Finite Group Databasebeta

← #10765654

Subgroups — #10765654

Order 768.

606 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
1C3×#3101C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic, complemented
3#82304C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic
6#70900C212yesmaximal, complemented
7#70899C212yesmaximal, complemented
8#70898C212yesmaximal, complemented
9#70896C212yesmaximal, complemented
10#70751C211yesmaximal, characteristic, complemented
11#70743C211yesmaximal, characteristic, complemented
12#17952—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C3×#483C412yes—
16C3×#476C412yes—
17C3×#461C412yes—
18C3×#450C412yes—
19C3×C4×SD16C2212yes—
20C3×C4×SD16C2212yes—
21C3×#431C2211yescharacteristic
22#5207—22no—
23#5024—22no—
24#5024C2212yescomplemented
25Q8×C3⋊C8C2212yescomplemented
26#5012—22no—
27#5012C2212yes—
28#5012C2212yes—
29#5000—22no—
30#4926C2211yescharacteristic
31#4925C2211yescharacteristic
32#4925C2211yescharacteristic
33C4×#208C2211yescharacteristic
34#4535C2212yescomplemented
35#4535C2212yescomplemented
36D4⋊(C3⋊C8)C2212yes—
37D4⋊(C3⋊C8)C2212yes—
38#4458C2212yescomplemented
39#4457C2212yescomplemented
40#4455C2211yescharacteristic, complemented
41#4454C2211yescharacteristic, complemented
42#4454C2211yescharacteristic, complemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3101D311yescharacteristic
45#2985—31no—
46#2975—31no—
47#1592—32no—
48#1590—32no—
49#1588—32no—
50#1583—32no—
51#1572—31no—
52#1490—31no—
53#1485—31no—
54#1453—31no—
55C22×C3×D4C2×C411yescharacteristic
56C23×C3×C4—21no—
57C2×C3×SD16C2×C412yes—
58C2×C3×SD16C2×C412yes—
59C3×C4⋊Q8C2×C411yescharacteristic
60C3×C4⋊Q8C2×C411yescharacteristic
61C3×#125—21no—
62C3×C22⋊Q8—21no—
63C3×C22⋊Q8C2×C412yes—
64C3×C4×Q8—21no—
65C3×C4×Q8C2312yes—
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C3×C4×D4C2×C412yes—
69C3×C4×D4C2312yes—
70C3×#117—21no—
71C2×C3×C4⋊C4C2×C411yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#208—21no—
75C2×#208—22no—
76C2×#208—22no—
77C3×#106C2×C412yes—
78C3×#106C2×C412yes—
79C3×C4⋊C8C2×C412yes—
80C3×#103C2×C412yes—
81C3×#103C2×C412yes—
82C3×#102C2×C412yes—
83C3×#102C2×C412yes—
84C3×C22⋊C8C2×C412yes—
85C3×C4×C8C2311yescharacteristic
86C3×C4×C8C2311yescharacteristic
87#830—21no—
88#830—21no—
89#830—22no—
90#830—22no—
91#830D412yes—
92C4⋊(C3⋊C8)—22no—
93C4⋊(C3⋊C8)—22no—
94C4⋊(C3⋊C8)D412yes—
95C4⋊(C3⋊C8)C2312yes—
96C4⋊(C3⋊C8)C2312yes—
97#803—21no—
98#803—21no—
99#803—22no—
100C4×C3⋊C8—21no—
101C4×C3⋊C8—22no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#533C2×D311yescharacteristic
107C2×C4×D4C2×D311yescharacteristic
108#483Dic312yes—
109#476Dic312yes—
110#461Dic312yes—
111#450Dic312yes—
112#446—32no—
113#446—62no—
114C8×Q8—32no—
115C4×SD16C2×D312yes—
116C4×SD16C2×D312yes—
117#436—32no—
118#436—32no—
119#436—62no—
120#433—62no—
121#432—31no—
122#431—31no—
123#431—31no—
124#431C2×D311yescharacteristic
125#407—62no—
126C4×M16—31no—
127#335—32no—
128#334—32no—
129Q8⋊C8—32no—
130Q8⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C24×C3—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8C22×C412yes—
140C2×C3×D4—21no—
141C2×C3×D4—42no—
142C2×C3×D4C22×C412yes—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—82no—
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C3×SD16—42no—
151C3×SD16—42no—
152C2×C3×C8C22×C412yes—
153C2×C3×C8C22×C412yes—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4C22×C412yes—
161C3×C4⋊C4C22×C412yes—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C42—22no—
166C3×C42C22×C411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4111yescharacteristic
169C3×C42C2411yescharacteristic
170#208—42no—
171#208—82no—
172#208—82no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8C2×D412yes—
178C2×C3⋊C8C2×D412yes—
179C22×D4C2×Dic311yescharacteristic
180C23×C4—21no—
181C2×SD16C2×Dic312yes—
182C2×SD16C2×Dic312yes—
183C2×M16—62no—
184C2×M16—62no—
