Finite Group Databasebeta

← #10763296

Subgroups — #10763296

Order 768.

972 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C211yesmaximal, characteristic
2#79331C212yesmaximal, complemented
3C3×#1537C211yesmaximal, characteristic, complemented
4#71823C211yesmaximal, characteristic, complemented
5#70811C212yesmaximal, complemented
6#68364C218yesmaximal, complemented
7#17673—31nomaximal, Sylow
8C3×#533C2211yescharacteristic
9#5728C412yes—
10#5725C412yes—
11#5691C2214yes—
12#5627—22no—
13#5624—22no—
14#5580—22no—
15C4×#239—22no—
16#5524C412yes—
17#5522C412yes—
18#5477C2211yescharacteristic
19C2×C42×D3C2211yescharacteristic
20C3×#432C2212yes—
21#4926C2212yes—
22#4837—22no—
23#4833—22no—
24#4726—22no—
25#4718—22no—
26#4707C2214yes—
27#4705C2214yes—
28C3×#330C2214yescomplemented
29#4543C2214yescomplemented
30#4464C2218yescomplemented
31#3444—31no—
32#2920—32no—
33#1537—31no—
34#1537D311yescharacteristic, complemented
35#1536—32no—
36#1529—38no—
37C2×#239—22no—
38C2×#235C2×C412yes—
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D412yes—
41C3×C4⋊Q8C2×C412yes—
42C3×#125C2312yes—
43C3×C22⋊Q8—22no—
44C3×C4×Q8—22no—
45C3×#117C2×C412yes—
46C2×C3×C42C2311yescharacteristic
47#947—22no—
48#946—22no—
49Q8×Dic3—22no—
50#930—22no—
51C2×C4×Dic3C2×C411yescharacteristic
52C2×C4×Dic3D412yes—
53C2×ΓL(1,𝔽25)—24no—
54#898—24no—
55#897C2×C414yes—
56#895—22no—
57#895C2×C414yes—
58#894—24no—
59#893—24no—
60#893—22no—
61#893C2×C414yes—
62#892—22no—
63#892—24no—
64#890—22no—
65#887C2×C414yes—
66#886—22no—
67#882—22no—
68#67⋊C4—24no—
69#875C2312yes—
70#874C2×C411yescharacteristic
71C4×D12—24no—
72C4×D12—22no—
73#872—22no—
74C42×D3—22no—
75C42×D3C2×C412yes—
76C42×D3D412yes—
77C42×D3D412yes—
78C42×D3C2312yes—
79#869C2×C411yescharacteristic
80C3×C4⋊C8—22no—
81C3×C22⋊C8—22no—
82C3×#97C2314yes—
83#830—22no—
84(C2×D3)⋊C8—24no—
85Dic3⋊C8—24no—
86C8×Dic3—24no—
87C4⋊(C3⋊C8)—22no—
88#803C2314yes—
89#533—32no—
90#533C2×D311yescharacteristic
91#532—32no—
92#532—31no—
93#529—34no—
94#529—62no—
95#529—62no—
96#525—32no—
97#525—62no—
98C4×#41—32no—
99C4×#41—62no—
100C22×C42—31no—
101#432—32no—
102#432—34no—
103#432C2×D312yes—
104#431—62no—
105#431—62no—
106#423—62no—
107#407—62no—
108C4×M16—34no—
109#330—34no—
110#330C2×D314yescomplemented
111#329—38no—
112C23×D3C22⋊C411yescharacteristic
113C2×C3×Q8—22no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D412yes—
116C2×#67—22no—
117C22×Dic3C22⋊C412yes—
118C22×Dic3C22⋊C411yescharacteristic
119#239—48no—
120#235—44no—
121C2×D12—24no—
122C2×D12—22no—
123C2×D12C22×C412yes—
124C2×C4×D3—22no—
125C2×C4×D3—24no—
126C2×C4×D3—44no—
127C2×C4×D3—42no—
128C2×C4×D3C22⋊C412yes—
129C2×C4×D3C22⋊C412yes—
130C2×C4×D3C22⋊C412yes—
131C2×C4×D3C22×C412yes—
132C2×C4×D3C2×D412yes—
133C2×C4×D3C2×D412yes—
134C2×Dic6C22×C412yes—
135C2×C3×C8—24no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4C22×C414yes—
139C3×C22⋊C4—22no—
140C3×C42—22no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D412yes—
143C3×C42C2411yescharacteristic
144(C2×D3)⋊C4—24no—
145(C2×D3)⋊C4—24no—
146(C2×D3)⋊C4—24no—
147(C2×D3)⋊C4—24no—
148C4⋊Dic3—22no—
149C4⋊Dic3—24no—
150Dic3⋊C4—24no—
151C4×Dic3—24no—
152C4×Dic3C22⋊C411yescharacteristic
153C4×Dic3C22⋊C412yes—
154C4×Dic3C22⋊C412yes—
155C4×Dic3C22⋊C411yescharacteristic
156C4×Dic3C22×C412yes—
157C4×Dic3C2×D412yes—
158C4×Dic3C2×D412yes—
159C2×C3⋊C8—24no—
160ΓL(1,𝔽25)—48no—
161C2×#41—32no—
162C2×#41—62no—
163C23×C4—31no—
164C23×C4—32no—
165C2×M16—64no—
166C4⋊Q8—31no—
167C4⋊Q8C4×D312yes—
168#127—31no—
169#126—64no—
170#125—62no—
171#125C22×D312yes—
172#124—32no—
173#124—62no—
174#123—62no—
175#123—64no—
176C22⋊Q8—22no—
177C22⋊Q8—34no—
178C22⋊Q8—62no—
179#121—34no—
180#121—62no—
181#121—62no—
182#120—62no—
183C4×Q8—22no—
184C4×Q8—34no—
185C4×Q8—62no—
186C4×D4—34no—
187C4×D4—64no—
188C4×D4—62no—
