Finite Group Databasebeta

← #10829275

Subgroups — #10829275

Order 768.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5726C211yesmaximal, characteristic
2C2×#5091C211yesmaximal, 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#68878C212yesmaximal, complemented
10#68874C212yesmaximal, complemented
11#68822C212yesmaximal, complemented
12#24132—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946—21no—
16C22×#204C2211yescharacteristic
17#5726—24no—
18#5726—24no—
19C2×#897C2212yes—
20C2×#895C2212yes—
21C2×#894—22no—
22C2×D3×C4⋊C4C2212yes—
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#5091—24no—
34#4987C2212yes—
35C2×#810C2212yes—
36C2×#809C2212yes—
37C2×#807C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4710—24no—
40#4672—21no—
41#4670—22no—
42#4661—21no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#477—31no—
48C2×#440—31no—
49#2536—31no—
50#2033—32no—
51#2000—32no—
52#1993D311yescharacteristic, complemented
53#1955—31no—
54#1872—32no—
55#1852—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—48no—
60C22×D12—22no—
61C22×D12D411yescharacteristic
62C22×D12C2311yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3—21no—
65C22×C4×D3C2312yes—
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42C2311yescharacteristic
75#946—28no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4D412yes—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—21no—
87C2×C4×Dic3—21no—
88C22×C3⋊C8C2312yes—
89C2×#204—22no—
90C2×#204—22no—
91C2×#204—22no—
92#897—24no—
93#897—42no—
94#895—24no—
95#895—24no—
96#894—216no—
97D3×C4⋊C4—24no—
98D3×C4⋊C4—24no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12D414yes—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#97—22no—
106C3×#95—21no—
107C3×#95—21no—
108C3⋊#95—22no—
109#821—22no—
110(C2×D3)⋊C8—24no—
111(C2×D3)⋊C8—42no—
112#818—22no—
113#816—22no—
114#810—24no—
115#809—42no—
116#807—24no—
117C4⋊(C3⋊C8)—22no—
118C22×#41—61no—
119C22×SD16—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#123—62no—
124C2×C22⋊Q8—32no—
125C2×C22⋊Q8—61no—
126C2×#121—32no—
127C2×C4×Q8—31no—
128C2×C4×D4—31no—
129C2×C4×D4—31no—
130C22×C4⋊C4—32no—
131#477—64no—
132#440—64no—
133C2×#107—31no—
134C2×#107—32no—
135C2×C4⋊C8—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97C2×D311yescharacteristic
143#398—61no—
144#394—61no—
145#388—62no—
146#382C2×D311yescharacteristic
147#380—32no—
148#340—32no—
149#340C2×D312yes—
150#340C2×D312yes—
151C23×D3—21no—
152C23×D3C2×D412yes—
153C2×C3×Q8—24no—
154C2×C3×Q8—44no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3C2×D412yes—
164#239—432no—
165C2×D12—24no—
166C2×D12—22no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—28no—
172C2×D12—21no—
173C2×D12—44no—
174C2×D12—44no—
175C2×D12C2×D412yes—
176C2×C4×D3—28no—
177C2×C4×D3—24no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—44no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6#4112yescomplemented
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8#4112yescomplemented
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4C2×D414yes—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—44no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—44no—
216C4⋊Dic3C2×D412yes—
217Dic3⋊C4—28no—
218Dic3⋊C4—42no—
219C4×Dic3—42no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223#204—28no—
224C2×#41—68no—
225C2×#41—128no—
226C22×Q8—21no—
227C22×Q8—31no—
228C22×D4—31no—
229C22×D4—31no—
230C22×D4—62no—
231C23×C4—32no—
232C23×C4—62no—
233C23×C4—61no—
234C2×SD16—62no—
235C2×SD16—62no—
236C2×SD16—62no—
237C22×C8—32no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240C4⋊Q8—22no—
241C4⋊Q8D1214yes—
242#123—616no—
243C22⋊Q8—68no—
244C22⋊Q8—64no—
245C22⋊Q8—122no—
246#121—64no—
247#121—64no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×D4—34no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—62no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4C22×D312yes—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—61no—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273#107—64no—
274C4⋊C8—62no—
275#103—62no—
276#103—64no—
277#102—62no—
278#102—122no—
279C22⋊C8—64no—
280C22⋊C8—122no—
