Finite Group Databasebeta

← #10816030

Subgroups — #10816030

Order 768.

1008 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#454C211yesmaximal, characteristic
2#84111C212yesmaximal, complemented
3#83930C212yesmaximal, complemented
4#72091C212yesmaximal, complemented
5#71725C211yesmaximal, characteristic, complemented
6#71723C211yesmaximal, characteristic, complemented
7#70369C212yesmaximal, complemented
8#70364C212yesmaximal, complemented
9#70106C212yesmaximal, complemented
10#22895—31nomaximal, Sylow
11C2×C3×2−1+4C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C2×C3×#124C2211yescharacteristic
14#5827C2212yes—
15C2×#948C2212yes—
16#5798C2211yescharacteristic
17C3×#454C418yescomplemented
18C2×C3×#104C2212yes—
19C2×C3×#101C2211yescharacteristic
20#5233—22no—
21#5233C2212yes—
22#5232—22no—
23#5232C2212yes—
24#5225—22no—
25#5225C2212yes—
26C2×#835—22no—
27C2×#835C2212yes—
28C2×#833C2211yescharacteristic
29C2×#832C2212yes—
30#5164—22no—
31Dic3×SD16—22no—
32#5152—22no—
33Dic3×D8—22no—
34#5122C2212yes—
35#5121C2212yes—
36#4936C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4558C2212yes—
39#4554C2212yes—
40#4523C2212yes—
41C2×#454D311yescharacteristic
42#2946—32no—
43#2898—32no—
44#1958—31no—
45#1930—32no—
46#1907—32no—
47#1885—31no—
48#1860—32no—
49#1764—32no—
50C3×2−1+4—24no—
51C3×2−1+4—41no—
52C3×2−1+4C2×C414yescomplemented
53C2×C3×#41—22no—
54C2×C3×#41—41no—
55C2×C3×#41C2312yes—
56C22×C3×Q8—22no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C3×AGL(1,ℤ/8)—24no—
60C3×AGL(1,ℤ/8)C2×C418yescomplemented
61C2×C3×SD16—22no—
62C2×C3×D8—22no—
63C2×C3×M16C2312yes—
64C3×#124—41no—
65C3×#124C2×C414yescomplemented
66C2×C3×C22⋊C4—41no—
67C2×C3×C42C2311yescharacteristic
68C2×#217—41no—
69#948—24no—
70#948—42no—
71Q8×Dic3—22no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74D4×Dic3—22no—
75D4×Dic3—42no—
76#926—22no—
77#926C2312yes—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3C2312yes—
80C2×#208—22no—
81C2×#208C2312yes—
82C22×C3⋊C8—22no—
83C22×C3⋊C8C2312yes—
84C3×#104C2×C418yescomplemented
85C3×#101C2×C414yescomplemented
86#835—22no—
87#835—22no—
88#835—22no—
89#835—22no—
90#835—22no—
91#835D412yes—
92#835D412yes—
93#833—41no—
94#832—22no—
95#832—22no—
96#832—22no—
97#832D412yes—
98#832D412yes—
99C3⋊#95—41no—
100#818—22no—
101#817—22no—
102#815—22no—
103C8×Dic3—22no—
104C4×C3⋊C8—41no—
105C2×2−1+4C2×D311yescharacteristic
106C2×AGL(1,ℤ/8)C2×D312yes—
107C2×#131—32no—
108C2×#124C2×D311yescharacteristic
109#520—32no—
110#518—31no—
111#454Dic318yescomplemented
112#442—62no—
113#440—62no—
114C4×SD16—62no—
115C4×D8—62no—
116#428—32no—
117C2×#104C2×D312yes—
118#419—32no—
119#419—62no—
120#417—32no—
121#417—62no—
122#416—32no—
123#416—62no—
124C2×#103—32no—
125C2×#103—62no—
126C2×#102—32no—
127C2×#101C2×D311yescharacteristic
128C2×#100—31no—
129#405—32no—
130C2×C4×C8—31no—
131#382—31no—
132#343—32no—
133#341—32no—
134#339—32no—
135C24×C3—21no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—28no—
139C3×#41—44no—
140C3×#41—44no—
