Finite Group Databasebeta

← #10765332

Subgroups — #10765332

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83351C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic, complemented
3#79286C211yesmaximal, characteristic
4C3×#1485C211yesmaximal, characteristic, complemented
5#70747C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic
7#69848C211yesmaximal, characteristic, complemented
8#68220C212yesmaximal, complemented
9#68216C212yesmaximal, complemented
10#68214C212yesmaximal, complemented
11#68212C212yesmaximal, complemented
12#17924—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×C4×M16C2211yescharacteristic
17#5084C412yes—
18#5083C412yes—
19#5046C412yes—
20#5043C412yes—
21C4×#215C2212yes—
22C4×#215C2212yes—
23#5024—22no—
24#5024C2212yescomplemented
25Q8×C3⋊C8C2212yescomplemented
26#5000—22no—
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4834C2212yes—
31#4834C2212yes—
32D3×C22⋊C8—22no—
33C8×D12—22no—
34C3×Q8⋊C8C2212yescomplemented
35C3×Q8⋊C8C2212yescomplemented
36#4480C2212yescomplemented
37#4479C2212yescomplemented
38#4459C2212yes—
39#4459C2212yes—
40#4455C2211yescharacteristic, complemented
41#4454C2211yescharacteristic, complemented
42#4454C2211yescharacteristic, complemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3101—31no—
45#2985—31no—
46#2972—31no—
47#1585—32no—
48#1582—32no—
49SD16⋊C8—32no—
50C8×SD16—32no—
51#1572—31no—
52#1490—31no—
53#1485D311yescharacteristic, complemented
54#1453—31no—
55C22×D12C2×C411yescharacteristic
56C22×C4×D3—21no—
57C2×C3×M16—21no—
58C3×C4⋊Q8C2×C411yescharacteristic
59C3×C4⋊Q8C2×C411yescharacteristic
60C3×#125—21no—
61C3×C22⋊Q8—21no—
62C3×C22⋊Q8C2×C412yes—
63C3×C4×Q8—21no—
64C3×C4×Q8C2312yes—
65C3×#117—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#215C2×C412yes—
68C2×#215C2×C412yes—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3C2×C411yescharacteristic
71C2×D3×C8—22no—
72C2×D3×C8—22no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12C2×C412yes—
76C4×D12C2312yes—
77C3×C4⋊C8C2312yes—
78C3×C22⋊C8—21no—
79C3×#97—21no—
80C3×C4×C8—21no—
81C3×C4×C8C2311yescharacteristic
82C3×C4×C8C2311yescharacteristic
83#835C2×C412yes—
84#835C2×C412yes—
85#830—21no—
86#830C2×C412yes—
87#830D412yes—
88(C2×D3)⋊C8—22no—
89(C2×D3)⋊C8—22no—
90#809C2×C412yes—
91#809C2×C412yes—
92#808C2×C412yes—
93#808C2×C412yes—
94C4⋊(C3⋊C8)—22no—
95C4⋊(C3⋊C8)—22no—
96C4⋊(C3⋊C8)C2×C412yes—
97C4⋊(C3⋊C8)D412yes—
98C4⋊(C3⋊C8)C2312yes—
99#803—21no—
100#803—22no—
101C4×C3⋊C8—22no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#533C2×D311yescharacteristic
107C2×C4×D4—31no—
108#483—32no—
109#476—32no—
110#461—32no—
111#450—32no—
112#446—32no—
113#446—62no—
114C8×Q8—32no—
115C4×SD16—32no—
116C4×SD16—32no—
117C8×D4—32no—
118C8×D4—32no—
119C8×D4—62no—
120#433—62no—
121#432—31no—
122#431—31no—
123#431—31no—
124#431C2×D311yescharacteristic
125C2×C22⋊C8—62no—
126C4×M16C2×D311yescharacteristic
127#335—32no—
128#334—32no—
129Q8⋊C8C2×D312yescomplemented
130Q8⋊C8C2×D312yescomplemented
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C23×D3—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8C22×C412yes—
140C22×C3×C4C22×C411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×Dic3—21no—
144C2×D12—21no—
145C2×D12—24no—
146C2×D12C22×C412yes—
147C2×C4×D3—22no—
148C2×C4×D3—22no—
149C2×C4×D3—82no—
150C3×M16—42no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C2×C3×C8—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4C22×C412yes—
160C3×C22⋊C4—22no—
161C3×C42—22no—
162C3×C42C22×C411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42#4111yescharacteristic
165C3×C42C2411yescharacteristic
166#215—24no—
167#215—24no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3—21no—
171C4⋊Dic3C22×C412yes—
172C2×C3⋊C8—22no—
173C2×C3⋊C8C22×C412yes—
174C2×C3⋊C8C22×C412yes—
175C2×C3⋊C8C2×D412yes—
176C2×C3⋊C8C2×D412yes—
177D3×C8—82no—
178D3×C8—82no—
179C22×D4—31no—
180C23×C4—61no—
181C2×SD16—32no—
182C2×SD16—32no—
183C2×M16—21no—
184C22×C8—62no—
185C22×C8—62no—
186C4⋊Q8C4×D311yescharacteristic
187C4⋊Q8C4×D311yescharacteristic
188#125—21no—
189C22⋊Q8—21no—
190C22⋊Q8C4×D312yes—
