Finite Group Databasebeta

← #10849327

Subgroups — #10849327

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5559C211yesmaximal, characteristic
2C2×#5008C211yesmaximal, characteristic, complemented
3C2×#4714C211yesmaximal, characteristic, complemented
4#73711C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic, complemented
6#70178C212yesmaximal, complemented
7#70177C212yesmaximal
8#69838C211yesmaximal, characteristic, complemented
9#69488C211yesmaximal, characteristic, complemented
10#68907C212yesmaximal, complemented
11#68895C212yesmaximal, complemented
12#25659—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#940C2211yescharacteristic
15C2×#936C2211yescharacteristic
16C2×#929—21no—
17C2×C4×#67—21no—
18#5559—24no—
19#5559—24no—
20C2×#893C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#885—21no—
23C2×#883—21no—
24C2×#875C2211yescharacteristic
25C2×#869—21no—
26C2×#869C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#97C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#832C2212yes—
32C2×#821C2212yes—
33C2×#816C2212yes—
34#5008—24no—
35#4984C2212yes—
36#4984C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×#808C2212yes—
40#4929—21no—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4714—24no—
43#4661—21no—
44#4661—21no—
45C3×#340C2212yes—
46#4550C2212yes—
47#4550C2212yes—
48C2×#559—31no—
49C2×#489—31no—
50C2×#476—31no—
51#2546—31no—
52#2042—32no—
53#2041—32no—
54#1985—31no—
55#1983—31no—
56#1865—32no—
57#1864—32no—
58#1762D311yescharacteristic, complemented
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×#67—21no—
62C22×#67—21no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×Dic6—21no—
66C22×Dic6D411yescharacteristic
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C3×C4×D4—22no—
70C3×C4×D4D414yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—21no—
77C2×#217—21no—
78#940—22no—
79#940—22no—
80#936—22no—
81#936—22no—
82#929—28no—
83C4×#67—28no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4D411yescharacteristic
94C2×Dic3⋊C4D411yescharacteristic
95C2×C4×Dic3—21no—
96C2×C4×Dic3C2311yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99#893—22no—
100#893—22no—
101#887—22no—
102#887—22no—
103#885—28no—
104#883—28no—
105#875—22no—
106#875D414yes—
107#869—22no—
108#869—22no—
109#869—44no—
110C4×Dic6—22no—
111C4×Dic6—22no—
112C3×#102—24no—
113C3×#97—22no—
114C3×#95—21no—
115#832—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#821—24no—
119#816—24no—
120#810—24no—
121#809—24no—
122#808—24no—
123C4⋊(C3⋊C8)—22no—
124#559—64no—
125#559—64no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—61no—
128C2×#127—31no—
129C2×#124—31no—
130C2×#124—31no—
131C2×#124—61no—
132C2×#123—61no—
133C2×#123—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8—31no—
136C2×C4×D4—31no—
137C2×C4×D4—61no—
138C2×C4×D4C2×D311yescharacteristic
139#489—64no—
140#476—64no—
141C2×#106—32no—
142C2×C4⋊C8—31no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102C2×D312yes—
149C2×#97C2×D311yescharacteristic
150#398—61no—
151#398—61no—
152#391—61no—
153#389—32no—
154#389—32no—
155C4×C4⋊C4C2×D311yescharacteristic
156#340C2×D312yes—
157#336—32no—
158#336—32no—
159C24×C3—21no—
160C23×D3—21no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—24no—
168C22×C3×C4—82no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—24no—
173C2×#67—24no—
174C2×#67—42no—
175C2×#67—42no—
176C2×#67—82no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12C2×D412yes—
187C2×C4×D3—24no—
188C2×C4×D3—82no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×Dic6—22no—
194C2×Dic6—41no—
195C2×Dic6—42no—
196C2×Dic6—42no—
197C2×Dic6—42no—
198C2×Dic6C2×D412yes—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4C2×D412yes—
205C3×C4⋊C4#4114yes—
206C3×C22⋊C4—24no—
207C3×C42—42no—
208C3×C42C2×D412yes—
209C3×C42C2×D412yes—
210#217—24no—
211#217—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—44no—
219C4⋊Dic3—42no—
220C4⋊Dic3—44no—
221Dic3⋊C4—24no—
222Dic3⋊C4—22no—
223Dic3⋊C4—22no—
224C4×Dic3—24no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C22×Q8—31no—
233C22×Q8—31no—
234C22×Q8—61no—
235C22×D4—31no—
236C22×D4—61no—
237C22×D4—61no—
238C22×D4C22×D311yescharacteristic
239C23×C4—21no—
240C23×C4—61no—
241C22×C8—32no—
242C22×C8C22×D312yes—
243C4⋊Q8—62no—
244C4⋊Q8—62no—
245C4⋊Q8—124no—
246#127—62no—
247#127—62no—
248#124—34no—
249#124—68no—
250#124—62no—
251#124—62no—
252#124—62no—
253#123—68no—
254#123—68no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—22no—
260C4×D4—62no—
261C4×D4—68no—
