Finite Group Databasebeta

← #10830421

Subgroups — #10830421

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3497C211yesmaximal, characteristic, complemented
2C2×#5245C212yesmaximal, complemented
3#83297C212yesmaximal, complemented
4#82455C211yesmaximal, characteristic, complemented
5#72143C212yesmaximal, complemented
6#71879C212yesmaximal, complemented
7#71876C212yesmaximal, complemented
8#71816C212yesmaximal, complemented
9#71637C211yesmaximal, characteristic, complemented
10#24227—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×#949C2214yes—
21C2×#947C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23#5245—28no—
24C2×#837C2214yes—
25C2×#837C2214yes—
26#5232C2214yes—
27#5229C2211yescharacteristic
28C2×#836C2212yes—
29#5083—22no—
30#5072—22no—
31#5057—24no—
32#4989—22no—
33#4988C2211yescharacteristic
34#4975C2211yescharacteristic
35#4940—22no—
36#4932C2211yescharacteristic
37#4924C2212yes—
38#4530C2212yes—
39#3497D311yescharacteristic, complemented
40#3036—32no—
41C2×#454—32no—
42#2326—31no—
43#2039—32no—
44#2006—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×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8D412yes—
60C3×#127—22no—
61C3×#127D412yes—
62C3×#124—24no—
63C3×#124D414yes—
64C3×C22⋊Q8—28no—
65C3×C22⋊Q8—24no—
66C3×#121—28no—
67C3×#121—24no—
68C3×#121—22no—
69C3×#121—24no—
70C3×#121—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×D4—24no—
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C2×C3×C4⋊C4D412yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C42D412yes—
80C2×C3×C42C2311yescharacteristic
81#949—28no—
82#949—44no—
83#947—22no—
84#947D414yescomplemented
85C2×#215—24no—
86C2×#213—24no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C2×#208C2314yes—
95C4×D12—22no—
96C4×D12—22no—
97#837—28no—
98#837—44no—
99#836—24no—
100#835—24no—
101#832—24no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—22no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×2−1+4C2×D312yes—
108C2×AGL(1,ℤ/8)—34no—
109#550—22no—
110D4×Q8—22no—
111#548—24no—
112#537—22no—
113#537C2×D312yes—
114#532—22no—
115#532C2×D312yes—
116C2×#124—31no—
117C2×C4×Q8C2×D311yescharacteristic
118C2×C4×D4—31no—
119#463—64no—
120#461—62no—
121#459—62no—
122#454—68no—
123#423—32no—
124C2×#104—34no—
125C2×#104—34no—
126#417—34no—
127C2×#101—32no—
128#391—31no—
129#390—31no—
130#388—62no—
131#385—62no—
132#384—31no—
133#376—31no—
134#343—32no—
135C23×D3—21no—
136C3×#41—24no—
137C3×#41—24no—
138C3×#41—24no—
139C3×#41—24no—
140C3×#41—48no—
141C3×#41—48no—
142C3×#41—48no—
143C3×#41—48no—
144C2×C3×Q8—24no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—48no—
148C2×C3×Q8—48no—
149C2×C3×Q8C2×D412yes—
150C2×C3×Q8C2×D414yes—
151C2×C3×Q8#4112yes—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—28no—
159C22×C3×C4—24no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12C2×D412yes—
172C2×C4×D3—42no—
173C2×C4×D3—82no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186#215—48no—
187#213—48no—
188(C2×D3)⋊C4—28no—
189(C2×D3)⋊C4—41no—
190(C2×D3)⋊C4—41no—
191(C2×D3)⋊C4—82no—
192C4⋊Dic3—22no—
193Dic3⋊C4—82no—
194C4×Dic3—21no—
195C4×Dic3—22no—
196C4×Dic3C2×D412yes—
197#208—24no—
198#208—24no—
199C2×C3⋊C8—24no—
2002−1+4—24no—
2012−1+4—24no—
2022−1+4—44no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41C22×D314yes—
206C22×Q8—24no—
207C22×Q8C22×D311yescharacteristic
208C22×D4—31no—
209C23×C4—61no—
210AGL(1,ℤ/8)—68no—
211AGL(1,ℤ/8)—124no—
212C2×SD16—64no—
213C2×D8—64no—
214C2×M16—34no—
215C4⋊Q8—22no—
216C4⋊Q8#6712yes—
217#127—22no—
218#127#6712yes—
219#124—24no—
220#124—34no—
221#124—62no—
222#124#6714yes—
223C22⋊Q8—28no—
224C22⋊Q8—24no—
225#121—28no—
226#121—24no—
227#121—22no—
228#121—24no—
229#121—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—62no—
237C4×D4—62no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4#6712yes—
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—62no—
245C2×C42—31no—
246C2×C42—31no—
247C2×C42#6712yes—
248C2×C42C22×D311yescharacteristic
249C4⋊C8—62no—
250C4⋊C8—62no—
251#104—68no—
252#104—124no—
253#103—64no—
254#102—64no—
255#101—64no—
256#95—61no—
257#95—61no—
258#95—62no—
259C23×C3—24no—
260C23×C3C22×D411yescharacteristic
