Finite Group Databasebeta

← #10829982

Subgroups — #10829982

Order 768.

1082 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
2#82482C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×C4×#204C211yesmaximal, characteristic
5C3×#1984C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic
7#69771C211yesmaximal, characteristic, complemented
8#68685C212yesmaximal, complemented
9#68684C212yesmaximal, complemented
10#68677C212yesmaximal
11#68630C212yesmaximal
12#24189—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C22×Dic12C2212yes—
16C22×#204C2211yescharacteristic
17#5584—24no—
18#5584—24no—
19C2×#898—22no—
20C2×#892—21no—
21C2×#875C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5229—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#817C2211yescharacteristic
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36#4982—22no—
37#4972—21no—
38#4935C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41#4849—24no—
42C4×#204—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×C4×SD16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984D311yescharacteristic, complemented
53#1956—31no—
54#1878—32no—
55#1865—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6C2312yes—
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D412yes—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71#946—28no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—21no—
87C2×Dic12—24no—
88C2×Dic12—24no—
89C2×Dic12—42no—
90C2×#204—22no—
91C2×#204—22no—
92C2×#204—22no—
93#898—216no—
94#892—28no—
95#875—24no—
96#875—24no—
97C4×D12—22no—
98C4×D12D414yes—
99#869—22no—
100#869D414yes—
101C4×Dic6—24no—
102C4×Dic6—24no—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×C4⋊C8—22no—
106C3×C4×C8—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110#821—22no—
111#821—42no—
112#817—22no—
113#816—24no—
114#816—24no—
115#816—22no—
116#816—42no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×#117—61no—
130#462—64no—
131C4×SD16—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C4×C8C2×D311yescharacteristic
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336C2×D312yes—
149#336C2×D312yes—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12C2×D412yes—
170C2×C4×D3—24no—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6—22no—
176C2×Dic6—21no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—21no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—21no—
183C2×Dic6C2×D414yes—
184C2×Dic6#4112yes—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—22no—
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—
203C4⋊Dic3—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—42no—
211C4⋊Dic3—81no—
212C4⋊Dic3C2×D412yes—
213Dic3⋊C4—28no—
214Dic3⋊C4—42no—
215Dic3⋊C4—82no—
216C4×Dic3—28no—
217C4×Dic3—42no—
218Dic12—48no—
219#204—28no—
220C22×Q8—32no—
221C22×Q8—31no—
222C22×Q8D1211yescharacteristic
223C22×D4—31no—
224C23×C4—61no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—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×Q8—62no—
245C4×Q8—62no—
246C4×D4—34no—
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—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4D1211yescharacteristic
260C2×C4⋊C4D1212yes—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—62no—
264C2×C42—62no—
265C2×C42—61no—
266C2×C42C22×D312yes—
267C2×C42C22×D311yescharacteristic
268#106—62no—
269C4⋊C8—22no—
270#103—62no—
271#103—64no—
272#103—64no—
273#103—122no—
274#102—62no—
275#102—122no—
276C4×C8—22no—
277#95—62no—
278#95—62no—
279#95—62no—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C3×Q8—44no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
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#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4#13512yes—
300C2×C3×C4#13712yes—
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—22no—
310C2×Dic3—41no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—41no—
322D12—22no—
323D12—22no—
324D12—41no—
325D12—41no—
326C4×D3—48no—
327Dic6—22no—
328Dic6—22no—
329Dic6—24no—
330Dic6—24no—
331Dic6—42no—
332Dic6—41no—
333Dic6—42no—
334Dic6—41no—
335C3×C8—22no—
336C3×C8—41no—
337C3×C8—42no—
338C24—61no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—34no—
343C2×Q8—32no—
344C2×Q8—62no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—62no—
349C2×Q8—61no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—61no—
354C2×D4—32no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—64no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—122no—
367C22×C4—122no—
368C22×C4C2×D1212yes—
369C22×C4C2×D1212yes—
370C22×C4C2×D1211yescharacteristic
371C22×C4C2×D1211yescharacteristic
372C22×C4C23×D311yescharacteristic
373Q16—128no—
374SD16—68no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8#23912yes—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—32no—
387C4⋊C4—68no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—122no—
396C4⋊C4—122no—
397C4⋊C4—242no—
398C4⋊C4—241no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—64no—
402C42—22no—
403C42—22no—
404C42—68no—
405C42—122no—
406C42D3×D412yes—
407C42D3×D412yes—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3C2×#13511yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57711yescharacteristic
414C22×C3#57811yescharacteristic
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—81no—
419C2×D3—81no—
420C2×D3—81no—
421C2×D3—81no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4#45214yes—
426C3×C4#52511yescharacteristic
427C3×C4#53711yescharacteristic
428C3×C4#54611yescharacteristic
429C3×C4#54711yescharacteristic
430Dic3—42no—
431Dic3—44no—
432Dic3—42no—
433Dic3—82no—
434Dic3—81no—
435C23—62no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23C22×D1211yescharacteristic
443Q8—44no—
444Q8—62no—
445Q8—64no—
446Q8—64no—
447Q8—62no—
448Q8—121no—
449Q8—122no—
450Q8—122no—
451Q8—121no—
452D4—62no—
453D4—62no—
454D4—121no—
455D4—121no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—62no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—128no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4#88212yes—
481C2×C4#88212yes—
482C2×C4#90412yes—
483C2×C4#90912yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23911yescharacteristic
487C2×C4#100711yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4#100911yescharacteristic
490C8—22no—
491C8—42no—
492C8—41no—
493C2×C3#240711yescharacteristic
494C2×C3C2×#45211yescharacteristic
495C2×C3#301311yescharacteristic
496C2×C3#336411yescharacteristic
497C2×C3#336911yescharacteristic
498C2×C3#339411yescharacteristic
499C2×C3#339811yescharacteristic
500D3—82no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22C2×#88211yescharacteristic
509C22#557611yescharacteristic
510C22#557711yescharacteristic
511C22C2×#90411yescharacteristic
512C22C2×#90911yescharacteristic
513C22#575111yescharacteristic
514C22#575311yescharacteristic
515C4—22no—
516C4—42no—
517C4—42no—
518C4—122no—
519C4—124no—
520C4—122no—
521C4—242no—
522C4—241no—
523C4#473314yes—
524C4#562711yescharacteristic
525C4#562911yescharacteristic
526C4#567611yescharacteristic
527C4#572611yescharacteristic
528C3#2418911yescharacteristic, complemented, Sylow
529C2—242no—
530C2#7661811yescharacteristic
531C2C2×#473311yescharacteristic
532C2#7868011yescharacteristic
533C2#7991311yescharacteristic
534C2#7993411yescharacteristic
535C2#7995111yescharacteristic
536C2#7998111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1984D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82482C2semidirect
3. C2×#4849C2semidirect
5. C3×#1984C2semidirect
7. #69771C2semidirect
8. #68685C2semidirect
9. #68684C2semidirect
52. #1984D3semidirect
528. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
#824822
C2×#48492
C2×C4×#2042
C3×#19842
#704792
#697712
#686852
#686842
#686772
#686302
#241893