Finite Group Databasebeta

← #10848681

Subgroups — #10848681

Order 768.

1438 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×C3×#532C211yesmaximal, characteristic
2C2×#5093C211yesmaximal, characteristic, complemented
3C2×#5076C211yesmaximal, characteristic, complemented
4#82076C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#71764C211yesmaximal, characteristic, complemented
7#71685C214yesmaximal, complemented
8#71612C214yesmaximal, complemented
9C4×#830C211yesmaximal, characteristic, complemented
10#25597—31nomaximal, Sylow
11C22×C3×#41—21no—
12C3×#532—24no—
13C3×#532—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×#214C2214yes—
23C2×#924C2212yes—
24C2×#869C2211yescharacteristic
25C2×#832C2214yes—
26#5212—21no—
27#5212—21no—
28C2×#830C2212yes—
29#5093—24no—
30#5076—24no—
31C2×#810C2214yes—
32#4935C2211yescharacteristic
33#4929C2212yes—
34#4928—21no—
35#4928—21no—
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#532D311yescharacteristic
41C2×#492—31no—
42C2×#487—31no—
43#2857—31no—
44#2000—34no—
45#1990—31no—
46#1957—31no—
47#1877—34no—
48C4×C22⋊C8—31no—
49C2×C3×#41—28no—
50C2×C3×#41—44no—
51C2×C3×#41—44no—
52C22×C3×Q8—21no—
53C22×C3×D4—22no—
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4C2312yes—
57C23×C3×C4—22no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×Dic6C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#127—22no—
65C3×#127D414yes—
66C3×#124—28no—
67C3×#121—28no—
68C3×#121—24no—
69C3×#121—24no—
70C3×C4×D4—28no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#217—22no—
84C2×#214—28no—
85C2×#214—28no—
86C2×#214—44no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2312yes—
92#924—44no—
93C2×Dic3⋊C4—22no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96#869—22no—
97#869—22no—
98#832—44no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#830—24no—
102#810—28no—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—22no—
105C22×#41—21no—
106C22×SD16—34no—
107#532—24no—
108#532—24no—
109C2×C4⋊Q8—31no—
110C2×C4⋊Q8C2×D311yescharacteristic
111C2×#127C2×D311yescharacteristic
112C2×#124—21no—
113C2×C22⋊Q8—32no—
114C2×#121—21no—
115C2×#121C2×D312yes—
116C2×C4×D4—21no—
117C2×C4×D4C2×D312yes—
118C22×C42C2×D311yescharacteristic
119#492—64no—
120#487—64no—
121C2×#103—34no—
122C2×#102—34no—
123C2×C22⋊C8—32no—
124C2×C4×C8—31no—
125C2×C4×C8—31no—
126#391—32no—
127#389—61no—
128#389—61no—
129#384—31no—
130#380—61no—
131#380—61no—
132#340—34no—
133#336—32no—
134C24×C3—22no—
135C24×C3C2×D411yescharacteristic
136C3×#41—432no—
137C2×C3×Q8—24no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×D4—28no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4C2×D414yes—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—28no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—44no—
165C22×C3×C4—44no—
166C22×C3×C4SD1614yescomplemented
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—24no—
178C2×Dic6—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—28no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190#217—44no—
191#214—416no—
192C4⋊Dic3—24no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4⋊Dic3—42no—
196C4⋊Dic3—42no—
197Dic3⋊C4—44no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8#4114yescomplemented
204C2×#41—28no—
205C2×#41—44no—
206C2×#41—44no—
207C22×Q8—21no—
208C22×Q8—32no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×D312yes—
213C23×C4—22no—
214C23×C4#6711yescharacteristic
215C23×C4#6711yescharacteristic
216C23×C4C22×D311yescharacteristic
217C2×SD16—68no—
218C2×SD16—68no—
219C2×SD16—124no—
220C22×C8—32no—
221C22×C8—32no—
222C4⋊Q8—22no—
223C4⋊Q8—62no—
224C4⋊Q8—62no—
225C4⋊Q8#6714yes—
226#127—22no—
227#127#6714yes—
228#124—28no—
229C22⋊Q8—124no—
230#121—28no—
231#121—24no—
232#121—24no—
233C4×D4—28no—
234C4×D4—24no—
235C4×D4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4C22×D312yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—62no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42#6711yescharacteristic
250C2×C42#6711yescharacteristic
251C2×C42C22×D311yescharacteristic
252C2×C42C22×D311yescharacteristic
253#103—68no—
254#102—124no—
