Finite Group Databasebeta

← #10846291

Subgroups — #10846291

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5116C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic
3C2×#4696C211yesmaximal, characteristic, complemented
4C2×C4×D24C211yesmaximal, characteristic, complemented
5#73486C212yesmaximal, complemented
6C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69775C211yesmaximal, characteristic, complemented
9#69770C211yesmaximal, characteristic, complemented
10#68672C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#25413—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×#930C2212yes—
15C2×#924C2212yes—
16C22×Dic12C2211yescharacteristic
17C22×D24C2211yescharacteristic
18C22×#204C2212yes—
19C2×#875C2212yes—
20C2×C4×D12C2212yes—
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#4956—22no—
35#4929C2212yes—
36#4928C2212yes—
37#4696—24no—
38C4×D24—24no—
39#4675—22no—
40#4667—22no—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44C2×#495—31no—
45C2×#467—31no—
46C2×#466—31no—
47C2×C4×D8—31no—
48#2427—32no—
49#2042—32no—
50#2000—32no—
51#1969—31no—
52#1942—31no—
53#1877—32no—
54C8×C22⋊C4D311yescharacteristic, complemented
55C23×C3×C4C2311yescharacteristic
56C22×#67—22no—
57C22×D12C2312yes—
58C22×C4×D3—22no—
59C22×Dic6C2312yes—
60C22×C3×C8—22no—
61C22×C3×C8—22no—
62C22×C3×C8—22no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C2×C3×C22⋊C4D411yescharacteristic
68C2×C3×C22⋊C4D411yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72#930—24no—
73#930—42no—
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×Dic12—22no—
85C2×Dic12—22no—
86C2×Dic12—22no—
87C2×D24—22no—
88C2×D24—22no—
89C2×D24D414yes—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93#875—24no—
94#875—42no—
95C4×D12—24no—
96C4×D12—42no—
97C3×C22⋊C8—22no—
98C3×C4×C8—22no—
99C3×C4×C8—22no—
100C3×#95D411yescharacteristic
101C3×#95D411yescharacteristic
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105#821—24no—
106#821—24no—
107#818—22no—
108#818D414yescomplemented
109#816—24no—
110#816—42no—
111C22×Q16—31no—
112C22×SD16—32no—
113C22×D8—31no—
114C23×C8C2×D311yescharacteristic
115C2×#124—32no—
116C2×C22⋊Q8—32no—
117C2×#121—32no—
118C2×C4×D4—32no—
119#495—64no—
120#467—64no—
121#466—64no—
122C4×D8—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#397—62no—
133#394—62no—
134#391—32no—
135#389—32no—
136#386—62no—
137#380—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×D12C2×D414yes—
168C2×C4×D3—44no—
169C2×C4×D3—44no—
170C2×C4×D3—82no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—44no—
186C2×C3×C8C2×D412yes—
187C2×C3×C8C2×D412yes—
188C2×C3×C8#4112yes—
189C2×C3×C8#4112yes—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—21no—
196#217—44no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—44no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3C2×D414yes—
205Dic3⋊C4—44no—
206Dic3⋊C4—82no—
207Dic12—44no—
208D24—28no—
209#204—48no—
210C22×Q8—32no—
211C22×D4—32no—
212C22×D4—62no—
213C23×C4—62no—
214C23×C4C22×D311yescharacteristic
215C2×Q16—62no—
216C2×Q16—62no—
217C2×Q16—62no—
218C2×SD16—64no—
219C2×SD16—64no—
220C2×SD16—122no—
221C2×D8—34no—
222C2×D8—62no—
223C2×D8—62no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
230C22×C8C22×D311yescharacteristic
231#124—64no—
232#124—122no—
233C22⋊Q8—64no—
234C22⋊Q8—122no—
235#121—64no—
236#121—122no—
237C4×D4—64no—
238C4×D4—122no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—32no—
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⋊C4D1211yescharacteristic
249C2×C22⋊C4D1211yescharacteristic
250C2×C42C22×D311yescharacteristic
251C2×C42C22×D311yescharacteristic
252#107—34no—
253#107—62no—
254#103—64no—
255#103—122no—
256#102—64no—
257#102—64no—
258C22⋊C8—22no—
259C4×C8—22no—
260C4×C8—22no—
261#95—62no—
262#95—62no—
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—24no—
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—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316D12—24no—
317D12—24no—
318D12—42no—
319D12—42no—
320C4×D3—88no—
