Finite Group Databasebeta

← #10764125

Subgroups — #10764125

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86243C211yesmaximal, characteristic
2#79051C212yesmaximal, complemented
3C3×#1560C211yesmaximal, characteristic, complemented
4#71805C211yesmaximal, characteristic, complemented
5#70507C212yesmaximal, complemented
6#68487C212yesmaximal, complemented
7D3×#333C212yesmaximal, complemented
8#68385C214yesmaximal, complemented
9#17779—31nomaximal, Sylow
10C2×#1010C412yes—
11C3×#532C2211yescharacteristic
12#5825C412yes—
13D3×C4⋊Q8C2212yes—
14#5716C2212yes—
15#5675C414yes—
16#5611—24no—
17#5551—22no—
18D4×Dic6—22no—
19C4×#235C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C3×#423C2212yes—
22#5127—24no—
23#4925C2212yes—
24#4835C2214yes—
25#4834—22no—
26D3×C4⋊C8C2214yes—
27#4831—22no—
28C3×#333C2212yes—
29C3×#331C2212yes—
30#4546C2211yescharacteristic
31#4546C2211yescharacteristic
32#4544C2212yes—
33#4481C2214yes—
34#4481C2214yes—
35#3499—31no—
36#2990—32no—
37#1560D311yescharacteristic, complemented
38#1558—32no—
39#1551—34no—
40#1548—32no—
41#1543—31no—
42C2×#333—32no—
43C2×C3×#41C2×C412yes—
44#1010—48no—
45C2×D3×Q8C2×C414yes—
46C2×#237—24no—
47C2×#237C2×C414yes—
48C2×#235—22no—
49C2×#235C2×C411yescharacteristic
50C22×Dic6—22no—
51C22×Dic6C2×C412yes—
52C2×C3×M16—22no—
53C3×C4⋊Q8C2311yescharacteristic
54C3×#127C2311yescharacteristic
55C3×#124C2×C412yes—
56C3×#121—22no—
57C3×C4×D4—22no—
58C2×C3×C42C2311yescharacteristic
59#944C2×C414yes—
60#938—24no—
61#936C2×C414yes—
62#935—24no—
63D4×Dic3—24no—
64C4×#67D414yes—
65#926C2×C411yescharacteristic
66C2×C4⋊Dic3—22no—
67#924—24no—
68C2×D3×C8—24no—
69D3×C4⋊C4—24no—
70#888—24no—
71#883—28no—
72#878—28no—
73C4×D12D412yes—
74#872C2×C412yes—
75C42×D3C2312yes—
76#869—22no—
77#869C2×C412yes—
78#869C2312yes—
79C4×Dic6D412yes—
80C3×C4⋊C8—22no—
81C3×C4⋊C8C2314yes—
82#830—22no—
83(C2×D3)⋊C8—24no—
84#815—24no—
85Dic3⋊C8—24no—
86C4⋊(C3⋊C8)C2312yes—
87C4⋊(C3⋊C8)C2312yes—
88C4×C3⋊C8—22no—
89C2×2−1+4—32no—
90D4×Q8—62no—
91#547—62no—
92#547—64no—
93#536—32no—
94#536—34no—
95#532—32no—
96#532C2×D311yescharacteristic
97C2×C4⋊Q8—32no—
98C2×C4⋊Q8—31no—
99C4×#41—31no—
100#446—62no—
101#435—64no—
102C8×D4—62no—
103#432—34no—
104#431—32no—
105#423C2×D312yes—
106C2×C4⋊C8—34no—
107#333—31no—
108#333—31no—
109#333—34no—
110#333—34no—
111#333C2×D312yes—
112#331—32no—
113#331C2×D312yes—
114C3×#41—44no—
115C2×C3×Q8C22×C412yes—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4C22×C412yes—
119C22×C3×C4—22no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D412yes—
122C2×#67—24no—
123C2×#67C22⋊C412yes—
124C22×Dic3—22no—
125C22×Dic3—24no—
126D3×Q8—48no—
127#237—48no—
128#237—416no—
129#235—24no—
130#235—48no—
131C2×D12C22⋊C411yescharacteristic
132C2×C4×D3—24no—
133C2×C4×D3C22×C412yes—
134C2×C4×D3C2×D414yes—
135C2×Dic6—22no—
136C2×Dic6—24no—
137C2×Dic6—48no—
138C2×Dic6—44no—
139C2×Dic6C22⋊C411yescharacteristic
140C2×Dic6C22×C414yes—
141C3×M16—82no—
142C2×C3×C8—24no—
143C3×C4⋊C4—22no—
144C3×C22⋊C4—24no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D412yes—
147C3×C42C2411yescharacteristic
148#217—28no—
149#217C22⋊C412yes—
150(C2×D3)⋊C4C22⋊C414yes—
151C4⋊Dic3—24no—
152C4⋊Dic3—24no—
153C4⋊Dic3C22⋊C412yes—
154Dic3⋊C4—28no—
155Dic3⋊C4C22⋊C414yes—
156C4×Dic3C22×C412yes—
157C4×Dic3C2×D414yes—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160D3×C8—84no—
1612−1+4—128no—
162C2×#41—31no—
163C2×#41—34no—
164C2×#41—62no—
165C2×#41—64no—
166C2×#41C4×D312yes—
167C22×Q8—34no—
168C22×Q8—32no—
169C22×Q8—62no—
170C2×M16—22no—
171C22×C8—64no—
172C4⋊Q8—34no—
173C4⋊Q8—32no—
174C4⋊Q8—32no—
175C4⋊Q8—64no—
176C4⋊Q8—62no—
177C4⋊Q8C22×D311yescharacteristic
178#127C22×D311yescharacteristic
179#124—34no—
180#124C4×D312yes—
181#123—68no—
182#123—64no—
183C22⋊Q8—68no—
184C22⋊Q8—64no—
185#121—22no—
