Finite Group Databasebeta

← #10763252

Subgroups — #10763252

Order 768.

1306 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#124C211yesmaximal, characteristic
2#79331C212yesmaximal, complemented
3C3×#1533C211yesmaximal, characteristic, complemented
4#71719C211yesmaximal, characteristic, complemented
5#70808C212yesmaximal, complemented
6#68343C218yesmaximal, complemented
7#17666—31nomaximal, Sylow
8C22×D3×Q8C411yescharacteristic
9C22×D3×D4C411yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#947C411yescharacteristic
12C2×#936C411yescharacteristic
13D3×#124—22no—
14D3×#124—22no—
15D3×#124C414yes—
16D3×#124C2214yes—
17C2×#885—22no—
18C2×D3×C22⋊C4—22no—
19C2×#875C2211yescharacteristic
20C2×C42×D3C2211yescharacteristic
21C3×#432C2212yescomplemented
22#4926C2212yescomplemented
23#4837—22no—
24#4833—22no—
25#4726—22no—
26#4718—22no—
27#4707C2214yescomplemented
28#4705C2214yescomplemented
29C3×#329C2214yescomplemented
30#4540C2214yescomplemented
31#4464C2218yescomplemented
32C22×#124—31no—
33#2920—32no—
34#1533—32no—
35#1533—31no—
36#1533D311yescharacteristic, complemented
37#1527—38no—
38C24×D3—21no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×#67—21no—
42C2×D3×Q8—22no—
43C2×D3×Q8—44no—
44C2×D3×Q8C2×C414yes—
45C2×D3×D4—22no—
46C2×D3×D4—44no—
47C2×D3×D4C2×C414yes—
48C22×D12C2×C411yescharacteristic
49C22×C4×D3—21no—
50C22×C4×D3C2×C411yescharacteristic
51C22×C4×D3D412yes—
52C22×Dic6—21no—
53C22×Dic6C2×C411yescharacteristic
54C3×#124—22no—
55C3×#124C2×C412yes—
56C3×#124C2312yes—
57C2×C3×C22⋊C4—22no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60#947—21no—
61#947—21no—
62#947C2×C414yes—
63#936—21no—
64#936—21no—
65#936C2×C414yes—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4×Dic3C2×C411yescharacteristic
69C2×C4×Dic3D412yes—
70C2×ΓL(1,𝔽25)—24no—
71#885—24no—
72#885—24no—
73#885—24no—
74#885—24no—
75D3×C22⋊C4—24no—
76D3×C22⋊C4—24no—
77D3×C22⋊C4—24no—
78D3×C22⋊C4—24no—
79#875—22no—
80#875C2×C412yes—
81#875C2312yes—
82C42×D3C2×C412yes—
83C42×D3D412yes—
84C42×D3D412yes—
85C42×D3C2312yes—
86C3×C4⋊C8—22no—
87C3×C22⋊C8—22no—
88C3×#97C2314yescomplemented
89#830—22no—
90(C2×D3)⋊C8—24no—
91Dic3⋊C8—24no—
92C8×Dic3—24no—
93C4⋊(C3⋊C8)—22no—
94#803C2314yescomplemented
95C23×Q8—31no—
96C23×D4—31no—
97C2×#124—31no—
98C2×#124—31no—
99C2×#124—34no—
100C2×#124—34no—
101C2×#124—31no—
102C2×#124—62no—
103C2×#124—62no—
104C2×#124—62no—
105C2×#124C2×D311yescharacteristic
106C22×C22⋊C4—62no—
107C22×C42—31no—
108#432—32no—
109#432—34no—
110#432C2×D312yescomplemented
111#431—62no—
112#431—62no—
113#423—62no—
114#407—62no—
115C4×M16—34no—
116#329—34no—
117#329—38no—
118#329C2×D314yescomplemented
119C24×C3—21no—
120C23×D3—21no—
121C23×D3—24no—
122C23×D3—44no—
123C23×D3—48no—
124C23×D3C22⋊C411yescharacteristic
125C2×C3×Q8—21no—
126C2×C3×Q8—42no—
127C2×C3×Q8C22×C412yes—
128C2×C3×D4—21no—
129C2×C3×D4—42no—
130C2×C3×D4C22×C412yes—
131C22×C3×C4—21no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D412yes—
134C2×#67—24no—
135C2×#67—44no—
136C22×Dic3—21no—
137C22×Dic3C22⋊C412yes—
138C22×Dic3C22⋊C411yescharacteristic
139D3×Q8—48no—
