Finite Group Databasebeta

← #10763243

Subgroups — #10763243

Order 768.

950 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#533C211yesmaximal, characteristic
2#78517C212yesmaximal
3C3×#1526C211yesmaximal, characteristic, complemented
4#71821C211yesmaximal, characteristic, complemented
5#69862C212yesmaximal, complemented
6#68369C214yesmaximal, complemented
7#68367C214yesmaximal, complemented
8#17664—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5728C412yes—
11D3×C4⋊Q8C412yes—
12#5688C2212yes—
13D3×#125C2212yes—
14D3×C22⋊Q8—22no—
15#5622—22no—
16#5578—22no—
17C4×D3×Q8—22no—
18D3×#117C412yes—
19#5517C412yes—
20C2×C42×D3C2211yescharacteristic
21#5470C2211yescharacteristic
22C3×C4×M16C2212yes—
23#5126—22no—
24#5118—22no—
25C4×#208C2212yes—
26#4705C2214yes—
27#4705C2214yes—
28#4685—22no—
29#4685—22no—
30C3×#330C2214yescomplemented
31#4543C2214yescomplemented
32#4465C2218yescomplemented
33C2×#533—31no—
34#2920—32no—
35#1537—34no—
36#1534—34no—
37#1526—32no—
38#1526—31no—
39#1526D311yescharacteristic, complemented
40C2×D3×Q8—22no—
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D412yes—
43C2×C3×M16—22no—
44C3×C4⋊Q8C2×C412yes—
45C3×#125C2312yes—
46C3×C22⋊Q8—22no—
47C3×C4×Q8—22no—
48C3×#117C2×C412yes—
49C2×C3×C42C2311yescharacteristic
50#946—22no—
51Q8×Dic3—22no—
52#944—22no—
53#926C2×C412yes—
54#924—22no—
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3D412yes—
57C2×#208—22no—
58#897—22no—
59#896—24no—
60#892C2×C414yes—
61D3×C4⋊C4—22no—
62D3×C4⋊C4—24no—
63D3×C4⋊C4C2×C414yes—
64#890C2×C414yes—
65#889—24no—
66#888—22no—
67#888C2×C414yes—
68#887—22no—
69#887—24no—
70D3×C22⋊C4—22no—
71#878—24no—
72#877—22no—
73#872—22no—
74C42×D3—22no—
75C42×D3C2×C411yescharacteristic
76C42×D3C2×C411yescharacteristic
77C42×D3D412yes—
78C42×D3D412yes—
79C42×D3C2311yescharacteristic
80C42×D3C2311yescharacteristic
81#870C2312yes—
82#869C2×C412yes—
83C4×Dic6—22no—
84C4×Dic6—24no—
85C3×#97C2314yes—
86C3×C4×C8—22no—
87(C2×D3)⋊C8—24no—
88(C2×D3)⋊C8—24no—
89Dic3⋊C8—24no—
90Dic3⋊C8—24no—
91#803C2314yes—
92C4×C3⋊C8—22no—
93#533—32no—
94#533—32no—
95#533—32no—
96#533—31no—
97#533—62no—
98#533—62no—
99#533C2×D311yescharacteristic
100C2×C4⋊Q8—32no—
101C2×#125—32no—
102C2×C22⋊Q8—62no—
103C2×C4×Q8—62no—
104C2×#117—32no—
105C22×C42—31no—
106#432—34no—
107#432—34no—
108#431—62no—
109#431—62no—
110#423—62no—
111#407—62no—
112C4×M16—32no—
113C4×M16C2×D312yes—
114#330—34no—
115#330—38no—
116#330C2×D314yescomplemented
117C23×D3C22⋊C411yescharacteristic
118C2×C3×Q8—22no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D412yes—
121C22×Dic3C22⋊C411yescharacteristic
122C22×Dic3C22⋊C412yes—
