Finite Group Databasebeta

← #10849343

Subgroups — #10849343

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5681C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic, complemented
3#83243C211yesmaximal, characteristic, complemented
4C2×#5087C211yesmaximal, characteristic, complemented
5#71721C211yesmaximal, characteristic, complemented
6#70279C212yesmaximal, complemented
7#70235C212yesmaximal, complemented
8#70211C212yesmaximal
9#70167C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#25660—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×#947C2211yescharacteristic
16C2×Q8×Dic3C2211yescharacteristic
17C2×#940C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19#5681—24no—
20#5681—24no—
21C2×#886—21no—
22C2×#885—21no—
23C2×#883—21no—
24C2×#67⋊C4—21no—
25C2×C4×D12C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#489—24no—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#97C2211yescharacteristic
31#5230—21no—
32#5230—21no—
33C2×#835C2212yes—
34C2×#832C2212yes—
35C2×#821C2212yes—
36#5087—24no—
37#4975—21no—
38#4973—21no—
39#4972C2212yes—
40#4972C2212yes—
41C2×#809C2212yes—
42C2×#807C2212yes—
43C4×C4⋊Dic3C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4551C2212yes—
46#4550C2212yes—
47#4550C2212yes—
48C2×#559—31no—
49C2×#489D311yescharacteristic, complemented
50C2×#474—31no—
51#2567—31no—
52#2042—32no—
53#2041—32no—
54#1985—31no—
55#1983—31no—
56#1863—32no—
57#1862—32no—
58#1762—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6D411yescharacteristic
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#124—22no—
72C3×#124D414yes—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42C2311yescharacteristic
78C2×#217—21no—
79#947—22no—
80#947—22no—
81Q8×Dic3—22no—
82Q8×Dic3—22no—
83#940—22no—
84#940—22no—
85D4×Dic3—22no—
86D4×Dic3—22no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2312yes—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D411yescharacteristic
97C2×Dic3⋊C4D411yescharacteristic
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3C2311yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103#886—28no—
104#885—28no—
105#883—28no—
106#67⋊C4—28no—
107C4×D12—22no—
108C4×D12D414yes—
109C4×Dic6—22no—
110C4×Dic6—22no—
111C3×#103—24no—
112C3×#102—24no—
113C3×#97—22no—
114#835—24no—
115#832—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#821—24no—
120#809—24no—
121#807—24no—
122C4⋊(C3⋊C8)—22no—
123#559—64no—
124#559—64no—
125C2×C4⋊Q8C2×D311yescharacteristic
126C2×#127—31no—
127C2×#124—31no—
128C2×#124—61no—
129C2×#124C2×D311yescharacteristic
130C2×#123—61no—
131C2×#123—61no—
132C2×C4×Q8—31no—
133C2×C4×Q8—31no—
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137#489—24no—
138#474—64no—
139C2×#107—32no—
140C2×C4⋊C8—31no—
141C2×#103—32no—
142C2×#103C2×D312yes—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102C2×D312yes—
147C2×#97C2×D311yescharacteristic
148#395—61no—
149#395—61no—
150#391—61no—
151#389—32no—
152#389—32no—
153#384—61no—
154C4×C4⋊C4—31no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C23×D3—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—42no—
164C2×C3×Q8—82no—
165C2×C3×Q8C2×D412yes—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4C2×D412yes—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—42no—
178C2×#67—42no—
179C2×#67—82no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—24no—
183C22×Dic3—82no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—21no—
191C2×D12C2×D412yes—
192C2×C4×D3—24no—
193C2×C4×D3—82no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—41no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207#217—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3C2×D412yes—
216C4⋊Dic3#4114yes—
217Dic3⋊C4—22no—
218Dic3⋊C4—22no—
219Dic3⋊C4—82no—
220Dic3⋊C4—82no—
221C4×Dic3—24no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C4×Dic3—42no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×Q8—21no—
231C22×Q8—31no—
232C22×Q8C22×D311yescharacteristic
233C22×D4—31no—
234C22×D4—61no—
235C22×D4—61no—
236C22×D4C22×D311yescharacteristic
237C23×C4—61no—
238C23×C4—61no—
239C22×C8—32no—
240C22×C8C22×D312yes—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243#127—62no—
244#127—62no—
245#124—22no—
246#124—68no—