261C22×D3—22no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—41no—
265C22×D3—41no—
266C3×Q8—24no—
267C3×Q8—24no—
268C3×Q8—24no—
269C3×Q8—24no—
270C3×Q8—48no—
271C3×D4—24no—
272C3×D4—24no—
273C3×D4—44no—
274C3×D4—48no—
275C3×D4—44no—
276C3×D4—44no—
277C3×D4—82no—
278C3×D4—84no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—416no—
286C2×C3×C4—44no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4#12011yescharacteristic
291C2×C3×C4#12012yes—
292C2×C3×C4#12114yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D412yes—
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4H(5,𝔽2)12yes—
297C2×Dic3—22no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305D12—24no—
306D12—42no—
307C4×D3—84no—
308C3⋊C8—44no—
309C24—61no—
310#41—24no—
311#41—24no—
312#41—24no—
313#41—24no—
314#41—48no—
315#41—48no—
316#41—48no—
317#41—48no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8D3×D412yes—
324C2×Q8#23712yes—
325C2×Q8C2×#6714yes—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—32no—
332C2×D4—61no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4C2×#6714yes—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—122no—
342C22×C4—242no—
343C22×C4C2×#6712yes—
344C22×C4C2×#6712yes—
345C22×C4C2×#6711yescharacteristic
346C22×C4C2×#6711yescharacteristic
347C22×C4C23×D311yescharacteristic
348SD16—128no—
349D8—128no—
350M16—64no—
351M16—64no—
352C2×C8—64no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—62no—
359C4⋊C4—242no—
360C22⋊C4—28no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—68no—
364C22⋊C4—121no—
365C22⋊C4—121no—
366C22⋊C4—242no—
367C42—22no—
368C42—22no—
369C42—32no—
370C42—61no—
371C42—62no—
372C42C2×#6712yes—
373C42C2×#6712yes—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3C2×#12011yescharacteristic
380C22×C3C2×#12111yescharacteristic
381C22×C3#53511yescharacteristic
382C2×D3—44no—
383C2×D3—42no—
384C2×D3—82no—
385C2×D3—81no—
386C2×D3—81no—
387C3×C4—24no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—22no—
391C3×C4—44no—
392C3×C4—42no—
393C3×C4D4212yes—
394C3×C4#54612yes—
395Dic3—42no—
396Dic3—81no—
397Dic3—81no—
398C23—24no—
399C23—62no—
400C23—122no—
401C23—122no—
402C23—121no—
403C23—121no—
404C23C22×#6711yescharacteristic
405Q8—24no—
406Q8—24no—
407Q8—24no—
408Q8—24no—
409Q8—48no—
410D4—24no—
411D4—24no—
412D4—48no—
413D4—44no—
414D4—44no—
415D4—44no—
416D4—64no—
417D4—82no—
418D4—84no—
419D4—122no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—416no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—62no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—244no—
439C2×C4#93814yes—
440C2×C4#95311yescharacteristic
441C2×C4#95311yescharacteristic
442C2×C4#95312yes—
443C2×C4C2×D3×D411yescharacteristic
444C2×C4C2×#23711yescharacteristic
445C2×C4#100412yes—
446C2×C4C22×#6712yes—
447C8—124no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—84no—
451C2×C3#240811yescharacteristic
452C2×C3#302011yescharacteristic
453C2×C3#302111yescharacteristic
454D3—82no—
455C22—21no—
456C22—21no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—124no—
461C22—122no—
462C22—242no—
463C22—241no—
464C22—241no—
465C22C2×#93811yescharacteristic
466C22#580511yescharacteristic
467C22C2×#95311yescharacteristic
468C4—24no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—44no—
473C4—42no—
474C4—122no—
475C4—241no—
476C4—241no—
477C4D4×#6712yes—
478C4#580612yes—
479C3#2422711yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—84no—
483C2—242no—
484C2#8405311yescharacteristic
485C2#8411611yescharacteristic
486C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3497C2
socleC22×C3C2×#120
hypercenter#3497D3
hypocenterC3#24227
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
1. C3×#3497C2semidirect
2. C2×#5245C2semidirect
3. #83297C2semidirect
4. #82455C2semidirect
5. #72143C2semidirect
6. #71879C2semidirect
7. #71876C2semidirect
8. #71816C2semidirect
9. #71637C2semidirect
39. #3497D3semidirect
84. #947D4semidirect
479. C3#24227semidirect

Maximal subgroups

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

subgroupindex
C3×#34972
C2×#52452
#832972
#824552
#721432
#718792
#718762
#718162
#716372
#242273