Finite Group Databasebeta

← #10830086

Subgroups — #10830086

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic, complemented
4C2×#4711C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69822C211yesmaximal, characteristic, complemented
8#68686C212yesmaximal, complemented
9#68684C212yesmaximal, complemented
10#68675C212yesmaximal
11#68630C212yesmaximal, complemented
12#24197—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C22×D24C2211yescharacteristic
16C22×#204C2212yes—
17#5584—24no—
18#5584—24no—
19C2×#898—22no—
20C2×#892—21no—
21C2×#875C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5231—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2212yes—
32C2×#817C2211yescharacteristic
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4993—21no—
37#4989—22no—
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41#4841—24no—
42#4711—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×#442—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985D311yescharacteristic, complemented
53#1960—31no—
54#1880—32no—
55#1865—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—21no—
62C22×Dic6C2312yes—
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72#946—28no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×C4×Dic3—22no—
87C2×D24—22no—
88C2×D24—22no—
89C2×D24—22no—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93#898—216no—
94#892—28no—
95#875—24no—
96#875—24no—
97C4×D12—22no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12D414yes—
101#869—22no—
102#869D414yes—
103C4×Dic6—24no—
104C4×Dic6—24no—
105C3×C4⋊C8—22no—
106C3×#97—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—22no—
111#821—22no—
112#821—42no—
113#817—22no—
114#816—24no—
115#816—24no—
116#816—42no—
117C22×SD16—32no—
118C22×D8—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—32no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×#117—61no—
130#461—64no—
131#442—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97C2×D311yescharacteristic
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336C2×D312yes—
149#336C2×D312yes—
150C23×D3—21no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12C2×D412yes—
174C2×D12#4112yes—
175C2×C4×D3—24no—
176C2×C4×D3—44no—
177C2×C4×D3—42no—
178C2×C4×D3—82no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6C2×D414yes—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8#4112yes—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
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—24no—
207C4⋊Dic3—24no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3C2×D412yes—
214Dic3⋊C4—28no—
215Dic3⋊C4—82no—
216C4×Dic3—28no—
217D24—28no—
218#204—48no—
219C22×Q8—32no—
220C22×Q8D1211yescharacteristic
221C22×D4—31no—
222C22×D4—31no—
223C23×C4—61no—
224C23×C4—61no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—62no—
231C22×C8C22×D312yes—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234C4⋊Q8—34no—
235C4⋊Q8—62no—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—64no—
243C4×Q8—64no—
244C4×D4—34no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4—62no—
248#117—68no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4D1211yescharacteristic
257C2×C4⋊C4D1212yes—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—62no—
265C2×C42C22×D312yes—
266C2×C42C22×D311yescharacteristic
267#106—62no—
268C4⋊C8—22no—
269#103—64no—
270#103—64no—
271#103—122no—
272#102—62no—
273#102—62no—
274#102—122no—
275#97—22no—
276#95—61no—
277#95—61no—
278#95—62no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C3×Q8—44no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4#12012yes—
300C2×C3×C4#12012yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4H(5,𝔽2)11yescharacteristic
306C2×C3×C42−1+411yescharacteristic
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317D12—22no—
318D12—22no—
319D12—22no—
320D12—22no—
321D12—41no—
322D12—41no—
323D12—41no—
324D12—41no—
325C4×D3—48no—
326C4×D3—84no—
327Dic6—24no—
328Dic6—24no—
329Dic6—42no—
330Dic6—42no—
331C3×C8—22no—
332C3×C8—41no—
333C3×C8—42no—
334C24—61no—
335C24—61no—
336C2×Q8—22no—
337C2×Q8—34no—
338C2×Q8—42no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×D4—32no—
345C2×D4—32no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—64no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4—124no—
362C22×C4—122no—
363C22×C4—242no—
364C22×C4C2×D1212yes—
365C22×C4C2×D1212yes—
366C22×C4C2×D1211yescharacteristic
367C22×C4C2×D1211yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—128no—
370D8—68no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—21no—
379C2×C8#23912yes—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—32no—
383C4⋊C4—68no—
384C4⋊C4—64no—
385C4⋊C4—64no—
386C4⋊C4—61no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—242no—
392C22⋊C4—68no—
393C22⋊C4—68no—
394C22⋊C4—64no—
395C22⋊C4—124no—
396C22⋊C4—122no—
397C22⋊C4—242no—
398C42—22no—
399C42—22no—
400C42—68no—
401C42D3×D412yes—
402C42D3×D412yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3#55011yescharacteristic
406C22×C3#57512yes—
407C22×C3#57712yes—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—82no—
414C2×D3—82no—
415C2×D3—82no—
416C2×D3—81no—
417C2×D3—81no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4#52511yescharacteristic
423C3×C4#53711yescharacteristic
424C3×C4#54611yescharacteristic
425C3×C4#54711yescharacteristic
426Dic3—44no—
427Dic3—42no—
428Dic3—82no—
429Dic3—81no—
430C23—62no—
431C23—62no—
432C23—121no—
433C23—122no—
434C23—122no—
435C23—122no—
436C23—122no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23C22×D1211yescharacteristic
441Q8—44no—
442Q8—64no—
443Q8—64no—
444Q8—122no—
445Q8—122no—
446D4—62no—
447D4—62no—
448D4—62no—
449D4—62no—
450D4—121no—
451D4—121no—
452D4—121no—
453D4—121no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—128no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—244no—
474C2×C4#88212yes—
475C2×C4#88212yes—
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×#23911yescharacteristic
479C2×C4#100711yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4#100911yescharacteristic
482C8—22no—
483C8—42no—
484C8—41no—
485C2×C3#240711yescharacteristic
486C2×C3#300511yescharacteristic
487C2×C3#301011yescharacteristic
488C2×C3#337612yes—
489C2×C3#340112yes—
490D3—82no—
491D3—82no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—124no—
497C22—241no—
498C22—242no—
499C22—242no—
500C22—241no—
501C22—242no—
502C22C2×#88211yescharacteristic
503C22#557611yescharacteristic
504C22#557711yescharacteristic
505C22#574812yes—
506C22#575112yes—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—122no—
512C4—124no—
513C4—242no—
514C4—241no—
515C4#562711yescharacteristic
516C4#562911yescharacteristic
517C4#567611yescharacteristic
518C4#572611yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—242no—
521C2—242no—
522C2#7661811yescharacteristic
523C2#7863811yescharacteristic
524C2#7867811yescharacteristic
525C2#7995412yes—
526C2#7997412yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1985D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
1. C2×#5584C2semidirect
2. #82501C2semidirect
3. C2×#4841C2semidirect
5. C3×#1985C2semidirect
6. #70480C2semidirect
7. #69822C2semidirect
8. #68686C2semidirect
9. #68684C2semidirect
11. #68630C2semidirect
52. #1985D3semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
#825012
C2×#48412
C2×#47112
C3×#19852
#704802
#698222
#686862
#686842
#686752
#686302
#241973