Finite Group Databasebeta

← #10846280

Subgroups — #10846280

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5116C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic
3C2×C4×Dic12C211yesmaximal, characteristic, complemented
4C2×#4696C211yesmaximal, characteristic, complemented
5#73485C212yesmaximal, complemented
6C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69776C211yesmaximal, characteristic, complemented
9#69770C211yesmaximal, characteristic, complemented
10#68686C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#25412—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×#930C2212yes—
15C2×#924C2212yes—
16C22×Dic12C2211yescharacteristic
17C22×D24C2211yescharacteristic
18C22×#204C2212yes—
19C2×#875C2212yes—
20C2×C4×Dic6C2212yes—
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×C4×C22⋊C4C2211yescharacteristic
25#5116—24no—
26#5115—24no—
27C2×#821C2212yes—
28C2×#821C2212yes—
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#816C2212yes—
32C2×#816C2212yes—
33#4959—22no—
34#4957—22no—
35#4939C2212yes—
36#4928C2212yes—
37C4×Dic12—24no—
38#4696—24no—
39#4676—22no—
40#4669—22no—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44C2×#495—31no—
45C2×#467—31no—
46C2×#466—31no—
47C2×C4×Q16—31no—
48#2426—32no—
49#2042—32no—
50#2000—32no—
51#1970—31no—
52#1942—31no—
53#1880—32no—
54C8×C22⋊C4D311yescharacteristic, complemented
55C23×C3×C4C2311yescharacteristic
56C22×#67—22no—
57C22×D12C2312yes—
58C22×Dic6C2312yes—
59C22×C3×C8—22no—
60C22×C3×C8—22no—
61C22×C3×C8—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C2×C3×C22⋊C4D411yescharacteristic
67C2×C3×C22⋊C4D411yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71#930—24no—
72#930—42no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2312yes—
80#924—24no—
81#924—42no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C2×Dic12—22no—
87C2×Dic12—22no—
88C2×Dic12D414yescomplemented
89C2×D24—22no—
90C2×D24—22no—
91C2×D24—22no—
92C2×#204—24no—
93C2×#204—24no—
94C2×#204—42no—
95#875—24no—
96#875—42no—
97C4×Dic6—24no—
98C4×Dic6—42no—
99C3×C22⋊C8—22no—
100C3×C4×C8—22no—
101C3×C4×C8—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3⋊#95—22no—
105#821—24no—
106#821—42no—
107#818—22no—
108#818D414yes—
109#816—24no—
110#816—24no—
111C22×Q16—31no—
112C22×SD16—32no—
113C22×D8—31no—
114C23×C8C2×D311yescharacteristic
115C2×#124—32no—
116C2×C22⋊Q8—32no—
117C2×#121—32no—
118C2×C4×Q8—32no—
119#495—64no—
120#467—64no—
121#466—64no—
122C4×Q16—64no—
123C2×#107—31no—
124C2×#107—31no—
125C2×#103—32no—
126C2×#103—32no—
127C2×#102—32no—
128C2×#102—32no—
129C2×C22⋊C8C2×D311yescharacteristic
130C2×C4×C8C2×D311yescharacteristic
131C2×C4×C8C2×D311yescharacteristic
132#396—62no—
133#394—62no—
134#390—32no—
135#390—62no—
136#389—32no—
137#388—62no—
138C4×C22⋊C4C2×D311yescharacteristic
139#340—32no—
140#336C2×D311yescharacteristic
141#336C2×D311yescharacteristic
142C24×C3C2×D411yescharacteristic
143C23×D3—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C2×#67—44no—
157C2×#67—44no—
158C2×#67—82no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6C2×D414yes—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—44no—
183C2×C3×C8C2×D412yes—
184C2×C3×C8C2×D412yes—
185C2×C3×C8#4112yes—
186C2×C3×C8#4112yes—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42—21no—
193#217—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—44no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—82no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3C2×D414yes—
203Dic3⋊C4—44no—
204Dic3⋊C4—44no—
205Dic3⋊C4—82no—
206C4×Dic3—44no—
207Dic12—28no—
208D24—44no—
209#204—48no—
210C22×Q8—32no—
211C22×D4—32no—
212C22×D4—62no—
213C23×C4C22×D311yescharacteristic
214C2×Q16—34no—
215C2×Q16—62no—
216C2×Q16—62no—
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—62no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8C22×D311yescharacteristic
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
230#124—64no—
231#124—122no—
232C22⋊Q8—64no—
233C22⋊Q8—122no—
234#121—64no—
235#121—122no—
236C4×Q8—64no—
237C4×Q8—122no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4D1211yescharacteristic
250C2×C22⋊C4D1211yescharacteristic
251C2×C42—62no—
252C2×C42C22×D311yescharacteristic
253C2×C42C22×D311yescharacteristic
254#107—34no—
255#107—62no—
256#103—64no—
257#103—64no—
258#102—64no—
259#102—122no—
260C22⋊C8—22no—
261C4×C8—22no—
262C4×C8—22no—
263#95—62no—
264#95D1211yescharacteristic
265#95D1211yescharacteristic
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3#13512yes—