191C4×Q8—21no—
192C4×Q8C22×D312yes—
193C4×D4—32no—
194C4×D4—32no—
195C4×D4—61no—
196C4×D4—61no—
197#117—21no—
198C2×C4⋊C4—31no—
199C2×C22⋊C4—61no—
200C2×C42C22×D311yescharacteristic
201#106—32no—
202#106—32no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—62no—
207C4⋊C8—62no—
208C4⋊C8C22×D312yes—
209#103—32no—
210#103—32no—
211#102—32no—
212#102—32no—
213C22⋊C8—21no—
214C22⋊C8—32no—
215C22⋊C8—32no—
216C22⋊C8—62no—
217C22⋊C8—62no—
218C22⋊C8—61no—
219#97—21no—
220#97—62no—
221#97—61no—
222C4×C8—21no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8—31no—
226C4×C8—31no—
227C4×C8—62no—
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C23×C3C4×D411yescharacteristic
231C22×D3—22no—
232C22×D3—24no—
233C22×D3—81no—
234C22×D3—81no—
235C3×Q8—22no—
236C3×Q8—22no—
237C3×Q8—81no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C23×C411yescharacteristic
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C2×#4111yescharacteristic
249C2×Dic3—22no—
250C2×Dic3—81no—
251D12—24no—
252D12—22no—
253D12#13114yes—
254C4×D3—42no—
255C4×D3—42no—
256C4×D3—82no—
257C3×C8—22no—
258C3×C8—41no—
259C3×C8—42no—
260C3⋊C8—22no—
261C3⋊C8—22no—
262C3⋊C8—22no—
263C3⋊C8—22no—
264C3⋊C8—42no—
265C24—61no—
266C2×Q8—21no—
267C2×Q8C2×C4×D312yes—
268C2×D4—32no—
269C2×D4—64no—
270C2×D4—61no—
271C22×C4—61no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—242no—
275C22×C4C2×C4×D311yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278SD16—64no—
279SD16—64no—
280M16—42no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—32no—
285C2×C8—32no—
286C2×C8—32no—
287C2×C8—32no—
288C2×C8—62no—
289C2×C8—242no—
290C2×C8—242no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—32no—
297C4⋊C4—61no—
298C4⋊C4C2×C4×D312yes—
299C22⋊C4—22no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C42—22no—
303C42C2×C4×D311yescharacteristic
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
306C42C23×D311yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3C2×C4×D411yescharacteristic
311C2×D3—24no—
312C2×D3—42no—
313C2×D3—42no—
314C2×D3—84no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—42no—
321C3×C4C2×#13112yes—
322C3×C4#57712yes—
323Dic3—42no—
324C23—64no—
325C23—62no—
326C23—241no—
327C23—241no—
328C23#67⋊C411yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—81no—
332D4—34no—
333D4—64no—
334D4—62no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—242no—
344C2×C4—241no—
345C2×C4#67⋊C411yescharacteristic
346C2×C4#67⋊C411yescharacteristic
347C2×C4#67⋊C411yescharacteristic
348C2×C4C22×C4×D311yescharacteristic
349C2×C4C2×D3×D411yescharacteristic
350C2×C4C2×#23711yescharacteristic
351C8—22no—
352C8—41no—
353C8—42no—
354C8—62no—
355C8—62no—
356C8—62no—
357C8—62no—
358C8—122no—
359C2×C3—41no—
360C2×C3#292911yescharacteristic
361C2×C3#294911yescharacteristic
362C2×C3#295611yescharacteristic
363D3—44no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—64no—
368C22—122no—
369C22—122no—
370C22—244no—
371C22C2×#67⋊C411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—122no—
379C4D3×#13112yes—
380C4#577812yes—
381C3#1792411yescharacteristic, complemented, Sylow
382C2—41no—
383C2—124no—
384C2#7865511yescharacteristic
385C2#7894711yescharacteristic
386C2#7898011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1485C2
socleC22×C3C2×C4×D4
prime radicalC22×D12C2×C4
cosocleC3×C42C24
hypercenter#1485D3
hypocenterC3#17924
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
2. #83331C2semidirect
4. C3×#1485C2semidirect
5. #70747C2semidirect
7. #69848C2semidirect
8. #68220C2semidirect
9. #68216C2semidirect
10. #68214C2semidirect
11. #68212C2semidirect
24. #5024C22semidirect
25. Q8×C3⋊C8C22semidirect
34. C3×Q8⋊C8C22semidirect
35. C3×Q8⋊C8C22semidirect
36. #4480C22semidirect
37. #4479C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
53. #1485D3semidirect
129. Q8⋊C8C2×D3semidirect
130. Q8⋊C8C2×D3semidirect
381. C3#17924semidirect

Maximal subgroups

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

subgroupindex
#833512
#833312
#792862
C3×#14852
#707472
#698552
#698482
#682202
#682162
#682142
#682122
#179243