189C4×D4—64no—
190C4×D4—64no—
191C4×D4—62no—
192#117—62no—
193#117—64no—
194#117—62no—
195#117C4×D312yes—
196C2×C42—31no—
197C2×C42—32no—
198C2×C42—32no—
199C2×C42—32no—
200C2×C42—32no—
201C2×C42—32no—
202C2×C42—62no—
203C2×C42C22×D311yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—64no—
206C4⋊C8—62no—
207C22⋊C8—22no—
208C22⋊C8—64no—
209C22⋊C8—62no—
210#97—34no—
211#97C22×D314yes—
212C4×C8—64no—
213C23×C3C2×C22⋊C411yescharacteristic
214C22×D3—22no—
215C22×D3—41no—
216C22×D3—42no—
217C22×D3C2×C22⋊C412yes—
218C3×Q8—44no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—42no—
222C2×C3×C4C2×C22⋊C412yes—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C23×C411yescharacteristic
225C2×C3×C4C22×D412yes—
226#67—44no—
227C2×Dic3—24no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3C2×C22⋊C412yes—
231C2×Dic3C2×C22⋊C412yes—
232C2×Dic3C2×C22⋊C412yes—
233D12—42no—
234D12—44no—
235D12—48no—
236C4×D3—24no—
237C4×D3—24no—
238C4×D3—24no—
239C4×D3—42no—
240C4×D3—42no—
241C4×D3—42no—
242C4×D3—44no—
243C4×D3—48no—
244Dic6—42no—
245C3×C8—44no—
246C3⋊C8—44no—
247C24—31no—
248#41—124no—
249#41—128no—
250C2×Q8—22no—
251C2×Q8—32no—
252C2×D4—32no—
253C2×D4—64no—
254C2×D4—62no—
255C2×D4—62no—
256C22×C4—32no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—64no—
265C22×C4—62no—
266C22×C4—122no—
267C22×C4—124no—
268C22×C4C2×C4×D311yescharacteristic
269C22×C4D3×D412yes—
270M16—128no—
271C2×C8—24no—
272C2×C8—64no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—64no—
276C4⋊C4—64no—
277C4⋊C4—62no—
278C4⋊C4C2×C4×D314yes—
279C22⋊C4—22no—
280C22⋊C4—64no—
281C22⋊C4—64no—
282C22⋊C4—64no—
283C22⋊C4—64no—
284C42—22no—
285C42—31no—
286C42—31no—
287C42—32no—
288C42—32no—
289C42—32no—
290C42—32no—
291C42—32no—
292C42—64no—
293C42C2×C4×D311yescharacteristic
294C42D3×D412yes—
295C42C23×D311yescharacteristic
296C22×C3—21no—
297C22×C3—22no—
298C22×C3C22×C22⋊C411yescharacteristic
299C2×D3—22no—
300C2×D3—22no—
301C2×D3—22no—
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—44no—
305C3×C4—21no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—44no—
309C3×C4#41312yes—
310Dic3—22no—
311Dic3—22no—
312Dic3—42no—
313Dic3#42114yes—
314C23—32no—
315C23—62no—
316C23—121no—
317C23—122no—
318C23D3×C22⋊C411yescharacteristic
319Q8—44no—
320Q8—122no—
321D4—122no—
322D4—124no—
323D4—128no—
324D4—124no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—42no—
331C2×C4—64no—
332C2×C4—64no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—62no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—128no—
340C2×C4—121no—
341C2×C4—124no—
342C2×C4D3×C22⋊C412yes—
343C2×C4D3×C22⋊C411yescharacteristic
344C2×C4C22×C4×D311yescharacteristic
345C2×C4C2×D3×D412yes—
346C8—44no—
347C8—124no—
348C2×C3—41no—
349C2×C3C2×#41311yescharacteristic
350C2×C3C2×#42112yes—
351D3—42no—
352D3—82no—
353C22—21no—
354C22—22no—
355C22—62no—
356C22—62no—
357C22—62no—
358C22—124no—
359C22—122no—
360C22—124no—
361C22C2×D3×C22⋊C411yescharacteristic
362C4—21no—
363C4—22no—
364C4—22no—
365C4—34no—
366C4—44no—
367C4—62no—
368C4—62no—
369C4—122no—
370C4#475112yes—
371C3#1767311yescharacteristic, complemented, Sylow
372C2—41no—
373C2—122no—
374C2—242no—
375C2C2×#475111yescharacteristic
376C2D3×#42112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1537C2
socleC22×C3C22×C22⋊C4
prime radical#86257C2
cosocleC3×C42C24
hypercenter#1537D3
hypocenterC3#17673
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C3×#1537C2semidirect
4. #71823C2semidirect
5. #70811C2semidirect
6. #68364C2semidirect
28. C3×#330C22semidirect
29. #4543C22semidirect
30. #4464C22semidirect
34. #1537D3semidirect
110. #330C2×D3semidirect
371. C3#17673semidirect

Maximal subgroups

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

subgroupindex
#862572
#793312
C3×#15372
#718232
#708112
#683642
#176733