Finite Group Databasebeta

← #10848609

Subgroups — #10848609

Order 768.

1188 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85848C211yesmaximal, characteristic
2C3×#3039C211yesmaximal, characteristic, complemented
3#83251C211yesmaximal, characteristic, complemented
4#83198C211yesmaximal, characteristic, complemented
5#72106C214yesmaximal, complemented
6C4×#837C211yesmaximal, characteristic, complemented
7#70521C211yesmaximal, characteristic, complemented
8#70514C211yesmaximal, characteristic, complemented
9#70343C214yesmaximal, complemented
10#25590—31nomaximal, Sylow
11C2×C3×#137C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C3×C4×#41C2211yescharacteristic
14#5830C2212yes—
15C2×#947C2214yes—
16#5585—22no—
17#5553—22no—
18#5550—24no—
19#5535—22no—
20#5521C2212yes—
21#5477—22no—
22C2×#874C2211yescharacteristic
23#5470C2211yescharacteristic
24C3×#462—22no—
25C3×#461—22no—
26C3×#423C2211yescharacteristic
27C3×#419C2212yes—
28C2×#837C2212yes—
29#5225C2214yes—
30C2×#823C2214yes—
31#5082—22no—
32#5079—22no—
33#4988—24no—
34C42×Dic3C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4555C2214yes—
37#4532C2212yes—
38#3508—31no—
39#3139—31no—
40#3039D311yescharacteristic, complemented
41#2858—31no—
42#2004—34no—
43#1994—31no—
44#1965—31no—
45#1885—34no—
46C4×#104—31no—
47C2×C3×#41C2311yescharacteristic
48C22×C3×Q8C2312yes—
49C2×#235—24no—
50C22×D12D414yes—
51C22×D12C2312yes—
52C3×#137—44no—
53C2×C3×Q16—22no—
54C2×C3×SD16—22no—
55C2×C3×M16C2312yes—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—21no—
59C3×C4×Q8D412yes—
60C3×C4×D4—22no—
61C3×C4×D4D412yes—
62C3×#117—21no—
63C3×#117C2311yescharacteristic
64C2×C3×C4⋊C4—22no—
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67#947—22no—
68#947—28no—
69#947—44no—
70#940—22no—
71#937—24no—
72#930—24no—
73#930—24no—
74C2×(C2×D3)⋊C4—28no—
75#926C2312yes—
76#924—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3C2312yes—
79C2×#208C2312yes—
80#894—28no—
81#893—24no—
82#884—28no—
83D3×C22⋊C4—24no—
84#875—24no—
85#874—21no—
86#874—21no—
87#874—22no—
88C4×D12—24no—
89C4×D12—24no—
90#870D412yes—
91#869D412yes—
92C4×Dic6—22no—
93C3×C4⋊C8—22no—
94C3×#103—24no—
95#837—24no—
96#835—28no—
97#823—44no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×#137C2×D312yes—
101#559—62no—
102#550—62no—
103#546—64no—
104#540—62no—
105#535—32no—
106#533—31no—
107#532—62no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#127—31no—
110C2×#124—34no—
111#518—32no—
112C4×#41C2×D311yescharacteristic
113#491—62no—
114#487—62no—
115#462—22no—
116#461—22no—
117#423C2×D311yescharacteristic
118C2×#104—32no—
119#419—34no—
120#419C2×D312yes—
121C2×#100—34no—
122C4×M16—31no—
123#390—64no—
124C43—31no—
125#339—34no—
126#339—32no—
127C23×D3C2×D414yes—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8C2×D411yescharacteristic
135C2×C3×D4—21no—
136C2×C3×D4C2×D411yescharacteristic
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2411yescharacteristic
142C2×#67—28no—
143C22×Dic3—22no—
144#235—88no—
145C2×D12—24no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—24no—
149C2×D12—24no—
150C2×D12—24no—
151C2×D12—48no—
152C2×C4×D3—28no—
153C2×Dic6—22no—
154C3×Q16—44no—
155C3×SD16—44no—
156C3×M16—24no—
157C2×C3×C8—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4C2×D412yes—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—22no—
164C3×C42—22no—
165C3×C42—21no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D412yes—
169#217—22no—
170(C2×D3)⋊C4—28no—
171(C2×D3)⋊C4—28no—
172(C2×D3)⋊C4—416no—
173C4⋊Dic3—22no—
174C4⋊Dic3C2×D414yes—
175Dic3⋊C4—24no—
176C4×Dic3—22no—
177C4×Dic3—22no—
178C4×Dic3—44no—
179C4×Dic3#4114yescomplemented
180#208—22no—
181#208—22no—
182C2×C3⋊C8—22no—
183C2×#41—64no—
184C2×#41C22×D311yescharacteristic
185C22×Q8C22×D312yes—
186C22×D4—34no—
187C22×D4—32no—
188#137—44no—
189C2×Q16—22no—
190C2×SD16—22no—
191C2×M16—32no—
192C2×M16C22×D312yes—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—22no—
196C4⋊Q8—32no—
197#127—62no—
198#127—61no—
199#127—61no—
200#127—62no—
201#125—32no—
202#124—64no—
203#124—68no—
204#124—62no—
205#124—124no—
206#123—68no—
