Finite Group Databasebeta

← #10830439

Subgroups — #10830439

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3497C211yesmaximal, characteristic
2C2×#5246C212yesmaximal, complemented
3#83314C212yesmaximal, complemented
4#82456C211yesmaximal, characteristic, complemented
5#72143C212yesmaximal, complemented
6#71881C212yesmaximal, complemented
7#71877C212yesmaximal, complemented
8#71816C212yesmaximal, complemented
9#71637C211yesmaximal, characteristic, complemented
10#24229—31nomaximal, Sylow
11C2×C3×2−1+4C2212yes—
12C3×#550—22no—
13C3×D4×Q8—22no—
14C3×#548—24no—
15C3×#537—22no—
16C3×#537C2212yes—
17C3×#532—22no—
18C3×#532C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#951C2214yes—
21C2×#944C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23#5246—28no—
24C2×#837C2214yes—
25C2×#837C2214yes—
26#5232C2214yes—
27#5229C2211yescharacteristic
28C2×#836C2212yes—
29#5090—22no—
30#5075—22no—
31#5054—24no—
32#4978—22no—
33#4977C2211yescharacteristic
34#4973C2211yescharacteristic
35#4933—22no—
36#4932C2211yescharacteristic
37#4924C2212yes—
38#4530C2212yes—
39#3497D311yescharacteristic
40#3037—32no—
41C2×#456—32no—
42#2327—31no—
43#2040—32no—
44#2007—32no—
45#1963—32no—
46#1801—31no—
47#1793—32no—
48C3×2−1+4—24no—
49C3×2−1+4—24no—
50C3×2−1+4—44no—
51C2×C3×#41—24no—
52C2×C3×#41—24no—
53C2×C3×#41C2314yes—
54C22×C3×Q8—24no—
55C22×C3×Q8C2311yescharacteristic
56C22×Dic6C2311yescharacteristic
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8D412yes—
59C3×#127—22no—
60C3×#127D412yes—
61C3×#124—24no—
62C3×#124D414yes—
63C3×C22⋊Q8—28no—
64C3×C22⋊Q8—24no—
65C3×#121—28no—
66C3×#121—24no—
67C3×#121—22no—
68C3×#121—24no—
69C3×#121—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C42D412yes—
79C2×C3×C42C2311yescharacteristic
80#951—28no—
81#951—44no—
82#944—22no—
83#944D414yescomplemented
84C2×#216—24no—
85C2×#214—24no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C2×#208C2314yes—
93C4×Dic6—22no—
94C4×Dic6—22no—
95#837—28no—
96#837—44no—
97#836—24no—
98#835—24no—
99#832—24no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×2−1+4C2×D312yes—
108C2×#137—34no—
109#550—22no—
110D4×Q8—22no—
111#548—24no—
112#537—22no—
113#537C2×D312yes—
114#532—22no—
115#532C2×D312yes—
116C2×C4⋊Q8—31no—
117C2×C4×Q8—31no—
118C2×C4×Q8C2×D311yescharacteristic
119#464—64no—
120#462—62no—
121#460—62no—
122#456—68no—
123#423—32no—
124C2×#104—34no—
125C2×#104—34no—
126#417—34no—
127C2×#101—32no—
128#391—31no—
129#391—31no—
130#387—62no—
131#387—62no—
132#382—31no—
133#376—31no—
134#343—32no—
135C3×#41—24no—
136C3×#41—24no—
137C3×#41—24no—
138C3×#41—24no—
139C3×#41—48no—
140C3×#41—48no—
141C3×#41—48no—
142C3×#41—48no—
143C2×C3×Q8—24no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—48no—
147C2×C3×Q8—48no—
148C2×C3×Q8C2×D412yes—
149C2×C3×Q8C2×D414yes—
150C2×C3×Q8#4112yes—
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4C2×D414yes—
157C22×C3×C4—28no—
158C22×C3×C4—24no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6—21no—
171C2×Dic6C2×D412yes—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42C2×D412yes—
183C3×C42C2×D412yes—
184#216—48no—
185#214—48no—
186C4⋊Dic3—22no—
187Dic3⋊C4—28no—
188Dic3⋊C4—41no—
189Dic3⋊C4—41no—
190C4×Dic3—21no—
191C4×Dic3—22no—
192C4×Dic3—42no—
193C4×Dic3C2×D412yes—
194#208—24no—
195#208—24no—
196C2×C3⋊C8—24no—
1972−1+4—24no—
1982−1+4—24no—
1992−1+4—44no—
200C2×#41—24no—
201C2×#41—24no—
202C2×#41C22×D314yes—
203C22×Q8—24no—
204C22×Q8—31no—
205C22×Q8C22×D311yescharacteristic
206#137—68no—
207#137—124no—
208C2×Q16—64no—
209C2×SD16—64no—
210C2×M16—34no—
211C4⋊Q8—22no—
212C4⋊Q8—34no—
213C4⋊Q8—62no—
214C4⋊Q8#6712yes—
215#127—22no—
216#127#6712yes—
217#124—24no—
218#124#6714yes—
219C22⋊Q8—28no—
220C22⋊Q8—24no—
221#121—28no—
222#121—24no—
223#121—22no—
224#121—24no—
225#121—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—62no—
229C4×Q8—62no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4#6712yes—
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—61no—
