Finite Group Databasebeta

← #10829286

Subgroups — #10829286

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5734C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3C2×#4710C211yesmaximal, characteristic, complemented
4#74240C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69837C211yesmaximal, characteristic, complemented
7#69556C211yesmaximal, characteristic, complemented
8#68887C212yesmaximal, complemented
9#68879C212yesmaximal, complemented
10#68877C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#24133—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947—22no—
15C22×#204C2211yescharacteristic
16#5734—24no—
17#5734—24no—
18C2×#897C2212yes—
19C2×#895—21no—
20C2×#895C2212yes—
21C2×#892—21no—
22C2×#892C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#5094—24no—
34#4987C2212yes—
35C2×#810C2212yes—
36C2×#809C2212yes—
37C2×#808C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4710—24no—
40#4672—21no—
41#4670—22no—
42#4661—21no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45#4529C2212yes—
46C2×#555—31no—
47C2×#475—31no—
48C2×#440—31no—
49#2536—31no—
50#2034—32no—
51#2000—32no—
52#1993D311yescharacteristic, complemented
53#1955—31no—
54#1873—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×D12—22no—
59C22×D12D411yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3C2312yes—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72#947—216no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4D412yes—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2312yes—
87C2×#204—22no—
88C2×#204—22no—
89C2×#204—22no—
90#897—24no—
91#897—42no—
92#895—28no—
93#895—24no—
94#895—24no—
95#892—28no—
96#892—24no—
97#892—24no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12D414yes—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C3×#97—22no—
105C3×#95—21no—
106C3×#95—21no—
107C3⋊#95—22no—
108#821—22no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—42no—
111#818—22no—
112#816—22no—
113#810—24no—
114#809—42no—
115#808—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16—31no—
118#555—64no—
119#555—64no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#124—62no—
122C2×C22⋊Q8—32no—
123C2×#121—32no—
124C2×#121—61no—
125C2×C4×Q8—31no—
126C2×C4×D4—31no—
127C2×C4×D4—31no—
128C2×#117—32no—
129C2×#117—61no—
130#475—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97C2×D311yescharacteristic
142#398—61no—
143#394—61no—
144#388—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340C2×D312yes—
149#340C2×D312yes—
150C23×D3—21no—
151C23×D3C2×D412yes—
152C2×C3×Q8—24no—
153C2×C3×Q8—82no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D412yes—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—22no—
168C2×D12—21no—
169C2×D12—28no—
170C2×D12—42no—
171C2×D12—82no—
172C2×D12—82no—
173C2×D12C2×D412yes—
174C2×C4×D3—24no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—44no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6#4112yes—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8#4112yes—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C2×D414yes—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201(C2×D3)⋊C4—28no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—44no—
207C4⋊Dic3—22no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—44no—
212C4⋊Dic3C2×D412yes—
213Dic3⋊C4—42no—
214C4×Dic3—28no—
215C4×Dic3—42no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219#204—28no—
220C22×Q8—21no—
221C22×Q8—31no—
222C22×D4—31no—
223C22×D4—31no—
224C22×D4—62no—
225C23×C4—32no—
226C23×C4—61no—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8—32no—
231C22×C8C22×D311yescharacteristic
232C22×C8C22×D311yescharacteristic
233C4⋊Q8—22no—
234C4⋊Q8D1214yes—
235#124—616no—
236C22⋊Q8—64no—
237C22⋊Q8—122no—
238#121—64no—
239#121—68no—
240#121—64no—
241C4×Q8—62no—
242C4×Q8—62no—
243C4×D4—34no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4—62no—
247#117—64no—
248#117—68no—
249#117—64no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4C22×D312yes—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—62no—
265C2×C42—61no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—62no—
