Finite Group Databasebeta

← #10691211

Subgroups — #10691211

Order 768.

524 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2621C211yesmaximal, characteristic
2#82204C211yesmaximal, characteristic
3#71773C212yesmaximal, complemented
4#71571C211yesmaximal, characteristic
5#67631C212yesmaximal, complemented
6#9195—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C2×C3×#117C2211yescharacteristic
10C22×#208—21no—
11C3×#398—21no—
12C3×#398C412yes—
13C3×#397—21no—
14C3×#397C412yes—
15C3×#391—21no—
16C3×#382C2211yescharacteristic
17#4999—24no—
18#4925C2212yes—
19#4924—21no—
20#4924C2212yes—
21#4547—24no—
22#4528—21no—
23#4525—21no—
24#2905—31no—
25#2621D311yescharacteristic
26#1546—32no—
27#1458—31no—
28#1286—32no—
29C22×C3×Q8—21no—
30C23×C3×C4D411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C22⋊Q8—42no—
34C3×C22⋊Q8—42no—
35C3×#117—22no—
36C3×#117—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×#208—22no—
49C2×#208—22no—
50C2×#208—42no—
51C2×#208—42no—
52C22×C3⋊C8—22no—
53C3×#95—22no—
54C3×#95—22no—
55C3×#95—21no—
56C3×#95C2×C412yes—
57#830—22no—
58C4⋊(C3⋊C8)—24no—
59C4⋊(C3⋊C8)—22no—
60C4⋊(C3⋊C8)—22no—
61C4⋊(C3⋊C8)D414yes—
62C4×C3⋊C8—22no—
63C22×M16—61no—
64C2×C4⋊Q8C2×D311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#117C2×D311yescharacteristic
67#431—32no—
68#424—64no—
69#423—32no—
70#423—61no—
71#398—21no—
72#398Dic312yes—
73#397—21no—
74#397Dic312yes—
75#391—21no—
76#382C2×D311yescharacteristic
77#332—64no—
78#324—61no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C2×C3×Q8—41no—
82C2×C3×Q8—41no—
83C2×C3×Q8—41no—
84C2×C3×Q8—41no—
85C2×C3×Q8—41no—
86C2×C3×Q8—41no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4D812yes—
97C22×C3×C4SD1612yes—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—24no—
104C3×C4⋊C4—44no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—44no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42C2×D412yes—
114C3×C42#4112yes—
115#208—44no—
116#208—42no—
117#208—42no—
118C2×C3⋊C8—24no—
119C2×C3⋊C8—44no—
120C2×C3⋊C8—42no—
121C22×Q8—21no—
122C23×C4#6711yescharacteristic
123C2×M16—62no—
124C2×M16—62no—
125C2×M16—122no—
126C2×M16—122no—
127C22×C8—62no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C2×Dic312yes—
138C2×C4⋊C4C2×Dic311yescharacteristic
139C2×C4⋊C4#6711yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4C2×Dic311yescharacteristic
142C2×C22⋊C4#6711yescharacteristic
143C2×C42#6711yescharacteristic
144C2×C42C22×D311yescharacteristic
145C4⋊C8—34no—
146C4⋊C8—64no—
147C4⋊C8—62no—
148C4⋊C8—62no—
149C22⋊C8—62no—
150C4×C8—62no—
151#95—22no—
152#95—22no—
153#95—21no—
154#95C2×Dic312yes—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3#10212yes—
160C23×C3C2×C22⋊C411yescharacteristic
161C3×Q8—81no—
162C3×Q8—81no—
163C3×Q8—81no—
164C3×Q8—81no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4#10212yes—
184C2×C3×C4C4×D412yes—
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12112yes—
187C2×C3×C4#12311yescharacteristic
188C2×C3×C4C2×D811yescharacteristic
189C2×C3×C4C2×SD1611yescharacteristic
190C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
191C2×C3×C4#13711yescharacteristic
192C3⋊C8—44no—
193C3⋊C8—82no—
194C24#21711yescharacteristic
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4#21312yes—
208C22×C4#21412yes—
209C22×C4#21711yescharacteristic
210C22×C4#21711yescharacteristic
211C22×C4#21711yescharacteristic
212C22×C4C22×Dic311yescharacteristic
213C22×C4C2×#6711yescharacteristic
214C22×C4C2×#6711yescharacteristic
215M16—124no—
216M16—122no—
217M16—122no—
218C2×C8—64no—
219C2×C8—124no—
220C2×C8—122no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—24no—
224C4⋊C4—44no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—44no—
231C42—21no—
232C42—21no—
233C42D3×D412yes—
234C42#23712yes—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—42no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—81no—
242C22×C3#38611yescharacteristic
243C22×C3C2×#10211yescharacteristic
244C22×C3#41711yescharacteristic
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—82no—
251C3×C4—82no—
252C3×C4#44711yescharacteristic
253C3×C4#44811yescharacteristic
254C3×C4#48011yescharacteristic
255C3×C4#48111yescharacteristic
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#83212yes—
261C23C2×#21711yescharacteristic
262Q8—81no—
263Q8—81no—
264Q8—81no—
265Q8—81no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4#83212yes—
285C2×C4C2×#21311yescharacteristic
286C2×C4C2×#21411yescharacteristic
287C2×C4D4×Dic312yes—
288C2×C4#93511yescharacteristic
289C2×C4#93711yescharacteristic
290C2×C4#93912yes—
291C2×C4#94911yescharacteristic
292C2×C4#95111yescharacteristic
293C8—124no—
294C8—242no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—42no—
298C2×C3#187411yescharacteristic
299C2×C3#190911yescharacteristic
300C2×C3#191511yescharacteristic
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#83211yescharacteristic
309C22#522011yescharacteristic
310C22#523211yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—82no—
318C4#503311yescharacteristic
319C4#503411yescharacteristic
320C4#503611yescharacteristic
321C4#504011yescharacteristic
322C3#919511yescharacteristic, complemented, Sylow
323C2—21no—
324C2—21no—
325C2—42no—
326C2#7167711yescharacteristic
327C2#7186511yescharacteristic
328C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2621C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2621D3
hypocenterC3#9195
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #71773C2semidirect
5. #67631C2semidirect
322. C3#9195semidirect

Maximal subgroups

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

subgroupindex
C3×#26212
#822042
#717732
#715712
#676312
#91953