Finite Group Databasebeta

← #10846285

Subgroups — #10846285

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×C3×C4×D8C211yesmaximal, characteristic, complemented
2C2×#5157C211yesmaximal, characteristic
3C2×#5155C211yesmaximal, characteristic, complemented
4#82408C212yesmaximal, complemented
5C2×#4887C211yesmaximal, characteristic, complemented
6#71612C212yesmaximal, complemented
7#70612C211yesmaximal, characteristic, complemented
8#70205C212yesmaximal, complemented
9#70178C212yesmaximal, complemented
10C8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11#69775C211yesmaximal, characteristic, complemented
12#25413—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×#938C2212yes—
16C2×#936C2212yes—
17C22×#214C2212yes—
18C22×Dic12C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#897C2212yes—
21C3×C4×D8—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#102C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C2×#832C2212yes—
26#5213—22no—
27#5212—22no—
28#5157—24no—
29#5155—24no—
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#816C2212yes—
33C2×C8×Dic3C2211yescharacteristic
34#4984C2212yes—
35C2×#810C2212yes—
36#4961—22no—
37#4953—22no—
38C4×(C2×D3)⋊C4C2211yescharacteristic
39#4929C2212yes—
40#4887—24no—
41#4550C2211yescharacteristic
42#4529C2212yes—
43#4524C2211yescharacteristic
44C2×#495—31no—
45C2×#467—31no—
46C2×#466—31no—
47C2×C4×D8D311yescharacteristic, complemented
48#2427—32no—
49#2042—32no—
50#2000—32no—
51#1969—31no—
52#1942—31no—
53#1877—32no—
54C8×C22⋊C4—31no—
55C22×C3×D4C2312yes—
56C23×C3×C4—22no—
57C22×#67—22no—
58C22×C4×D3C2311yescharacteristic
59C22×Dic6C2312yes—
60C2×C3×D8—22no—
61C2×C3×D8—22no—
62C2×C3×D8D414yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4×D4—24no—
66C3×C4×D4—42no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73#938—24no—
74#938—42no—
75#936—24no—
76#936—42no—
77C2×#214—24no—
78C2×#214—24no—
79C2×#214—42no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4D411yescharacteristic
82C2×(C2×D3)⋊C4D411yescharacteristic
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3C2312yes—
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89C2×Dic12—22no—
90C2×Dic12—22no—
91C2×Dic12—22no—
92C2×D3×C8—22no—
93C2×D3×C8—22no—
94C2×D3×C8—22no—
95#897—24no—
96#897—42no—
97C3×#107D414yescomplemented
98C3×#102—24no—
99C3×C4×C8—22no—
100#832—24no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95D411yescharacteristic
105C3⋊#95D411yescharacteristic
106(C2×D3)⋊C8—22no—
107#818—22no—
108#816—42no—
109C8×Dic3—22no—
110#810—24no—
111C22×Q16—31no—
112C22×SD16—32no—
113C22×D8C2×D311yescharacteristic
114C23×C8—31no—
115C2×#124—32no—
116C2×C22⋊Q8—32no—
117C2×#121—32no—
118C2×C4×D4C2×D312yes—
119#495—64no—
120#467—64no—
121#466—64no—
122C4×D8—24no—
123C2×#107—31no—
124C2×#107C2×D311yescharacteristic
125C2×#103—32no—
126C2×#103—32no—
127C2×#102—32no—
128C2×#102C2×D312yes—
129C2×C22⋊C8—31no—
130C2×C4×C8—31no—
131C2×C4×C8C2×D311yescharacteristic
132#397—62no—
133#394—62no—
134#391—32no—
135#389—32no—
136#386—62no—
137#380—62no—
138C4×C22⋊C4—31no—
139#340—32no—
140#336—31no—
141#336—31no—
142C24×C3—22no—
143C23×D3C2×D411yescharacteristic
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4C2×D414yes—
150C22×C3×C4—22no—
151C22×C3×C4—44no—
152C22×C3×C4—44no—
153C22×C3×C4—82no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C2×#67—44no—
158C2×#67—44no—
159C2×#67—82no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C2×C4×D3—21no—
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C3×D8—28no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8C2×D412yes—
182C2×C3×C8C2×D412yes—
183C2×C3×C8#4112yes—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×D414yes—
186C3×C22⋊C4—44no—
187C3×C42—21no—
188C3×C42—21no—
189#217—44no—
190#217—44no—
191#217—44no—
192#214—48no—
193(C2×D3)⋊C4—22no—
194(C2×D3)⋊C4—22no—
195(C2×D3)⋊C4—44no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—44no—
199C4⋊Dic3—82no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8#4112yes—
208Dic12—44no—
209D3×C8—44no—
210C22×Q8—32no—
211C22×D4—62no—
212C22×D4C22×D312yes—
213C23×C4—22no—
214C23×C4—31no—
215C2×Q16—62no—
216C2×Q16—62no—
217C2×Q16—62no—
218C2×SD16—64no—
219C2×SD16—64no—
220C2×SD16—122no—
221C2×D8—22no—
222C2×D8—22no—
223C2×D8D1214yes—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8—62no—
227C22×C8—62no—
228C22×C8—62no—
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—24no—
238C4×D4—42no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C42—31no—
251C2×C42C22×D311yescharacteristic
252#107—62no—
253#107#6714yescomplemented
254#103—64no—
