Finite Group Databasebeta

← #10692962

Subgroups — #10692962

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2486C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#71553C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9470—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×C4×D4C2211yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C3×#398C412yescomplemented
11C3×#397C412yescomplemented
12C3×#389—21no—
13C3×#381—21no—
14C3×#380—21no—
15C3×#380C2211yescharacteristic
16C2×#834—21no—
17#5212C2212yes—
18C2×#830—21no—
19C2×#830C2212yes—
20D4×C3⋊C8—24no—
21#4925—21no—
22#4529—24no—
23#4526—21no—
24#2972—31no—
25#2486D311yescharacteristic
26#1823—32no—
27#1799—31no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×#123—42no—
34C3×#123—42no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#217—22no—
50C2×C4⋊Dic3—22no—
51C22×C3⋊C8—22no—
52C22×C3⋊C8D414yes—
53C3×#95—22no—
54C3×#95—22no—
55C3×#95C2×C412yes—
56#834—42no—
57C3⋊#95—22no—
58#830—22no—
59#830—22no—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4×C3⋊C8—22no—
63C2×#123—21no—
64C2×C4×D4C2×D311yescharacteristic
65C22×C4⋊C4C2×D311yescharacteristic
66C8×D4—64no—
67#431—61no—
68C2×#99—61no—
69C2×C22⋊C8—32no—
70C2×C22⋊C8—61no—
71#398Dic312yescomplemented
72#397Dic312yescomplemented
73#389—21no—
74#381—21no—
75#380—21no—
76#380—32no—
77#380C2×D311yescharacteristic
78#342—61no—
79#340—64no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C2×D411yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4D812yes—
85C2×C3×D4SD1612yes—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—42no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4112yes—
103C22×Dic3—42no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—24no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—44no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—44no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C42—21no—
116C3×C42—21no—
117#217—42no—
118#217—42no—
119C4⋊Dic3—44no—
120C2×C3⋊C8—24no—
121C2×C3⋊C8—44no—
122C2×C3⋊C8—82no—
123C22×D4#6711yescharacteristic
124C23×C4—21no—
125C23×C4#6711yescharacteristic
126C23×C4C22×D311yescharacteristic
127C22×C8—34no—
128C22×C8—62no—
129#123—42no—
130#123—42no—
131C4×D4—22no—
132C4×D4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4C2×Dic312yes—
140C2×C4⋊C4C2×Dic311yescharacteristic
141C2×C22⋊C4—22no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4C2×Dic311yescharacteristic
145C2×C22⋊C4#6711yescharacteristic
146C2×C42#6711yescharacteristic
147C4⋊C8—62no—
148#99—122no—
149C22⋊C8—62no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152C4×C8—62no—
153#95—22no—
154#95—22no—
155#95—62no—
156#95C2×Dic312yes—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3#10212yes—
164C23×C3C2×C22⋊C411yescharacteristic
165C23×C3#12112yes—
166C23×C3C2×D811yescharacteristic
167C23×C3C2×SD1611yescharacteristic
168C23×C3#13512yes—
169C3×D4—24no—
170C3×D4—24no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—24no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—81no—
189C2×C3×C4—82no—
190C2×C3×C4—82no—
191C2×C3×C4#10212yes—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×Dic3—82no—
196C2×Dic3—82no—
197C2×Dic3—82no—
198C3⋊C8—44no—
199C24#21711yescharacteristic
200C24C2×#6711yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4#21312yes—
204C2×D4#21412yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—122no—
216C22×C4#21711yescharacteristic
217C22×C4#21711yescharacteristic
218C22×C4#21711yescharacteristic
219C22×C4D3×D412yes—
220C22×C4#23712yes—
221C22×C4C22×Dic311yescharacteristic
222C22×C4C2×#6711yescharacteristic
223C2×C8—64no—
224C2×C8—124no—
225C2×C8—242no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—124no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—44no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—122no—
239C22⋊C4—122no—
240C42—21no—
241C42—21no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—81no—
254C22×C3#38611yescharacteristic
255C22×C3C2×#10211yescharacteristic
256C22×C3#41611yescharacteristic
257C22×C3#46512yes—
258C22×C3#46612yes—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—82no—
265C3×C4—82no—
266C3×C4#45812yes—
267Dic3—162no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#83212yes—
275C23C2×#21311yescharacteristic
276C23C2×#21411yescharacteristic
277C23#93912yes—
278C23#95012yes—
279C23C2×#21711yescharacteristic
280D4—24no—
281D4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—242no—
303C2×C4—242no—
304C2×C4—242no—
305C2×C4#83212yes—
306C2×C4D4×Dic312yes—
307C2×C4#93511yescharacteristic
308C2×C4#93711yescharacteristic
309C8—124no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—42no—
315C2×C3#189011yescharacteristic
316C2×C3#189311yescharacteristic
317C2×C3#190911yescharacteristic
318C22—21no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22#503812yes—
331C22#504112yes—
332C22C2×#83211yescharacteristic
333C22#522011yescharacteristic
334C22#523311yescharacteristic
335C4—22no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—82no—
341C4—82no—
342C4—482no—
343C4#505312yes—
344C3#947011yescharacteristic, complemented, Sylow
345C2—21no—
346C2—22no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#7168111yescharacteristic
351C2#7170411yescharacteristic
352C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2486C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2486D3
hypocenterC3#9470
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #71553C2semidirect
5. #67625C2semidirect
10. C3×#398C4semidirect
11. C3×#397C4semidirect
71. #398Dic3semidirect
72. #397Dic3semidirect
344. C3#9470semidirect

Maximal subgroups

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

subgroupindex
C3×#24862
#823012
#716192
#715532
#676252
#94703