262C4×D4—62no—
263C4×D4D1214yes—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—21no—
281C2×C42—31no—
282C2×C42—31no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#106—64no—
286C4⋊C8—62no—
287#103—64no—
288#103—64no—
289#102—24no—
290#102—64no—
291#102—64no—
292#102—64no—
293#97—22no—
294#95—21no—
295#95—62no—
296#95—62no—
297C23×C3—22no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C22×D3—22no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—42no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—42no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—84no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4112yes—
325C2×C3×C42−1+412yes—
326#67—48no—
327#67—84no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—22no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3AGL(1,ℤ/8)12yes—
343D12—24no—
344D12—41no—
345D12—41no—
346C4×D3—84no—
347Dic6—24no—
348Dic6—44no—
349Dic6—41no—
350Dic6—41no—
351Dic6—84no—
352C3×C8—42no—
353C3⋊C8—42no—
354C24—21no—
355C24—61no—
356C2×Q8—32no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—61no—
360C2×Q8—62no—
361C2×Q8—62no—
362C2×Q8—121no—
363C2×Q8—122no—
364C2×Q8—122no—
365C2×Q8—122no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—32no—
370C2×D4—64no—
371C2×D4—64no—
372C2×D4—61no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—122no—
376C2×D4—122no—
377C2×D4—242no—
378C2×D4C2×D1212yes—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—64no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—82no—
390C22×C4—242no—
391C22×C4D3×D411yescharacteristic
392C22×C4D3×D411yescharacteristic
393C22×C4C23×D311yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—64no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—124no—
409C4⋊C4—124no—
410C4⋊C4—122no—
411C4⋊C4C2×D1212yes—
412C4⋊C4#23514yes—
413C22⋊C4—24no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C42—42no—
420C42—64no—
421C42—61no—
422C42—61no—
423C42—61no—
424C42—61no—
425C42C2×D1212yes—
426C42D3×D412yes—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54812yes—
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3#57511yescharacteristic
436C22×C3#57712yes—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—82no—
441C2×D3—82no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—44no—
445C3×C4—42no—
446C3×C4#53211yescharacteristic
447C3×C4#53611yescharacteristic
448C3×C4#54712yes—
449Dic3—22no—
450Dic3—42no—
451Dic3—41no—
452Dic3—41no—
453Dic3—41no—
454Dic3—82no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—62no—
461C23—121no—
462C23—122no—
463C23—121no—
464C23—122no—
465C23C2×D3×D411yescharacteristic
466Q8—64no—
467Q8—124no—
468Q8—121no—
469Q8—121no—
470Q8—244no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—64no—
475D4—128no—
476D4—121no—
477D4—121no—
478D4—244no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—32no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—62no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—61no—
498C2×C4—84no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—244no—
504C2×C4#88212yes—
505C2×C4#88212yes—
506C2×C4C22×D1211yescharacteristic
507C2×C4C2×#23512yes—
508C2×C4C2×D3×D411yescharacteristic
509C2×C4#101012yes—
510C8—42no—
511C8—122no—
512C2×C3—82no—
513C2×C3#243211yescharacteristic
514C2×C3#300611yescharacteristic
515C2×C3#300911yescharacteristic
516C2×C3#337112yes—
517C2×C3#337312yes—
518D3—82no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—242no—
528C22—242no—
529C22C2×#88211yescharacteristic
530C22#555212yes—
531C22D3×AGL(1,ℤ/8)11yescharacteristic
532C22#577512yes—
533C22#577711yescharacteristic
534C4—22no—
535C4—22no—
536C4—42no—
537C4—44no—
538C4—62no—
539C4—122no—
540C4—121no—
541C4—121no—
542C4—121no—
543C4—242no—
544C4#547711yescharacteristic
545C4#552611yescharacteristic
546C4#555112yes—
547C3#2565911yescharacteristic, complemented, Sylow
548C2—82no—
549C2—242no—
550C2#7348911yescharacteristic
551C2#7888411yescharacteristic
552C2#7892911yescharacteristic
553C2#7975412yes—
554C2#7985312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1762D3
hypocenterC3#25659
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#5008C2semidirect
3. C2×#4714C2semidirect
4. #73711C2semidirect
5. C3×#1762C2semidirect
6. #70178C2semidirect
8. #69838C2semidirect
9. #69488C2semidirect
10. #68907C2semidirect
11. #68895C2semidirect
58. #1762D3semidirect
547. C3#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#55592
C2×#50082
C2×#47142
#737112
C3×#17622
#701782
#701772
#698382
#694882
#689072
#688952
#256593