270#103—62no—
271#103—64no—
272#102—62no—
273#102—122no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—22no—
277#95—21no—
278#95—21no—
279#95—62no—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C3×Q8—84no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320D12—22no—
321D12—22no—
322D12—44no—
323D12—41no—
324D12—41no—
325D12—88no—
326C4×D3—48no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—42no—
331Dic6—22no—
332Dic6—22no—
333Dic6—41no—
334Dic6—41no—
335C3×C8—22no—
336C3×C8—41no—
337C3⋊C8—42no—
338C24—32no—
339C24—61no—
340C2×Q8—24no—
341C2×Q8—32no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—82no—
348C2×D4—32no—
349C2×D4—68no—
350C2×D4—62no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—122no—
357C2×D4—242no—
358C2×D4—242no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—32no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—64no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—124no—
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375C22×C4C23×D311yescharacteristic
376SD16—68no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8#23912yes—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—32no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—124no—
396C4⋊C4—122no—
397C4⋊C4C2×D1214yes—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—124no—
403C22⋊C4—124no—
404C42—68no—
405C42—122no—
406C42C2×D1212yes—
407C42D3×D412yes—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54611yescharacteristic
410C22×C3#54711yescharacteristic
411C22×C3#57512yes—
412C22×C3#57712yes—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—82no—
424C2×D3—82no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—44no—
428C3×C4—82no—
429C3×C4#52511yescharacteristic
430C3×C4#53711yescharacteristic
431C3×C4#54811yescharacteristic
432C3×C4#55011yescharacteristic
433Dic3—42no—
434Dic3—42no—
435Dic3—42no—
436Dic3—81no—
437C23—62no—
438C23—62no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23—62no—
443C23—62no—
444C23—62no—
445C23—121no—
446C23—121no—
447C23—122no—
448C23—122no—
449C23C2×D3×D411yescharacteristic
450Q8—62no—
451Q8—62no—
452Q8—84no—
453Q8—121no—
454Q8—121no—
455D4—62no—
456D4—62no—
457D4—124no—
458D4—121no—
459D4—121no—
460D4—248no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—128no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4#88212yes—
485C2×C4#88212yes—
486C2×C4C22×D1211yescharacteristic
487C2×C4C2×D3×D411yescharacteristic
488C2×C4C2×#23911yescharacteristic
489C2×C4#100711yescharacteristic
490C2×C4D3×#4111yescharacteristic
491C2×C4#100911yescharacteristic
492C8—22no—
493C8—41no—
494C8—122no—
495C2×C3#240711yescharacteristic
496C2×C3#300511yescharacteristic
497C2×C3#300911yescharacteristic
498C2×C3#332112yes—
499C2×C3#334812yes—
500D3—82no—
501D3—82no—
502C22—122no—
503C22—122no—
504C22—122no—
505C22—122no—
506C22—122no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—242no—
513C22—242no—
514C22C2×#88211yescharacteristic
515C22#567611yescharacteristic
516C22#572611yescharacteristic
517C22#577512yes—
518C22#577712yes—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—82no—
523C4—122no—
524C4—122no—
525C4—122no—
526C4—241no—
527C4#557611yescharacteristic
528C4#557711yescharacteristic
529C4#562711yescharacteristic
530C4#562911yescharacteristic
531C3#2413311yescharacteristic, complemented, Sylow
532C2—242no—
533C2—242no—
534C2#7661811yescharacteristic
535C2#7892911yescharacteristic
536C2#7895911yescharacteristic
537C2#8170312yes—
538C2#8199512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC2×#5734C2
hypercenter#1993D3
hypocenterC3#24133
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
3. C2×#4710C2semidirect
4. #74240C2semidirect
5. C3×#1993C2semidirect
6. #69837C2semidirect
7. #69556C2semidirect
8. #68887C2semidirect
9. #68879C2semidirect
10. #68877C2semidirect
11. #68823C2semidirect
52. #1993D3semidirect
531. C3#24133semidirect

Maximal subgroups

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

subgroupindex
C2×#57342
C2×#50942
C2×#47102
#742402
C3×#19932
#698372
#695562
#688872
#688792
#688772
#688232
#241333