255#103—122no—
256#102—24no—
257#102—64no—
258C22⋊C8—62no—
259C4×C8—22no—
260C4×C8—62no—
261#95—31no—
262#95—31no—
263#95—62no—
264#95—62no—
265#95—62no—
266C23×C3—24no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3C22×D411yescharacteristic
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3#13512yes—
281C3×D4—24no—
282C3×D4—24no—
283C3×D4—42no—
284C3×D4—42no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—88no—
294C2×C3×C4#13512yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4H(5,𝔽2)11yescharacteristic
300C2×C3×C4H(5,𝔽2)11yescharacteristic
301#67—88no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C4×D3—41no—
318C4×D3—41no—
319C4×D3—41no—
320C4×D3—41no—
321Dic6—42no—
322Dic6—42no—
323Dic6—42no—
324Dic6—42no—
325C3×C8—41no—
326C3×C8#12114yes—
327C3⋊C8—22no—
328C3⋊C8—41no—
329C24—22no—
330C24—31no—
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—124no—
343C2×D4—124no—
344C2×D4—242no—
345C2×D4C2×D1214yes—
346C22×C4—22no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—82no—
361C22×C4D3×D411yescharacteristic
362C22×C4D3×D411yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—124no—
365SD16—128no—
366D8—28no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—32no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—61no—
376C2×C8—124no—
377C2×C8C2×D1212yes—
378C2×C8#23512yes—
379C2×C8C2×#6712yes—
380C4⋊C4—22no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—124no—
384C4⋊C4—242no—
385C4⋊C4C2×#6714yes—
386C22⋊C4—44no—
387C22⋊C4—62no—
388C22⋊C4—62no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C22⋊C4—124no—
393C42—21no—
394C42—21no—
395C42—61no—
396C42—61no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3#53211yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3D4211yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×#13511yescharacteristic
409C22×C3#57611yescharacteristic
410C22×C3#57811yescharacteristic
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—41no—
415C2×D3—41no—
416C2×D3—41no—
417C2×D3—41no—
418C3×C4—22no—
419C3×C4—44no—
420C3×C4—41no—
421C3×C4C2×#12111yescharacteristic
422C3×C4#54011yescharacteristic
423C3×C4#54611yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—41no—
426Dic3—41no—
427Dic3—41no—
428Dic3—82no—
429Dic3—82no—
430C23—24no—
431C23—32no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23—61no—
443C23—61no—
444C23C2×D3×D411yescharacteristic
445Q8—122no—
446Q8—122no—
447Q8—122no—
448Q8—122no—
449D4—24no—
450D4—24no—
451D4—42no—
452D4—42no—
453D4—248no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—88no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4#91212yes—
483C2×C4C22×D1211yescharacteristic
484C2×C4C2×#23511yescharacteristic
485C2×C4#100411yescharacteristic
486C2×C4D3×#4111yescharacteristic
487C2×C4#100911yescharacteristic
488C2×C4C22×#6711yescharacteristic
489C8—41no—
490C8—62no—
491C8—121no—
492C8#93014yes—
493C2×C3—84no—
494C2×C3#242811yescharacteristic
495C2×C3#316811yescharacteristic
496C2×C3#317511yescharacteristic
497C2×C3#328011yescharacteristic
498C2×C3#328011yescharacteristic
499C2×C3#328311yescharacteristic
500C2×C3#328511yescharacteristic
501D3—42no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—121no—
513C22—121no—
514C22—121no—
515C22—121no—
516C22#554811yescharacteristic
517C22D4×D1211yescharacteristic
518C22C2×#91211yescharacteristic
519C22#575711yescharacteristic
520C22D3×#13511yescharacteristic
521C22#577111yescharacteristic
522C22D4×#6711yescharacteristic
523C4—22no—
524C4—44no—
525C4—41no—
526C4—121no—
527C4—121no—
528C4—121no—
529C4—242no—
530C4—242no—
531C4#555411yescharacteristic
532C4#556211yescharacteristic
533C4C2×#93011yescharacteristic
534C4#583411yescharacteristic
535C3#2541311yescharacteristic, complemented, Sylow
536C2—84no—
537C2—122no—
538C2#7958311yescharacteristic
539C2#7960011yescharacteristic
540C2#7960611yescharacteristic
541C2#7960711yescharacteristic
542C2#8240911yescharacteristic
543C2#8383911yescharacteristic
544C2#8392011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC2×#5157C2
hypercenterC2×C4×D8D3
hypocenterC3#25413
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D8C2semidirect
3. C2×#5155C2semidirect
4. #82408C2semidirect
5. C2×#4887C2semidirect
6. #71612C2semidirect
7. #70612C2semidirect
8. #70205C2semidirect
9. #70178C2semidirect
10. C8×(C2×D3)⋊C4C2semidirect
11. #69775C2semidirect
47. C2×C4×D8D3semidirect
97. C3×#107D4semidirect
253. #107#67semidirect
535. C3#25413semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
C2×#51572
C2×#51552
#824082
C2×#48872
#716122
#706122
#702052
#701782
C8×(C2×D3)⋊C42
#697752
#254133