Finite Group Databasebeta

← #10765331

Subgroups — #10765331

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#79298C211yesmaximal, characteristic
2#79286C211yesmaximal, characteristic
3#79266C211yesmaximal, characteristic, complemented
4C3×#1453C211yesmaximal, characteristic, complemented
5#70751C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#69855C211yesmaximal, characteristic
8#68080C212yesmaximal, complemented
9#68079C212yesmaximal, complemented
10#68076C212yesmaximal, complemented
11C8×#204C212yesmaximal, complemented
12#17924—31nomaximal, Sylow
13C2×C4×D12C2211yescharacteristic
14#5470C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×C4×M16C2211yescharacteristic
18#4926C2211yescharacteristic
19#4925C2211yescharacteristic
20#4842C412yes—
21#4841C412yes—
22#4834—22no—
23#4828—22no—
24#4731C412yes—
25D3×C22⋊C8—22no—
26#4721C412yes—
27#4720—22no—
28C4×#204C2212yes—
29C4×#204C2212yes—
30C8×D12C2212yes—
31C8×D12C2212yes—
32#4679C2212yescomplemented
33C8×Dic6C2212yescomplemented
34C3×#322C2211yescharacteristic, complemented
35C3×#322C2211yescharacteristic, complemented
36C3×#322C2211yescharacteristic, complemented
37C3×C82C2211yescharacteristic, complemented
38#4459C2212yes—
39#4459C2212yes—
40#4458C2212yescomplemented
41#4457C2212yescomplemented
42#4456C2212yescomplemented
43#4456C2212yescomplemented
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#1485—31no—
54#1453D311yescharacteristic, complemented
55C22×D12C2×C411yescharacteristic
56C22×C4×D3—21no—
57C2×C3×M16—21no—
58C2×C3×C42C2311yescharacteristic
59C2×(C2×D3)⋊C4—21no—
60#926—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62#924—21no—
63#924C2×C412yes—
64C2×#204C2×C412yes—
65C2×#204C2×C412yes—
66C2×D3×C8—22no—
67C2×D3×C8—22no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12C2×C412yes—
71C4×D12C2312yes—
72#870—21no—
73#869C2×C411yescharacteristic
74#869C2×C411yescharacteristic
75C4×Dic6—21no—
76C4×Dic6C2312yes—
77C3×C4⋊C8C2×C412yes—
78C3×C4⋊C8D412yes—
79C3×C22⋊C8C2×C412yes—
80C3×C22⋊C8D412yes—
81C3×#97—21no—
82C3×C4×C8C2311yescharacteristic
83C3×C4×C8C2311yescharacteristic
84C3×C4×C8C2311yescharacteristic
85C3×C4×C8C2311yescharacteristic
86C3×C4×C8C2311yescharacteristic
87C3×C4×C8C2311yescharacteristic
88#830—21no—
89#830—21no—
90#821C2×C412yes—
91#821C2×C412yes—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94#817C2×C412yes—
95#817C2×C412yes—
96#816C2×C412yes—
97#816C2×C412yes—
98#815—22no—
99Dic3⋊C8—22no—
100Dic3⋊C8—22no—
101C8×Dic3—22no—
102C4⋊(C3⋊C8)C2312yes—
103C4⋊(C3⋊C8)C2312yes—
104#803—21no—
105C4×C3⋊C8—21no—
106#533—31no—
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#431C2×D311yescharacteristic
124#431C2×D311yescharacteristic
125C2×C22⋊C8—62no—
126C4×M16C2×D311yescharacteristic
127#335—32no—
128#334—32no—
129Q8⋊C8—32no—
130Q8⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322C2×D311yescharacteristic, complemented
134#322C2×D311yescharacteristic, complemented
135#322C2×D311yescharacteristic, complemented
136C82C2×D311yescharacteristic, complemented
137C23×D3—21no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C2×D12—21no—
143C2×D12—24no—
144C2×D12C22×C412yes—
145C2×C4×D3—22no—
146C2×C4×D3—22no—
147C2×C4×D3—82no—
148C2×Dic6—21no—
149C2×Dic6C22×C412yes—
150C3×M16—42no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8C22×C412yes—
154C2×C3×C8C22×C412yes—
155C2×C3×C8C2×D412yes—
156C2×C3×C8C2×D412yes—
157C3×C42C22×C411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160C3×C42C2411yescharacteristic
161#217—22no—
162(C2×D3)⋊C4—22no—
163(C2×D3)⋊C4—22no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3C22×C412yes—
169C4⋊Dic3C22×C412yes—
170Dic3⋊C4—22no—
171Dic3⋊C4—22no—
172C4×Dic3—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175#204—24no—
176#204—24no—
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⋊Q8—31no—
187C4⋊Q8—31no—
188#125—61no—
