Finite Group Databasebeta

← #10763293

Subgroups — #10763293

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#86219C211yesmaximal, characteristic
2#79331C212yesmaximal, complemented
3C3×#1536C211yesmaximal, characteristic, complemented
4#71799C211yesmaximal, characteristic, complemented
5#70815C212yesmaximal, complemented
6#68344C218yesmaximal, complemented
7#17673—31nomaximal, Sylow
8C3×#532C2211yescharacteristic
9#5828C412yes—
10Dic3×#41C412yes—
11#5728C412yes—
12#5716C412yes—
13#5680C2214yes—
14#5606—22no—
15#5600—22no—
16#5557—22no—
17C4×#237—22no—
18C2×C42×D3C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#432C2212yes—
21#4926C2212yes—
22#4837—22no—
23#4833—22no—
24#4726—22no—
25#4718—22no—
26#4707C2214yes—
27#4705C2214yes—
28C3×#329C2214yescomplemented
29#4540C2214yescomplemented
30#4465C2218yescomplemented
31#3444—31no—
32#2920—32no—
33#1537—32no—
34#1536—31no—
35#1536D311yescharacteristic, complemented
36#1529—38no—
37C2×C3×#41C2×C412yes—
38C2×#237—22no—
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D412yes—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#127C2×C411yescharacteristic
43C3×#124C2312yes—
44C3×#121—22no—
45C3×C4×D4—22no—
46C2×C3×C42C2311yescharacteristic
47#946C2×C414yes—
48Q8×Dic3C2×C414yes—
49#938—22no—
50#938C2×C414yes—
51#937—22no—
52#936—22no—
53#935—22no—
54D4×Dic3—24no—
55D4×Dic3—22no—
56D4×Dic3C2×C414yes—
57C4×#67—24no—
58#926—22no—
59#926C2×C412yes—
60#924—22no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3C2×C411yescharacteristic
63C2×C4×Dic3D412yes—
64C2×ΓL(1,𝔽25)—24no—
65#897—22no—
66#892—22no—
67#890—22no—
68#887—22no—
69#883—24no—
70#67⋊C4—24no—
71#879—24no—
72#877—24no—
73C42×D3C2×C412yes—
74C42×D3D412yes—
75C42×D3D412yes—
76C42×D3C2312yes—
77#870C2312yes—
78#869C2×C412yes—
79C4×Dic6—22no—
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#533—31no—
91#532—32no—
92#532C2×D311yescharacteristic
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—38no—
110#329—34no—
111#329C2×D314yescomplemented
112C23×D3C22⋊C411yescharacteristic
113C3×#41—44no—
114C2×C3×Q8C22×C412yes—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4C22×C412yes—
118C22×C3×C4—22no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D412yes—
121C2×#67—24no—
122C22×Dic3—24no—
123C22×Dic3C22⋊C412yes—
124C22×Dic3C22⋊C411yescharacteristic
125#237—48no—
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×Dic6—22no—
135C2×C3×C8—24no—
136C3×C4⋊C4—22no—
137C3×C22⋊C4—24no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D412yes—
140C3×C42C2411yescharacteristic
141#217—22no—
142#217—24no—
143#217—24no—
144(C2×D3)⋊C4—24no—
145C4⋊Dic3—22no—
146C4⋊Dic3—24no—
147C4⋊Dic3C22×C414yes—
148Dic3⋊C4—24no—
149Dic3⋊C4—24no—
150C4×Dic3—22no—
151C4×Dic3—24no—
152C4×Dic3C22⋊C412yes—
153C4×Dic3C22⋊C412yes—
154C4×Dic3C22⋊C411yescharacteristic
155C4×Dic3C22⋊C411yescharacteristic
156C4×Dic3C22×C412yes—
157C4×Dic3C2×D412yes—
158C4×Dic3C2×D412yes—
159C2×C3⋊C8—24no—
160ΓL(1,𝔽25)—48no—
161C2×#41—62no—
162C2×#41C4×D312yes—
163C23×C4—31no—
164C23×C4—32no—
165C2×M16—64no—
166C4⋊Q8—32no—
167C4⋊Q8C4×D311yescharacteristic
168#127C4×D311yescharacteristic
169#126—64no—
170#125—32no—
171#125—62no—
172#124—62no—
173#124C22×D312yes—
174#123—64no—
175#123—62no—
176C22⋊Q8—34no—
177C22⋊Q8—62no—
178C22⋊Q8—62no—
179#121—22no—
180#121—34no—
181#121—62no—
182#120—62no—
183C4×Q8—34no—
184C4×Q8—62no—
185C4×Q8—62no—
186C4×D4—22no—
187C4×D4—34no—
188C4×D4—62no—
189C4×D4—64no—
