Finite Group Databasebeta

← #10694016

Subgroups — #10694016

Order 768.

656 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4528C214yesmaximal
3#71530C211yesmaximal, characteristic
4C4×#830C211yesmaximal, characteristic
5#9650—31nomaximal, Sylow
6C22×C3×C4⋊C4C411yescharacteristic
7C22×C3×C4⋊C4C411yescharacteristic
8C22×C3×C4⋊C4C2212yes—
9C22×C3×C42C2211yescharacteristic
10C3×#389—21no—
11C3×#389—21no—
12C3×#389—21no—
13C3×#389—21no—
14C2×C3×#95C412yes—
15C2×#830C2212yes—
16C2×#830C2212yes—
17C2×C4⋊(C3⋊C8)—21no—
18C2×C4×C3⋊C8—21no—
19#4528—28no—
20#4524—21no—
21#4524—21no—
22C2×#389D311yescharacteristic
23#1797—31no—
24C4×C22⋊C8—31no—
25C2×#324—34no—
26C23×C3×C4C2×C412yes—
27C23×C3×C4C2×C411yescharacteristic
28C23×C3×C4C2×C411yescharacteristic
29C23×C3×C4C2×C412yes—
30C23×C3×C4C2311yescharacteristic
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—24no—
39C2×C3×C4⋊C4—24no—
40C2×C3×C4⋊C4—42no—
41C2×C3×C4⋊C4C814yescomplemented
42C2×C3×C42—42no—
43C2×C3×C42D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C22×C3⋊C8—22no—
48C22×C3⋊C8—22no—
49C3×#95—22no—
50C3×#95—22no—
51#830—22no—
52#830—24no—
53#830D414yes—
54C4⋊(C3⋊C8)—42no—
55C4×C3⋊C8—42no—
56C22×C4⋊C4Dic311yescharacteristic
57C22×C4⋊C4Dic311yescharacteristic
58C22×C4⋊C4C2×D312yes—
59C22×C42C2×D311yescharacteristic
60C2×C4⋊C8—61no—
61C2×C22⋊C8—32no—
62C2×C22⋊C8—32no—
63C2×C4×C8—61no—
64#389—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68C2×#95Dic312yes—
69#336—61no—
70#336—61no—
71#324—68no—
72C24×C3C22×C411yescharacteristic
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—24no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—24no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—24no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C2×C814yes—
93C22×C3×C4C2×C812yes—
94C22×C3×C4M1612yes—
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4112yes—
99C3×C4⋊C4—24no—
100C3×C4⋊C4—24no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—44no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—42no—
106C3×C42—22no—
107C3×C42—22no—
108C3×C42—41no—
109C3×C42—41no—
110C2×C3⋊C8—24no—
111C2×C3⋊C8—42no—
112C2×C3⋊C8—42no—
113C2×C3⋊C8—42no—
114C2×C3⋊C8—42no—
115C2×C3⋊C8—42no—
116C23×C4C2×Dic312yes—
117C23×C4C2×Dic311yescharacteristic
118C23×C4C2×Dic311yescharacteristic
119C23×C4C2×Dic312yes—
120C23×C4C22×D311yescharacteristic
121C22×C8—62no—
122C22×C8—62no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C3⋊C814yescomplemented
134C2×C42—42no—
135C2×C42#6711yescharacteristic
136C2×C42#6711yescharacteristic
137C2×C42#6711yescharacteristic
138C2×C42#6711yescharacteristic
139C4⋊C8—122no—
140C22⋊C8—34no—
141C22⋊C8—62no—
142C22⋊C8—64no—
143C4×C8—122no—
144#95—22no—
145#95—22no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3C2×C22⋊C411yescharacteristic
151C23×C3C4×D412yes—
152C23×C3C22×C811yescharacteristic
153C23×C3C2×M1611yescharacteristic
154C23×C3#13112yes—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—24no—
164C2×C3×C4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—28no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—44no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—44no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—84no—
180C2×C3×C4C22⋊C812yes—
181C2×C3×C4C22⋊C812yes—
182C2×C3×C4#9912yes—
183C2×C3×C4#10112yes—
184C2×C3×C4#10112yes—
185C2×C3×C4#12112yes—
186C2×C3×C4#12411yescharacteristic
187C2×C3×C4#12711yescharacteristic
188C3⋊C8—44no—
189C3⋊C8—82no—
190C24C22×Dic311yescharacteristic
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4C2×C3⋊C814yes—
207C22×C4C2×C3⋊C812yes—
208C22×C4#20812yes—
209C22×C4#21711yescharacteristic
210C22×C4#21711yescharacteristic
211C22×C4#21711yescharacteristic
212C22×C4#21711yescharacteristic
213C22×C4D3×D412yes—
214C22×C4#23712yes—
215C22×C4C2×#6711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217C2×C8—64no—
218C2×C8—122no—
219C2×C8—122no—
220C2×C8—122no—
221C2×C8—122no—
222C2×C8—122no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—41no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3#39011yescharacteristic
243C22×C3C2×C22⋊C811yescharacteristic
244C22×C3#40811yescharacteristic
245C22×C3C2×#9911yescharacteristic
246C22×C3#41011yescharacteristic
247C22×C3C2×#10111yescharacteristic
248C22×C3C2×#10111yescharacteristic
249C22×C3C8×D412yes—
250C22×C3#43612yes—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—44no—
257C3×C4—41no—
258C3×C4—82no—
259C3×C4#32414yescomplemented
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C22×C3⋊C811yescharacteristic
265C23C2×#20811yescharacteristic
266C23D4×Dic312yes—
267C23#94812yes—
268C23C2×#21711yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—28no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—84no—
294C2×C4#83012yes—
295C2×C4#83012yes—
296C2×C4#83412yes—
297C2×C4#83612yes—
298C2×C4#83612yes—
299C2×C4#93611yescharacteristic
300C2×C4#93812yes—
301C2×C4#94011yescharacteristic
302C8—124no—
303C8—242no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3C2×#32411yescharacteristic
307C2×C3#146111yescharacteristic
308C2×C3#147011yescharacteristic
309C2×C3#147211yescharacteristic
310C2×C3#195411yescharacteristic
311C2×C3#196411yescharacteristic
312C2×C3#196611yescharacteristic
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22D4×C3⋊C812yes—
322C22#501212yes—
323C22C2×#83011yescharacteristic
324C22C2×#83411yescharacteristic
325C22#522111yescharacteristic
326C22C2×#83611yescharacteristic
327C22C2×#83611yescharacteristic
328C22#523411yescharacteristic
329C22#524011yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—22no—
334C4—22no—
335C4—41no—
336C4—44no—
337C4—82no—
338C4#452814yescomplemented
339C3#965011yescharacteristic, complemented, Sylow
340C2—22no—
341C2—22no—
342C2C2×#452811yescharacteristic
343C2#7161411yescharacteristic
344C2#7162411yescharacteristic
345C2#7168611yescharacteristic
346C2#7169111yescharacteristic
347C2#7173711yescharacteristic
348C2#7178911yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC2×C3×#389C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC23×C3×C4C23
hypercenterC2×#389D3
hypocenterC3#9650
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
41. C2×C3×C4⋊C4C8semidirect
133. C2×C4⋊C4C3⋊C8semidirect
259. C3×C4#324semidirect
338. C4#4528semidirect
339. C3#9650semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#45282
#715302
C4×#8302
#96503