189C22⋊Q8—32no—
190C22⋊Q8—61no—
191C4×Q8—32no—
192C4×Q8—61no—
193C4×D4—32no—
194C4×D4—32no—
195C4×D4—61no—
196C4×D4—61no—
197#117—61no—
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—62no—
206C4⋊C8—62no—
207C4⋊C8C4×D312yes—
208C4⋊C8D1212yes—
209#103—32no—
210#103—32no—
211#102—32no—
212#102—32no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—62no—
216C22⋊C8—62no—
217C22⋊C8C4×D312yes—
218C22⋊C8D1212yes—
219#97—21no—
220#97—62no—
221#97—61no—
222C4×C8—62no—
223C4×C8—61no—
224C4×C8C22×D311yescharacteristic
225C4×C8C22×D311yescharacteristic
226C4×C8C22×D311yescharacteristic
227C4×C8C22×D311yescharacteristic
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C23×C3C4×D411yescharacteristic
231C22×D3—22no—
232C22×D3—24no—
233C22×D3—81no—
234C22×D3—81no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4C23×C411yescharacteristic
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—81no—
248D12—24no—
249D12—22no—
250D12#13114yes—
251C4×D3—42no—
252C4×D3—42no—
253C4×D3—82no—
254Dic6—22no—
255Dic6—22no—
256Dic6—81no—
257C3×C8—22no—
258C3×C8—22no—
259C3×C8—22no—
260C3×C8—22no—
261C3×C8—22no—
262C3×C8—41no—
263C3⋊C8—42no—
264C3⋊C8—42no—
265C24—61no—
266C2×Q8—32no—
267C2×Q8—61no—
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×C4C2×D1211yescharacteristic
277C22×C4#23511yescharacteristic
278SD16—64no—
279SD16—64no—
280M16—42no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—242no—
286C2×C8—242no—
287C2×C8C2×C4×D312yes—
288C2×C8C2×C4×D312yes—
289C2×C8C2×D1212yes—
290C2×C8C2×D1212yes—
291C4⋊C4—32no—
292C4⋊C4—32no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—62no—
297C4⋊C4—61no—
298C4⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C42—62no—
303C42C2×C4×D311yescharacteristic
304C42C2×D1211yescharacteristic
305C42#23511yescharacteristic
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×C4C2×#13112yes—
320C3×C4#57712yes—
321Dic3—42no—
322Dic3—42no—
323Dic3—42no—
324C23—64no—
325C23—62no—
326C23—241no—
327C23—241no—
328C23C4×D1211yescharacteristic
329Q8—62no—
330Q8—62no—
331Q8—241no—
332D4—34no—
333D4—64no—
334D4—62no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—242no—
344C2×C4—241no—
345C2×C4C4×D1211yescharacteristic
346C2×C4C4×D1211yescharacteristic
347C2×C4C4×D1211yescharacteristic
348C2×C4C22×C4×D311yescharacteristic
349C2×C4C22×D1211yescharacteristic
350C2×C4C2×#23511yescharacteristic
351C8—22no—
352C8—22no—
353C8—22no—
354C8—22no—
355C8—22no—
356C8—41no—
357C8—122no—
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×C4×D1211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—122no—
377C4—122no—
378C4—122no—
379C4D3×#13112yes—
380C4#575112yes—
381C3#1792411yescharacteristic, complemented, Sylow
382C2—41no—
383C2—124no—
384C2#7854111yescharacteristic
385C2#7855311yescharacteristic
386C2#7857211yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1453C2
socleC22×C3C2×C4×D4
prime radicalC22×D12C2×C4
cosocleC3×C42C24
hypercenter#1453D3
hypocenterC3#17924
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1453

Complements

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

NH ≅ G/Ntype
3. #79266C2semidirect
4. C3×#1453C2semidirect
5. #70751C2semidirect
6. #69856C2semidirect
8. #68080C2semidirect
9. #68079C2semidirect
10. #68076C2semidirect
11. C8×#204C2semidirect
32. #4679C22semidirect
33. C8×Dic6C22semidirect
34. C3×#322C22semidirect
35. C3×#322C22semidirect
36. C3×#322C22semidirect
37. C3×C82C22semidirect
40. #4458C22semidirect
41. #4457C22semidirect
42. #4456C22semidirect
43. #4456C22semidirect
54. #1453D3semidirect
133. #322C2×D3semidirect
134. #322C2×D3semidirect
135. #322C2×D3semidirect
136. C82C2×D3semidirect
381. C3#17924semidirect

Maximal subgroups

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

subgroupindex
#792982
#792862
#792662
C3×#14532
#707512
#698562
#698552
#680802
#680792
#680762
C8×#2042
#179243