242C2×C42#6712yes—
243C2×C42C22×D311yescharacteristic
244C4⋊C8—62no—
245C4⋊C8—62no—
246#104—68no—
247#104—124no—
248#103—64no—
249#102—64no—
250#101—64no—
251#95—61no—
252#95—61no—
253#95—62no—
254#95—62no—
255#95—62no—
256C23×C3—24no—
257C23×C3C22×D411yescharacteristic
258C3×Q8—24no—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—48no—
263C3×D4—24no—
264C3×D4—24no—
265C3×D4—44no—
266C3×D4—48no—
267C3×D4—44no—
268C3×D4—44no—
269C3×D4—82no—
270C3×D4—84no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—416no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4#12012yes—
282C2×C3×C4#12011yescharacteristic
283C2×C3×C4#12011yescharacteristic
284C2×C3×C4#12114yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D412yes—
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4H(5,𝔽2)12yes—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300Dic6—24no—
301Dic6—42no—
302C3⋊C8—44no—
303#41—24no—
304#41—24no—
305#41—24no—
306#41—24no—
307#41—48no—
308#41—48no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—32no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8D3×D412yes—
321C2×Q8#23712yes—
322C2×Q8C2×#6714yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4C2×#6714yes—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4C2×#6712yes—
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6711yescharacteristic
339C22×C4C23×D311yescharacteristic
340Q16—128no—
341SD16—128no—
342M16—64no—
343M16—64no—
344C2×C8—64no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—68no—
351C4⋊C4—62no—
352C4⋊C4—121no—
353C4⋊C4—121no—
354C22⋊C4—28no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C42—22no—
358C42—22no—
359C42—32no—
360C42—61no—
361C42—62no—
362C42—122no—
363C42C2×#6712yes—
364C42C2×#6712yes—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—44no—
368C22×C3—44no—
369C22×C3—44no—
370C22×C3C2×#12011yescharacteristic
371C22×C3C2×#12111yescharacteristic
372C22×C3#53511yescharacteristic
373C3×C4—24no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—22no—
377C3×C4—44no—
378C3×C4—42no—
379C3×C4D4212yes—
380C3×C4#54612yes—
381Dic3—42no—
382Dic3—42no—
383Dic3—81no—
384Dic3—81no—
385C23—24no—
386C23C22×#6711yescharacteristic
387Q8—24no—
388Q8—24no—
389Q8—24no—
390Q8—24no—
391Q8—48no—
392Q8—64no—
393Q8—122no—
394D4—24no—
395D4—24no—
396D4—48no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—82no—
401D4—84no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—416no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4#93814yes—
424C2×C4#95311yescharacteristic
425C2×C4#95311yescharacteristic
426C2×C4#95312yes—
427C2×C4C2×D3×D411yescharacteristic
428C2×C4C2×#23711yescharacteristic
429C2×C4#100412yes—
430C2×C4C22×#6712yes—
431C8—124no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—84no—
435C2×C3#240811yescharacteristic
436C2×C3#302011yescharacteristic
437C2×C3#302211yescharacteristic
438C22—21no—
439C22—21no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22C2×#93811yescharacteristic
444C22#580511yescharacteristic
445C22C2×#95311yescharacteristic
446C4—24no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—44no—
451C4—42no—
452C4—122no—
453C4—122no—
454C4—241no—
455C4—241no—
456C4D4×#6712yes—
457C4#580612yes—
458C3#2422911yescharacteristic, complemented, Sylow
459C2—21no—
460C2—21no—
461C2—84no—
462C2#8405311yescharacteristic
463C2#8411611yescharacteristic
464C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3497C2
socleC22×C3C2×#120
hypercenter#3497D3
hypocenterC3#24229
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
2. C2×#5246C2semidirect
3. #83314C2semidirect
4. #82456C2semidirect
5. #72143C2semidirect
6. #71881C2semidirect
7. #71877C2semidirect
8. #71816C2semidirect
9. #71637C2semidirect
83. #944D4semidirect
458. C3#24229semidirect

Maximal subgroups

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

subgroupindex
C3×#34972
C2×#52462
#833142
#824562
#721432
#718812
#718772
#718162
#716372
#242293