141C3×#41—44no—
142C3×#41C22×C414yes—
143C3×#41#4114yes—
144C2×C3×Q8—21no—
145C2×C3×Q8—28no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—41no—
149C2×C3×Q8—44no—
150C2×C3×Q8C22×C412yes—
151C2×C3×Q8C2×D412yes—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—21no—
156C2×C3×D4—41no—
157C2×C3×D4C22×C412yes—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—42no—
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—22no—
165C22×Dic3—42no—
166C22×Dic3—42no—
167C3×SD16—28no—
168C3×D8—28no—
169C3×M16—22no—
170C3×M16C22×C414yes—
171C2×C3×C8—22no—
172C3×C22⋊C4—44no—
173C3×C42—21no—
174C3×C42C22×C412yes—
175#217—22no—
176#217—41no—
177#217—41no—
178#217—42no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3—42no—
182C4⋊Dic3C2×D412yes—
183C4⋊Dic3C2×D412yes—
184C4×Dic3—22no—
185C4×Dic3—42no—
186#208—22no—
187#208—22no—
188#208—44no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—44no—
193C2×C3⋊C8C2×D412yes—
194C2×C3⋊C8C2×D412yes—
1952−1+4—24no—
1962−1+4—41no—
1972−1+4C2×Dic314yescomplemented
198C2×#41—22no—
199C2×#41—41no—
200C2×#41C22×D312yes—
201C22×Q8—22no—
202C22×Q8C22×D311yescharacteristic
203C22×D4C22×D311yescharacteristic
204AGL(1,ℤ/8)—24no—
205AGL(1,ℤ/8)C2×Dic318yescomplemented
206C2×SD16—22no—
207C2×D8—22no—
208#131—64no—
209#131—122no—
210C2×M16—32no—
211C2×M16—62no—
212C2×M16C22×D312yes—
213C22×C8—32no—
214C22×C8—62no—
215#124—41no—
216#124C2×Dic314yescomplemented
217C4×Q8—62no—
218C4×D4—62no—
219C4×D4—62no—
220C4×D4—62no—
221C4×D4—122no—
222#117—32no—
223#117—62no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—62no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—121no—
228C2×C42C22×D311yescharacteristic
229#107—62no—
230#106—62no—
231#104C2×Dic318yescomplemented
232#103—32no—
233#103—32no—
234#103—62no—
235#103—62no—
236#103—62no—
237#103—62no—
238#103—62no—
239#102—32no—
240#102—32no—
241#102—62no—
242#102—62no—
243#102—62no—
244#101C2×Dic314yescomplemented
245#100—121no—
246#97—62no—
247C4×C8—62no—
248C4×C8—121no—
249#95—121no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—82no—
255C23×C3H(5,𝔽2)11yescharacteristic
256C3×Q8—22no—
257C3×Q8—24no—
258C3×Q8—44no—
259C3×Q8—42no—
260C3×Q8C4×D414yes—
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—24no—
264C3×D4—22no—
265C3×D4—44no—
266C3×D4—42no—
267C3×D4C4×D414yes—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—48no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4C23×C411yescharacteristic
278C2×C3×C4C22×D412yes—
279C2×C3×C4C2×#4112yes—
280C2×C3×C4H(5,𝔽2)11yescharacteristic
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—42no—
284C2×Dic3—82no—
285C2×Dic3—162no—
286C2×Dic3—162no—
287C3×C8—24no—
288C3⋊C8—22no—
289C3⋊C8—22no—
290C3⋊C8—42no—
291C24—21no—
292#41—22no—
293#41—22no—
294#41—28no—
295#41—44no—
296#41—44no—
297#41—44no—
298#41#23714yes—
299#41C22×Dic314yes—
300C2×Q8—21no—
301C2×Q8—28no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—41no—
305C2×Q8—44no—
306C2×Q8D3×D412yes—
