Finite Group Databasebeta

← #10829262

Subgroups — #10829262

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic, complemented
3C2×#4842C211yesmaximal, characteristic, complemented
4#74239C211yesmaximal, characteristic
5C3×#1805C211yesmaximal, characteristic, complemented
6#70471C211yesmaximal, characteristic, complemented
7#70272C212yesmaximal, complemented
8#70243C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68881C212yesmaximal, complemented
11#68876C212yesmaximal, complemented
12#24131—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2212yes—
15C22×#215C2211yescharacteristic
16#5584—24no—
17#5584—24no—
18C2×#898—22no—
19C2×#895C2212yes—
20C2×#892C2212yes—
21C2×#875—21no—
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C3×#382C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#817C2212yes—
34#5026—24no—
35#4987C2212yes—
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4842—24no—
40#4674—21no—
41#4671—21no—
42#4665—22no—
43C3×#340C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#476—31no—
48C2×#440—31no—
49#2535—31no—
50#2035—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1875—32no—
55#1854—32no—
56#1805D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12—22no—
59C22×D12D411yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3C2312yes—
63C22×Dic6—21no—
64C22×C3×C8C2312yes—
65C3×C4×Q8—22no—
66C3×C4×Q8D414yescomplemented
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72#946—24no—
73#946—24no—
74C2×#215—22no—
75C2×#215—22no—
76C2×#215—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4D412yes—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C2×C4×Dic3—22no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91#898—216no—
92#895—24no—
93#895—42no—
94#892—24no—
95#892—24no—
96#875—28no—
97C4×D12—22no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101#869—22no—
102#869D414yescomplemented
103C4×Dic6—28no—
104C3×C4⋊C8—22no—
105C3×#103—42no—
106C3×#95—21no—
107C3×#95—21no—
108#835—22no—
109C3⋊#95—22no—
110#821—24no—
111(C2×D3)⋊C8—24no—
112(C2×D3)⋊C8—42no—
113#817—24no—
114#809—22no—
115#807—22no—
116#803—22no—
117C22×SD16—31no—
118#556—64no—
119#556—64no—
120C2×C4⋊Q8—31no—
121C2×#126—62no—
122C2×#124—61no—
123C2×C22⋊Q8—32no—
124C2×#121—32no—
125C2×C4×Q8—61no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×#117—32no—
130#476—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#103C2×D312yes—
137C2×#102—32no—
138C2×#102—31no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#397—61no—
143#392—61no—
144#386—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340—32no—
149#340C2×D312yes—
150C23×D3—21no—
151C23×D3C2×D412yes—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D412yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3C2×D412yes—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—22no—
172C2×D12—21no—
173C2×D12—42no—
174C2×D12—44no—
175C2×D12—44no—
176C2×D12—82no—
177C2×D12#4112yes—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—42no—
185C2×C4×D3—82no—
186C2×Dic6—24no—
187C2×Dic6—44no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4C2×D412yes—
196C3×C42—24no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#215—28no—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—28no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—42no—
205(C2×D3)⋊C4—82no—
206C4⋊Dic3—22no—
207C4⋊Dic3—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3C2×D414yes—
212Dic3⋊C4—28no—
213C4×Dic3—28no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—61no—
221C22×Q8C22×D311yescharacteristic
222C22×D4—31no—
223C22×D4—31no—
224C22×D4—62no—
225C23×C4—32no—
226C23×C4—61no—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8C22×D312yes—
233C4⋊Q8—34no—
234C4⋊Q8—62no—
235#126—616no—
236#124—68no—
237C22⋊Q8—64no—
238C22⋊Q8—64no—
239#121—64no—
240#121—122no—
241C4×Q8—22no—
242C4×Q8—68no—
243C4×Q8D1214yescomplemented
244C4×D4—62no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4—62no—
248#117—64no—
249#117—64no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4C22×D312yes—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—32no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—62no—
264C2×C42—21no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—22no—
270#103—42no—
271#103—62no—
272#102—64no—