273C23×C3C22×D411yescharacteristic
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4#13512yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4H(5,𝔽2)11yescharacteristic
305C2×C3×C4H(5,𝔽2)11yescharacteristic
306#67—88no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316D12—42no—
317D12—42no—
318D12—42no—
319D12—42no—
320Dic6—24no—
321Dic6—24no—
322Dic6—42no—
323Dic6—42no—
324C3×C8—22no—
325C3×C8—41no—
326C3×C8—41no—
327C3×C8#12114yes—
328C24—62no—
329C24C2×D1211yescharacteristic
330C2×Q8—34no—
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—124no—
343C2×D4—124no—
344C2×D4—242no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4C2×D1211yescharacteristic
355C22×C4C2×D1211yescharacteristic
356C22×C4C2×D1211yescharacteristic
357C22×C4C2×D1211yescharacteristic
358C22×C4C2×D1211yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—68no—
361SD16—128no—
362D8—124no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—44no—
372C2×C8#23512yes—
373C2×C8#23512yes—
374C2×C8D3×D412yes—
375C2×C8D3×D412yes—
376C4⋊C4—34no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—124no—
381C4⋊C4—124no—
382C4⋊C4—242no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—124no—
386C22⋊C4—124no—
387C22⋊C4—124no—
388C22⋊C4—124no—
389C22⋊C4—124no—
390C22⋊C4—242no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—124no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3#53211yescharacteristic
404C22×C3D4211yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3C2×#13511yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3#57611yescharacteristic
409C22×C3#57811yescharacteristic
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—82no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—82no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4C2×#12111yescharacteristic
423C3×C4#54011yescharacteristic
424C3×C4#54611yescharacteristic
425C3×C4#55011yescharacteristic
426Dic3—44no—
427Dic3—44no—
428Dic3—82no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—122no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23#90412yes—
443C23C22×D1211yescharacteristic
444Q8—64no—
445Q8—64no—
446Q8—122no—
447Q8—122no—
448D4—122no—
449D4—122no—
450D4—122no—
451D4—122no—
452D4—248no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—64no—
472C2×C4—64no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4#90412yes—
481C2×C4C2×#23511yescharacteristic
482C2×C4C2×#23511yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×D3×D411yescharacteristic
485C2×C4#100911yescharacteristic
486C2×C4#100911yescharacteristic
487C8—22no—
488C8—41no—
489C8—41no—
490C8#88414yes—
491C2×C3—42no—
492C2×C3#242811yescharacteristic
493C2×C3#316811yescharacteristic
494C2×C3#317511yescharacteristic
495C2×C3#328011yescharacteristic
496C2×C3#328511yescharacteristic
497C2×C3#328511yescharacteristic
498C2×C3#328711yescharacteristic
499D3—162no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—242no—
508C22—242no—
509C22—242no—
510C22—242no—
511C22—242no—
512C22—242no—
513C22—242no—
514C22#547711yescharacteristic
515C22D4×D1211yescharacteristic
516C22D4×D1211yescharacteristic
517C22C2×#90411yescharacteristic
518C22C2×#90411yescharacteristic
519C22#575211yescharacteristic
520C22#575311yescharacteristic
521C4—22no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—124no—
527C4—124no—
528C4—242no—
529C4C2×#88411yescharacteristic
530C4#553511yescharacteristic
531C4#555011yescharacteristic
532C4#555311yescharacteristic
533C3#2541211yescharacteristic, complemented, Sylow
534C2—42no—
535C2—482no—
536C2#7348711yescharacteristic
537C2#7846911yescharacteristic
538C2#7856811yescharacteristic
539C2#7943311yescharacteristic
540C2#7944411yescharacteristic
541C2#7944411yescharacteristic
542C2#7944611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C8×C22⋊C4C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
hypercenterC8×C22⋊C4D3
hypocenterC3#25412
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#5116C2semidirect
3. C2×C4×Dic12C2semidirect
4. C2×#4696C2semidirect
5. #73485C2semidirect
6. C3×C8×C22⋊C4C2semidirect
7. #69962C2semidirect
8. #69776C2semidirect
9. #69770C2semidirect
10. #68686C2semidirect
11. #68630C2semidirect
54. C8×C22⋊C4D3semidirect
88. C2×Dic12D4semidirect
533. C3#25412semidirect

Maximal subgroups

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

subgroupindex
C2×#51162
C2×#51152
C2×C4×Dic122
C2×#46962
#734852
C3×C8×C22⋊C42
#699622
#697762
#697702
#686862
#686302
#254123