Finite Group Databasebeta

← #10830081

Subgroups — #10830081

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5732C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3C2×#4711C211yesmaximal, characteristic
4#74190C211yesmaximal, characteristic, complemented
5C3×#1960C211yesmaximal, characteristic, complemented
6#70277C212yesmaximal
7#70245C212yesmaximal, complemented
8#70240C212yesmaximal, complemented
9#70238C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69556C211yesmaximal, characteristic, complemented
12#24197—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×#215C2212yes—
17C22×D24C2211yescharacteristic
18#5732—24no—
19#5732—24no—
20C2×#898—22no—
21C2×#897—21no—
22C2×#892—21no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#97C2211yescharacteristic
29C3×#382C2211yescharacteristic
30C2×#835C2212yes—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#817C2211yescharacteristic
34#5083—24no—
35#4988C2212yes—
36#4984C2212yes—
37C2×#809C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4711—24no—
40#4677—21no—
41#4673—21no—
42#4668—22no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×#442—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1960D311yescharacteristic, complemented
54#1880—32no—
55#1865—32no—
56#1804—31no—
57C22×C3×Q8C2312yes—
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—21no—
62C22×Dic6D411yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71#947—24no—
72#947—24no—
73Q8×Dic3—24no—
74Q8×Dic3—24no—
75C2×#215—24no—
76C2×#215—24no—
77C2×#215—42no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3—22no—
91C2×C4×Dic3C2312yes—
92C22×C3⋊C8C2312yes—
93C2×D24—22no—
94C2×D24—22no—
95C2×D24—22no—
96#898—216no—
97#897—28no—
98#892—28no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12D414yes—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#103—24no—
106C3×#103—42no—
107C3×#97—22no—
108C3×#95—21no—
109C3×#95—21no—
110#835—24no—
111C3⋊#95—22no—
112#821—22no—
113#821—22no—
114#817—22no—
115#809—42no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16—32no—
118C22×D8—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×#117—61no—
130#461—64no—
131#442—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103C2×D312yes—
136C2×#103C2×D312yes—
137C2×#102—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97C2×D311yescharacteristic
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—21no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3C2×D412yes—
166C22×Dic3C2×D412yes—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12C2×D412yes—
178C2×D12#4112yes—
179C2×C4×D3—24no—
180C2×C4×D3—44no—
181C2×C4×D3—42no—
182C2×C4×D3—82no—
183C2×Dic6—22no—
184C2×Dic6—42no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8#4112yes—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197#215—48no—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—42no—
203(C2×D3)⋊C4—82no—
204C4⋊Dic3—28no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—82no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—24no—
212C4×Dic3—28no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218D24—28no—
219C22×Q8—31no—
220C22×Q8C22×D312yes—
221C22×D4—31no—
222C22×D4—31no—
223C23×C4—61no—
224C23×C4—61no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—62no—
231C22×C8—32no—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234C4⋊Q8—22no—
235C4⋊Q8D1214yes—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×D4—34no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4—62no—
248#117—68no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—62no—
264C2×C42—32no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#106—62no—
268C4⋊C8—62no—
269#103—24no—
270#103—42no—
271#103—64no—
272#102—62no—
273#102—62no—
274#102—122no—
275#97—22no—
276#95—21no—
277#95—21no—
278#95—62no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C3×Q8—24no—
291C3×Q8—24no—
292C3×Q8—42no—
293C3×Q8—42no—
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—22no—
316C2×Dic3—22no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320D12—22no—
321D12—22no—
322D12—22no—
323D12—22no—
324D12—41no—
325D12—41no—
326D12—41no—
327D12—41no—
328C4×D3—48no—
329C4×D3—84no—
330Dic6—44no—
331C3×C8—22no—
332C3×C8—41no—
333C3⋊C8—42no—
334C24—61no—
335C24—61no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—62no—
342C2×Q8—122no—
343C2×Q8C2×D1214yes—
344C2×D4—32no—
345C2×D4—32no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—32no—
359C22×C4—32no—
360C22×C4—61no—
361C22×C4—64no—
362C22×C4—61no—
363C22×C4—124no—
364C22×C4—122no—
365C22×C4—242no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—128no—
370D8—68no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8#23712yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—32no—
385C4⋊C4—68no—
386C4⋊C4—62no—
387C4⋊C4—64no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—242no—
392C22⋊C4—68no—
393C22⋊C4—68no—
394C22⋊C4—64no—
395C22⋊C4—124no—
396C22⋊C4—122no—
397C22⋊C4—242no—
398C42—68no—
399C42—62no—
400C42—62no—
401C42C2×D1212yes—
402C42D3×D412yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3#55011yescharacteristic
406C22×C3#57512yes—
407C22×C3#57712yes—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—82no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—81no—
417C2×D3—81no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—82no—
422C3×C4#52511yescharacteristic
423C3×C4#53711yescharacteristic
424C3×C4#54611yescharacteristic
425C3×C4#54711yescharacteristic
426Dic3—42no—
427Dic3—42no—
428Dic3—42no—
429Dic3—81no—
430C23—62no—
431C23—62no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23C2×D3×D411yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—42no—
444Q8—42no—
445Q8—124no—
446D4—62no—
447D4—62no—
448D4—62no—
449D4—62no—
450D4—121no—
451D4—121no—
452D4—121no—
453D4—121no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—128no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—244no—
474C2×C4#88212yes—
475C2×C4#88212yes—
476C2×C4C22×D1211yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×#23711yescharacteristic
479C2×C4#100411yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4#101011yescharacteristic
482C8—22no—
483C8—41no—
484C8—122no—
485C2×C3#240711yescharacteristic
486C2×C3#300511yescharacteristic
487C2×C3#301011yescharacteristic
488C2×C3#337612yes—
489C2×C3#340112yes—
490D3—82no—
491D3—82no—
492C22—124no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—242no—
498C22—241no—
499C22—241no—
500C22—242no—
501C22—242no—
502C22C2×#88211yescharacteristic
503C22#567711yescharacteristic
504C22#571811yescharacteristic
505C22#577712yes—
506C22#577812yes—
507C4—22no—
508C4—22no—
509C4—44no—
510C4—82no—
511C4—122no—
512C4—122no—
513C4—122no—
514C4—241no—
515C4#555411yescharacteristic
516C4#555511yescharacteristic
517C4#560611yescharacteristic
518C4#560811yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—242no—
521C2—242no—
522C2#7643511yescharacteristic
523C2#7895911yescharacteristic
524C2#7896011yescharacteristic
525C2#8173412yes—
526C2#8186612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1960C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1960D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#5732C2semidirect
2. C2×#5083C2semidirect
4. #74190C2semidirect
5. C3×#1960C2semidirect
7. #70245C2semidirect
8. #70240C2semidirect
9. #70238C2semidirect
10. #69838C2semidirect
11. #69556C2semidirect
53. #1960D3semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×#50832
C2×#47112
#741902
C3×#19602
#702772
#702452
#702402
#702382
#698382
#695562
#241973