186#121—64no—
187C4×Q8—32no—
188C4×D4—22no—
189C4×D4—34no—
190C4×D4—32no—
191C4×D4—64no—
192#117—31no—
193#117—32no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—62no—
196C2×C42—32no—
197C2×C42C22×D311yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—32no—
200C4⋊C8—32no—
201C4⋊C8—64no—
202C4⋊C8C22×D314yes—
203C22⋊C8—64no—
204C22⋊C8—62no—
205#97—64no—
206C4×C8—62no—
207C23×C3—22no—
208C23×C3C2×C22⋊C411yescharacteristic
209C22×D3C2×C22⋊C412yes—
210C3×Q8—42no—
211C3×D4—42no—
212C3×D4—44no—
213C3×D4—42no—
214C3×D4—42no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—44no—
220C2×C3×C4C2×C22⋊C412yes—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C23×C411yescharacteristic
223C2×C3×C4C22×D412yes—
224#67—24no—
225#67—48no—
226C2×Dic3—24no—
227C2×Dic3—48no—
228C2×Dic3—44no—
229C2×Dic3C2×C22⋊C414yes—
230C2×Dic3C2×C22⋊C412yes—
231D12#10014yes—
232C4×D3—24no—
233C4×D3—24no—
234C4×D3—24no—
235C4×D3—48no—
236Dic6—22no—
237Dic6—48no—
238Dic6—44no—
239Dic6—44no—
240C3×C8—44no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243#41—44no—
244#41—64no—
245#41—128no—
246#41—128no—
247#41—1216no—
248C2×Q8—31no—
249C2×Q8—34no—
250C2×Q8—64no—
251C2×Q8—62no—
252C2×Q8—124no—
253C2×Q8—128no—
254C2×Q8—128no—
255C2×Q8C2×C4×D312yes—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—32no—
259C2×D4—31no—
260C2×D4—64no—
261C2×D4C2×C4×D312yes—
262C22×C4—22no—
263C22×C4—32no—
264C22×C4—34no—
265C22×C4—64no—
266C22×C4—64no—
267C22×C4—62no—
268C22×C4C2×C4×D311yescharacteristic
269C22×C4D3×D412yes—
270M16—82no—
271C2×C8—24no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—244no—
275C4⋊C4—22no—
276C4⋊C4—34no—
277C4⋊C4—32no—
278C4⋊C4—68no—
279C4⋊C4—64no—
280C4⋊C4—64no—
281C22⋊C4—24no—
282C22⋊C4—32no—
283C22⋊C4—34no—
284C22⋊C4—68no—
285C42—32no—
286C42—34no—
287C42C2×C4×D311yescharacteristic
288C42D3×D412yes—
289C42C23×D311yescharacteristic
290C22×C3—21no—
291C22×C3—22no—
292C22×C3—42no—
293C22×C3—44no—
294C22×C3C22×C22⋊C411yescharacteristic
295C2×D3—24no—
296C2×D3C2×#10014yes—
297C3×C4—21no—
298C3×C4—22no—
299C3×C4—42no—
300C3×C4C2×#10012yes—
301C3×C4#57712yes—
302C3×C4#57812yes—
303Dic3—22no—
304Dic3—24no—
305Dic3—44no—
306C23—22no—
307C23—32no—
308C23D3×C22⋊C411yescharacteristic
309Q8—42no—
310Q8—62no—
311Q8—124no—
312Q8—128no—
313Q8—124no—
314D4—34no—
315D4—42no—
316D4—44no—
317D4—42no—
318D4—42no—
319D4—64no—
320D4—128no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—34no—
326C2×C4—32no—
327C2×C4—44no—
328C2×C4—64no—
329C2×C4—64no—
330C2×C4—64no—
331C2×C4—64no—
332C2×C4—128no—
333C2×C4—128no—
334C2×C4—124no—
335C2×C4D3×C22⋊C412yes—
336C2×C4D3×C22⋊C411yescharacteristic
337C2×C4C22×C4×D311yescharacteristic
338C2×C4C2×D3×D412yes—
339C8—44no—
340C8—122no—
341C8—122no—
342C2×C3—41no—
343C2×C3—82no—
344C2×C3C22×#10011yescharacteristic
345C2×C3#289812yes—
346D3—24no—
347C22—22no—
348C22—21no—
349C22—34no—
350C22—44no—
351C22—42no—
352C22—64no—
353C22C2×D3×C22⋊C411yescharacteristic
354C4—22no—
355C4—21no—
356C4—42no—
357C4—64no—
358C4—62no—
359C4—124no—
360C4D3×#10012yes—
361C4#577512yes—
362C4#577912yes—
363C3#1777911yescharacteristic, complemented, Sylow
364C2—41no—
365C2—64no—
366C2—82no—
367C2C2×D3×#10011yescharacteristic
368C2#7888312yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1560C2
socleC22×C3C22×C22⋊C4
prime radical#86243C2
cosocleC3×C42C24
hypercenter#1560D3
hypocenterC3#17779
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
2. #79051C2semidirect
3. C3×#1560C2semidirect
4. #71805C2semidirect
5. #70507C2semidirect
6. #68487C2semidirect
7. D3×#333C2semidirect
8. #68385C2semidirect
37. #1560D3semidirect
363. C3#17779semidirect

Maximal subgroups

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

subgroupindex
#862432
#790512
C3×#15602
#718052
#705072
#684872
D3×#3332
#683852
#177793