281#97—22no—
282#95—21no—
283#95—21no—
284#95—62no—
285C23×C3C22×D411yescharacteristic
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3AGL(1,ℤ/8)14yes—
298C3×Q8—48no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4H(5,𝔽2)11yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325D12—28no—
326D12—22no—
327D12—22no—
328D12—416no—
329D12—41no—
330D12—41no—
331C4×D3—48no—
332C4×D3—416no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—42no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341C3×C8—22no—
342C3×C8—41no—
343C3⋊C8—42no—
344C24—32no—
345C24—61no—
346#41—1232no—
347C2×Q8—24no—
348C2×Q8—32no—
349C2×Q8—44no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×Q8—61no—
355C2×D4—32no—
356C2×D4—64no—
357C2×D4—68no—
358C2×D4—62no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—124no—
365C2×D4—124no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—32no—
369C22×C4—68no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—64no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386SD16—68no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8#23912yescomplemented
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—32no—
401C4⋊C4—68no—
402C4⋊C4—62no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—124no—
407C4⋊C4—122no—
408C4⋊C4C2×D1214yes—
409C22⋊C4—68no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—124no—
413C22⋊C4—124no—
414C42—122no—
415C42C2×D1212yes—
416C42D3×D412yes—
417C22×C3C2×#12011yescharacteristic
418C22×C3#54611yescharacteristic
419C22×C3#54711yescharacteristic
420C22×C3C2×AGL(1,ℤ/8)12yes—
421C22×C3#57812yes—
422C2×D3—24no—
423C2×D3—24no—
424C2×D3—24no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—82no—
433C2×D3—82no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—44no—
437C3×C4—82no—
438C3×C4#52511yescharacteristic
439C3×C4#53711yescharacteristic
440C3×C4#54811yescharacteristic
441C3×C4#55011yescharacteristic
442Dic3—42no—
443Dic3—42no—
444Dic3—42no—
445Dic3—81no—
446C23—34no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—121no—
455C23—121no—
456C23—122no—
457C23—122no—
458C23C2×D3×D411yescharacteristic
459Q8—48no—
460Q8—62no—
461Q8—62no—
462Q8—121no—
463Q8—121no—
464D4—68no—
465D4—62no—
466D4—62no—
467D4—1216no—
468D4—121no—
469D4—121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—128no—
484C2×C4—1216no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4#88212yes—
495C2×C4#88212yes—
496C2×C4C22×D1211yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×#23911yescharacteristic
499C2×C4#100711yescharacteristic
500C2×C4D3×#4111yescharacteristic
501C2×C4#100911yescharacteristic
502C8—22no—
503C8—41no—
504C8—122no—
505C2×C3#240711yescharacteristic
506C2×C3#300311yescharacteristic
507C2×C3#301111yescharacteristic
508C2×C3#330812yes—
509C2×C3#334312yes—
510D3—44no—
511D3—82no—
512C22—64no—
513C22—64no—
514C22—64no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—242no—
523C22—242no—
524C22C2×#88211yescharacteristic
525C22#567611yescharacteristic
526C22#572611yescharacteristic
527C22D3×AGL(1,ℤ/8)12yes—
528C22#577912yes—
529C4—22no—
530C4—22no—
531C4—44no—
532C4—82no—
533C4—122no—
534C4—122no—
535C4—122no—
536C4—241no—
537C4#557611yescharacteristic
538C4#557711yescharacteristic
539C4#562711yescharacteristic
540C4#562911yescharacteristic
541C3#2413211yescharacteristic, complemented, Sylow
542C2—124no—
543C2—242no—
544C2#7661811yescharacteristic
545C2#7892611yescharacteristic
546C2#7896111yescharacteristic
547C2#8169712yes—
548C2#8199012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC2×#5726C2
hypercenter#1993D3
hypocenterC3#24132
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×#5091C2semidirect
3. C2×#4710C2semidirect
4. #74240C2semidirect
5. C3×#1993C2semidirect
6. #69837C2semidirect
7. #69556C2semidirect
8. #68887C2semidirect
9. #68878C2semidirect
10. #68874C2semidirect
11. #68822C2semidirect
52. #1993D3semidirect
192. C2×Dic6#41semidirect
198. C2×C3×C8#41semidirect
395. C2×C8#239semidirect
541. C3#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#57262
C2×#50912
C2×#47102
#742402
C3×#19932
#698372
#695562
#688872
#688782
#688742
#688222
#241323