Finite Group Databasebeta

← #10763250

Subgroups — #10763250

Order 768.

972 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C211yesmaximal, characteristic
2#78517C212yesmaximal, complemented
3C3×#1526C211yesmaximal, characteristic, complemented
4#71821C211yesmaximal, characteristic, complemented
5#69859C212yesmaximal, complemented
6#68368C218yesmaximal, complemented
7#17665—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×C4×M16C2212yes—
21#5126—22no—
22#5118—22no—
23C4×#208C2212yes—
24#4705C2214yes—
25#4705C2214yes—
26#4685—22no—
27#4685—22no—
28C3×#330C2214yescomplemented
29#4543C2214yescomplemented
30#4464C2218yescomplemented
31#3444—31no—
32#2920—32no—
33#1535—38no—
34#1526—31no—
35#1526D311yescharacteristic, complemented
36#1525—32no—
37C2×#239—22no—
38C2×#235C2×C412yes—
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D412yes—
41C2×C3×M16—22no—
42C3×C4⋊Q8C2×C412yes—
43C3×#125C2312yes—
44C3×C22⋊Q8—22no—
45C3×C4×Q8—22no—
46C3×#117C2×C412yes—
47C2×C3×C42C2311yescharacteristic
48#947—22no—
49#946—22no—
50Q8×Dic3—22no—
51#930—22no—
52C2×C4×Dic3C2×C411yescharacteristic
53C2×C4×Dic3D412yes—
54C2×#208—22no—
55#898—24no—
56#897C2×C414yes—
57#895—22no—
58#895C2×C414yes—
59#894—24no—
60#893—24no—
61#893—22no—
62#893C2×C414yes—
63#892—22no—
64#892—24no—
65#890—22no—
66#887C2×C414yes—
67#886—22no—
68#882—22no—
69#67⋊C4—24no—
70#875C2312yes—
71#874C2×C411yescharacteristic
72C4×D12—24no—
73C4×D12—22no—
74#872—22no—
75C42×D3—22no—
76C42×D3C2×C412yes—
77C42×D3D412yes—
78C42×D3D412yes—
79C42×D3C2312yes—
80#869C2×C411yescharacteristic
81C3×#97C2314yes—
82C3×C4×C8—22no—
83(C2×D3)⋊C8—24no—
84(C2×D3)⋊C8—24no—
85Dic3⋊C8—24no—
86Dic3⋊C8—24no—
87#803C2314yes—
88C4×C3⋊C8—22no—
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—34no—
102#432—34no—
103#431—62no—
104#431—62no—
105#423—62no—
106#407—62no—
107C4×M16—32no—
108C4×M16C2×D312yes—
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—42no—
127C2×C4×D3—44no—
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—
135C3×M16—44no—
136C2×C3×C8—24no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4C22×C414yes—
140C3×C22⋊C4—22no—
141C3×C42—22no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D412yes—
144C3×C42C2411yescharacteristic
145(C2×D3)⋊C4—24no—
146(C2×D3)⋊C4—24no—
147(C2×D3)⋊C4—24no—
148(C2×D3)⋊C4—24no—
149C4⋊Dic3—22no—
150C4⋊Dic3—24no—
151Dic3⋊C4—24no—
152C4×Dic3—24no—
153C4×Dic3C22⋊C411yescharacteristic
154C4×Dic3C22⋊C412yes—
155C4×Dic3C22⋊C412yes—
156C4×Dic3C22⋊C411yescharacteristic
157C4×Dic3C22×C412yes—
158C4×Dic3C2×D412yes—
159C4×Dic3C2×D412yes—
160#208—44no—
161C2×C3⋊C8—24no—
162C2×#41—32no—
163C2×#41—62no—
164C23×C4—31no—
165C23×C4—32no—
166C2×M16—22no—
167C2×M16—62no—
168C4⋊Q8—31no—
169C4⋊Q8C4×D312yes—
170#127—31no—
171#126—64no—
172#125—62no—
173#125C22×D312yes—
174#124—32no—
175#124—62no—
176#123—62no—
177#123—64no—
178C22⋊Q8—22no—
179C22⋊Q8—34no—
180C22⋊Q8—62no—
181#121—34no—
182#121—62no—
183#121—62no—
184#120—62no—
185C4×Q8—22no—
186C4×Q8—34no—
187C4×Q8—62no—
188C4×D4—34no—
189C4×D4—64no—
190C4×D4—62no—
191C4×D4—64no—
192C4×D4—64no—
193C4×D4—62no—
194#117—62no—
195#117—64no—
196#117—62no—
197#117C4×D312yes—
198C2×C42—31no—
199C2×C42—32no—
200C2×C42—32no—
201C2×C42—32no—
202C2×C42—32no—
203C2×C42—32no—
204C2×C42—62no—
205C2×C42C22×D311yescharacteristic
206C4⋊C8—64no—
207C4⋊C8—64no—
208C22⋊C8—64no—
209C22⋊C8—64no—
210#97—34no—