255C22⋊C8—64no—
256C4×C8—62no—
257C4×C8—62no—
258#95—62no—
259#95—62no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—24no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3C2×SD1612yes—
274C23×C3C22×D411yescharacteristic
275C3×Q8—48no—
276C3×D4—28no—
277C3×D4—416no—
278C3×D4—44no—
279C3×D4—48no—
280C3×D4—48no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—28no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—416no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—44no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4C2×SD1612yes—
300C2×C3×C4C2×SD1612yes—
301C2×C3×C4#13512yes—
302C2×C3×C4AGL(1,ℤ/8)12yes—
303C2×C3×C4#13712yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4112yes—
307C2×C3×C4H(5,𝔽2)12yes—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314Dic6—44no—
315Dic6—44no—
316C3⋊C8—24no—
317C3⋊C8—42no—
318C24—22no—
319C24C2×#6711yescharacteristic
320#41—432no—
321C2×Q8—24no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—64no—
325C2×Q8—64no—
326C2×Q8—62no—
327C2×Q8—62no—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4C2×#6714yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4#21414yescomplemented
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—1216no—
364C2×C8—34no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—64no—
373C4⋊C4—124no—
374C4⋊C4—122no—
375C4⋊C4—122no—
376C4⋊C4—122no—
377C4⋊C4—122no—
378C4⋊C4C2×#6714yes—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—124no—
382C42—22no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42C2×#6712yes—
387C42C2×#6712yes—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#53211yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3C22×SD1611yescharacteristic
406C22×C3C2×#13511yescharacteristic
407C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
408C22×C3C2×#13711yescharacteristic
409C3×C4—22no—
410C3×C4—41no—
411C3×C4—44no—
412C3×C4—41no—
413C3×C4#45014yes—
414C3×C4#45214yes—
415C3×C4#54612yes—
416C3×C4#55012yes—
417Dic3—82no—
418Dic3—82no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—24no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23C2×#21412yes—
433C23C22×#6711yescharacteristic
434Q8—48no—
435Q8—124no—
436Q8—124no—
437D4—28no—
438D4—416no—
439D4—48no—
440D4—48no—
441D4—44no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—416no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4#93212yes—
465C2×C4C2×#21412yes—
466C2×C4C2×#21412yes—
467C2×C4#95012yes—
468C2×C4#95112yes—
469C2×C4#95312yes—
470C2×C4#95312yes—
471C2×C4D3×#4112yes—
472C2×C4#100912yes—
473C2×C4C22×#6711yescharacteristic
474C2×C4C22×#6711yescharacteristic
475C8—64no—
476C8—122no—
477C2×C3—42no—
478C2×C3—84no—
479C2×C3#243111yescharacteristic
480C2×C3C2×#45011yescharacteristic
481C2×C3C2×#45211yescharacteristic
482C2×C3#316711yescharacteristic
483C2×C3#318211yescharacteristic
484C2×C3#320611yescharacteristic
485C2×C3#320711yescharacteristic
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22C2×#93211yescharacteristic
501C22C22×#21411yescharacteristic
502C22C2×#95011yescharacteristic
503C22C2×#95111yescharacteristic
504C22#583011yescharacteristic
505C22#583211yescharacteristic
506C22C2×#95311yescharacteristic
507C4—22no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—242no—
512C4—242no—
513C4#521814yes—
514C4#523914yes—
515C4#582912yes—
516C4#583412yes—
517C3#2559711yescharacteristic, complemented, Sylow
518C2—42no—
519C2—84no—
520C2#8329011yescharacteristic
521C2#8329411yescharacteristic
522C2#8330411yescharacteristic
523C2#8331711yescharacteristic
524C2C2×#521811yescharacteristic
525C2C2×#523911yescharacteristic
526C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#25597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5093C2semidirect
3. C2×#5076C2semidirect
4. #82076C2semidirect
5. #71787C2semidirect
6. #71764C2semidirect
7. #71685C2semidirect
8. #71612C2semidirect
9. C4×#830C2semidirect
166. C22×C3×C4SD16semidirect
203. C2×C3⋊C8#41semidirect
356. C22×C4#214semidirect
517. C3#25597semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50932
C2×#50762
#820762
#717872
#717642
#716852
#716122
C4×#8302
#255973