Finite Group Databasebeta

← #10828837

Subgroups — #10828837

Order 768.

1080 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5513C211yesmaximal, characteristic
2C22×#808C211yesmaximal, characteristic, complemented
3C2×#4722C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic
5C3×#1823C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic
7#69514C211yesmaximal, characteristic, complemented
8#68907C212yesmaximal, complemented
9#68877C212yesmaximal, complemented
10#68829C212yesmaximal, complemented
11#68824C212yesmaximal, complemented
12#24095—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#929—21no—
15C2×C4×#67C2211yescharacteristic
16C2×#926—21no—
17C22×C4⋊Dic3C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20#5513—24no—
21#5513—24no—
22C2×#898—21no—
23C2×#897C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#889—21no—
27C2×#887C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C2×(C2×D3)⋊C8C2212yes—
31C2×#818C2212yes—
32C2×#816C2212yes—
33#4987C2212yes—
34#4984C2212yes—
35C2×#810C2212yes—
36C2×#808—24no—
37C2×#808C2212yes—
38C2×#808C2212yes—
39#4958—21no—
40#4950—21no—
41#4722—24no—
42#4677—21no—
43#4659—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#552—31no—
49C2×#484—31no—
50C22×#106—31no—
51#2487—31no—
52#2059—32no—
53#2035—32no—
54#1892—31no—
55#1869—31no—
56#1865—32no—
57#1854—32no—
58#1823D311yescharacteristic, complemented
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67D411yescharacteristic
62C23×Dic3—21no—
63C22×C4×D3C2311yescharacteristic
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2312yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D414yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×#217—21no—
74C2×#217D411yescharacteristic
75#929—28no—
76C4×#67—22no—
77C4×#67—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4D411yescharacteristic
82#926—28no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91#924—22no—
92#924D414yes—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4D411yescharacteristic
97C2×C4×Dic3—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8—22no—
101C22×C3⋊C8—24no—
102C22×C3⋊C8C2312yes—
103#898—28no—
104#897—22no—
105#897—22no—
106#892—22no—
107#892—22no—
108#890—22no—
109#890—22no—
110#889—28no—
111#887—22no—
112#887—22no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117(C2×D3)⋊C8—24no—
118#818—24no—
119#816—24no—
120#810—24no—
121#808—24no—
122#808—24no—
123C23×C8—31no—
124#552—64no—
125#552—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#125—61no—
129C2×#123—61no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8—31no—
132C2×C4×Q8—31no—
133C2×C4×D4—31no—
134C2×#117—31no—
135C2×#117—61no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#484—64no—
139C2×#107—32no—
140C2×#106—32no—
141C2×#106—32no—
142C2×#106—64no—
143C2×#103—32no—
144C2×#103—32no—
145C2×C22⋊C8—32no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#399—61no—
149#389—32no—
150#386—61no—
151#385—61no—
152#380—32no—
153#380C2×D311yescharacteristic
154#340—32no—
155#340C2×D312yes—
156#340C2×D312yes—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3C2×D411yescharacteristic
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C2×#67—22no—
173C2×#67—22no—
174C2×#67—22no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—42no—
179C22×Dic3—82no—
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6C2×D412yes—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4C2×D412yes—
200C3×C4⋊C4C2×D412yes—
201C3×C22⋊C4—42no—
202#217—24no—
203#217—22no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—22no—
207(C2×D3)⋊C4—82no—
208C4⋊Dic3—24no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4⋊Dic3—82no—
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—22no—
219Dic3⋊C4—24no—
220C4×Dic3—24no—
221C4×Dic3—24no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8#4114yes—
228C22×Q8—31no—
229C22×D4—31no—
230C23×C4—21no—
231C23×C4—31no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—64no—
236C22×C8—62no—
237C22×C8C22×D312yes—
238#126—68no—
239#125—62no—
240#125—68no—
241#125—62no—
242#123—68no—
243C22⋊Q8—34no—
244C22⋊Q8—62no—
245C22⋊Q8—62no—
246C22⋊Q8—62no—
247C4×Q8—62no—
248C4×Q8—62no—
249C4×D4—62no—
250C4×D4—62no—
