Finite Group Databasebeta

← #10900965

Subgroups — #10900965

Order 768.

1064 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×#5728C211yesmaximal, characteristic, complemented
2C2×#4866C212yesmaximal
3#76286C211yesmaximal, characteristic
4C3×#2058C211yesmaximal, characteristic, complemented
5#71792C211yesmaximal, characteristic, complemented
6#70566C212yesmaximal, complemented
7#69604C211yesmaximal, characteristic, complemented
8#68899C212yesmaximal, complemented
9#68873C212yesmaximal, complemented
10#68678C211yesmaximal, characteristic, complemented
11#68664C211yesmaximal, characteristic, complemented
12#29644—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—21no—
15C2×Q8×Dic3—21no—
16#5728—28no—
17C2×#897—21no—
18C2×#895C2211yescharacteristic
19C2×#894—21no—
20C2×#892C2212yes—
21C2×#890C2212yes—
22C2×#887—21no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#106C2212yes—
26C3×#389C2211yescharacteristic
27C2×#821C2212yes—
28C2×(C2×D3)⋊C8C2212yes—
29#4990—21no—
30#4988C2211yescharacteristic
31#4987C2211yescharacteristic
32#4972C2211yescharacteristic
33C2×#809C2212yes—
34C2×#807C2212yes—
35#4939C2211yescharacteristic
36#4866—28no—
37#4669—21no—
38#4665—21no—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#482—32no—
48#2854—31no—
49#2059—32no—
50#2058D311yescharacteristic, complemented
51#2057—31no—
52#2025—32no—
53#2023—31no—
54#2023—31no—
55#1825—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×D12—21no—
59C22×D12C2311yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3D411yescharacteristic
62C22×C4×D3C2311yescharacteristic
63C22×Dic6—21no—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72#946—28no—
73Q8×Dic3—28no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—21no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92#897—28no—
93#895—24no—
94#894—44no—
95#892—24no—
96#892D418yes—
97#890—24no—
98#890—24no—
99#887—28no—
100C42×D3—24no—
101#869—24no—
102C3×#106—24no—
103C3×#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106#821—24no—
107(C2×D3)⋊C8—24no—
108#809—24no—
109#807—24no—
110#533—68no—
111C2×C4⋊Q8—31no—
112C2×C4⋊Q8C2×D311yescharacteristic
113C2×#125—32no—
114C2×#123—61no—
115C2×C22⋊Q8—61no—
116C2×C22⋊Q8—61no—
117C2×#121—31no—
118C2×C4×Q8—61no—
119C2×C4×Q8—61no—
120C2×#117—32no—
121C22×C42—31no—
122#482—68no—
123C2×#107—32no—
124C2×#106C2×D312yes—
125C2×#102—32no—
126C2×#102—32no—
127C2×C22⋊C8—32no—
128#390—31no—
129#390—31no—
130#389—31no—
131#389C2×D311yescharacteristic
132#388—61no—
133#388—61no—
134#386—61no—
135#380—31no—
136#340—32no—
137#340—32no—
138#340—32no—
139#340C2×D312yes—
140#336—32no—
141#336—31no—
142#336C2×D311yescharacteristic
143C23×D3—21no—
144C23×D3C2×D411yescharacteristic
145C2×C3×Q8—24no—
146C2×C3×Q8—42no—
147C2×C3×Q8—42no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—22no—
155C22×Dic3—21no—
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C2×D12—22no—
160C2×D12—21no—
161C2×D12—22no—
162C2×D12—21no—
163C2×D12—42no—
164C2×D12—42no—
165C2×D12—42no—
166C2×C4×D3—21no—
167C2×C4×D3—22no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—41no—
174C2×C4×D3—41no—
175C2×C4×D3C2×D414yes—
176C2×Dic6—24no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4C2×D414yes—
189C3×C42—22no—
190(C2×D3)⋊C4—28no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—42no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—42no—
197C4⋊Dic3—24no—
198C4⋊Dic3—28no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—42no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C4×Dic3—28no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—21no—
212C22×Q8—61no—
213C22×D4—31no—
214C22×D4—61no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—31no—
218C22×C8—32no—
219C22×C8C22×D312yes—
220C4⋊Q8—24no—
221C4⋊Q8—64no—
222#125—64no—
223#125—64no—
224#123—124no—
225C22⋊Q8—68no—
226C22⋊Q8—68no—
227#121—64no—
228C4×Q8—68no—
229C4×Q8—68no—
230#117—38no—
231#117—64no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C42—31no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—64no—
