Finite Group Databasebeta

← #10690798

Subgroups — #10690798

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2851C211yesmaximal, characteristic
2C2×C4×#208C211yesmaximal, characteristic
3C2×#4543C212yesmaximal, complemented
4#71529C211yesmaximal, characteristic
5#67630C212yesmaximal, complemented
6#9122—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9C22×#208—21no—
10C3×#389C411yescharacteristic
11C3×#389C411yescharacteristic
12C3×#387—21no—
13C3×#387—21no—
14C3×#385—21no—
15C3×#385—21no—
16C3×#382C2211yescharacteristic
17C3×#380C411yescharacteristic
18C3×#380C411yescharacteristic
19C2×#830—21no—
20C4×#208—24no—
21C2×#803C2212yes—
22C2×#803C2212yes—
23C2×C4×C3⋊C8—21no—
24#4543—24no—
25#4524—21no—
26C2×C4×M16—31no—
27#2851D311yescharacteristic
28#1749—31no—
29C2×#330—32no—
30#1295—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C3×#125—22no—
35C3×#125—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C42—22no—
47C2×C3×C42—22no—
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#208—24no—
53C2×#208—24no—
54C2×#208—24no—
55C22×C3⋊C8—22no—
56C22×C3⋊C8—22no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63#830—24no—
64#803—22no—
65#803—22no—
66#803—22no—
67#803D414yes—
68C4×C3⋊C8—24no—
69C22×M16—61no—
70C2×#125C2×D311yescharacteristic
71C22×C42C2×D311yescharacteristic
72C2×C22⋊C8—61no—
73C4×M16—64no—
74C2×#97—32no—
75C2×#97—32no—
76C2×C4×C8—61no—
77#389Dic311yescharacteristic
78#389Dic311yescharacteristic
79#387—21no—
80#387—21no—
81#385—21no—
82#385—21no—
83#382C2×D311yescharacteristic
84#380Dic311yescharacteristic
85#380Dic311yescharacteristic
86#336—61no—
87#330—64no—
88C24×C3C22⋊C411yescharacteristic
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—42no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—42no—
117C3×C4⋊C4—41no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C42—22no—
121C3×C42—21no—
122C3×C42—22no—
123C3×C42—21no—
124C3×C42C2×D412yes—
125C3×C42#4112yes—
126#208—216no—
127C2×C3⋊C8—24no—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—24no—
130C2×C3⋊C8—24no—
131C2×C3⋊C8—42no—
132C2×C3⋊C8—42no—
133C23×C4C2×Dic311yescharacteristic
134C23×C4#6711yescharacteristic
135C23×C4#6711yescharacteristic
136C2×M16—64no—
137C2×M16—64no—
138C2×M16—64no—
139C22×C8—62no—
140C22×C8—62no—
141#125—22no—
142#125—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic311yescharacteristic
148C2×C4⋊C4C2×Dic311yescharacteristic
149C2×C4⋊C4C2×Dic311yescharacteristic
150C2×C4⋊C4C2×Dic311yescharacteristic
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42C2×Dic311yescharacteristic
156C2×C42#6711yescharacteristic
157C2×C42#6711yescharacteristic
158C2×C42C22×D311yescharacteristic
159C22⋊C8—64no—
160#97—34no—
161#97—62no—
162#97—62no—
163#97—62no—
164C4×C8—64no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—44no—
194C2×C3×C4—44no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4#10112yes—
208C2×C3×C4#10412yes—
209C2×C3×C4#10412yes—
210C2×C3×C4C4×D412yes—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12112yes—
213C2×C3×C4#12311yescharacteristic
214C3⋊C8—28no—
215C3⋊C8—44no—
216C24#21711yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4#21711yescharacteristic
232C22×C4#21711yescharacteristic
233C22×C4#21711yescharacteristic
234C22×C4C22×Dic311yescharacteristic
235C22×C4C2×#6711yescharacteristic
236C22×C4C2×#6711yescharacteristic
237M16—616no—
238C2×C8—64no—
239C2×C8—64no—
240C2×C8—64no—
241C2×C8—64no—
242C2×C8—122no—
243C2×C8—122no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—22no—
256C42—21no—
257C42—21no—
258C42—22no—
259C42D3×D412yes—
260C42#23712yes—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3#38611yescharacteristic
269C22×C3C2×#10111yescharacteristic
270C22×C3#41311yescharacteristic
271C22×C3C2×#10411yescharacteristic
272C22×C3C2×#10411yescharacteristic
273C22×C3C2×#10411yescharacteristic
274C22×C3#42111yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—81no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4—81no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23#83712yes—
293C23C2×#21711yescharacteristic
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4#83612yes—
323C2×C4#83712yes—
324C2×C4#83712yes—
325C2×C4D4×Dic312yes—
326C2×C4#93511yescharacteristic
327C2×C4#93711yescharacteristic
328C2×C4#93912yes—
329C8—68no—
330C8—124no—
331C2×C3—42no—
332C2×C3#152411yescharacteristic
333C2×C3#152611yescharacteristic
334C2×C3#152811yescharacteristic
335C2×C3#152911yescharacteristic
336C2×C3#187111yescharacteristic
337C2×C3#189611yescharacteristic
338C2×C3#190111yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22#522011yescharacteristic
347C22C2×#83611yescharacteristic
348C22C2×#83711yescharacteristic
349C22C2×#83711yescharacteristic
350C22C2×#83711yescharacteristic
351C22#523611yescharacteristic
352C22#523711yescharacteristic
353C4—22no—
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—22no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C3#912211yescharacteristic, complemented, Sylow
365C2—42no—
366C2#7169811yescharacteristic
367C2#7171111yescharacteristic
368C2#7171311yescharacteristic
369C2#7173511yescharacteristic
370C2#7174611yescharacteristic
371C2#7174711yescharacteristic
372C2#7182111yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2851C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2851D3
hypocenterC3#9122
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. C2×#4543C2semidirect
5. #67630C2semidirect
364. C3#9122semidirect

Maximal subgroups

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

subgroupindex
C3×#28512
C2×C4×#2082
C2×#45432
#715292
#676302
#91223