Finite Group Databasebeta

← #10900970

Subgroups — #10900970

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×#5470C211yesmaximal, characteristic, complemented
2C3×#2854C211yesmaximal, characteristic
3C2×#5035C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#69639C212yesmaximal, complemented
6#69638C211yesmaximal, characteristic, complemented
7#69638C211yesmaximal, characteristic, complemented
8#69606C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#69604C211yesmaximal, characteristic, complemented
11#69503C212yesmaximal, complemented
12#29644—31nomaximal, Sylow
13C2×C3×#123—21no—
14C2×C3×#121C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#926C2212yes—
17C2×#924—21no—
18C2×#924—21no—
19#5470—28no—
20C2×#870C2212yes—
21C2×#869C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6—21no—
25C3×#390C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#388—21no—
30C3×#388—21no—
31C3×#386—21no—
32C3×#380C2211yescharacteristic
33C2×#832C2212yes—
34C2×#832C2212yes—
35C2×#830C2212yes—
36#5035—28no—
37C2×#808C2212yes—
38C2×#807C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#533—31no—
47C2×#482—32no—
48#2854D311yescharacteristic
49#2059—32no—
50#2058—31no—
51#2057—31no—
52#2025—32no—
53#2023—31no—
54#2023—31no—
55#1825—32no—
56#1817—31no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6—21no—
64C3×#123—44no—
65C3×#121—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42—24no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83#926—24no—
84#926D418yes—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89#924—28no—
90#924—28no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—22no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96#870—24no—
97#870—24no—
98#869—24no—
99#869—24no—
100C4×Dic6—28no—
101C4×Dic6—28no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105#832—24no—
106#832—24no—
107#830—24no—
108#808—24no—
109#807—24no—
110#533—68no—
111C2×C4⋊Q8—31no—
112C2×C4⋊Q8—31no—
113C2×#125—32no—
114C2×#123—21no—
115C2×C22⋊Q8—61no—
116C2×C22⋊Q8—61no—
117C2×#121C2×D311yescharacteristic
118C2×C4×Q8—61no—
119C2×C4×Q8—61no—
120C2×#117—32no—
121C22×C42C2×D311yescharacteristic
122#482—68no—
123C2×#107—32no—
124C2×#106—32no—
125C2×#102—32no—
126C2×#102—32no—
127C2×C22⋊C8—32no—
128#390C2×D311yescharacteristic
129#390C2×D311yescharacteristic
130#389C2×D311yescharacteristic
131#389C2×D311yescharacteristic
132#388—21no—
133#388—21no—
134#386—21no—
135#380C2×D311yescharacteristic
136#340—32no—
137#340—32no—
138#340—32no—
139#340—32no—
140#336—31no—
141#336—31no—
142#336—32no—
143C24×C3—21no—
144C24×C3C2×D411yescharacteristic
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3—22no—
173C2×Dic6—24no—
174C2×Dic6—24no—
175C2×Dic6—42no—
176C2×Dic6—42no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—42no—
190C3×C22⋊C4—42no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195#217—28no—
196C4⋊Dic3—24no—
197C4⋊Dic3—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3C2×D414yes—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—28no—
204C4×Dic3—28no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—61no—
212C22×Q8—61no—
213C22×D4—21no—
214C22×D4C22×D311yescharacteristic
215C23×C4#6711yescharacteristic
216C23×C4#6711yescharacteristic
217C23×C4C22×D311yescharacteristic
218C22×C8—32no—
219C22×C8—32no—
220C4⋊Q8—64no—
221C4⋊Q8—64no—
222#125—64no—
223#125—64no—
224#123—44no—
225C22⋊Q8—68no—
226C22⋊Q8—68no—
227#121—24no—
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—32no—
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×D311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—62no—
250C2×C42—24no—
251C2×C42—62no—
252C2×C42#6711yescharacteristic
253C2×C42#6711yescharacteristic
254C2×C42C22×D311yescharacteristic
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#102—64no—
259#102—64no—
260C22⋊C8—64no—
261#95—21no—
262#95—21no—
263#95—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3#12012yes—
278C23×C3C22×D411yescharacteristic
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—84no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4#12012yes—
308C2×C3×C4#13512yes—
309C2×C3×C4AGL(1,ℤ/8)12yes—
310C2×C3×C4C22×D412yes—
311C2×C3×C42−1+412yes—
312C2×C3×C42−1+411yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320Dic6—48no—
321Dic6—48no—
322C3⋊C8—42no—
323C3⋊C8—42no—
324C24—21no—
325C24C2×#6711yescharacteristic
326C2×Q8—64no—
327C2×Q8—64no—
328C2×Q8—122no—
329C2×Q8—122no—
330C2×Q8—122no—
331C2×Q8—122no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4D3×D414yes—
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C23×D311yescharacteristic
360C2×C8—62no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—34no—
371C4⋊C4—34no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—68no—
375C4⋊C4—68no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—68no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—68no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3C2×#12011yescharacteristic
407C22×C3#53611yescharacteristic
408C22×C3#53611yescharacteristic
409C22×C3C2×#13511yescharacteristic
410C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C3#57511yescharacteristic
412C22×C3#57511yescharacteristic
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4—81no—
421C3×C4#53612yes—
422C3×C4#53612yes—
423Dic3—44no—
424Dic3—44no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23#93712yes—
439C23C22×#6711yescharacteristic
440Q8—128no—
441Q8—128no—
442D4—42no—
443D4—42no—
444D4—84no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4#93212yes—
477C2×C4#93712yes—
478C2×C4#95012yes—
479C2×C4C2×D3×D412yes—
480C2×C4#101012yes—
481C2×C4#101011yescharacteristic
482C2×C4#101011yescharacteristic
483C8—122no—
484C8—122no—
485C2×C3—42no—
486C2×C3—161no—
487C2×C3#261411yescharacteristic
488C2×C3#300211yescharacteristic
489C2×C3#300611yescharacteristic
490C2×C3#322811yescharacteristic
491C2×C3#323311yescharacteristic
492C2×C3#323511yescharacteristic
493C2×C3#324011yescharacteristic
494C22—22no—
495C22—22no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C2×#93211yescharacteristic
509C22C2×#93711yescharacteristic
510C22C2×#95011yescharacteristic
511C22#582211yescharacteristic
512C22#582211yescharacteristic
513C22#582511yescharacteristic
514C22#582511yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—124no—
524C4—124no—
525C4#564812yes—
526C4#564812yes—
527C3#2964411yescharacteristic, complemented, Sylow
528C2—42no—
529C2—161no—
530C2#8262411yescharacteristic
531C2#8286411yescharacteristic
532C2#8292611yescharacteristic
533C2#8312211yescharacteristic
534C2#8312511yescharacteristic
535C2#8327211yescharacteristic
536C2#8327411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2854C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2854D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
4. #71531C2semidirect
5. #69639C2semidirect
6. #69638C2semidirect
7. #69638C2semidirect
8. #69606C2semidirect
9. #69605C2semidirect
10. #69604C2semidirect
11. #69503C2semidirect
527. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×#28542
C2×#50352
#715312
#696392
#696382
#696062
#696052
#696042
#695032
#296443