Finite Group Databasebeta

← #10848648

Subgroups — #10848648

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3509C211yesmaximal, characteristic, complemented
2#83338C211yesmaximal, characteristic, complemented
3#83264C211yesmaximal, characteristic, complemented
4#82527C211yesmaximal, characteristic
5#71868C212yesmaximal, complemented
6#71867C212yesmaximal
7#71840C211yesmaximal, characteristic, complemented
8#71789C211yesmaximal, characteristic, complemented
9#71661C212yesmaximal, complemented
10#71660C212yesmaximal, complemented
11#71643C211yesmaximal, characteristic, complemented
12#25594—31nomaximal, Sylow
13C3×#557—22no—
14C3×#556—22no—
15C3×#552—24no—
16C3×#541—22no—
17C3×#536C2212yes—
18C3×#533—22no—
19C3×#533C2211yescharacteristic
20C2×C3×C4⋊Q8C2211yescharacteristic
21C3×#519C2212yes—
22C3×C4×#41C2211yescharacteristic
23C2×#951C2211yescharacteristic
24C2×#949C2211yescharacteristic
25C2×#947C2212yes—
26C2×#944C2212yes—
27C2×#875C2211yescharacteristic
28C2×#837C2212yes—
29C2×#836C2212yes—
30C2×#836C2212yes—
31#5089—22no—
32#5068—22no—
33#5028—22no—
34#5016—22no—
35#5004C2211yescharacteristic
36#5004C2212yes—
37#5001C2211yescharacteristic
38#5001C2212yes—
39#4993—24no—
40#4932C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#208C2211yescharacteristic
43#4532C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#3509D311yescharacteristic, complemented
47#3140—31no—
48#3040—31no—
49#2658—31no—
50#2005—32no—
51#2003—32no—
52#1995—31no—
53#1966—31no—
54#1888—32no—
55#1886—32no—
56#1759—31no—
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8C2312yes—
59C22×D12C2311yescharacteristic
60C22×Dic6C2311yescharacteristic
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D412yes—
66C3×#126—28no—
67C3×#125—24no—
68C3×#125—28no—
69C3×#125D412yes—
70C3×#124—22no—
71C3×#123—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—24no—
75C3×C4×Q8—24no—
76C3×C4×Q8—24no—
77C3×C4×Q8—24no—
78C3×C4×Q8—22no—
79C3×C4×Q8D412yes—
80C3×C4×D4—22no—
81C3×C4×D4D412yes—
82C3×#117—24no—
83C3×#117—24no—
84C3×#117—21no—
85C3×#117C2311yescharacteristic
86C3×#117C2312yes—
87C2×C3×C4⋊C4D414yes—
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90#951—42no—
91#949—42no—
92#947—24no—
93#947—42no—
94#944—24no—
95#944—42no—
96C2×#216—21no—
97C2×#215—21no—
98C2×#214—21no—
99C2×#213—21no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×C4×Dic3C2312yes—
104C2×#208C2312yes—
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107#875—22no—
108#875—21no—
109#875—21no—
110#837—24no—
111#836—42no—
112#836—42no—
113C3⋊#95—22no—
114#810—22no—
115#810—24no—
116#809—22no—
117#809—24no—
118C4⋊(C3⋊C8)—22no—
119#803—21no—
120C4×C3⋊C8—21no—
121C2×#137—31no—
122C2×AGL(1,ℤ/8)—31no—
123#557—22no—
124#556—22no—
125#552—24no—
126#541—22no—
127#536C2×D312yes—
128#533—22no—
129#533C2×D311yescharacteristic
130C2×C4⋊Q8—32no—
131C2×C4⋊Q8C2×D311yescharacteristic
132C2×#124—32no—
133C2×#124—31no—
134#519C2×D312yes—
135C4×#41C2×D311yescharacteristic
136#489—62no—
137#488—62no—
138#464—62no—
139#463—62no—
140#423—31no—
141C2×#104—32no—
142#419—32no—
143#419—31no—
144#418—32no—
145#418—31no—
146C2×#101—32no—
147C2×#101—32no—
148C4×M16—31no—
149#399—64no—
150#376—31no—
151#339—32no—
152#339—32no—
153#339—32no—
154C23×D3—21no—
155C3×#41—22no—
156C3×#41—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8C2×D411yescharacteristic
165C2×C3×D4—21no—
166C2×C3×D4C2×D411yescharacteristic
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—22no—
177C2×D12—22no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4C2×D412yes—
190C3×C4⋊C4C2×D414yes—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C42—28no—
196C3×C42—22no—
197C3×C42—21no—
198C3×C42—22no—
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D412yes—
202#216—42no—
203#215—42no—
204#214—42no—
205#213—42no—
206(C2×D3)⋊C4—48no—
207(C2×D3)⋊C4—42no—
208(C2×D3)⋊C4—42no—
209Dic3⋊C4—48no—
210C4×Dic3—22no—
211C4×Dic3#4114yes—
212#208—22no—
213#208—22no—
214#208—22no—
215#208—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×#41C22×D311yescharacteristic
220C22×Q8—31no—
221C22×Q8C22×D312yes—
222C22×D4—31no—
223#137—122no—
224AGL(1,ℤ/8)—122no—
225C2×Q16—61no—
226C2×SD16—61no—
227C2×SD16—61no—
228C2×D8—61no—
229C2×M16—31no—
230C2×M16—31no—
231C2×M16—32no—
232C4⋊Q8—22no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—22no—
236C4⋊Q8—64no—
