×#91411yescharacteristic
504C22C2×#91511yescharacteristic
505C22C2×#91611yescharacteristic
506C22C2×#93711yescharacteristic
507C4—62no—
508C4—82no—
509C4—124no—
510C4—121no—
511C4—121no—
512C4—242no—
513C4#516914yes—
514C4#517314yes—
515C4#571812yes—
516C4#580612yes—
517C3#2559711yescharacteristic, complemented, Sylow
518C2—84no—
519C2—122no—
520C2#8175211yescharacteristic
521C2#8175411yescharacteristic
522C2#8176411yescharacteristic
523C2#8176511yescharacteristic
524C2#8318711yescharacteristic
525C2C2×#516911yescharacteristic
526C2C2×#517311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#492C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenterC2×#492D3
hypocenterC3#25597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
2. C2×C3×#492C2semidirect
3. #83269C2semidirect
4. C2×#5079C2semidirect
5. #72054C2semidirect
6. #72047C2semidirect
7. #71764C2semidirect
8. #69777C2semidirect
9. C4×(C2×D3)⋊C8C2semidirect
41. C2×#492D3semidirect
175. C2×C4×D3SD16semidirect
182. C2×C3×C8#41semidirect
369. C2×C8#237semidirect
517. C3#25597semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×C3×#4922
#832692
C2×#50792
#720542
#720472
#717642
#697772
C4×(C2×D3)⋊C82
#255973