254C2×C42—62no—
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—24no—
258#102—64no—
259#102—64no—
260C22⋊C8—64no—
261#95—21no—
262#95—61no—
263#95—61no—
264C23×C3C22×D411yescharacteristic
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—21no—
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3#12012yes—
279C3×Q8—48no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—24no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4#13512yes—
294C2×C3×C4AGL(1,ℤ/8)12yes—
295C2×C3×C4C22×D412yes—
296C2×C3×C42−1+411yescharacteristic
297C2×C3×C42−1+412yes—
298C2×C3×C42−1+411yescharacteristic
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—24no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3#12012yes—
314D12—42no—
315D12—42no—
316D12—84no—
317C4×D3—24no—
318C4×D3—24no—
319C4×D3—44no—
320C4×D3—44no—
321Dic6—48no—
322C3×C8—42no—
323C3⋊C8—42no—
324C24—31no—
325C24—61no—
326C2×Q8—24no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—64no—
330C2×Q8—122no—
331C2×Q8—122no—
332C2×D4—62no—
333C2×D4—61no—
334C2×D4—62no—
335C2×D4—61no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—122no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—31no—
343C22×C4—34no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—34no—
372C4⋊C4—42no—
373C4⋊C4—68no—
374C4⋊C4—68no—
375C4⋊C4—64no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—122no—
379C4⋊C4D3×D414yes—
380C22⋊C4—68no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C42—22no—
388C42—62no—
389C42—62no—
390C42—68no—
391C42—62no—
392C22×C3C2×#12011yescharacteristic
393C22×C3#53611yescharacteristic
394C22×C3#53611yescharacteristic
395C22×C3C2×#13511yescharacteristic
396C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
397C22×C3#57511yescharacteristic
398C22×C3#57511yescharacteristic
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—44no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4#53612yes—
419C3×C4#53612yes—
420Dic3—22no—
421Dic3—41no—
422Dic3—44no—
423Dic3—41no—
424Dic3—81no—
425C23—32no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23C2×D3×D411yescharacteristic
440Q8—48no—
441Q8—128no—
442D4—122no—
443D4—122no—
444D4—244no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—32no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—64no—
470C2×C4—124no—
471C2×C4—124no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4#91412yes—
478C2×C4#91612yes—
479C2×C4C2×D3×D412yes—
480C2×C4#100711yescharacteristic
481C2×C4#100712yes—
482C2×C4#101011yescharacteristic
483C8—42no—
484C8—122no—
485C2×C3#261411yescharacteristic
486C2×C3#300211yescharacteristic
487C2×C3#300611yescharacteristic
488C2×C3#322811yescharacteristic
489C2×C3#323311yescharacteristic
490C2×C3#323511yescharacteristic
491C2×C3#324011yescharacteristic
492D3—42no—
493D3—161no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22D3×#12011yescharacteristic
509C22#562811yescharacteristic
510C22#572411yescharacteristic
511C22C2×#91411yescharacteristic
512C22C2×#91611yescharacteristic
513C22#577311yescharacteristic
514C22#577711yescharacteristic
515C4—22no—
516C4—22no—
517C4—44no—
518C4—62no—
519C4—81no—
520C4—81no—
521C4—121no—
522C4—121no—
523C4—124no—
524C4—241no—
525C4#562812yes—
526C4#572412yes—
527C3#2964411yescharacteristic, complemented, Sylow
528C2—122no—
529C2—481no—
530C2#7725611yescharacteristic
531C2#8004011yescharacteristic
532C2#8014511yescharacteristic
533C2#8101611yescharacteristic
534C2#8106611yescharacteristic
535C2#8189511yescharacteristic
536C2#8198411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2058C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2058D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
4. C3×#2058C2semidirect
5. #71792C2semidirect
6. #70566C2semidirect
7. #69604C2semidirect
8. #68899C2semidirect
9. #68873C2semidirect
10. #68678C2semidirect
11. #68664C2semidirect
50. #2058D3semidirect
527. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#48662
#762862
C3×#20582
#717922
#705662
#696042
#688992
#688732
#686782
#686642
#296443