237C4⋊Q8—122no—
238C4⋊Q8#6712yes—
239#126—28no—
240#125—24no—
241#125—28no—
242#125#6712yes—
243#124—22no—
244#124—64no—
245#124—62no—
246#124—61no—
247#124—61no—
248#124—122no—
249#123—24no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—22no—
257C4×Q8#6712yes—
258C4×D4—22no—
259C4×D4#6712yes—
260#117—24no—
261#117—24no—
262#117—21no—
263#117C22×D311yescharacteristic
264#117C22×D312yes—
265C2×C4⋊C4—64no—
266C2×C4⋊C4#6714yes—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—62no—
269C2×C42—21no—
270C2×C42—32no—
271C2×C42C22×D311yescharacteristic
272C4⋊C8—62no—
273#104—64no—
274#103—62no—
275#103—64no—
276#102—62no—
277#102—64no—
278#101—122no—
279#101—122no—
280#97—61no—
281C4×C8—61no—
282#95—62no—
283C23×C3—21no—
284C23×C3C22×D411yescharacteristic
285C22×D3—44no—
286C22×D3—41no—
287C22×D3—42no—
288C3×Q8—21no—
289C3×Q8—21no—
290C3×Q8—44no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×D4—21no—
296C3×D4—21no—
297C3×D4—42no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—21no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—42no—
305C2×C3×C4—44no—
306C2×C3×C4—42no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C4H(5,𝔽2)12yes—
313C2×Dic3—24no—
314C2×Dic3—44no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317D12—41no—
318D12—41no—
319D12—42no—
320Dic6—41no—
321Dic6—41no—
322Dic6—42no—
323C3⋊C8—41no—
324C3⋊C8—41no—
325C3⋊C8—42no—
326C24—61no—
327#41—22no—
328#41—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8C2×#6711yescharacteristic
341C2×D4—21no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4C2×#6711yescharacteristic
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4C2×#6714yes—
352C22×C4C2×#6712yes—
353C22×C4C23×D311yescharacteristic
354Q16—122no—
355SD16—122no—
356SD16—122no—
357D8—122no—
358M16—62no—
359M16—62no—
360M16—62no—
361M16—62no—
362C2×C8—62no—
363C2×C8—61no—
364C2×C8—61no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—128no—
373C4⋊C4C2×#6712yes—
374C4⋊C4C2×#6714yes—
375C22⋊C4—28no—
376C22⋊C4—21no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—128no—
380C22⋊C4—122no—
381C22⋊C4—122no—
382C42—28no—
383C42—22no—
384C42—21no—
385C42—22no—
386C42—34no—
387C42—62no—
388C42C2×#6711yescharacteristic
389C42C2×#6711yescharacteristic
390C42C2×#6712yes—
391C22×C3—22no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#54612yes—
396C2×D3—84no—
397C2×D3—81no—
398C2×D3—82no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—44no—
402C3×C4—41no—
403C3×C4—42no—
404C3×C4—44no—
405C3×C4#53211yescharacteristic
406C3×C4#53511yescharacteristic
407C3×C4#55012yes—
408Dic3—44no—
409Dic3—81no—
410C23—21no—
411C23—124no—
412C23—122no—
413C23—121no—
414C23C22×#6711yescharacteristic
415Q8—21no—
416Q8—21no—
417Q8—44no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—121no—
423Q8—121no—
424Q8—122no—
425D4—21no—
426D4—21no—
427D4—42no—
428D4—121no—
429D4—121no—
430D4—122no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—64no—
441C2×C4—124no—
442C2×C4—122no—
443C2×C4—121no—
444C2×C4#95312yes—
445C2×C4#95312yes—
446C2×C4D3×#4112yes—
447C2×C4#100912yes—
448C2×C4C22×#6711yescharacteristic
449C2×C4C22×#6711yescharacteristic
450C8—121no—
451C8—121no—
452C8—122no—
453C2×C3—22no—
454C2×C3—81no—
455C2×C3#243111yescharacteristic
456C2×C3#302012yes—
457D3—161no—
458C22—22no—
459C22—41no—
460C22—42no—
461C22—244no—
462C22—241no—
463C22—242no—
464C22#582912yes—
465C22C2×#95311yescharacteristic
466C4—22no—
467C4—22no—
468C4—44no—
469C4—41no—
470C4—42no—
471C4—44no—
472C4—124no—
473C4—241no—
474C4#583011yescharacteristic
475C4#583211yescharacteristic
476C4#583412yes—
477C3#2559411yescharacteristic, complemented, Sylow
478C2—22no—
479C2—81no—
480C2—481no—
481C2#8409211yescharacteristic
482C2#8411612yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3509C2
socleC22×C3C2×#120
prime radicalC2×#949C22
hypercenter#3509D3
hypocenterC3#25594
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. C3×#3509C2semidirect
2. #83338C2semidirect
3. #83264C2semidirect
5. #71868C2semidirect
7. #71840C2semidirect
8. #71789C2semidirect
9. #71661C2semidirect
10. #71660C2semidirect
11. #71643C2semidirect
46. #3509D3semidirect
477. C3#25594semidirect

Maximal subgroups

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

subgroupindex
C3×#35092
#833382
#832642
#825272
#718682
#718672
#718402
#717892
#716612
#716602
#716432
#255943