273#102—62no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—62no—
277#95—21no—
278#95—21no—
279#95—62no—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—41no—
296C3×Q8—41no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4H(5,𝔽2)11yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—44no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322D12—22no—
323D12—22no—
324D12—44no—
325D12—41no—
326D12—41no—
327D12—88no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—84no—
333Dic6—48no—
334C3×C8—42no—
335C3⋊C8—22no—
336C3⋊C8—41no—
337C24—32no—
338C24—61no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—64no—
345C2×Q8—124no—
346C2×Q8C2×D1212yes—
347C2×D4—32no—
348C2×D4—62no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—122no—
355C2×D4—124no—
356C2×D4—124no—
357C2×D4—242no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—32no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—122no—
371C22×C4—242no—
372C22×C4D3×D411yescharacteristic
373C22×C4D3×D411yescharacteristic
374C22×C4C23×D311yescharacteristic
375SD16—68no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—32no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—61no—
385C4⋊C4—24no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—34no—
390C4⋊C4—68no—
391C4⋊C4—62no—
392C4⋊C4—64no—
393C4⋊C4—62no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4C2×D1212yes—
397C22⋊C4—68no—
398C22⋊C4—68no—
399C22⋊C4—64no—
400C22⋊C4—124no—
401C22⋊C4—122no—
402C22⋊C4—242no—
403C42—24no—
404C42—68no—
405C42C2×D1212yes—
406C42D3×D412yes—
407C22×C3C2×#12011yescharacteristic
408C22×C3#54611yescharacteristic
409C22×C3#54711yescharacteristic
410C22×C3#57512yes—
411C22×C3#57712yes—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—44no—
420C2×D3—42no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—82no—
429C3×C4#52511yescharacteristic
430C3×C4#53711yescharacteristic
431C3×C4#54811yescharacteristic
432C3×C4#55011yescharacteristic
433Dic3—42no—
434Dic3—44no—
435Dic3—81no—
436C23—62no—
437C23—62no—
438C23—62no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23—62no—
443C23—62no—
444C23—121no—
445C23—122no—
446C23—122no—
447C23—121no—
448C23C2×D3×D411yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453Q8—128no—
454D4—62no—
455D4—62no—
456D4—124no—
457D4—121no—
458D4—121no—
459D4—248no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—62no—
471C2×C4—64no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—124no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—121no—
480C2×C4—122no—
481C2×C4—244no—
482C2×C4#88212yes—
483C2×C4#88212yes—
484C2×C4C22×D1211yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4#100411yescharacteristic
487C2×C4#100711yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4D3×#4111yescharacteristic
490C8—42no—
491C8—62no—
492C8—121no—
493C2×C3#240711yescharacteristic
494C2×C3#300511yescharacteristic
495C2×C3#301011yescharacteristic
496C2×C3#331112yes—
497C2×C3#335012yes—
498D3—82no—
499D3—82no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—122no—
505C22—122no—
506C22—124no—
507C22—122no—
508C22—122no—
509C22—242no—
510C22—241no—
511C22—241no—
512C22C2×#88211yescharacteristic
513C22#561211yescharacteristic
514C22#563111yescharacteristic
515C22#577012yes—
516C22#577712yes—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—122no—
523C4—124no—
524C4—241no—
525C4#551011yescharacteristic
526C4#552411yescharacteristic
527C4#559911yescharacteristic
528C4#563311yescharacteristic
529C3#2413111yescharacteristic, complemented, Sylow
530C2—242no—
531C2—242no—
532C2#7401511yescharacteristic
533C2#7883611yescharacteristic
534C2#7895911yescharacteristic
535C2#8061712yes—
536C2#8125812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radical#74239C2
hypercenter#1805D3
hypocenterC3#24131
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#5584C2semidirect
2. C2×#5026C2semidirect
3. C2×#4842C2semidirect
5. C3×#1805C2semidirect
6. #70471C2semidirect
7. #70272C2semidirect
8. #70243C2semidirect
9. #69555C2semidirect
10. #68881C2semidirect
11. #68876C2semidirect
56. #1805D3semidirect
66. C3×C4×Q8D4semidirect
102. #869D4semidirect
243. C4×Q8D12semidirect
529. C3#24131semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C2×#50262
C2×#48422
#742392
C3×#18052
#704712
#702722
#702432
#695552
#688812
#688762
#241313