Finite Group Databasebeta

← #10828561

Subgroups — #10828561

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×#5044C211yesmaximal, characteristic
3C2×#5043C211yesmaximal, characteristic
4#82234C211yesmaximal, characteristic
5#71781C212yesmaximal, complemented
6#71739C212yesmaximal, complemented
7#71693C211yesmaximal, characteristic
8#71693C211yesmaximal, characteristic
9#71629C212yesmaximal, complemented
10#71628C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic
12#24071—31nomaximal, Sylow
13C3×#541—24no—
14C3×#541—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#126—21no—
18C2×C3×#125—21no—
19C2×C3×#125—21no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32C2×#830C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#5044—24no—
35#5043—24no—
36#4977—21no—
37#4975—21no—
38C2×#808C2212yes—
39C2×#807C2212yes—
40#4946—21no—
41#4946—21no—
42#4928C2212yes—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#541D311yescharacteristic
49C2×#484—31no—
50C2×#483—31no—
51#2317—31no—
52#2059—32no—
53#2043—32no—
54#1873—31no—
55#1873—31no—
56#1865—32no—
57#1863—32no—
58#1818—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×#126—28no—
66C3×#126—28no—
67C3×#125—28no—
68C3×#125—28no—
69C3×#125—22no—
70C3×#125—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8D414yes—
74C3×C22⋊Q8D414yes—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×#117—22no—
80C3×#117—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102#926—24no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2312yes—
108C2×Dic3⋊C4—22no—
109C2×Dic3⋊C4—22no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#835—24no—
114#835—24no—
115#835—24no—
116#835—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—22no—
120#830—22no—
121#808—24no—
122#807—24no—
123#541—24no—
124#541—24no—
125C2×C4⋊Q8C2×D311yescharacteristic
126C2×#126—21no—
127C2×#126—21no—
128C2×#125—21no—
129C2×#125—21no—
130C2×#125C2×D311yescharacteristic
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C22⋊Q8C2×D311yescharacteristic
133C2×C4×Q8C2×D311yescharacteristic
134C2×C4×Q8C2×D311yescharacteristic
135C2×#117—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138#484—64no—
139#483—64no—
140C2×#107—32no—
141C2×#106—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×C22⋊C8—31no—
147C2×C22⋊C8—31no—
148#389—32no—
149#389—32no—
150#385—61no—
151#385—61no—
152#384—61no—
153#382—61no—
154#340—32no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8C2×D412yes—
166C2×C3×Q8C2×D412yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—42no—
184C22×Dic3—82no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C42—24no—
204C3×C42—24no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42—21no—
208C3×C42—21no—
209#217—24no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4⋊Dic3#4114yes—
215Dic3⋊C4—82no—
216Dic3⋊C4—82no—
217C4×Dic3—24no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C22×Q8C22×D311yescharacteristic
225C22×Q8C22×D311yescharacteristic
226C23×C4—61no—
227C23×C4C22×D311yescharacteristic
228C22×C8—32no—
229C22×C8—32no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232#126—28no—
233#126—28no—
234#125—28no—
235#125—28no—
236#125—22no—
237#125—22no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8#6714yes—
241C22⋊Q8#6714yes—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246#117—22no—
247#117—22no—
248#117—64no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4#6711yescharacteristic
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4#6711yescharacteristic
272C2×C22⋊C4#6711yescharacteristic
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—61no—
276C2×C42C22×D311yescharacteristic
277C2×C42C22×D311yescharacteristic
278#107—64no—
279#106—64no—
280#103—64no—
281#103—64no—
282#103—64no—
283#103—64no—
284C22⋊C8—62no—
285C22⋊C8—62no—
286#95—62no—
287#95—62no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3#12012yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—24no—
295C3×Q8—24no—
296C3×Q8—42no—
297C3×Q8—42no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12012yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4112yes—
324C2×C3×C42−1+412yes—
325C2×Dic3—24no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—84no—
332C3⋊C8—42no—
333C3⋊C8—42no—
334C24C2×#6711yescharacteristic
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8C2×#6712yes—
342C2×Q8C2×#6712yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—122no—
352C22×C4—242no—
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6712yes—
359C22×C4C2×#6712yes—
360C22×C4C23×D311yescharacteristic
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—62no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—34no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—122no—
384C4⋊C4—242no—
385C4⋊C4—242no—
386C4⋊C4C2×#6712yes—
387C4⋊C4C2×#6712yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—64no—
393C42—24no—
394C42—24no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—64no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3C2×#12011yescharacteristic
406C22×C3#52511yescharacteristic
407C22×C3#53611yescharacteristic
408C22×C3#57712yes—
409C22×C3#57812yes—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—42no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—42no—
419C3×C4#54712yes—
420C3×C4#54712yes—
421Dic3—44no—
422Dic3—82no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23#95312yes—
428C23C22×#6711yescharacteristic
429Q8—24no—
430Q8—24no—
431Q8—42no—
432Q8—42no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—64no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—244no—
462C2×C4#95312yes—
463C2×C4D3×#4112yes—
464C2×C4#101012yes—
465C2×C4C22×#6711yescharacteristic
466C2×C4C22×#6711yescharacteristic
467C8—122no—
468C8—122no—
469C2×C3—42no—
470C2×C3#240611yescharacteristic
471C2×C3#301011yescharacteristic
472C2×C3#301111yescharacteristic
473C2×C3#312012yes—
474C2×C3#324212yes—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—42no—
479C22—42no—
480C22#582511yescharacteristic
481C22#582811yescharacteristic
482C22#583712yes—
483C22#583812yes—
484C22C2×#95311yescharacteristic
485C4—22no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—42no—
491C4—41no—
492C4—41no—
493C4—42no—
494C4—124no—
495C4—242no—
496C4#582412yes—
497C4#582412yes—
498C3#2407111yescharacteristic, complemented, Sylow
499C2—42no—
500C2#8344212yes—
501C2#8344612yes—
502C2#8409011yescharacteristic
503C2#8411311yescharacteristic
504C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#541C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#541D3
hypocenterC3#24071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
5. #71781C2semidirect
6. #71739C2semidirect
9. #71629C2semidirect
10. #71628C2semidirect
498. C3#24071semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#50442
C2×#50432
#822342
#717812
#717392
#716932
#716292
#716282
#715842
#240713