247#124—62no—
248#124—62no—
249#124#6714yes—
250#123—68no—
251#123—68no—
252C4×Q8—62no—
253C4×Q8—62no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×D4—34no—
257C4×D4—68no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42—61no—
280C2×C42—61no—
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—62no—
284#103—24no—
285#103—64no—
286#102—24no—
287#102—64no—
288#102—64no—
289#102—64no—
290#97—22no—
291#95—62no—
292#95—62no—
293#95—61no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C3×Q8—24no—
306C3×Q8—41no—
307C3×Q8—41no—
308C3×Q8—84no—
309C3×D4—24no—
310C3×D4—41no—
311C3×D4—41no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
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—24no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—84no—
343C2×Dic3#13712yes—
344D12—22no—
345D12—22no—
346D12—42no—
347C4×D3—84no—
348Dic6—44no—
349C3×C8—42no—
350C3⋊C8—42no—
351C24—21no—
352C24—61no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—42no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—82no—
360C2×Q8—121no—
361C2×Q8C2×#6712yes—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—32no—
366C2×D4—64no—
367C2×D4—64no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—61no—
371C2×D4—122no—
372C2×D4—122no—
373C2×D4—242no—
374C2×D4C2×#6712yes—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—64no—
384C22×C4—64no—
385C22×C4—242no—
386C22×C4—242no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—32no—
399C4⋊C4—34no—
400C4⋊C4—42no—
401C4⋊C4—64no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—242no—
408C4⋊C4—242no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C42—64no—
416C42—61no—
417C42—61no—
418C42—61no—
419C42—61no—
420C42—122no—
421C42D3×D412yes—
422C42C2×#6712yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3C2×#12011yescharacteristic
429C22×C3#54812yes—
430C22×C3C2×#13711yescharacteristic
431C22×C3#57511yescharacteristic
432C22×C3#57612yes—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—82no—
437C2×D3—82no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—42no—
441C3×C4—82no—
442C3×C4#53211yescharacteristic
443C3×C4#53611yescharacteristic
444C3×C4#54712yes—
445Dic3—22no—
446Dic3—44no—
447Dic3—41no—
448Dic3—42no—
449Dic3—41no—
450Dic3—41no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—62no—
457C23—122no—
458C23—122no—
459C23—121no—
460C23—121no—
461C23C2×D3×D411yescharacteristic
462Q8—24no—
463Q8—41no—
464Q8—41no—
465Q8—84no—
466Q8—124no—
467D4—24no—
468D4—41no—
469D4—41no—
470D4—62no—
471D4—62no—
472D4—128no—
473D4—122no—
474D4—244no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—32no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—244no—
499C2×C4—244no—
500C2×C4#93712yes—
501C2×C4#93712yes—
502C2×C4C2×D3×D411yescharacteristic
503C2×C4D3×#4112yes—
504C2×C4#101012yes—
505C2×C4C22×#6711yescharacteristic
506C8—42no—
507C8—122no—
508C2×C3—82no—
509C2×C3#243211yescharacteristic
510C2×C3#300711yescharacteristic
511C2×C3#300811yescharacteristic
512C2×C3#337012yes—
513C2×C3#337512yes—
514D3—82no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—82no—
519C22—82no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—242no—
524C22—242no—
525C22#567712yes—
526C22#577412yes—
527C22D3×#13711yescharacteristic
528C22#577711yescharacteristic
529C22C2×#93711yescharacteristic
530C4—22no—
531C4—22no—
532C4—42no—
533C4—62no—
534C4—82no—
535C4—124no—
536C4—121no—
537C4—121no—
538C4—121no—
539C4—122no—
540C4#566811yescharacteristic
541C4#567511yescharacteristic
542C4#582412yes—
543C3#2566011yescharacteristic, complemented, Sylow
544C2—82no—
545C2—242no—
546C2#8167212yes—
547C2#8167512yes—
548C2#8295911yescharacteristic
549C2#8394711yescharacteristic
550C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenterC2×#489D3
hypocenterC3#25660
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. C2×C3×#489C2semidirect
3. #83243C2semidirect
4. C2×#5087C2semidirect
5. #71721C2semidirect
6. #70279C2semidirect
7. #70235C2semidirect
9. #70167C2semidirect
10. #69838C2semidirect
11. #69812C2semidirect
49. C2×#489D3semidirect
543. C3#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#56812
C2×C3×#4892
#832432
C2×#50872
#717212
#702792
#702352
#702112
#701672
#698382
#698122
#256603