211#97C22×D314yes—
212C4×C8—22no—
213C4×C8—62no—
214C23×C3C2×C22⋊C411yescharacteristic
215C22×D3—22no—
216C22×D3—41no—
217C22×D3—42no—
218C22×D3C2×C22⋊C412yes—
219C3×Q8—44no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4C2×C22⋊C412yes—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C23×C411yescharacteristic
226C2×C3×C4C22×D412yes—
227#67—44no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3C2×C22⋊C412yes—
232C2×Dic3C2×C22⋊C412yes—
233C2×Dic3C2×C22⋊C412yes—
234D12—42no—
235D12—48no—
236D12—44no—
237C4×D3—24no—
238C4×D3—24no—
239C4×D3—24no—
240C4×D3—42no—
241C4×D3—42no—
242C4×D3—42no—
243C4×D3—48no—
244C4×D3—44no—
245Dic6—42no—
246C3×C8—44no—
247C3⋊C8—44no—
248C24—31no—
249#41—124no—
250#41—128no—
251C2×Q8—22no—
252C2×Q8—32no—
253C2×D4—32no—
254C2×D4—64no—
255C2×D4—62no—
256C2×D4—62no—
257C22×C4—32no—
258C22×C4—31no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—32no—
265C22×C4—64no—
266C22×C4—62no—
267C22×C4—122no—
268C22×C4—124no—
269C22×C4C2×C4×D311yescharacteristic
270C22×C4D3×D412yes—
271M16—44no—
272M16—124no—
273C2×C8—24no—
274C2×C8—64no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—64no—
278C4⋊C4—64no—
279C4⋊C4—62no—
280C4⋊C4C2×C4×D314yes—
281C22⋊C4—22no—
282C22⋊C4—64no—
283C22⋊C4—64no—
284C22⋊C4—64no—
285C22⋊C4—64no—
286C42—22no—
287C42—32no—
288C42—32no—
289C42—31no—
290C42—31no—
291C42—32no—
292C42—32no—
293C42—32no—
294C42—64no—
295C42C2×C4×D311yescharacteristic
296C42D3×D412yes—
297C42C23×D311yescharacteristic
298C22×C3—21no—
299C22×C3—22no—
300C22×C3C22×C22⋊C411yescharacteristic
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—22no—
304C2×D3—44no—
305C2×D3—42no—
306C2×D3—44no—
307C3×C4—21no—
308C3×C4—22no—
309C3×C4—44no—
310C3×C4#41312yes—
311C3×C4#42114yes—
312Dic3—22no—
313Dic3—22no—
314Dic3—22no—
315Dic3—42no—
316C23—32no—
317C23—62no—
318C23—121no—
319C23—122no—
320C23D3×C22⋊C411yescharacteristic
321Q8—44no—
322Q8—122no—
323D4—122no—
324D4—128no—
325D4—124no—
326D4—124no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4—62no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—128no—
341C2×C4—121no—
342C2×C4—122no—
343C2×C4—124no—
344C2×C4D3×C22⋊C412yes—
345C2×C4D3×C22⋊C411yescharacteristic
346C2×C4C22×C4×D311yescharacteristic
347C2×C4C2×D3×D412yes—
348C8—44no—
349C8—124no—
350C2×C3—41no—
351C2×C3C2×#41311yescharacteristic
352C2×C3C2×#42112yes—
353D3—42no—
354D3—82no—
355C22—21no—
356C22—22no—
357C22—62no—
358C22—62no—
359C22—62no—
360C22—124no—
361C22—122no—
362C22—124no—
363C22C2×D3×C22⋊C411yescharacteristic
364C4—21no—
365C4—22no—
366C4—44no—
367C4—62no—
368C4—62no—
369C4—62no—
370C4—122no—
371C4#475112yes—
372C4#482114yes—
373C3#1766511yescharacteristic, complemented, Sylow
374C2—41no—
375C2—122no—
376C2—242no—
377C2C2×#475111yescharacteristic
378C2C2×#482112yes—
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1526C2
socleC22×C3C22×C22⋊C4
prime radical#86257C2
cosocleC3×C42C24
hypercenter#1526D3
hypocenterC3#17665
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #78517C2semidirect
3. C3×#1526C2semidirect
4. #71821C2semidirect
5. #69859C2semidirect
6. #68368C2semidirect
28. C3×#330C22semidirect
29. #4543C22semidirect
30. #4464C22semidirect
35. #1526D3semidirect
110. #330C2×D3semidirect
373. C3#17665semidirect

Maximal subgroups

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

subgroupindex
#862572
#785172
C3×#15262
#718212
#698592
#683682
#176653