Finite Group Databasebeta

← #10829270

Subgroups — #10829270

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×#5555C211yesmaximal, characteristic
2#82375C211yesmaximal, characteristic, complemented
3C2×#4850C211yesmaximal, characteristic, complemented
4C2×#4710C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69934C212yesmaximal, complemented
9#69930C212yesmaximal, complemented
10#69927C212yesmaximal, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24132—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#930C2212yes—
16C22×C4⋊Dic3C2212yes—
17C2×#924C2212yes—
18C22×#204C2211yescharacteristic
19#5555—24no—
20#5555—24no—
21C2×#897—21no—
22C2×#886—22no—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30#5222—21no—
31#5212C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2212yes—
34C2×#818C2211yescharacteristic
35C2×#818C2212yes—
36C2×#816C2211yescharacteristic
37C2×#816C2212yes—
38#4983—21no—
39#4944—22no—
40#4936C2211yescharacteristic
41#4850—24no—
42#4710—24no—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#547—31no—
47C2×#477—31no—
48C2×#440—31no—
49#2536—31no—
50#2033—32no—
51#2000—32no—
52#1993—31no—
53#1955D311yescharacteristic, complemented
54#1872—32no—
55#1852—32no—
56#1804—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C23×Dic3—22no—
61C2×#237—28no—
62C2×#237—48no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×Dic6—21no—
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77#930—24no—
78#930—24no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89#924—24no—
90#924—42no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—21no—
94C2×#204—22no—
95C2×#204—22no—
96C2×#204—22no—
97#897—28no—
98#886—216no—
99C4×D12—22no—
100C4×D12D414yes—
101#869—22no—
102#869D414yes—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×C4⋊C8—22no—
106C3×C22⋊C8—24no—
107C3×C22⋊C8—42no—
108C3×#97—22no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112#821—22no—
113#821—42no—
114#818—22no—
115#818—24no—
116#816—22no—
117#816—24no—
118C22×#41—61no—
119C22×SD16—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8—31no—
123C2×#123—62no—
124C2×C22⋊Q8—32no—
125C2×C22⋊Q8—61no—
126C2×#121—32no—
127C2×C4×Q8—31no—
128C2×C4×D4—31no—
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#477—64no—
132#440—64no—
133C2×#107—32no—
134C2×#107—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—32no—
137C2×#103—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8C2×D312yes—
141C2×C22⋊C8C2×D312yes—
142C2×#97C2×D311yescharacteristic
143#398—61no—
144#394—61no—
145#388—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3C2×D412yes—
152C23×D3—21no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—28no—
166C2×#67—44no—
167C2×#67—44no—
168C22×Dic3—28no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—44no—
175C22×Dic3—44no—
176#237—432no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12C2×D412yes—
183C2×C4×D3—24no—
184C2×C4×D3—44no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—24no—
190C2×Dic6—21no—
191C2×Dic6—44no—
192C2×Dic6#4112yescomplemented
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8#4112yescomplemented
202C3×C4⋊C4—22no—
203C3×C22⋊C4—24no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206#217—44no—
207#217—44no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—24no—
210C4⋊Dic3—28no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3C2×D414yes—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—44no—
221Dic3⋊C4—42no—
222C4×Dic3—42no—
223#204—28no—
224C2×#41—68no—
225C2×#41—128no—
226C22×Q8—31no—
227C22×Q8—61no—
228C22×D4—31no—
229C22×D4—62no—
230C22×D4D1211yescharacteristic
231C23×C4—62no—
232C23×C4—61no—
233C23×C4C22×D312yes—
234C2×SD16—62no—
235C2×SD16—62no—
236C2×SD16—62no—
237C22×C8C22×D311yescharacteristic
238C22×C8C22×D312yes—
239C22×C8C22×D311yescharacteristic
240C4⋊Q8—34no—
241C4⋊Q8—62no—
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—22no—
251C4×D4—22no—
252C4×D4—34no—
253C4×D4—62no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4D1211yescharacteristic
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4D1212yes—
270C2×C42—61no—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273#107—64no—
274C4⋊C8—22no—
275#103—62no—
276#103—64no—
277#102—62no—
278#102—122no—
279C22⋊C8—24no—
280C22⋊C8—42no—