207C22⋊Q8—62no—
208#121—68no—
209#121—64no—
210#121—64no—
211#120—64no—
212C4×Q8—62no—
213C4×Q8#6712yes—
214C4×D4—22no—
215C4×D4—64no—
216C4×D4—64no—
217C4×D4—64no—
218C4×D4#6712yes—
219#117—21no—
220#117—32no—
221#117C22×D311yescharacteristic
222C2×C4⋊C4—22no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—68no—
225C2×C42—21no—
226C2×C42—32no—
227C2×C42—62no—
228C2×C42C22×D311yescharacteristic
229C4⋊C8—22no—
230#104—64no—
231#103—24no—
232#103—68no—
233#100—124no—
234#97—61no—
235C4×C8—61no—
236C23×C3—21no—
237C23×C3C22×D411yescharacteristic
238C22×D3—24no—
239C22×D3—28no—
240C22×D3—48no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—42no—
244C3×Q8—42no—
245C3×Q8—44no—
246C3×D4—21no—
247C3×D4—21no—
248C3×D4—42no—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4#12012yes—
257C2×C3×C4#12012yes—
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4112yes—
261C2×C3×C4H(5,𝔽2)12yes—
262#67—88no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—44no—
266D12—42no—
267D12—42no—
268D12—44no—
269D12—48no—
270D12—84no—
271C4×D3—88no—
272Dic6—44no—
273C3×C8—42no—
274C3⋊C8—42no—
275C24—34no—
276#41—22no—
277#41—22no—
278#41—248no—
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—62no—
284C2×Q8C2×#6711yescharacteristic
285C2×D4—21no—
286C2×D4—68no—
287C2×D4—64no—
288C2×D4—64no—
289C2×D4—62no—
290C2×D4—64no—
291C2×D4—64no—
292C2×D4—62no—
293C2×D4—128no—
294C2×D4C2×#6711yescharacteristic
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—68no—
299C22×C4—62no—
300C22×C4D3×D412yes—
301C22×C4C23×D311yescharacteristic
302Q16—44no—
303SD16—44no—
304M16—24no—
305M16—62no—
306M16—62no—
307C2×C8—22no—
308C2×C8—62no—
309C4⋊C4—22no—
310C4⋊C4—34no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—62no—
314C4⋊C4—64no—
315C4⋊C4C2×#6712yes—
316C22⋊C4—21no—
317C22⋊C4—22no—
318C22⋊C4—68no—
319C22⋊C4—68no—
320C22⋊C4—62no—
321C22⋊C4—1216no—
322C42—22no—
323C42—21no—
324C42—34no—
325C42—62no—
326C42—62no—
327C42—124no—
328C42D3×D411yescharacteristic
329C42D3×D411yescharacteristic
330C42C2×#6712yes—
331C22×C3—22no—
332C22×C3—41no—
333C22×C3—42no—
334C22×C3C2×#12011yescharacteristic
335C22×C3#54612yes—
336C2×D3—44no—
337C2×D3—48no—
338C2×D3—88no—
339C3×C4—22no—
340C3×C4—41no—
341C3×C4—42no—
342C3×C4—82no—
343C3×C4#45514yes—
344C3×C4#53211yescharacteristic
345C3×C4#53511yescharacteristic
346C3×C4#55012yes—
347Dic3—44no—
348Dic3—44no—
349C23—21no—
350C23—64no—
351C23—68no—
352C23—128no—
353C23C2×D3×D411yescharacteristic
354Q8—21no—
355Q8—21no—
356Q8—42no—
357Q8—42no—
358Q8—44no—
359Q8—124no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—128no—
364D4—122no—
365D4—122no—
366D4—124no—
367D4—244no—
368D4—248no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—64no—
377C2×C4—64no—
378C2×C4—124no—
379C2×C4—248no—
380C2×C4#93712yes—
381C2×C4#93712yes—
382C2×C4C2×D3×D411yescharacteristic
383C2×C4#100412yes—
384C2×C4D3×#4112yes—
385C2×C4C22×#6711yescharacteristic
386C8—42no—
387C8—122no—
388C2×C3—22no—
389C2×C3—81no—
390C2×C3#243111yescharacteristic
391C2×C3C2×#45512yes—
392D3—84no—
393C22—22no—
394C22—42no—
395C22—41no—
396C22—128no—
397C22—124no—
398C22—248no—
399C22#561212yes—
400C22C2×#93711yescharacteristic
401C4—22no—
402C4—42no—
403C4—41no—
404C4—82no—
405C4—124no—
406C4—124no—
407C4#520214yes—
408C4#559912yes—
409C4#580511yescharacteristic
410C4#583211yescharacteristic
411C3#2559011yescharacteristic, complemented, Sylow
412C2—22no—
413C2—81no—
414C2—244no—
415C2#8265211yescharacteristic
416C2C2×#520212yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3039C2
socleC22×C3C2×#120
prime radical#85848C2
hypercenter#3039D3
hypocenterC3#25590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
2. C3×#3039C2semidirect
3. #83251C2semidirect
4. #83198C2semidirect
5. #72106C2semidirect
6. C4×#837C2semidirect
7. #70521C2semidirect
8. #70514C2semidirect
9. #70343C2semidirect
40. #3039D3semidirect
179. C4×Dic3#41semidirect
411. C3#25590semidirect

Maximal subgroups

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

subgroupindex
#858482
C3×#30392
#832512
#831982
#721062
C4×#8372
#705212
#705142
#703432
#255903