Finite Group Databasebeta

← #10877202

Subgroups — #10877202

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#384C211yesmaximal, characteristic, complemented
2C4×#895C211yesmaximal, characteristic
3C3×#1948C211yesmaximal, characteristic, complemented
4#70587C212yesmaximal, complemented
5#70579C212yesmaximal, complemented
6#70479C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic, complemented
8#69584C211yesmaximal, characteristic, complemented
9#69467C211yesmaximal, characteristic, complemented
10#68880C212yesmaximal
11#68843C212yesmaximal, complemented
12#27678—31nomaximal, Sylow
13C2×#895C2211yescharacteristic
14C2×D3×C4⋊C4—21no—
15C2×D3×C4⋊C4—21no—
16C2×D3×C4⋊C4C2212yes—
17C2×C4×D12—21no—
18C2×C4×D12C2211yescharacteristic
19C2×C42×D3C2211yescharacteristic
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C3×#384C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#821C2212yes—
25C2×(C2×D3)⋊C8C2212yes—
26#4988C2211yescharacteristic
27#4987C2211yescharacteristic
28#4984C2211yescharacteristic
29#4975—21no—
30#4975—21no—
31#4975C2212yes—
32C2×#809C2212yes—
33C2×#808C2212yes—
34C4×(C2×D3)⋊C4—21no—
35#4935C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4670—21no—
38#4665—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×#121—31no—
45C2×#384—31no—
46#2100—32no—
47#2094—32no—
48#1948—31no—
49#1948D311yescharacteristic, complemented
50#1874—32no—
51#1853—32no—
52#1802—31no—
53#1802—31no—
54#1797—31no—
55C22×D12—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—22no—
58C22×C4×D3—21no—
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×C3×C8C2312yes—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87#895—22no—
88#895Q814yes—
89D3×C4⋊C4—42no—
90D3×C4⋊C4—42no—
91D3×C4⋊C4—44no—
92D3×C4⋊C4—44no—
93C4×D12—22no—
94C4×D12—22no—
95C4×D12—44no—
96C42×D3—22no—
97C42×D3—22no—
98C3×#106—42no—
99C3×C4⋊C8—22no—
100C3×#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103#821—42no—
104(C2×D3)⋊C8—24no—
105#809—42no—
106#808—42no—
107C4⋊(C3⋊C8)—22no—
108C2×#121—31no—
109C2×C4×D4—31no—
110C2×C4×D4—61no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—61no—
113C22×C4⋊C4—61no—
114C22×C42—31no—
115C2×#106—32no—
116C2×#106C2×D312yes—
117C2×C4⋊C8—31no—
118C2×C4⋊C8C2×D311yescharacteristic
119C2×#102—32no—
120C2×#102—32no—
121C2×C22⋊C8—32no—
122#390—31no—
123#389—31no—
124#388—61no—
125#386—61no—
126#384—32no—
127#384—31no—
128#384—61no—
129#384—61no—
130#384C2×D311yescharacteristic
131#380—31no—
132C4×C4⋊C4C2×D311yescharacteristic
133C4×C22⋊C4—61no—
134#340—32no—
135#340—32no—
136#340—32no—
137#340C2×D312yes—
138#336—32no—
139C23×D3—21no—
140C23×D3C2×D411yescharacteristic
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—21no—
148C22×Dic3—22no—
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C2×D12—21no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—24no—
157C2×D12—21no—
158C2×D12—42no—
159C2×D12—42no—
160C2×D12C2×Q812yes—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—21no—
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—44no—
171C2×C4×D3—44no—
172C2×C4×D3—44no—
173C2×C4×D3—42no—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×C4×D3SD1614yes—
177C2×C4×D3C2×Q812yes—
178C2×C4×D3#4112yes—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×Q812yes—
190C3×C42—42no—
191C3×C42#4112yes—
192C3×C42#4112yes—
193(C2×D3)⋊C4—24no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—42no—
197(C2×D3)⋊C4—42no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—42no—
203C4⋊Dic3—42no—
204C4⋊Dic3—42no—
205Dic3⋊C4—44no—
206Dic3⋊C4—44no—
207Dic3⋊C4—44no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C4×Dic3—21no—
211C4×Dic3—21no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C22×D4—31no—
216C22×D4—61no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—62no—
221C23×C4—61no—
222C22×C8—32no—
223C22×C8C22×D312yes—
224#121—34no—
225#121—62no—
226C4×D4—62no—
227C4×D4—62no—
228C4×D4—124no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—122no—
240C2×C4⋊C4—122no—
241C2×C4⋊C4—124no—
242C2×C4⋊C4—124no—
243C2×C4⋊C4C22×D312yes—
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—61no—
251C2×C42—21no—
252C2×C42—31no—
253C2×C42—31no—
254C2×C42—31no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42C22×D311yescharacteristic
258#106—42no—
259#106—122no—
260C4⋊C8—22no—
261C4⋊C8—62no—
262#102—122no—
263#102—122no—
264C22⋊C8—64no—
265#95—21no—
266#95—61no—
267#95—61no—
268C23×C3C22×D411yescharacteristic
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—22no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3C22⋊Q812yes—
281C22×D3C2×SD1612yes—
282C22×D3AGL(1,ℤ/8)12yes—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4#12312yes—