190C4×D4—64no—
191C4×D4—64no—
192#117—32no—
193#117—64no—
194#117—62no—
195#117—62no—
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×C3—22no—
214C23×C3C2×C22⋊C411yescharacteristic
215C22×D3—41no—
216C22×D3—42no—
217C22×D3C2×C22⋊C412yes—
218C3×Q8—42no—
219C3×D4—42no—
220C3×D4—44no—
221C3×D4—44no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—42no—
225C2×C3×C4—44no—
226C2×C3×C4C2×C22⋊C412yes—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C23×C411yescharacteristic
229C2×C3×C4C22×D412yes—
230#67—48no—
231C2×Dic3—24no—
232C2×Dic3—24no—
233C2×Dic3—41no—
234C2×Dic3—48no—
235C2×Dic3C2×C22⋊C412yes—
236C2×Dic3C2×C22⋊C412yes—
237C2×Dic3C2×C22⋊C412yes—
238C4×D3—24no—
239C4×D3—24no—
240C4×D3—24no—
241C4×D3—42no—
242C4×D3—42no—
243C4×D3—42no—
244Dic6—44no—
245C3×C8—44no—
246C3⋊C8—44no—
247C24—31no—
248#41—44no—
249#41—128no—
250C2×Q8—62no—
251C2×Q8C2×C4×D312yes—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—64no—
255C2×D4C2×C4×D312yes—
256C22×C4—22no—
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—122no—
267C22×C4—124no—
268C22×C4C2×C4×D311yescharacteristic
269C22×C4D3×D412yes—
270M16—128no—
271C2×C8—24no—
272C2×C8—64no—
273C4⋊C4—22no—
274C4⋊C4—34no—
275C4⋊C4—64no—
276C4⋊C4—64no—
277C4⋊C4—62no—
278C4⋊C4—64no—
279C22⋊C4—24no—
280C22⋊C4—64no—
281C22⋊C4—64no—
282C22⋊C4—64no—
283C22⋊C4—62no—
284C42—31no—
285C42—31no—
286C42—32no—
287C42—32no—
288C42—32no—
289C42—32no—
290C42—32no—
291C42—64no—
292C42—62no—
293C42C2×C4×D311yescharacteristic
294C42D3×D412yes—
295C42C23×D311yescharacteristic
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—42no—
299C22×C3—44no—
300C22×C3C22×C22⋊C411yescharacteristic
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—22no—
304C2×D3—44no—
305C3×C4—21no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4#41312yes—
310Dic3—22no—
311Dic3—22no—
312Dic3—44no—
313Dic3#42114yes—
314C23—22no—
315C23—32no—
316C23—121no—
317C23—122no—
318C23D3×C22⋊C411yescharacteristic
319Q8—42no—
320Q8—124no—
321D4—42no—
322D4—44no—
323D4—44no—
324D4—128no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—64no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4—64no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—128no—
341C2×C4—121no—
342C2×C4D3×C22⋊C412yes—
343C2×C4D3×C22⋊C411yescharacteristic
344C2×C4C22×C4×D311yescharacteristic
345C2×C4C2×D3×D412yes—
346C8—44no—
347C8—124no—
348C2×C3—41no—
349C2×C3—82no—
350C2×C3C2×#41311yescharacteristic
351C2×C3C2×#42112yes—
352D3—42no—
353C22—21no—
354C22—22no—
355C22—42no—
356C22—44no—
357C22—62no—
358C22—62no—
359C22—62no—
360C22—124no—
361C22C2×D3×C22⋊C411yescharacteristic
362C4—21no—
363C4—22no—
364C4—22no—
365C4—34no—
366C4—42no—
367C4—62no—
368C4—62no—
369C4—124no—
370C4#474512yes—
371C3#1767311yescharacteristic, complemented, Sylow
372C2—41no—
373C2—82no—
374C2—122no—
375C2C2×#474511yescharacteristic
376C2D3×#42112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1536C2
socleC22×C3C22×C22⋊C4
prime radical#86219C2
cosocleC3×C42C24
hypercenter#1536D3
hypocenterC3#17673
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C3×#1536C2semidirect
4. #71799C2semidirect
5. #70815C2semidirect
6. #68344C2semidirect
28. C3×#329C22semidirect
29. #4540C22semidirect
30. #4465C22semidirect
35. #1536D3semidirect
111. #329C2×D3semidirect
371. C3#17673semidirect

Maximal subgroups

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

subgroupindex
#862192
#793312
C3×#15362
#717992
#708152
#683442
#176733