Finite Group Databasebeta

← #10829933

Subgroups — #10829933

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5732C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3C2×C4×D24C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic, complemented
5C3×#1956C211yesmaximal, characteristic, complemented
6#70277C212yesmaximal
7#70245C212yesmaximal, complemented
8#70240C212yesmaximal, complemented
9#70236C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic, complemented
12#24185—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×#215C2212yes—
17C22×D24C2211yescharacteristic
18#5732—24no—
19#5732—24no—
20C2×#898—22no—
21C2×#897—21no—
22C2×#897—21no—
23C2×#892—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#835C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35#5083—24no—
36#4994—22no—
37#4988C2212yes—
38#4987—21no—
39#4984C2212yes—
40C2×#809C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C4×D24—24no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×C4×D8—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1956D311yescharacteristic, complemented
54#1880—32no—
55#1863—32no—
56#1802—31no—
57C22×C3×Q8C2312yes—
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—21no—
62C22×Dic6D411yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42C2311yescharacteristic
73#947—24no—
74#947—24no—
75Q8×Dic3—24no—
76Q8×Dic3—24no—
77C2×#215—24no—
78C2×#215—24no—
79C2×#215—42no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4D412yes—
92C2×C4×Dic3—22no—
93C2×C4×Dic3C2312yes—
94C22×C3⋊C8C2312yes—
95C2×D24—22no—
96C2×D24—22no—
97C2×D24—22no—
98#898—216no—
99#897—28no—
100#897—44no—
101#892—28no—
102C4×D12—22no—
103C4×D12—22no—
104C4×D12—22no—
105C4×D12D414yes—
106C4×Dic6—22no—
107C4×Dic6—22no—
108C3×#103—24no—
109C3×#103—42no—
110C3×C4×C8—22no—
111#835—24no—
112C3⋊#95—22no—
113#821—22no—
114#821—22no—
115#818—22no—
116#809—42no—
117C4⋊(C3⋊C8)—22no—
118C22×SD16—32no—
119C22×D8—31no—
120#556—64no—
121#556—64no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#126—62no—
124C2×#124—32no—
125C2×C22⋊Q8—61no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×D4—31no—
130C2×C4×D4—31no—
131C2×#117—61no—
132#461—64no—
133C4×D8—64no—
134C2×#107—31no—
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103C2×D312yes—
138C2×#103C2×D312yes—
139C2×#102—31no—
140C2×#102—31no—
141C2×#102—32no—
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384C2×D311yescharacteristic
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C23×D3—21no—
152C23×D3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8C2×D414yes—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3C2×D412yes—
167C22×Dic3C2×D412yes—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C2×D412yes—
179C2×D12#4112yes—
180C2×C4×D3—24no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8#4112yes—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—81no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203#215—48no—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—82no—
210C4⋊Dic3—28no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—44no—
216C4⋊Dic3C2×D412yes—
217Dic3⋊C4—24no—
218C4×Dic3—28no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224D24—28no—
225C22×Q8—31no—
226C22×Q8C22×D312yes—
227C22×D4—31no—
228C22×D4—31no—
229C23×C4—61no—
230C23×C4—61no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C2×D8—62no—
235C2×D8—62no—
236C2×D8—62no—
237C22×C8—32no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240C4⋊Q8—22no—
241C4⋊Q8D1214yes—
242#126—616no—
243#124—64no—
244#124—64no—
245C22⋊Q8—68no—
246C22⋊Q8—124no—
247C4×Q8—64no—
248C4×Q8—64no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—34no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4—62no—
255#117—68no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D312yes—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—62no—
273C2×C42—32no—
274C2×C42—62no—
275C2×C42C22×D311yescharacteristic
276#107—62no—
277C4⋊C8—62no—
278#103—24no—
279#103—42no—
280#103—64no—
281#102—62no—
282#102—62no—
283#102—122no—
284C4×C8—22no—
285#95—62no—
286C23×C3C22×D411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C3×Q8—24no—
302C3×Q8—24no—
303C3×Q8—42no—
304C3×Q8—42no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—24no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12012yes—
