Finite Group Databasebeta

← #10901418

Subgroups — #10901418

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5009C212yesmaximal, complemented
3#82074C211yesmaximal, characteristic, complemented
4#71783C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#71762C211yesmaximal, characteristic, complemented
7#71762C211yesmaximal, characteristic, complemented
8#71680C212yesmaximal, complemented
9#71679C212yesmaximal, complemented
10#71605C212yesmaximal, complemented
11#71530C211yesmaximal, characteristic, complemented
12#29694—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#532—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×#832C2212yes—
25C2×#832C2212yes—
26C2×#832C2212yes—
27C2×#832C2212yes—
28#5212C2211yescharacteristic
29#5212C2211yescharacteristic
30#5210—21no—
31#5210—21no—
32C2×#830C2212yes—
33#5009—28no—
34C2×#808C2212yes—
35C2×#807C2212yes—
36#4936C2211yescharacteristic
37#4933—21no—
38#4933—21no—
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#532D311yescharacteristic
47C2×#478—32no—
48#2851—31no—
49#2058—31no—
50#2057—31no—
51#2041—31no—
52#2041—31no—
53#2034—32no—
54#2033—32no—
55#1804—32no—
56#1797—31no—
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C3×C4⋊Q8—24no—
70C3×#127—24no—
71C3×#124—28no—
72C3×#121—28no—
73C3×#121—24no—
74C3×#121—24no—
75C3×C4×D4—28no—
76C3×C4×D4—24no—
77C3×C4×D4D418yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—24no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C22×C3⋊C8C2312yes—
98C22×C3⋊C8C2312yes—
99#870—24no—
100#832—24no—
101#832—24no—
102#832—42no—
103#832—42no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#830—42no—
111#808—24no—
112#807—24no—
113C4⋊(C3⋊C8)—24no—
114C22×#41—21no—
115#532—28no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#127C2×D311yescharacteristic
118C2×#125—31no—
119C2×#124—21no—
120C2×#121—21no—
121C2×#121C2×D312yes—
122C2×C4×D4—21no—
123C2×C4×D4C2×D312yes—
124C22×C42C2×D311yescharacteristic
125#478—68no—
126C2×#107—32no—
127C2×#106—32no—
128C2×C4⋊C8—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8—32no—
134#389—31no—
135#389—31no—
136#387—61no—
137#387—61no—
138#385—61no—
139#385—61no—
140#382—31no—
141#380—31no—
142#380—31no—
143#340—32no—
144#340—32no—
145#336—31no—
146#336—31no—
147C24×C3—22no—
148C24×C3C2×D411yescharacteristic
149C3×#41—432no—
150C2×C3×Q8—24no—
151C2×C3×Q8—42no—
152C2×C3×Q8—42no—
153C2×C3×D4—24no—
154C2×C3×D4—28no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4C2×D414yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—28no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—41no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42C2×D414yes—
201#217—42no—
202#217—42no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209Dic3⋊C4—42no—
210Dic3⋊C4—42no—
211Dic3⋊C4—42no—
212Dic3⋊C4—42no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×#41—28no—
220C2×#41—44no—
221C2×#41—44no—
222C22×Q8—21no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C22×D312yes—
227C23×C4—22no—
228C23×C4#6711yescharacteristic
229C23×C4#6711yescharacteristic
230C23×C4C22×D311yescharacteristic
231C22×C8—32no—
232C22×C8—32no—
233C4⋊Q8—24no—
234#127—24no—
235#125—64no—
236#124—28no—
237#121—28no—
238#121—24no—
239#121—24no—
240C4×D4—28no—
241C4×D4—24no—
242C4×D4#6718yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—61no—
257C2×C42—24no—
258C2×C42#6711yescharacteristic
259C2×C42#6711yescharacteristic
260C2×C42C22×D311yescharacteristic
261C2×C42C22×D311yescharacteristic
262#107—64no—
263#106—64no—
264C4⋊C8—64no—
265#102—64no—
266#102—64no—
267#102—122no—
268#102—122no—
269C22⋊C8—122no—
270#95—61no—
271#95—61no—
272#95—61no—
273#95—61no—
274#95—61no—
275#95—61no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—24no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3AGL(1,ℤ/8)12yes—
290C23×C3C22×D411yescharacteristic
291C3×Q8—48no—
292C3×D4—24no—
293C3×D4—24no—
294C3×D4—48no—
295C3×D4—48no—
296C3×D4—416no—
297C3×D4—42no—
298C3×D4—42no—
299C2×C3×C4—28no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—24no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—44no—
313C2×C3×C4—416no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4#13512yes—
321C2×C3×C4AGL(1,ℤ/8)12yes—
322C2×C3×C4C22×D412yes—
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C2×C3×C42−1+411yescharacteristic
325C2×C3×C42−1+412yes—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C3⋊C8—42no—
339C3⋊C8—42no—
340C24—22no—
341C24C2×#6711yescharacteristic
342#41—432no—
343C2×Q8—24no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4C2×#6714yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C23×D311yescharacteristic
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—61no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—122no—
400C4⋊C4—122no—
401C4⋊C4—122no—
402C4⋊C4—122no—
403C4⋊C4C2×#6714yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42C2×#6714yes—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3C2×#12011yescharacteristic
427C22×C3#53611yescharacteristic
428C22×C3#53711yescharacteristic
429C22×C3C2×#13511yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
432C22×C3#57511yescharacteristic
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—44no—
439C3×C4#53612yes—
440C3×C4#53712yes—
441Dic3—81no—
442Dic3—81no—
443Dic3—81no—
444Dic3—81no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23#93212yes—
459C23C22×#6711yescharacteristic
460Q8—48no—
461D4—24no—
462D4—24no—
463D4—48no—
464D4—48no—
465D4—416no—
466D4—42no—
467D4—42no—
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—416no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4#93212yes—
500C2×C4#95012yes—
501C2×C4#95312yes—
502C2×C4#95312yes—
503C2×C4#100911yescharacteristic
504C2×C4#101011yescharacteristic
505C2×C4#101012yes—
506C2×C4C22×#6712yes—
507C8—122no—
508C8—122no—
509C2×C3—42no—
510C2×C3—84no—
511C2×C3#261611yescharacteristic
512C2×C3#300211yescharacteristic
513C2×C3#300311yescharacteristic
514C2×C3#322811yescharacteristic
515C2×C3#323411yescharacteristic
516C2×C3#324711yescharacteristic
517C2×C3#325111yescharacteristic
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C2×#93211yescharacteristic
533C22C2×#93211yescharacteristic
534C22C2×#95011yescharacteristic
535C22#582211yescharacteristic
536C22#582511yescharacteristic
537C22#583311yescharacteristic
538C22C2×#95311yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—41no—
545C4—241no—
546C4—241no—
547C4—241no—
548C4—241no—
549C4#582512yes—
550C4#583312yes—
551C3#2969411yescharacteristic, complemented, Sylow
552C2—42no—
553C2—84no—
554C2#8327211yescharacteristic
555C2#8327311yescharacteristic
556C2#8331111yescharacteristic
557C2#8331211yescharacteristic
558C2#8403811yescharacteristic
559C2#8408011yescharacteristic
560C2#8409411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#29694
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×#5009C2semidirect
3. #82074C2semidirect
4. #71783C2semidirect
5. #71782C2semidirect
6. #71762C2semidirect
7. #71762C2semidirect
8. #71680C2semidirect
9. #71679C2semidirect
10. #71605C2semidirect
11. #71530C2semidirect
551. C3#29694semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50092
#820742
#717832
#717822
#717622
#716802
#716792
#716052
#715302
#296943