Finite Group Databasebeta

← #10830430

Subgroups — #10830430

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic
2C2×#5242C212yesmaximal, complemented
3#83314C212yesmaximal, complemented
4#82328C211yesmaximal, characteristic, complemented
5#72140C212yesmaximal, complemented
6#71881C212yesmaximal, complemented
7#71876C212yesmaximal, complemented
8#71816C212yesmaximal, complemented
9#71617C211yesmaximal, characteristic, complemented
10#24228—31nomaximal, Sylow
11C2×C3×H(5,𝔽2)C2212yes—
12C3×#547—22no—
13C3×#546—24no—
14C3×D42—22no—
15C3×#537C2212yes—
16C3×#534—22no—
17C3×#532C2212yes—
18C2×C3×#121—22no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×#951C2214yescomplemented
21C2×#936C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic, complemented
23#5242—28no—
24C2×#837C2214yescomplemented
25C2×#837C2214yescomplemented
26#5232C2214yescomplemented
27#5221C2211yescharacteristic
28C2×#833C2212yes—
29#5210—22no—
30#5090—22no—
31#5075—22no—
32#5054—24no—
33#4975C2211yescharacteristic, complemented
34#4973C2211yescharacteristic, complemented
35#4944—22no—
36#4932C2211yescharacteristic, complemented
37#4924C2212yescomplemented
38#4530C2212yescomplemented
39#3496D311yescharacteristic
40#3037—32no—
41C2×#455—32no—
42#2326—31no—
43#2039—32no—
44#2007—32no—
45#1963—32no—
46#1800—31no—
47#1792—32no—
48C3×H(5,𝔽2)—24no—
49C3×H(5,𝔽2)—24no—
50C3×H(5,𝔽2)—44no—
51C2×C3×#41—24no—
52C2×C3×#41C2314yes—
53C22×C3×D4—24no—
54C22×C3×D4—22no—
55C22×C3×D4—24no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C22×Dic6C2311yescharacteristic
59C3×C4⋊Q8D412yes—
60C3×#127D412yes—
61C3×#124D414yes—
62C3×#123—28no—
63C3×C22⋊Q8—24no—
64C3×#121—24no—
65C3×#121—24no—
66C3×#121—24no—
67C3×#121—22no—
68C3×#121—24no—
69C3×#121—22no—
70C3×#120—28no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4D412yes—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82#951—28no—
83#951—44no—
84C2×#216—24no—
85#936—22no—
86#936D414yescomplemented
87C2×#214—24no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C2×#208C2314yes—
95C4×Dic6—22no—
96C4×Dic6—22no—
97#837—28no—
98#837—44no—
99#835—24no—
100#833—24no—
101#832—24no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—22no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×H(5,𝔽2)C2×D312yes—
108C2×#137—34no—
109#547—22no—
110#546—24no—
111D42—22no—
112#537C2×D312yes—
113#534—22no—
114#532C2×D312yes—
115C2×#124—31no—
116C2×#121—22no—
117C2×C4×Q8—31no—
118C2×C4×D4C2×D311yescharacteristic
119#464—64no—
120#462—62no—
121#460—62no—
122#455—68no—
123#423—32no—
124C2×#104—34no—
125C2×#104—34no—
126#417—34no—
127C2×#100—32no—
128#391—31no—
129#390—31no—
130#388—62no—
131#385—62no—
132#384—31no—
133#376—31no—
134#343—32no—
135C24×C3—24no—
136C24×C3C2×D412yes—
137C3×#41—24no—
138C3×#41—24no—
139C3×#41—24no—
140C3×#41—24no—
141C3×#41—48no—
142C3×#41—48no—
143C2×C3×Q8C2×D414yes—
144C2×C3×D4—28no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—48no—
152C2×C3×D4—48no—
153C2×C3×D4—48no—
154C2×C3×D4—48no—
155C2×C3×D4—84no—
156C2×C3×D4C2×D412yes—
157C2×C3×D4C2×D414yes—
158C2×C3×D4#4112yes—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—82no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—21no—
175C2×Dic6C2×D412yescomplemented
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C22⋊C4—28no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186#217—28no—
187#217—82no—
188#216—48no—
189#214—48no—
190C4⋊Dic3—22no—
191Dic3⋊C4—41no—
192Dic3⋊C4—41no—
193Dic3⋊C4—82no—
194C4×Dic3—21no—
195C4×Dic3—22no—
196C4×Dic3—42no—
197C4×Dic3C2×D412yescomplemented
198#208—24no—
199#208—24no—
200C2×C3⋊C8—24no—
201H(5,𝔽2)—24no—
202H(5,𝔽2)—24no—
203H(5,𝔽2)—44no—
204C2×#41—24no—
205C2×#41C22×D314yes—
206C22×Q8—31no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4C22×D311yescharacteristic
211C23×C4#6712yes—
212#137—68no—
213#137—124no—
214C2×Q16—64no—
215C2×SD16—64no—
216C2×M16—34no—
217C4⋊Q8#6712yes—
218#127#6712yes—
219#124—34no—
220#124—62no—
221#124#6714yes—
222#123—28no—
223C22⋊Q8—24no—
224#121—24no—
225#121—24no—
226#121—24no—
227#121—22no—
228#121—24no—
229#121—22no—
230#120—28no—
231C4×Q8—62no—
232C4×Q8—62no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C22⋊C4—24no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4#6712yes—
246C2×C42—31no—
247C2×C42—31no—
248C2×C42—61no—
249C2×C42C22×D311yescharacteristic
250C4⋊C8—62no—
251C4⋊C8—62no—