123D3×Q8—48no—
124C2×C4×D3—24no—
125C2×C4×D3—24no—
126C2×C4×D3—44no—
127C2×C4×D3C22⋊C412yes—
128C2×C4×D3C22⋊C412yes—
129C2×C4×D3C22⋊C411yescharacteristic
130C2×C4×D3C22⋊C411yescharacteristic
131C2×C4×D3C22×C411yescharacteristic
132C2×C4×D3C22×C411yescharacteristic
133C2×C4×D3C2×D412yes—
134C2×C4×D3C2×D412yes—
135C2×Dic6—22no—
136C2×Dic6—24no—
137C3×M16—44no—
138C2×C3×C8—24no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4C22×C414yes—
142C3×C22⋊C4—22no—
143C3×C42—22no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D412yes—
146C3×C42C2411yescharacteristic
147#217—22no—
148(C2×D3)⋊C4—24no—
149C4⋊Dic3—22no—
150C4⋊Dic3—24no—
151C4⋊Dic3C22×C414yes—
152Dic3⋊C4—24no—
153Dic3⋊C4—24no—
154Dic3⋊C4—24no—
155Dic3⋊C4—24no—
156C4×Dic3—22no—
157C4×Dic3—24no—
158C4×Dic3C22⋊C412yes—
159C4×Dic3C22⋊C412yes—
160C4×Dic3C22⋊C412yes—
161C4×Dic3C22×C411yescharacteristic
162C4×Dic3C22×C411yescharacteristic
163C4×Dic3C2×D412yes—
164C4×Dic3C2×D412yes—
165#208—44no—
166C2×C3⋊C8—24no—
167C22×Q8—62no—
168C23×C4—31no—
169C23×C4—32no—
170C2×M16—22no—
171C2×M16—62no—
172C4⋊Q8—34no—
173C4⋊Q8—32no—
174C4⋊Q8—62no—
175C4⋊Q8—62no—
176C4⋊Q8C4×D312yes—
177#125—34no—
178#125—32no—
179#125—64no—
180#125C22×D312yes—
181C22⋊Q8—22no—
182C22⋊Q8—64no—
183C22⋊Q8—64no—
184C22⋊Q8—62no—
185C22⋊Q8—62no—
186C22⋊Q8—62no—
187C4×Q8—22no—
188C4×Q8—64no—
189C4×Q8—62no—
190C4×Q8—64no—
191C4×Q8—62no—
192C4×Q8—62no—
193#117—32no—
194#117—34no—
195#117—62no—
196#117—62no—
197#117C4×D312yes—
198C2×C4⋊C4—34no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C42—31no—
203C2×C42—32no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—32no—
207C2×C42—32no—
208C2×C42—31no—
209C2×C42—31no—
210C2×C42—62no—
211C2×C42C22×D311yescharacteristic
212C4⋊C8—64no—
213C4⋊C8—64no—
214C22⋊C8—64no—
215C22⋊C8—64no—
216#97—34no—
217#97C22×D314yes—
218C4×C8—22no—
219C4×C8—62no—
220C23×C3C2×C22⋊C411yescharacteristic
221C22×D3—22no—
222C22×D3—42no—
223C22×D3C2×C22⋊C411yescharacteristic
224C22×D3C2×C22⋊C411yescharacteristic
225C3×Q8—44no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4C2×C22⋊C412yes—
230C2×C3×C4C2×C22⋊C411yescharacteristic
231C2×C3×C4C23×C411yescharacteristic
232C2×C3×C4C22×D412yes—
233C2×Dic3—22no—
234C2×Dic3—24no—
235C2×Dic3—42no—
236C2×Dic3C2×C22⋊C412yes—
237C2×Dic3C2×C22⋊C412yes—
238C2×Dic3C2×C22⋊C411yescharacteristic
239C2×Dic3C2×C22⋊C411yescharacteristic
240C4×D3—22no—
241C4×D3—22no—
242C4×D3—24no—
243C4×D3—24no—
244C4×D3—22no—
245C4×D3—42no—
246C4×D3—42no—
247C4×D3—48no—
248C4×D3#10114yes—
249Dic6—48no—
250Dic6—44no—
251C3×C8—44no—
252C3⋊C8—44no—
253C24—31no—
254C2×Q8—22no—
255C2×Q8—64no—
256C2×Q8—62no—