185C2×M16—61no—
186C4⋊Q8C2×Dic311yescharacteristic
187C4⋊Q8C2×Dic311yescharacteristic
188#125—21no—
189C22⋊Q8—21no—
190C22⋊Q8C2×Dic312yes—
191C4×Q8—21no—
192C4×Q8C22×D312yes—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4C2×Dic312yes—
196C4×D4C22×D312yes—
197#117—21no—
198C2×C4⋊C4C2×Dic311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C22×D311yescharacteristic
201#106C2×Dic312yes—
202#106C2×Dic312yes—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—62no—
207C4⋊C8—62no—
208C4⋊C8C2×Dic312yes—
209#103C2×Dic312yes—
210#103C2×Dic312yes—
211#102C2×Dic312yes—
212#102C2×Dic312yes—
213C22⋊C8—32no—
214C22⋊C8—62no—
215C22⋊C8—62no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8C2×Dic312yes—
219#97—62no—
220#97—61no—
221#97—61no—
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—22no—
231C23×C3—42no—
232C23×C3—81no—
233C23×C3—81no—
234C23×C3C4×D411yescharacteristic
235C3×Q8—22no—
236C3×Q8—22no—
237C3×Q8—81no—
238C3×D4—22no—
239C3×D4—22no—
240C3×D4—42no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4—81no—
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C23×C411yescharacteristic
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
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—21no—
266C2×Q8—21no—
267C2×Q8C22×Dic312yes—
268C2×D4—21no—
269C2×D4—42no—
270C2×D4C22×Dic312yes—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—82no—
275C22×C4D3×D411yescharacteristic
276C22×C4#23711yescharacteristic
277C22×C4C22×Dic311yescharacteristic
278SD16—42no—
279SD16—42no—
280M16—122no—
281M16—242no—
282M16—242no—
283C2×C8—32no—
284C2×C8—32no—
285C2×C8—61no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—61no—
289C2×C8C22×Dic312yes—
290C2×C8C22×Dic312yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4C22×Dic312yes—
298C4⋊C4C22×Dic312yes—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C42—22no—
303C42D3×D411yescharacteristic
304C42#23711yescharacteristic
305C42C22×Dic311yescharacteristic
306C42C23×D311yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—42no—
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×C4—42no—
321C3×C4—42no—
322C3×C4#56812yes—
323C3×C4#57712yes—
324C23—22no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23D4×Dic311yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—81no—
332D4—22no—
333D4—22no—
334D4—42no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4D4×Dic311yescharacteristic
346C2×C4D4×Dic311yescharacteristic
347C2×C4D4×Dic311yescharacteristic
348C2×C4C2×D3×D411yescharacteristic
349C2×C4C2×#23711yescharacteristic
350C2×C4C23×Dic311yescharacteristic
351C8—22no—
352C8—22no—
353C8—62no—
354C8—62no—
355C8—62no—
356C8—121no—
357C8—122no—
358C8—122no—
359C2×C3—41no—
360C2×C3—82no—
361C2×C3#293011yescharacteristic
362C2×C3#294911yescharacteristic
363C2×C3#295611yescharacteristic
364C22—21no—
365C22—21no—
366C22—21no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—84no—
371C22C2×D4×Dic311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4#576212yes—
380C4#581912yes—
381C3#1795211yescharacteristic, complemented, Sylow
382C2—41no—
383C2—82no—
384C2#8291711yescharacteristic
385C2#8384311yescharacteristic
386C2#8400011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#3101C2
socleC22×C3C2×C4×D4
prime radicalC22×C3×D4C2×C4
cosocleC3×C42C24
hypercenter#3101D3
hypocenterC3#17952
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #83331C2semidirect
4. #71825C2semidirect
6. #70900C2semidirect
7. #70899C2semidirect
8. #70898C2semidirect
9. #70896C2semidirect
10. #70751C2semidirect
11. #70743C2semidirect
24. #5024C22semidirect
25. Q8×C3⋊C8C22semidirect
34. #4535C22semidirect
35. #4535C22semidirect
38. #4458C22semidirect
39. #4457C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
381. C3#17952semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
#833312
#823042
#718252
#715992
#709002
#708992
#708982
#708962
#707512
#707432
#179523