Finite Group Databasebeta

← #10765323

Subgroups — #10765323

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#82314C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3#79316C211yesmaximal, characteristic, complemented
4C3×#1490C211yesmaximal, characteristic
5#70796C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#69848C211yesmaximal, characteristic, complemented
8#68156C212yesmaximal, complemented
9#68154C212yesmaximal, complemented
10#68150C212yesmaximal, complemented
11#68149C212yesmaximal, complemented
12#17924—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14#5470C2211yescharacteristic
15C3×#432C2211yescharacteristic
16C3×C4×M16C2211yescharacteristic
17#5218C412yes—
18C2×#830—22no—
19#5019C412yes—
20C4×#214C2212yes—
21C4×#214C2212yes—
22D4×C3⋊C8—22no—
23D4×C3⋊C8C2212yes—
24D4×C3⋊C8C2212yes—
25#5008C412yes—
26#4971C412yes—
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4831C2212yescomplemented
31#4830C2212yescomplemented
32#4720—22no—
33#4701—22no—
34C3×D4⋊C8C2212yes—
35C3×D4⋊C8C2212yes—
36#4480C2212yescomplemented
37#4479C2212yescomplemented
38#4456C2212yescomplemented
39#4456C2212yescomplemented
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#1490D311yescharacteristic
53#1485—31no—
54#1453—31no—
55C22×C3×D4C2×C411yescharacteristic
56C23×C3×C4—21no—
57C2×C3×M16—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4C2×C412yes—
61C3×C4×D4C2312yes—
62C2×C3×C4⋊C4C2×C411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#214C2×C412yes—
66C2×#214C2×C412yes—
67#926—21no—
68#924—21no—
69#924C2×C412yes—
70C22×C3⋊C8—22no—
71C22×C3⋊C8—22no—
72#870—21no—
73#869C2×C411yescharacteristic
74#869C2×C411yescharacteristic
75C4×Dic6—21no—
76C4×Dic6C2312yes—
77C3×C4⋊C8C2312yes—
78C3×C22⋊C8—21no—
79C3×#97—21no—
80C3×#97—21no—
81C3×C4×C8C2311yescharacteristic
82C3×C4×C8C2311yescharacteristic
83#832C2×C412yes—
84#832C2×C412yes—
85#830—21no—
86#830—22no—
87#830—22no—
88#830C2×C412yes—
89#830D412yes—
90#815—22no—
91Dic3⋊C8—22no—
92Dic3⋊C8—22no—
93C8×Dic3—22no—
94#810C2×C412yes—
95#810C2×C412yes—
96#808C2×C412yes—
97#808C2×C412yes—
98C4⋊(C3⋊C8)C2×C412yes—
99C4⋊(C3⋊C8)D412yes—
100C4⋊(C3⋊C8)C2312yes—
101C4×C3⋊C8—21no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#533—31no—
107C2×C4×D4C2×D311yescharacteristic
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#432C2×D311yescharacteristic
122#431—31no—
123#431—31no—
124#431—31no—
125C2×C22⋊C8—62no—
126C4×M16C2×D311yescharacteristic
127#335—32no—
128#334—32no—
129Q8⋊C8—32no—
130Q8⋊C8—32no—
131D4⋊C8C2×D312yes—
132D4⋊C8C2×D312yes—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C24×C3—21no—
138C2×C3×D4—24no—
139C2×C3×D4—21no—
140C2×C3×D4C22×C412yes—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—82no—
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C2×Dic6—21no—
149C2×Dic6C22×C412yes—
150C3×M16—42no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C2×C3×C8—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4C22×C412yes—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—22no—
158C3×C42C22×C411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161C3×C42C2411yescharacteristic
162#217—22no—
163#214—24no—
164#214—24no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3C22×C412yes—
169Dic3⋊C4—22no—
170Dic3⋊C4—22no—
171C4×Dic3—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—82no—
174C2×C3⋊C8—82no—
175C2×C3⋊C8C22×C412yes—
176C2×C3⋊C8C22×C412yes—
177C2×C3⋊C8C2×D412yes—
178C2×C3⋊C8C2×D412yes—
179C22×D4C4×D311yescharacteristic
180C23×C4—21no—
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—21no—
194C4×D4—21no—
195C4×D4C4×D312yes—
196C4×D4C22×D312yes—
197#117—61no—
198C2×C4⋊C4C4×D311yescharacteristic
199C2×C22⋊C4—21no—
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—21no—
221#97—62no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8—31no—
226C4×C8—62no—
227C4×C8—61no—
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—81no—
233C23×C3—81no—
234C23×C3C4×D411yescharacteristic
235C3×D4—24no—
236C3×D4—22no—
237C3×D4#13114yes—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—82no—
245C2×C3×C4—81no—
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4C23×C411yescharacteristic
250C2×C3×C4C22×D411yescharacteristic
251C2×C3×C4C2×#4111yescharacteristic
252C2×Dic3—22no—
253C2×Dic3—22no—
254Dic6—22no—
255Dic6—22no—
256Dic6—81no—
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—21no—
266C2×Q8—32no—
267C2×Q8—61no—
268C2×D4—24no—
269C2×D4—21no—
270C2×D4C2×C4×D312yes—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—82no—
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—32no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—62no—
296C4⋊C4—61no—
297C4⋊C4—62no—
298C4⋊C4C2×C4×D312yes—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—62no—
302C42—62no—
303C42C2×C4×D311yescharacteristic
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
306C42C23×D311yescharacteristic
307C22×C3—21no—
308C22×C3—24no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—84no—
314C22×C3C2×C4×D411yescharacteristic
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4C2×#13112yes—
321C3×C4#57712yes—
322Dic3—42no—
323Dic3—42no—
324C23—24no—
325C23—22no—
326C23—81no—
327C23—81no—
328C23#67⋊C411yescharacteristic
329Q8—62no—
330Q8—62no—
331Q8—241no—
332D4—24no—
333D4—22no—
334D4#90714yes—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—82no—
344C2×C4—81no—
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—44no—
361C2×C3#292911yescharacteristic
362C2×C3#294911yescharacteristic
363C2×C3#295611yescharacteristic
364C22—21no—
365C22—24no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—42no—
370C22—84no—
371C22C2×#67⋊C411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—122no—
378C4—122no—
379C4C2×#90712yes—
380C4#577512yes—
381C3#1792411yescharacteristic, complemented, Sylow
382C2—41no—
383C2—44no—
384C2#7862611yescharacteristic
385C2#7891611yescharacteristic
386C2#7898011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1490C2
socleC22×C3C2×C4×D4
prime radicalC22×C3×D4C2×C4
cosocleC3×C42C24
hypercenter#1490D3
hypocenterC3#17924
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
3. #79316C2semidirect
5. #70796C2semidirect
6. #69856C2semidirect
7. #69848C2semidirect
8. #68156C2semidirect
9. #68154C2semidirect
10. #68150C2semidirect
11. #68149C2semidirect
30. #4831C22semidirect
31. #4830C22semidirect
36. #4480C22semidirect
37. #4479C22semidirect
38. #4456C22semidirect
39. #4456C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
381. C3#17924semidirect

Maximal subgroups

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

subgroupindex
#823142
#823012
#793162
C3×#14902
#707962
#698562
#698482
#681562
#681542
#681502
#681492
#179243