140D3×Q8—48no—
141D3×D4—48no—
142D3×D4—48no—
143C2×D12—21no—
144C2×D12—42no—
145C2×D12C22×C412yes—
146C2×C4×D3—24no—
147C2×C4×D3—24no—
148C2×C4×D3—44no—
149C2×C4×D3—44no—
150C2×C4×D3C22⋊C412yes—
151C2×C4×D3C22⋊C412yes—
152C2×C4×D3C22⋊C412yes—
153C2×C4×D3C22×C412yes—
154C2×C4×D3C2×D412yes—
155C2×C4×D3C2×D412yes—
156C2×Dic6—21no—
157C2×Dic6—24no—
158C2×Dic6—44no—
159C2×Dic6—42no—
160C2×Dic6C22×C412yes—
161C2×C3×C8—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—24no—
164C3×C42C22×C411yescharacteristic
165C3×C42C2×D412yes—
166C3×C42C2411yescharacteristic
167#217—24no—
168#217—24no—
169(C2×D3)⋊C4—24no—
170(C2×D3)⋊C4—24no—
171(C2×D3)⋊C4—24no—
172(C2×D3)⋊C4—24no—
173C4×Dic3C22⋊C412yes—
174C4×Dic3C22⋊C412yes—
175C4×Dic3C22⋊C411yescharacteristic
176C4×Dic3C22⋊C411yescharacteristic
177C4×Dic3C22×C412yes—
178C4×Dic3C2×D412yes—
179C4×Dic3C2×D412yes—
180C2×C3⋊C8—24no—
181ΓL(1,𝔽25)—48no—
182C25—61no—
183C22×Q8—34no—
184C22×Q8—31no—
185C22×Q8—61no—
186C22×Q8—62no—
187C22×Q8—124no—
188C22×Q8C4×D311yescharacteristic
189C22×D4—34no—
190C22×D4—31no—
191C22×D4—62no—
192C22×D4—61no—
193C22×D4—124no—
194C22×D4C4×D311yescharacteristic
195C23×C4—31no—
196C23×C4—32no—
197C23×C4—61no—
198C2×M16—64no—
199#124—22no—
200#124—34no—
201#124—34no—
202#124—32no—
203#124—32no—
204#124—64no—
205#124—64no—
206#124—64no—
207#124—64no—
208#124—61no—
209#124—61no—
210#124—61no—
211#124—61no—
212#124—62no—
213#124C4×D312yes—
214#124C22×D312yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4—62no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4—62no—
223C2×C42—31no—
224C2×C42—32no—
225C2×C42—32no—
226C2×C42—32no—
227C2×C42—32no—
228C2×C42—32no—
229C2×C42C22×D311yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—64no—
232C4⋊C8—62no—
233C22⋊C8—22no—
234C22⋊C8—64no—
235C22⋊C8—62no—
236#97—34no—
237#97C22×D314yescomplemented
238C4×C8—64no—
239C23×C3—22no—
240C23×C3—44no—
241C23×C3—42no—
242C23×C3C2×C22⋊C411yescharacteristic
243C22×D3—22no—
244C22×D3—24no—
245C22×D3—22no—
246C22×D3—48no—
247C22×D3—48no—
248C22×D3—44no—
249C22×D3—42no—
250C22×D3—48no—
251C22×D3—816no—
252C22×D3C2×C22⋊C412yes—
253C3×Q8—42no—
254C3×Q8—42no—
255C3×D4—42no—
256C3×D4—42no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4C2×C22⋊C412yes—
262C2×C3×C4C2×C22⋊C411yescharacteristic
263C2×C3×C4C23×C411yescharacteristic
264C2×C3×C4C22×D412yes—
265#67—48no—
266C2×Dic3—22no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3#10014yes—
270C2×Dic3C2×C22⋊C412yes—
271C2×Dic3C2×C22⋊C412yes—
272C2×Dic3C2×C22⋊C412yes—
273D12—42no—
274D12—42no—
275C4×D3—24no—
276C4×D3—24no—
277C4×D3—24no—
278C4×D3—24no—
279C4×D3—42no—
280C4×D3—42no—
281C4×D3—48no—
282Dic6—42no—
283Dic6—48no—
284Dic6—42no—
285C3×C8—44no—
286C3⋊C8—44no—
287C24—21no—
288C24—31no—
289C24—64no—
290C24—61no—
291C24—124no—
292C24—128no—
293C2×Q8—21no—
294C2×Q8—32no—
295C2×Q8—42no—
296C2×Q8—64no—
297C2×Q8—61no—
298C2×Q8—122no—
299C2×Q8—124no—
300C2×Q8—128no—
301C2×Q8—128no—
302C2×Q8C2×C4×D312yes—
303C2×D4—21no—
304C2×D4—32no—
305C2×D4—42no—
306C2×D4—64no—
307C2×D4—61no—
308C2×D4—122no—
309C2×D4—124no—
310C2×D4—128no—
311C2×D4—128no—