257C2×Q8—128no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—64no—
269C22×C4—64no—
270C22×C4—124no—
271C22×C4C2×C4×D311yescharacteristic
272C22×C4D3×D412yes—
273M16—44no—
274M16—124no—
275C2×C8—24no—
276C2×C8—64no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—34no—
280C4⋊C4—64no—
281C4⋊C4—64no—
282C4⋊C4—64no—
283C4⋊C4—64no—
284C4⋊C4—62no—
285C4⋊C4—64no—
286C4⋊C4C2×C4×D314yes—
287C22⋊C4—22no—
288C22⋊C4—64no—
289C22⋊C4—62no—
290C42—22no—
291C42—32no—
292C42—32no—
293C42—32no—
294C42—31no—
295C42—31no—
296C42—32no—
297C42—32no—
298C42—64no—
299C42—62no—
300C42C2×C4×D311yescharacteristic
301C42D3×D412yes—
302C42C23×D311yescharacteristic
303C22×C3—21no—
304C22×C3—22no—
305C22×C3C22×C22⋊C411yescharacteristic
306C2×D3—22no—
307C2×D3—21no—
308C2×D3—22no—
309C2×D3—44no—
310C2×D3C2×#10112yes—
311C3×C4—21no—
312C3×C4—22no—
313C3×C4—44no—
314C3×C4C2×#10112yes—
315C3×C4#42114yes—
316Dic3—21no—
317Dic3—22no—
318Dic3—22no—
319Dic3—44no—
320Dic3C2×#10112yes—
321C23—31no—
322C23—31no—
323C23—62no—
324C23—122no—
325C23D3×C22⋊C411yescharacteristic
326Q8—44no—
327Q8—128no—
328Q8—124no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—31no—
333C2×C4—34no—
334C2×C4—31no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—64no—
342C2×C4—64no—
343C2×C4—64no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—128no—
347C2×C4—122no—
348C2×C4D3×C22⋊C412yes—
349C2×C4D3×C22⋊C411yescharacteristic
350C2×C4C22×C4×D311yescharacteristic
351C2×C4C2×D3×D412yes—
352C8—44no—
353C8—124no—
354C2×C3—41no—
355C2×C3C22×#10111yescharacteristic
356C2×C3C2×#42112yes—
357D3—22no—
358D3—41no—
359C22—21no—
360C22—22no—
361C22—32no—
362C22—62no—
363C22—61no—
364C22—62no—
365C22—124no—
366C22C2×D3×C22⋊C411yescharacteristic
367C4—21no—
368C4—22no—
369C4—32no—
370C4—44no—
371C4—61no—
372C4—62no—
373C4—62no—
374C4—124no—
375C4D3×#10112yes—
376C4#482114yes—
377C3#1766411yescharacteristic, complemented, Sylow
378C2—41no—
379C2—62no—
380C2—121no—
381C2C2×D3×#10111yescharacteristic
382C2C2×#482112yes—
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1526C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1526D3
hypocenterC3#17664
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
3. C3×#1526C2semidirect
4. #71821C2semidirect
5. #69862C2semidirect
6. #68369C2semidirect
7. #68367C2semidirect
30. C3×#330C22semidirect
31. #4543C22semidirect
32. #4465C22semidirect
39. #1526D3semidirect
116. #330C2×D3semidirect
377. C3#17664semidirect

Maximal subgroups

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

subgroupindex
D3×#5332
#785172
C3×#15262
#718212
#698622
#683692
#683672
#176643