314C2×C3×C4#12012yes—
315C2×C3×C4#13512yes—
316C2×C3×C4AGL(1,ℤ/8)12yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322C2×C3×C42−1+411yescharacteristic
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335D12—22no—
336D12—22no—
337D12—22no—
338D12—22no—
339D12—41no—
340D12—41no—
341D12—41no—
342D12—41no—
343C4×D3—48no—
344C4×D3—84no—
345Dic6—44no—
346C3×C8—22no—
347C3×C8—41no—
348C3⋊C8—42no—
349C24—61no—
350C24—61no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8C2×D1214yes—
360C2×D4—32no—
361C2×D4—32no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—61no—
377C22×C4—64no—
378C22×C4—61no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4—122no—
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386C22×C4C23×D311yescharacteristic
387SD16—128no—
388D8—68no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8#23712yes—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—42no—
404C4⋊C4—68no—
405C4⋊C4—62no—
406C4⋊C4—64no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—81no—
411C4⋊C4—124no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C22⋊C4—64no—
415C22⋊C4—124no—
416C22⋊C4—122no—
417C22⋊C4—242no—
418C42—68no—
419C42—62no—
420C42—62no—
421C42C2×D1212yes—
422C42D3×D412yes—
423C22×C3C2×#12011yescharacteristic
424C22×C3#54811yescharacteristic
425C22×C3#55011yescharacteristic
426C22×C3C2×#13511yescharacteristic
427C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
428C22×C3#57711yescharacteristic
429C22×C3#57811yescharacteristic
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—81no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—22no—
445C3×C4—44no—
446C3×C4—82no—
447C3×C4#44914yes—
448C3×C4#52511yescharacteristic
449C3×C4#53711yescharacteristic
450C3×C4#54611yescharacteristic
451C3×C4#54711yescharacteristic
452Dic3—42no—
453Dic3—42no—
454Dic3—42no—
455Dic3—81no—
456C23—62no—
457C23—62no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23C2×D3×D411yescharacteristic
471Q8—24no—
472Q8—24no—
473Q8—42no—
474Q8—42no—
475Q8—124no—
476D4—62no—
477D4—62no—
478D4—62no—
479D4—62no—
480D4—121no—
481D4—121no—
482D4—121no—
483D4—121no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—128no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—244no—
506C2×C4#88212yes—
507C2×C4#88212yes—
508C2×C4#91412yes—
509C2×C4#91912yes—
510C2×C4C22×D1211yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×#23711yescharacteristic
513C2×C4#100411yescharacteristic
514C2×C4D3×#4111yescharacteristic
515C2×C4#101011yescharacteristic
516C8—22no—
517C8—41no—
518C8—122no—
519C2×C3#240711yescharacteristic
520C2×C3C2×#44911yescharacteristic
521C2×C3#301311yescharacteristic
522C2×C3#336011yescharacteristic
523C2×C3#336711yescharacteristic
524C2×C3#339311yescharacteristic
525C2×C3#339711yescharacteristic
526D3—82no—
527D3—82no—
528C22—122no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22—241no—
535C22—241no—
536C22—241no—
537C22—241no—
538C22—241no—
539C22—241no—
540C22—241no—
541C22—241no—
542C22C2×#88211yescharacteristic
543C22#567711yescharacteristic
544C22#571811yescharacteristic
545C22C2×#91411yescharacteristic
546C22#576211yescharacteristic
547C22C2×#91911yescharacteristic
548C22#576611yescharacteristic
549C4—22no—
550C4—44no—
551C4—82no—
552C4—122no—
553C4—122no—
554C4—122no—
555C4—241no—
556C4#481014yes—
557C4#555411yescharacteristic
558C4#555511yescharacteristic
559C4#560611yescharacteristic
560C4#560811yescharacteristic
561C3#2418511yescharacteristic, complemented, Sylow
562C2—242no—
563C2—242no—
564C2#7643511yescharacteristic
565C2C2×#481011yescharacteristic
566C2#7879511yescharacteristic
567C2#8162111yescharacteristic
568C2#8162211yescharacteristic
569C2#8175711yescharacteristic
570C2#8179511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1956D3
hypocenterC3#24185
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#5732C2semidirect
2. C2×#5083C2semidirect
4. #74369C2semidirect
5. C3×#1956C2semidirect
7. #70245C2semidirect
8. #70240C2semidirect
9. #70236C2semidirect
10. #69788C2semidirect
11. #69584C2semidirect
53. #1956D3semidirect
561. C3#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×#50832
C2×C4×D242
#743692
C3×#19562
#702772
#702452
#702402
#702362
#697882
#695842
#241853