312C2×D4C2×C4×D312yes—
313C22×C4—21no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—64no—
323C22×C4—64no—
324C22×C4—61no—
325C22×C4—124no—
326C22×C4—124no—
327C22×C4C2×C4×D311yescharacteristic
328C22×C4D3×D412yes—
329M16—128no—
330C2×C8—24no—
331C2×C8—64no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—64no—
335C22⋊C4—64no—
336C22⋊C4—64no—
337C22⋊C4—64no—
338C22⋊C4—64no—
339C22⋊C4—64no—
340C42—32no—
341C42—32no—
342C42—31no—
343C42—31no—
344C42—32no—
345C42—32no—
346C42—32no—
347C42C2×C4×D311yescharacteristic
348C42D3×D412yes—
349C42C23×D311yescharacteristic
350C22×C3—22no—
351C22×C3—42no—
352C22×C3—44no—
353C22×C3—84no—
354C22×C3C2×#10012yes—
355C22×C3C22×C22⋊C411yescharacteristic
356C2×D3—28no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—42no—
360C2×D3—44no—
361C2×D3—816no—
362C2×D3—84no—
363C2×D3#42114yes—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—42no—
369Dic3—22no—
370Dic3—22no—
371Dic3—42no—
372Dic3C2×#10014yes—
373C23—22no—
374C23—32no—
375C23—44no—
376C23—42no—
377C23—64no—
378C23—62no—
379C23—62no—
380C23—124no—
381C23—128no—
382C23—128no—
383C23—128no—
384C23—122no—
385C23—2416no—
386C23D3×C22⋊C411yescharacteristic
387Q8—42no—
388Q8—42no—
389Q8—122no—
390Q8—122no—
391Q8—128no—
392D4—42no—
393D4—42no—
394D4—122no—
395D4—122no—
396D4—128no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—32no—
400C2×C4—34no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—64no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—64no—
409C2×C4—62no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—128no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4D3×C22⋊C412yes—
416C2×C4D3×C22⋊C411yescharacteristic
417C2×C4C22×C4×D311yescharacteristic
418C2×C4C2×D3×D412yes—
419C8—44no—
420C8—124no—
421C2×C3—22no—
422C2×C3—82no—
423C2×C3C22×#10011yescharacteristic
424C2×C3C2×#42112yes—
425D3—24no—
426D3—82no—
427C22—22no—
428C22—34no—
429C22—42no—
430C22—44no—
431C22—68no—
432C22—62no—
433C22—62no—
434C22—84no—
435C22—122no—
436C22—124no—
437C22—244no—
438C22—2416no—
439C22D3×#10012yes—
440C22C2×D3×C22⋊C411yescharacteristic
441C4—21no—
442C4—21no—
443C4—22no—
444C4—22no—
445C4—34no—
446C4—42no—
447C4—62no—
448C4—62no—
449C4—122no—
450C3#1766611yescharacteristic, complemented, Sylow
451C2—22no—
452C2—64no—
453C2—82no—
454C2—242no—
455C2C2×D3×#10011yescharacteristic
456C2D3×#42112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1533C2
socleC22×C3C22×C22⋊C4
prime radicalC22×D3×D4C4
cosocleC3×C42C24
hypercenter#1533D3
hypocenterC3#17666
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C3×#1533C2semidirect
4. #71719C2semidirect
5. #70808C2semidirect
6. #68343C2semidirect
21. C3×#432C22semidirect
22. #4926C22semidirect
27. #4707C22semidirect
28. #4705C22semidirect
29. C3×#329C22semidirect
30. #4540C22semidirect
31. #4464C22semidirect
36. #1533D3semidirect
88. C3×#97C23semidirect
94. #803C23semidirect
110. #432C2×D3semidirect
118. #329C2×D3semidirect
237. #97C22×D3semidirect
450. C3#17666semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1242
#793312
C3×#15332
#717192
#708082
#683432
#176663