Finite Group Databasebeta

← #10690868

Subgroups — #10690868

Order 768.

568 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C2×#4926C211yesmaximal, characteristic
3C2×#4546C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67634C212yesmaximal, complemented
6#9136—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8C412yes—
9C22×C3×C42C2211yescharacteristic
10C3×#391C412yes—
11C3×#389—21no—
12C3×#389—21no—
13C3×#384C2211yescharacteristic
14C3×#380—21no—
15C3×#380—21no—
16C2×#830—21no—
17C2×#830—21no—
18#4926—24no—
19C2×C4⋊(C3⋊C8)C2212yes—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×#803—21no—
22#4546—24no—
23#4524—21no—
24C2×#432—31no—
25#2857D311yescharacteristic
26#1797—31no—
27C2×#333—32no—
28#1296—32no—
29C22×C3×Q8C2×C412yes—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C3×C22⋊Q8—44no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C22⋊C4—22no—
43C2×C3×C42—22no—
44C2×C3×C42—22no—
45C2×C3×C42C2×C411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49C22×C3⋊C8—22no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53#830—24no—
54#830—42no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C4⋊(C3⋊C8)D414yes—
59#803—24no—
60C2×C4⋊Q8C2×D311yescharacteristic
61C2×C22⋊Q8Dic312yes—
62C22×C42C2×D311yescharacteristic
63#432—64no—
64C2×C4⋊C8—32no—
65C2×C4⋊C8—32no—
66C2×C22⋊C8—61no—
67C2×C22⋊C8—61no—
68C2×#97—61no—
69#391Dic312yes—
70#389—21no—
71#389—21no—
72#384C2×D311yescharacteristic
73#380—21no—
74#380—21no—
75#336—61no—
76#333—64no—
77C24×C3C22⋊C411yescharacteristic
78C2×C3×Q8—24no—
79C2×C3×Q8—42no—
80C2×C3×Q8—42no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—21no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—41no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C3×C4⋊C4—42no—
101C3×C4⋊C4—44no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—42no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C22⋊C4—44no—
109C3×C42—22no—
110C3×C42—21no—
111C3×C42—22no—
112C3×C42—21no—
113C3×C42C2×D412yes—
114C3×C42#4112yes—
115C2×C3⋊C8—24no—
116C2×C3⋊C8—42no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—42no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—42no—
121C22×Q8C2×Dic312yes—
122C23×C4C2×Dic311yescharacteristic
123C23×C4#6711yescharacteristic
124C23×C4#6711yescharacteristic
125C22×C8—62no—
126C22×C8—62no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C22⋊Q8—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4C2×Dic312yes—
136C2×C22⋊C4—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42C2×Dic311yescharacteristic
140C2×C42#6711yescharacteristic
141C2×C42#6711yescharacteristic
142C2×C42C22×D311yescharacteristic
143C4⋊C8—34no—
144C4⋊C8—62no—
145C4⋊C8—62no—
146C4⋊C8—62no—
147C22⋊C8—64no—
148C22⋊C8—122no—
149#97—64no—
150#95—22no—
151#95—22no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3#10012yes—
159C23×C3C2×C22⋊C411yescharacteristic
160C3×Q8—84no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—44no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
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×C4C4×D412yes—
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12112yes—
186C2×C3×C4#12311yescharacteristic
187C2×C3×C4#13512yes—
188C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
189C2×C3×C4#13712yes—
190C2×C3×C4#13712yes—
191C2×C3×C4#13711yescharacteristic
192C3⋊C8—44no—
193C3⋊C8—82no—
194C24#21711yescharacteristic
195C2×Q8—24no—
196C2×Q8—42no—
197C2×Q8—42no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4#21711yescharacteristic
212C22×C4#21711yescharacteristic
213C22×C4#21711yescharacteristic
214C22×C4C22×Dic311yescharacteristic
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—42no—
224C4⋊C4—44no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C22⋊C4—44no—
232C42—22no—
233C42—21no—
234C42—21no—
235C42—22no—
236C42D3×D412yes—
237C42#23712yes—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3#38611yescharacteristic
246C22×C3C2×#10011yescharacteristic
247C22×C3#41311yescharacteristic
248C22×C3#41611yescharacteristic
249C22×C3#41711yescharacteristic
250C22×C3#41911yescharacteristic
251C22×C3#41911yescharacteristic
252C3×C4—22no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—82no—
257C3×C4—82no—
258C3×C4#45212yes—
259C3×C4#46812yes—
260C3×C4#47012yes—
261C3×C4#48512yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#83312yes—
269C23C2×#21711yescharacteristic
270Q8—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4D4×Dic312yes—
294C2×C4#93511yescharacteristic
295C2×C4#93711yescharacteristic
296C2×C4#93912yes—
297C2×C4#94212yes—
298C2×C4#94212yes—
299C2×C4#94911yescharacteristic
300C2×C4#95012yes—
301C2×C4#95111yescharacteristic
302C8—124no—
303C8—242no—
304C2×C3—42no—
305C2×C3#155111yescharacteristic
306C2×C3#155411yescharacteristic
307C2×C3#155811yescharacteristic
308C2×C3#155911yescharacteristic
309C2×C3#187311yescharacteristic
310C2×C3#189411yescharacteristic
311C2×C3#190011yescharacteristic
312C22—22no—
313C22—22no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22C2×#83311yescharacteristic
320C22#522011yescharacteristic
321C22#522511yescharacteristic
322C22#522511yescharacteristic
323C22#523211yescharacteristic
324C22#523311yescharacteristic
325C22#523711yescharacteristic
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—82no—
331C4—82no—
332C4#504512yes—
333C4#504712yes—
334C4#504912yes—
335C4#505212yes—
336C3#913611yescharacteristic, complemented, Sylow
337C2—42no—
338C2#7168311yescharacteristic
339C2#7169311yescharacteristic
340C2#7169711yescharacteristic
341C2#7177711yescharacteristic
342C2#7180711yescharacteristic
343C2#7183211yescharacteristic
344C2#7183311yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2857C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2857D3
hypocenterC3#9136
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. C2×#4546C2semidirect
5. #67634C2semidirect
336. C3#9136semidirect

Maximal subgroups

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

subgroupindex
C3×#28572
C2×#49262
C2×#45462
#715302
#676342
#91363