Finite Group Databasebeta

← #10690187

Subgroups — #10690187

Order 768.

800 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×#395C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic
3#72132C212yesmaximal
4C2×#4528C211yesmaximal, characteristic
5#67612C212yesmaximal
6#9015—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C22×C3×C4⋊C4—21no—
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#926—21no—
11C3×#395—22no—
12C3×#395C414yescomplemented
13C2×C3×#95—21no—
14C2×C3×#95C2211yescharacteristic
15C2×#835—24no—
16#5212—21no—
17#5212C2212yes—
18C2×#830C2212yes—
19#4984—24no—
20#4528—22no—
21C2×#395D311yescharacteristic
22#2316—31no—
23#1787—32no—
24C2×#324—31no—
25#1294—32no—
26C22×C3×Q8—21no—
27C22×C3×Q8—22no—
28C22×C3×Q8—22no—
29C22×C3×Q8—44no—
30C22×C3×Q8C2×C412yes—
31C23×C3×C4—21no—
32C23×C3×C4—21no—
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—24no—
39C2×C3×C4⋊C4—44no—
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4D412yes—
42C2×#217—22no—
43#926—42no—
44#926—42no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3D414yes—
48C2×C4×Dic3—22no—
49C22×C3⋊C8—22no—
50C3×#95—24no—
51C3×#95—21no—
52C3×#95C2×C412yes—
53#835—48no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56#830—22no—
57#830—22no—
58C23×Q8C2×D311yescharacteristic
59C2×#117—61no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×D311yescharacteristic
62C2×#103—64no—
63C2×C22⋊C8—32no—
64#395—22no—
65#395Dic314yescomplemented
66#389—64no—
67#380—32no—
68#380—61no—
69C2×#95—21no—
70C2×#95C2×D311yescharacteristic
71#324—62no—
72C24×C3C2×D411yescharacteristic
73C2×C3×Q8—28no—
74C2×C3×Q8—24no—
75C2×C3×Q8—44no—
76C2×C3×Q8—44no—
77C2×C3×Q8—44no—
78C2×C3×Q8—48no—
79C2×C3×Q8—42no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—24no—
85C22×C3×C4—24no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—42no—
98C22×C3×C4C22⋊C412yes—
99C22×C3×C4C22⋊C412yes—
100C22×C3×C4SD1612yes—
101C22×C3×C4Q1612yes—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4112yes—
106C22×Dic3—24no—
107C3×C4⋊C4—24no—
108C3×C4⋊C4—24no—
109C3×C4⋊C4—48no—
110#217—42no—
111#217—42no—
112C4⋊Dic3—44no—
113C4⋊Dic3—44no—
114C4⋊Dic3—82no—
115C4×Dic3—44no—
116C2×C3⋊C8—44no—
117C2×C3⋊C8—42no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8C2×Dic312yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—62no—
129#117—122no—
130#117—122no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—34no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4—62no—
138C2×C4⋊C4C2×Dic312yes—
139C2×C4⋊C4#6712yes—
140C2×C22⋊C4—62no—
141C2×C42—62no—
142#103—128no—
143C22⋊C8—62no—
144C22⋊C8—62no—
145#95—24no—
146#95—21no—
147#95—62no—
148#95—62no—
149#95C2×Dic312yes—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3C2×C22⋊C411yescharacteristic
153C23×C3#12011yescharacteristic
154C23×C3#12311yescharacteristic
155C23×C3C2×SD1611yescharacteristic
156C23×C3C2×Q1611yescharacteristic
157C23×C3AGL(1,ℤ/8)11yescharacteristic
158C23×C3#13711yescharacteristic
159C3×Q8—28no—
160C3×Q8—48no—
161C3×Q8—44no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—24no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—24no—
170C2×C3×C4—24no—
171C2×C3×C4—22no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—44no—
177C2×C3×C4—44no—
178C2×C3×C4—42no—
179C2×C3×C4—44no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—44no—
184C2×C3×C4—42no—
185C2×C3×C4—81no—
186C2×C3×C4#10314yescomplemented
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12112yes—
189C2×Dic3—44no—
190C2×Dic3—44no—
191C2×Dic3—44no—
192C3⋊C8—82no—
193C24C2×#6711yescharacteristic
194C2×Q8—28no—
195C2×Q8—24no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—48no—
200C2×Q8—42no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—64no—
220C22×C4#21512yes—
221C22×C4#21612yes—
222C22×C4#21712yes—
223C22×C4#21712yes—
224C22×C4D3×D412yes—
225C22×C4#23712yes—
226C22×C4C22×Dic311yescharacteristic
227C22×C4C2×#6711yescharacteristic
228C2×C8—124no—
229C2×C8—122no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—48no—
233C4⋊C4—124no—
234C4⋊C4—124no—
235C4⋊C4—242no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C42—124no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3#38611yescharacteristic
248C22×C3C2×#10311yescharacteristic
249C22×C3#41711yescharacteristic
250C22×C3#45011yescharacteristic
251C22×C3#45111yescharacteristic
252C22×C3#45612yes—
253C22×C3B(4,𝔽2)12yes—
254C22×C3#48311yescharacteristic
255C22×C3#48411yescharacteristic
256C3×C4—24no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—44no—
260C3×C4—44no—
261C3×C4—42no—
262Dic3—84no—
263C23—22no—
264C23—22no—
265C23#93511yescharacteristic
266C23#93711yescharacteristic
267C23C2×#21511yescharacteristic
268C23C2×#21611yescharacteristic
269C23#94911yescharacteristic
270C23#95111yescharacteristic
271C23C2×#21711yescharacteristic
272Q8—28no—
273Q8—48no—
274Q8—44no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—124no—
300C2×C4—124no—
301C2×C4—124no—
302C2×C4#83514yescomplemented
303C2×C4D4×Dic312yes—
304C2×C4#93912yes—
305C8—242no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3#160412yes—
311C2×C3#160512yes—
312C2×C3#187211yescharacteristic
313C2×C3#188611yescharacteristic
314C2×C3#188911yescharacteristic
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#503212yes—
324C22#504311yescharacteristic
325C22#504411yescharacteristic
326C22#504611yescharacteristic
327C22#505011yescharacteristic
328C22#509512yes—
329C22#522011yescharacteristic
330C22C2×#83511yescharacteristic
331C22#523211yescharacteristic
332C4—24no—
333C4—22no—
334C4—22no—
335C4—44no—
336C4—44no—
337C4—42no—
338C4—244no—
339C3#901511yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#6970612yes—
345C2#6970712yes—
346C2#7166911yescharacteristic
347C2#7169211yescharacteristic
348C2#7178411yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#395C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#395D3
hypocenterC3#9015
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
12. C3×#395C4semidirect
65. #395Dic3semidirect
186. C2×C3×C4#103semidirect
302. C2×C4#835semidirect
339. C3#9015semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3952
#821562
#721322
C2×#45282
#676122
#90153