251#117—62no—
252#117—68no—
253#117—62no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4D1214yes—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—31no—
274C2×C22⋊C4—31no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279C2×C42—31no—
280C2×C42—61no—
281C2×C42—61no—
282#107—64no—
283#106—64no—
284#106—64no—
285#103—64no—
286#103—64no—
287C22⋊C8—22no—
288C22⋊C8—64no—
289#95—21no—
290#95—62no—
291#95—62no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#12012yes—
295C23×C3#13512yes—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C22×D3—21no—
299C22×D3—22no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—48no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4112yes—
321C2×C3×C42−1+412yes—
322#67—44no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—84no—
338C4×D3—41no—
339C4×D3—41no—
340C4×D3—42no—
341Dic6—24no—
342Dic6—41no—
343Dic6—41no—
344C3×C8—42no—
345C3⋊C8—24no—
346C24—31no—
347C24D3×D411yescharacteristic
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—62no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—122no—
374C22×C4—242no—
375C22×C4C2×D1212yes—
376C22×C4D3×D411yescharacteristic
377C22×C4D3×D412yes—
378C22×C4C23×D311yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—34no—
383C2×C8—68no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—32no—
392C4⋊C4—64no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—62no—
397C4⋊C4—64no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—122no—
402C4⋊C4—242no—
403C4⋊C4C2×D1212yes—
404C4⋊C4C2×D1212yes—
405C22⋊C4—42no—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22⋊C4—62no—
409C22⋊C4—64no—
410C22⋊C4—62no—
411C22⋊C4—242no—
412C42—64no—
413C42—64no—
414C42—61no—
415C42—61no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3#45214yes—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54712yes—
423C22×C3C2×#13511yescharacteristic
424C22×C3C2×#13711yescharacteristic
425C22×C3#57712yes—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—41no—
429C2×D3—41no—
430C2×D3—41no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—82no—
435C3×C4#52511yescharacteristic
436C3×C4#53611yescharacteristic
437C3×C4#54712yes—
438Dic3—42no—
439Dic3—41no—
440Dic3—41no—
441Dic3—41no—
442Dic3—42no—
443Dic3—82no—
444C23—22no—
445C23—22no—
446C23—62no—
447C23—61no—
448C23—61no—
449C23—61no—
450C23—62no—
451C23#88212yes—
452C23#91212yes—
453C23#91512yes—
454C23C2×D3×D411yescharacteristic
455Q8—64no—
456Q8—121no—
457Q8—121no—
458D4—124no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—62no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—62no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—244no—
491C2×C4#88212yes—
492C2×C4C22×D1211yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4D3×#4112yes—
495C2×C4#101012yes—
496C8—42no—
497C8—64no—
498C2×C3—24no—
499C2×C3#240611yescharacteristic
500C2×C3C2×#45211yescharacteristic
501C2×C3#301011yescharacteristic
502C2×C3#333712yes—
503C2×C3#334412yes—
504D3—81no—
505C22—24no—
506C22—22no—
507C22—22no—
508C22—24no—
509C22—122no—
510C22—121no—
511C22—121no—
512C22—122no—
513C22—121no—
514C22#478314yes—
515C22C2×#88211yescharacteristic
516C22#565512yes—
517C22C2×#91211yescharacteristic
518C22C2×#91511yescharacteristic
519C22#577812yes—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—121no—
525C4—122no—
526C4—121no—
527C4—122no—
528C4—121no—
529C4—242no—
530C4#552411yescharacteristic
531C4#552611yescharacteristic
532C4#565512yes—
533C3#2409511yescharacteristic, complemented, Sylow
534C2—24no—
535C2—241no—
536C2#7406311yescharacteristic
537C2C2×#478311yescharacteristic
538C2#7896011yescharacteristic
539C2#8129212yes—
540C2#8143712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#1823D3
hypocenterC3#24095
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. C22×#808C2semidirect
3. C2×#4722C2semidirect
5. C3×#1823C2semidirect
7. #69514C2semidirect
8. #68907C2semidirect
9. #68877C2semidirect
10. #68829C2semidirect
11. #68824C2semidirect
58. #1823D3semidirect
533. C3#24095semidirect

Maximal subgroups

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

subgroupindex
C2×#55132
C22×#8082
C2×#47222
#740282
C3×#18232
#699522
#695142
#689072
#688772
#688292
#688242
#240953