281#97—22no—
282#95—62no—
283#95—61no—
284#95—61no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3AGL(1,ℤ/8)14yes—
292C23×C3C22×D411yescharacteristic
293C22×D3—22no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—42no—
297C22×D3—42no—
298C3×D4—28no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316#67—416no—
317C2×Dic3—24no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—416no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331D12—22no—
332D12—22no—
333D12—41no—
334D12—41no—
335C4×D3—48no—
336Dic6—22no—
337Dic6—22no—
338Dic6—48no—
339Dic6—41no—
340Dic6—41no—
341C3×C8—22no—
342C3×C8—41no—
343C3×C8—42no—
344C24—61no—
345C24C2×D1212yes—
346#41—1232no—
347C2×Q8—32no—
348C2×Q8—64no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×Q8—124no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—32no—
358C2×D4—68no—
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—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—68no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—62no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—124no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4C2×D1212yes—
383C22×C4C2×D1211yescharacteristic
384C22×C4C2×D1211yescharacteristic
385C22×C4C23×D311yescharacteristic
386SD16—68no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8#23712yescomplemented
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—34no—
399C4⋊C4—68no—
400C4⋊C4—64no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—124no—
408C4⋊C4—122no—
409C22⋊C4—24no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—124no—
413C22⋊C4—124no—
414C42—122no—
415C42D3×D412yes—
416C42D3×D412yes—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C2×#12011yescharacteristic
425C22×C3#54611yescharacteristic
426C22×C3#54711yescharacteristic
427C22×C3C2×AGL(1,ℤ/8)12yes—
428C22×C3#57812yes—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—82no—
433C2×D3—82no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4#52511yescharacteristic
438C3×C4#53711yescharacteristic
439C3×C4#54811yescharacteristic
440C3×C4#55011yescharacteristic
441Dic3—42no—
442Dic3—44no—
443Dic3—42no—
444Dic3—81no—
445Dic3—82no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—62no—
453C23—122no—
454C23—122no—
455C23—121no—
456C23—121no—
457C23#90814yes—
458C23C22×D1211yescharacteristic
459Q8—62no—
460Q8—62no—
461Q8—128no—
462Q8—121no—
463Q8—121no—
464D4—28no—
465D4—62no—
466D4—62no—
467D4—1216no—
468D4—121no—
469D4—121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—62no—
482C2×C4—128no—
483C2×C4—1216no—
484C2×C4—121no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4#88212yes—
495C2×C4#88212yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×#23711yescharacteristic
499C2×C4#100411yescharacteristic
500C2×C4D3×#4111yescharacteristic
501C2×C4#101011yescharacteristic
502C8—22no—
503C8—42no—
504C8—41no—
505C2×C3—44no—
506C2×C3#240711yescharacteristic
507C2×C3#300311yescharacteristic
508C2×C3#301111yescharacteristic
509C2×C3#330812yes—
510C2×C3#334312yes—
511D3—82no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—242no—
523C22—242no—
524C22C2×#88211yescharacteristic
525C22#555411yescharacteristic
526C22#555511yescharacteristic
527C22C2×#90812yes—
528C22#575312yes—
529C4—22no—
530C4—22no—
531C4—42no—
532C4—122no—
533C4—124no—
534C4—122no—
535C4—241no—
536C4—242no—
537C4#560611yescharacteristic
538C4#560811yescharacteristic
539C4#567711yescharacteristic
540C4#571811yescharacteristic
541C3#2413211yescharacteristic, complemented, Sylow
542C2—44no—
543C2—242no—
544C2#7643511yescharacteristic
545C2#7863711yescharacteristic
546C2#7867911yescharacteristic
547C2#7946912yes—
548C2#7950912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#1955D3
hypocenterC3#24132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. #82375C2semidirect
3. C2×#4850C2semidirect
4. C2×#4710C2semidirect
5. C3×#1955C2semidirect
6. #70480C2semidirect
7. #69962C2semidirect
8. #69934C2semidirect
9. #69930C2semidirect
10. #69927C2semidirect
11. #69828C2semidirect
53. #1955D3semidirect
192. C2×Dic6#41semidirect
201. C2×C3×C8#41semidirect
395. C2×C8#237semidirect
541. C3#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
#823752
C2×#48502
C2×#47102
C3×#19552
#704802
#699622
#699342
#699302
#699272
#698282
#241323