252#104—68no—
253#104—124no—
254#103—64no—
255#102—64no—
256#100—64no—
257#95—61no—
258#95—61no—
259#95—62no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—48no—
266C23×C3—84no—
267C23×C3—84no—
268C23×C3#12114yes—
269C23×C3C22×D411yescharacteristic
270C3×Q8—24no—
271C3×Q8—24no—
272C3×D4—24no—
273C3×D4—24no—
274C3×D4—24no—
275C3×D4—24no—
276C3×D4—44no—
277C3×D4—416no—
278C3×D4—44no—
279C3×D4—44no—
280C3×D4—82no—
281C3×D4—84no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—44no—
287C2×C3×C4—48no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—82no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12011yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D412yes—
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4H(5,𝔽2)12yes—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310Dic6—24no—
311Dic6—42no—
312C3⋊C8—44no—
313C24—24no—
314C24C2×#6712yes—
315#41—24no—
316#41—24no—
317#41—24no—
318#41—24no—
319#41—48no—
320#41—48no—
321C2×Q8—32no—
322C2×Q8—61no—
323C2×Q8—62no—
324C2×Q8—62no—
325C2×Q8C2×#6714yes—
326C2×D4—28no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—84no—
338C2×D4D3×D412yes—
339C2×D4#23712yes—
340C2×D4C2×#6714yes—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—82no—
350C22×C4C2×#6712yes—
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C22×C4C23×D311yescharacteristic
354Q16—128no—
355SD16—128no—
356M16—64no—
357M16—64no—
358C2×C8—64no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—62no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—242no—
367C22⋊C4—28no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—68no—
372C22⋊C4—242no—
373C42—32no—
374C42—61no—
375C42—62no—
376C42—122no—
377C42C2×#6712yes—
378C42C2×#6712yes—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—24no—
382C22×C3—44no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—82no—
389C22×C3—168no—
390C22×C3C2×#12011yescharacteristic
391C22×C3C2×#12111yescharacteristic
392C22×C3#53511yescharacteristic
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—22no—
396C3×C4—44no—
397C3×C4—42no—
398C3×C4D4212yes—
399C3×C4#54612yes—
400Dic3—42no—
401Dic3—42no—
402Dic3—81no—
403Dic3—81no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—42no—
408C23—42no—
409C23—48no—
410C23—84no—
411C23—84no—
412C23#93814yes—
413C23C22×#6711yescharacteristic
414Q8—24no—
415Q8—24no—
416Q8—64no—
417Q8—122no—
418D4—24no—
419D4—24no—
420D4—24no—
421D4—24no—
422D4—44no—
423D4—416no—
424D4—44no—
425D4—44no—
426D4—82no—
427D4—84no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—82no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—122no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4#95312yes—
450C2×C4#95311yescharacteristic
451C2×C4#95311yescharacteristic
452C2×C4C2×D3×D411yescharacteristic
453C2×C4C2×#23711yescharacteristic
454C2×C4#100412yes—
455C2×C4C22×#6712yes—
456C8—124no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—44no—
460C2×C3—84no—
461C2×C3#240811yescharacteristic
462C2×C3#301911yescharacteristic
463C2×C3#302211yescharacteristic
464C22—21no—
465C22—21no—
466C22—24no—
467C22—44no—
468C22—42no—
469C22—42no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—82no—
474C22—168no—
475C22C2×#93811yescharacteristic
476C22#580511yescharacteristic
477C22C2×#95311yescharacteristic
478C4—21no—
479C4—21no—
480C4—22no—
481C4—44no—
482C4—42no—
483C4—122no—
484C4—122no—
485C4—241no—
486C4—241no—
487C4D4×#6712yes—
488C4#580612yes—
489C3#2422811yescharacteristic, complemented, Sylow
490C2—21no—
491C2—21no—
492C2—44no—
493C2—84no—
494C2#8405311yescharacteristic
495C2#8410411yescharacteristic
496C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#24228
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
2. C2×#5242C2semidirect
3. #83314C2semidirect
4. #82328C2semidirect
5. #72140C2semidirect
6. #71881C2semidirect
7. #71876C2semidirect
8. #71816C2semidirect
9. #71617C2semidirect
20. C2×#951C22semidirect
22. C2×C4×Dic6C22semidirect
24. C2×#837C22semidirect
25. C2×#837C22semidirect
26. #5232C22semidirect
33. #4975C22semidirect
34. #4973C22semidirect
36. #4932C22semidirect
37. #4924C22semidirect
38. #4530C22semidirect
86. #936D4semidirect
175. C2×Dic6C2×D4semidirect
197. C4×Dic3C2×D4semidirect
489. C3#24228semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
C2×#52422
#833142
#823282
#721402
#718812
#718762
#718162
#716172
#242283