295C2×C3×C4#12312yes—
296C2×C3×C4C2×SD1612yes—
297C2×C3×C4AGL(1,ℤ/8)12yes—
298C2×C3×C4C22×Q811yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—41no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3C22⋊Q812yes—
317C2×Dic3C2×SD1612yes—
318C2×Dic3#13712yes—
319D12—22no—
320D12—22no—
321D12—48no—
322D12—41no—
323D12—41no—
324C4×D3—28no—
325C4×D3—22no—
326C4×D3—22no—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—28no—
330C4×D3—88no—
331C4×D3—84no—
332C3×C8—42no—
333C3⋊C8—42no—
334C24—31no—
335C24—61no—
336C2×D4—32no—
337C2×D4—64no—
338C2×D4—61no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—61no—
342C2×D4—61no—
343C2×D4—122no—
344C2×D4—122no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—34no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—124no—
366C22×C4—124no—
367C22×C4—124no—
368C22×C4—122no—
369C22×C4—122no—
370C22×C4—122no—
371C22×C4D3×D411yescharacteristic
372C22×C4D3×D411yescharacteristic
373C22×C4C23×D311yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—61no—
388C4⋊C4—62no—
389C4⋊C4—61no—
390C4⋊C4—62no—
391C4⋊C4—122no—
392C4⋊C4—122no—
393C4⋊C4—124no—
394C4⋊C4—124no—
395C4⋊C4—124no—
396C4⋊C4—122no—
397C4⋊C4D3×Q812yes—
398C22⋊C4—64no—
399C22⋊C4—122no—
400C22⋊C4—122no—
401C22⋊C4—122no—
402C22⋊C4—122no—
403C42—42no—
404C42—62no—
405C42—62no—
406C42—61no—
407C42—61no—
408C42#23512yes—
409C42#23912yes—
410C22×C3C2×C22⋊Q811yescharacteristic
411C22×C3C2×#12311yescharacteristic
412C22×C3#53511yescharacteristic
413C22×C3C22×SD1611yescharacteristic
414C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
415C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C3C2×#13711yescharacteristic
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—81no—
427C2×D3—81no—
428C2×D3—81no—
429C2×D3—81no—
430C2×D3#47614yes—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—81no—
434C3×C4—82no—
435C3×C4#48114yes—
436C3×C4#55311yescharacteristic
437C3×C4#55311yescharacteristic
438C3×C4#55411yescharacteristic
439C3×C4#55411yescharacteristic
440Dic3—22no—
441Dic3—22no—
442Dic3—22no—
443Dic3—81no—
444Dic3—82no—
445C23—32no—
446C23—32no—
447C23—32no—
448C23—61no—
449C23—61no—
450C23—61no—
451C23—61no—
452C23—62no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23C2×D3×D411yescharacteristic
460D4—62no—
461D4—62no—
462D4—128no—
463D4—121no—
464D4—121no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—32no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—68no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—68no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—248no—
498C2×C4—244no—
499C2×C4#89412yes—
500C2×C4#89412yes—
501C2×C4D3×SD1612yes—
502C2×C4#91412yes—
503C2×C4C2×#23511yescharacteristic
504C2×C4#100411yescharacteristic
505C2×C4C2×D3×Q811yescharacteristic
506C2×C4C2×#23911yescharacteristic
507C2×C4D3×#4111yescharacteristic
508C2×C4#100911yescharacteristic
509C8—42no—
510C8—122no—
511C2×C3#248711yescharacteristic
512C2×C3C2×#47611yescharacteristic
513C2×C3#307911yescharacteristic
514C2×C3C2×#48111yescharacteristic
515C2×C3#309211yescharacteristic
516C2×C3#320611yescharacteristic
517C2×C3#321711yescharacteristic
518D3—24no—
519D3—82no—
520C22—34no—
521C22—62no—
522C22—62no—
523C22—62no—
524C22—62no—
525C22—62no—
526C22—62no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—241no—
531C22—241no—
532C22—241no—
533C22—241no—
534C22C2×#89411yescharacteristic
535C22#560711yescharacteristic
536C22D3×C22⋊Q811yescharacteristic
537C22C2×D3×SD1611yescharacteristic
538C22C2×#91411yescharacteristic
539C22D3×AGL(1,ℤ/8)11yescharacteristic
540C22D3×#13711yescharacteristic
541C4—22no—
542C4—42no—
543C4—62no—
544C4—62no—
545C4—62no—
546C4—82no—
547C4—81no—
548C4—242no—
549C4—241no—
550C4#486314yes—
551C4#555811yescharacteristic
552C4#557911yescharacteristic
553C4#569511yescharacteristic
554C4#573011yescharacteristic
555C3#2767811yescharacteristic, complemented, Sylow
556C2—64no—
557C2—242no—
558C2#7623211yescharacteristic
559C2C2×#486311yescharacteristic
560C2#7921511yescharacteristic
561C2#8039011yescharacteristic
562C2#8043211yescharacteristic
563C2#8101211yescharacteristic
564C2D3×#47611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1948D3
hypocenterC3#27678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. D3×#384C2semidirect
3. C3×#1948C2semidirect
4. #70587C2semidirect
5. #70579C2semidirect
6. #70479C2semidirect
7. #70475C2semidirect
8. #69584C2semidirect
9. #69467C2semidirect
11. #68843C2semidirect
49. #1948D3semidirect
555. C3#27678semidirect

Maximal subgroups

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

subgroupindex
D3×#3842
C4×#8952
C3×#19482
#705872
#705792
#704792
#704752
#695842
#694672
#688802
#688432
#276783