321Dic6—42no—
322Dic6—42no—
323Dic6—42no—
324Dic6—42no—
325C3×C8—22no—
326C3×C8—41no—
327C3×C8—41no—
328C3×C8#12114yes—
329C24—62no—
330C24C2×D1211yescharacteristic
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×D4—34no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—124no—
344C2×D4—124no—
345C2×D4—242no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—124no—
356C22×C4—124no—
357C22×C4—242no—
358C22×C4C2×D1211yescharacteristic
359C22×C4C2×D1211yescharacteristic
360C22×C4C2×D1211yescharacteristic
361C22×C4C2×D1211yescharacteristic
362C22×C4C2×D1211yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—124no—
365SD16—128no—
366D8—68no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—44no—
376C2×C8#23512yes—
377C2×C8#23512yes—
378C2×C8D3×D412yes—
379C2×C8D3×D412yes—
380C4⋊C4—34no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—124no—
385C4⋊C4—242no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—124no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C22⋊C4—124no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3#53211yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3D4211yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×#13511yescharacteristic
409C22×C3#57611yescharacteristic
410C22×C3#57811yescharacteristic
411C2×D3—44no—
412C2×D3—44no—
413C2×D3—44no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—82no—
417C2×D3—82no—
418C3×C4—22no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4C2×#12111yescharacteristic
424C3×C4#54011yescharacteristic
425C3×C4#54611yescharacteristic
426C3×C4#55011yescharacteristic
427Dic3—44no—
428Dic3—82no—
429Dic3—82no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—64no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23—122no—
443C23#90412yes—
444C23C22×D1211yescharacteristic
445Q8—122no—
446Q8—122no—
447Q8—122no—
448Q8—122no—
449D4—64no—
450D4—64no—
451D4—122no—
452D4—122no—
453D4—248no—
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—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
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—122no—
481C2×C4—248no—
482C2×C4#90412yes—
483C2×C4C2×#23511yescharacteristic
484C2×C4C2×#23511yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×D3×D411yescharacteristic
487C2×C4#100911yescharacteristic
488C2×C4#100911yescharacteristic
489C8—22no—
490C8—41no—
491C8—41no—
492C8#88414yes—
493C2×C3—42no—
494C2×C3#242811yescharacteristic
495C2×C3#316811yescharacteristic
496C2×C3#317511yescharacteristic
497C2×C3#328011yescharacteristic
498C2×C3#328011yescharacteristic
499C2×C3#328311yescharacteristic
500C2×C3#328511yescharacteristic
501D3—84no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—242no—
513C22—242no—
514C22—242no—
515C22—242no—
516C22#547711yescharacteristic
517C22D4×D1211yescharacteristic
518C22D4×D1211yescharacteristic
519C22C2×#90411yescharacteristic
520C22C2×#90411yescharacteristic
521C22#575211yescharacteristic
522C22#575311yescharacteristic
523C4—22no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—124no—
529C4—242no—
530C4—242no—
531C4C2×#88411yescharacteristic
532C4#553511yescharacteristic
533C4#555011yescharacteristic
534C4#555311yescharacteristic
535C3#2541311yescharacteristic, complemented, Sylow
536C2—42no—
537C2—244no—
538C2#7348711yescharacteristic
539C2#7846911yescharacteristic
540C2#7856811yescharacteristic
541C2#7943311yescharacteristic
542C2#7943311yescharacteristic
543C2#7943611yescharacteristic
544C2#7944411yescharacteristic
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#25413
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×#4696C2semidirect
4. C2×C4×D24C2semidirect
5. #73486C2semidirect
6. C3×C8×C22⋊C4C2semidirect
7. #69962C2semidirect
8. #69775C2semidirect
9. #69770C2semidirect
10. #68672C2semidirect
11. #68630C2semidirect
54. C8×C22⋊C4D3semidirect
108. #818D4semidirect
535. C3#25413semidirect

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×#46962
C2×C4×D242
#734862
C3×C8×C22⋊C42
#699622
#697752
#697702
#686722
#686302
#254133