Finite Group Databasebeta

← #10848700

Subgroups — #10848700

Order 768.

1578 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic, complemented
2C2×C3×#493C211yesmaximal, characteristic, complemented
3#83192C211yesmaximal, characteristic, complemented
4C2×#5070C211yesmaximal, characteristic, complemented
5#72043C214yesmaximal, complemented
6#72037C214yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8#69777C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
10#25599—31nomaximal, Sylow
11C22×#237—21no—
12C22×C3×D8C2214yes—
13C2×C3×#127C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×#938—21no—
16C2×#938—21no—
17C2×#938—21no—
18C2×#938C2212yes—
19C2×#938C2212yes—
20C2×#936—21no—
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2212yes—
23#5716—24no—
24#5716—24no—
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C3×#493—24no—
28C2×C3×C4×C8C2211yescharacteristic
29#5221—21no—
30#5221—21no—
31#5221C2212yes—
32C2×#832C2214yes—
33C2×#832C2214yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5070—24no—
36#4935C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4551C2214yes—
39#4550C2212yes—
40C2×#532—31no—
41C2×#493D311yescharacteristic, complemented
42C2×#486—31no—
43#2856—31no—
44#1999—34no—
45#1990—31no—
46#1957—31no—
47#1879—34no—
48C4×C22⋊C8—31no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4C2312yes—
54C22×C3×D4C2312yes—
55C22×#67—22no—
56C22×#67—22no—
57C23×Dic3—22no—
58C2×#237—28no—
59C2×#237—44no—
60C2×#237—44no—
61C22×C4×D3D411yescharacteristic
62C22×C4×D3D411yescharacteristic
63C22×C4×D3C2311yescharacteristic
64C22×Dic6—21no—
65C2×C3×D8—28no—
66C2×C3×D8—28no—
67C2×C3×D8—44no—
68C22×C3×C8C2312yes—
69C3×#127—22no—
70C3×#127—22no—
71C3×#127—22no—
72C3×#127D414yescomplemented
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—22no—
79#938—28no—
80#938—24no—
81#938—24no—
82#938—44no—
83#938—44no—
84#938—44no—
85#936—28no—
86D4×Dic3—28no—
87D4×Dic3—24no—
88D4×Dic3—24no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C42×D3—22no—
99C42×D3—22no—
100#869—22no—
101#869D414yescomplemented
102C3×C4×C8—22no—
103#832—28no—
104#832—44no—
105(C2×D3)⋊C8—24no—
106C4×C3⋊C8—22no—
107C22×#41—61no—
108C22×D8C2×D314yes—
109#532—64no—
110#532—64no—
111C2×C4⋊Q8—31no—
112C2×#127C2×D311yescharacteristic
113C2×#127C2×D311yescharacteristic
114C2×#124—61no—
115C2×#121—32no—
116C2×#121—32no—
117C2×#121—61no—
118C2×#121—61no—
119C2×#121—61no—
120C2×C4×D4—32no—
121C2×C4×D4—61no—
122C22×C42—31no—
123#493—24no—
124#486—64no—
125C2×#102—34no—
126C2×#102—34no—
127C2×C22⋊C8—32no—
128C2×C4×C8—31no—
129C2×C4×C8C2×D311yescharacteristic
130#390—32no—
131#390—61no—
132#390—61no—
133#384—31no—
134#340—34no—
135#336—32no—
136C24×C3—22no—
137C24×C3—22no—
138C23×D3C2×D411yescharacteristic
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4C2×D414yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C2×#67—28no—
163C2×#67—44no—
164C2×#67—44no—
165C2×#67—44no—
166C2×#67—44no—
167C2×#67—44no—
168C22×Dic3—28no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—44no—
172C22×Dic3—44no—
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176#237—432no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3D814yescomplemented
187C2×Dic6—24no—
188C2×Dic6—42no—
189C2×Dic6—42no—
190C3×D8—416no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8#4114yescomplemented
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#217—28no—
197#217—28no—
198#217—44no—
199#217—44no—
200#217—44no—
201C4⋊Dic3—24no—
202C4⋊Dic3—22no—
203C4⋊Dic3—24no—
204C4⋊Dic3—42no—
205C4⋊Dic3—42no—
206C4⋊Dic3C2×D414yes—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—21no—
210C4×Dic3—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×#41—68no—
215C2×#41—124no—
216C2×#41—124no—
217C22×Q8—61no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—62no—
223C22×D4—62no—
224C22×D4C22×D312yes—
225C22×D4C22×D312yes—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—62no—
230C2×D8—28no—
231C2×D8—28no—
232C2×D8—44no—
233C22×C8—32no—
234C22×C8C22×D312yes—
235C4⋊Q8—34no—
236C4⋊Q8—62no—
237#127—22no—
238#127—22no—
239#127—22no—
240#127#6714yescomplemented
241#124—68no—
242#121—64no—
243#121—68no—
244#121—64no—
245#121—124no—
246#121—124no—
247#121—124no—
248C4×D4—64no—
249C4×D4—68no—
250C4×D4—64no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42—31no—
262C2×C42—31no—
263C2×C42—31no—
264C2×C42—62no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#102—68no—
268#102—124no—
269C22⋊C8—64no—
270C4×C8—22no—
271C4×C8—62no—
272C23×C3—24no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3C22×D411yescharacteristic
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3C2×D812yes—