307C2×Q8C22×Dic312yes—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4D3×D412yes—
314C2×D4C22×Dic312yes—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—42no—
319C22×C4—62no—
320C22×C4—122no—
321C22×C4—122no—
322C22×C4C23×D311yescharacteristic
323SD16—28no—
324D8—28no—
325M16—22no—
326M16—62no—
327M16—62no—
328M16—124no—
329M16C22×Dic314yes—
330C2×C8—22no—
331C2×C8—32no—
332C2×C8—32no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—124no—
337C4⋊C4—32no—
338C4⋊C4—32no—
339C4⋊C4—62no—
340C4⋊C4—62no—
341C4⋊C4—122no—
342C22⋊C4—44no—
343C22⋊C4—62no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—122no—
347C42—21no—
348C42—62no—
349C42—122no—
350C42C22×Dic312yes—
351C22×C3—22no—
352C22×C3—24no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—161no—
359C22×C3#51811yescharacteristic
360C22×C3D4211yescharacteristic
361C22×C3#55111yescharacteristic
362C3×C4—24no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4C2×C4×D412yes—
366C3×C4#54612yes—
367Dic3—42no—
368Dic3—42no—
369Dic3—82no—
370C23—22no—
371C23—22no—
372C23—41no—
373C23—41no—
374C23—82no—
375C23#100411yescharacteristic
376Q8—22no—
377Q8—24no—
378Q8—44no—
379Q8—42no—
380Q8D4×Dic314yes—
381D4—24no—
382D4—24no—
383D4—24no—
384D4—22no—
385D4—44no—
386D4—42no—
387D4D4×Dic314yes—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—122no—
400C2×C4—242no—
401C2×C4—482no—
402C2×C4—482no—
403C2×C4C2×D3×D412yes—
404C2×C4C2×#23712yes—
405C2×C4#100411yescharacteristic
406C2×C4C23×Dic311yescharacteristic
407C8—24no—
408C8—62no—
409C8—62no—
410C8—122no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—44no—
414C2×C3—82no—
415C2×C3#235811yescharacteristic
416C2×C3#329012yes—
417C22—22no—
418C22—24no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—82no—
423C22—82no—
424C22—161no—
425C22#558711yescharacteristic
426C22#559011yescharacteristic
427C22#579811yescharacteristic
428C4—24no—
429C4—42no—
430C4—42no—
431C4—122no—
432C4—122no—
433C4—242no—
434C4#559112yes—
435C4C2×D4×Dic312yes—
436C3#2289511yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—44no—
440C2—82no—
441C2#8028912yes—
442C2#8257911yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#454C2
socleC22×C3D42
cosocleC22×C3×C4C24
hypercenterC2×#454D3
hypocenterC3#22895
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#454

Complements

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

NH ≅ G/Ntype
2. #84111C2semidirect
3. #83930C2semidirect
4. #72091C2semidirect
5. #71725C2semidirect
6. #71723C2semidirect
7. #70369C2semidirect
8. #70364C2semidirect
9. #70106C2semidirect
17. C3×#454C4semidirect
52. C3×2−1+4C2×C4semidirect
60. C3×AGL(1,ℤ/8)C2×C4semidirect
65. C3×#124C2×C4semidirect
84. C3×#104C2×C4semidirect
85. C3×#101C2×C4semidirect
111. #454Dic3semidirect
197. 2−1+4C2×Dic3semidirect
205. AGL(1,ℤ/8)C2×Dic3semidirect
216. #124C2×Dic3semidirect
231. #104C2×Dic3semidirect
244. #101C2×Dic3semidirect
436. C3#22895semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4542
#841112
#839302
#720912
#717252
#717232
#703692
#703642
#701062
#228953