294C3×D4—28no—
295C3×D4—44no—
296C3×D4—44no—
297C3×D4—44no—
298C3×D4—48no—
299C3×D4—48no—
300C3×D4—84no—
301C3×D4—84no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4C2×D812yes—
305C2×C3×C4#13512yes—
306C2×C3×C4AGL(1,ℤ/8)12yes—
307C2×C3×C4AGL(1,ℤ/8)12yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4112yes—
311C2×C3×C4H(5,𝔽2)12yes—
312#67—416no—
313#67—88no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—24no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—416no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3C2×D812yes—
330C4×D3—28no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—44no—
334Dic6—48no—
335C3×C8—24no—
336C3⋊C8—42no—
337C24—22no—
338C24—22no—
339C24—31no—
340#41—1232no—
341C2×Q8—64no—
342C2×Q8—122no—
343C2×Q8—122no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—68no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4C2×#6714yes—
370C22×C4—31no—
371C22×C4—34no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—68no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—124no—
387C22×C4—124no—
388C22×C4D3×D411yescharacteristic
389C22×C4D3×D411yescharacteristic
390C22×C4C23×D311yescharacteristic
391D8—416no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8#23714yescomplemented
398C4⋊C4—34no—
399C4⋊C4—64no—
400C4⋊C4—62no—
401C4⋊C4—64no—
402C4⋊C4—122no—
403C4⋊C4—122no—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—124no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C42—62no—
410C42—62no—
411C42—61no—
412C42—61no—
413C42D3×D412yes—
414C42C2×#6712yes—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#53211yescharacteristic
431C22×C3#53511yescharacteristic
432C22×C3C22×D811yescharacteristic
433C22×C3C2×#13511yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—41no—
440C2×D3—41no—
441C2×D3—41no—
442C2×D3—41no—
443C3×C4#44714yes—
444C3×C4#44914yes—
445C3×C4#54612yes—
446C3×C4#55012yes—
447Dic3—22no—
448Dic3—41no—
449Dic3—44no—
450Dic3—41no—
451Dic3—82no—
452C23—24no—
453C23—32no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23C2×D3×D411yescharacteristic
474Q8—128no—
475D4—28no—
476D4—48no—
477D4—48no—
478D4—44no—
479D4—44no—
480D4—44no—
481D4—84no—
482D4—84no—
483D4—1216no—
484D4—248no—
485C2×C4—32no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—68no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—64no—
496C2×C4—1216no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—124no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4D3×D812yes—
506C2×C4#91112yes—
507C2×C4#91112yes—
508C2×C4#91212yes—
509C2×C4#93712yes—
510C2×C4#93712yes—
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×#23712yes—
513C2×C4#100412yes—
514C2×C4C22×#6711yescharacteristic
515C8—24no—
516C8—122no—
517C2×C3—84no—
518C2×C3—162no—
519C2×C3#243111yescharacteristic
520C2×C3C2×#44711yescharacteristic
521C2×C3C2×#44911yescharacteristic
522C2×C3#316911yescharacteristic
523C2×C3#318411yescharacteristic
524C2×C3#320511yescharacteristic
525C2×C3#320711yescharacteristic
526D3—42no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—62no—
531C22—62no—
532C22—62no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22—121no—
548C22#571511yescharacteristic
549C22#571611yescharacteristic
550C22C2×D3×D811yescharacteristic
551C22C2×#91111yescharacteristic
552C22C2×#91111yescharacteristic
553C22C2×#91211yescharacteristic
554C22C2×#93711yescharacteristic
555C4—62no—
556C4—124no—
557C4—121no—
558C4—121no—
559C4—242no—
560C4#515614yes—
561C4#515814yes—
562C4#571812yes—
563C4#580612yes—
564C3#2559911yescharacteristic, complemented, Sylow
565C2—84no—
566C2—122no—
567C2—162no—
568C2#8177511yescharacteristic
569C2#8177711yescharacteristic
570C2#8178511yescharacteristic
571C2#8178611yescharacteristic
572C2#8318711yescharacteristic
573C2C2×#515611yescharacteristic
574C2C2×#515811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#493C2
socleC23×C3C22×D4
hypercenterC2×#493D3
hypocenterC3#25599
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#493

Complements

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

NH ≅ G/Ntype
1. C2×#5716C2semidirect
2. C2×C3×#493C2semidirect
3. #83192C2semidirect
4. C2×#5070C2semidirect
5. #72043C2semidirect
6. #72037C2semidirect
7. #71764C2semidirect
8. #69777C2semidirect
9. C4×(C2×D3)⋊C8C2semidirect
41. C2×#493D3semidirect
72. C3×#127D4semidirect
101. #869D4semidirect
186. C2×C4×D3D8semidirect
193. C2×C3×C8#41semidirect
240. #127#67semidirect
397. C2×C8#237semidirect
564. C3#25599semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×C3×#4932
#831922
C2×#50702
#720432
#720372
#717642
#697772
C4×(C2×D3)⋊C82
#255993