Finite Group Databasebeta

← #10690928

Subgroups — #10690928

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2854C211yesmaximal, characteristic
2C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#4537C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#67624C212yesmaximal, complemented
6#9148—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C22×C4⋊Dic3—21no—
11C22×C4×Dic3—21no—
12C3×#390C412yescomplemented
13C3×#389C412yescomplemented
14C3×#388—22no—
15C3×#386—21no—
16C3×#380C2211yescharacteristic
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19C4×C4⋊Dic3—24no—
20#4537—24no—
21#4524—22no—
22#2854D311yescharacteristic
23C2×C4×C4⋊C4—31no—
24#1817—31no—
25C2×#327—32no—
26#1294—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—22no—
33C3×#123—42no—
34C3×#123—42no—
35C3×#121—22no—
36C3×#121—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C42—22no—
45C2×C3×C42—22no—
46C2×C3×C42C2×C412yes—
47C2×C3×C42D412yes—
48C2×C4⋊Dic3—24no—
49C2×C4⋊Dic3—28no—
50C2×C4×Dic3—24no—
51C2×C4×Dic3—42no—
52C2×C4×Dic3—42no—
53C22×C3⋊C8—22no—
54C3×#95—22no—
55C3×#95—21no—
56C3⋊#95—22no—
57C3⋊#95D414yes—
58#830—22no—
59#830—22no—
60C2×#123—21no—
61C2×#121C2×D311yescharacteristic
62C22×C4⋊C4—61no—
63C22×C42—61no—
64C22×C42C2×D311yescharacteristic
65C2×C22⋊C8—32no—
66#390Dic312yescomplemented
67#389Dic312yescomplemented
68#388—22no—
69#386—21no—
70#380C2×D311yescharacteristic
71C4×C4⋊C4—64no—
72C2×#95—32no—
73#336—62no—
74#327—64no—
75C24×C3—21no—
76C24×C3C2×D411yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—42no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C2×C3×D4—41no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—24no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—42no—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4C22⋊C412yes—
97C22×C3×C4D812yes—
98C22×C3×C4SD1612yes—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4112yes—
103C22×Dic3—28no—
104C22×Dic3—24no—
105C22×Dic3—44no—
106C22×Dic3—42no—
107C22×Dic3—42no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—44no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—44no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—42no—
116C3×C42—24no—
117C3×C42—24no—
118C4⋊Dic3—216no—
119C4×Dic3—48no—
120C2×C3⋊C8—44no—
121C2×C3⋊C8—42no—
122C22×D4—21no—
123C22×D4C2×Dic311yescharacteristic
124C23×C4—62no—
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—62no—
129#123—42no—
130#123—42no—
131#121—22no—
132#121—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—64no—
135C2×C4⋊C4—68no—
136C2×C4⋊C4C2×Dic311yescharacteristic
137C2×C4⋊C4C2×Dic312yes—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—64no—
145C2×C42—122no—
146C2×C42—122no—
147C2×C42C2×Dic312yes—
148C2×C42#6712yes—
149C22⋊C8—62no—
150C22⋊C8—62no—
151#95—22no—
152#95—21no—
153#95—34no—
154#95—62no—
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—41no—
158C23×C3—42no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—42no—
162C23×C3C2×C22⋊C411yescharacteristic
163C23×C3#12011yescharacteristic
164C23×C3#12311yescharacteristic
165C23×C3C2×D811yescharacteristic
166C23×C3C2×SD1611yescharacteristic
167C23×C3AGL(1,ℤ/8)11yescharacteristic
168C23×C3#13711yescharacteristic
169C3×D4—82no—
170C3×D4—81no—
171C3×D4—81no—
172C3×D4—81no—
173C3×D4—81no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4#10214yes—
195C2×C3×C4#10414yes—
196C2×C3×C4C4×D412yes—
197C2×C3×C4#12112yes—
198C2×Dic3—28no—
199C2×Dic3—44no—
200C2×Dic3—48no—
201C2×Dic3—48no—
202C2×Dic3—44no—
203C3⋊C8—82no—
204C24—21no—
205C24C2×#6711yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—68no—
225C22×C4—64no—
226C22×C4—124no—
227C22×C4—122no—
228C22×C4—122no—
229C22×C4#21312yes—
230C22×C4#21412yes—
231C22×C4#21712yes—
232C22×C4#21712yes—
233C22×C4D3×D412yes—
234C22×C4#23712yes—
235C22×C4C22×Dic311yescharacteristic
236C22×C4C2×#6711yescharacteristic
237C2×C8—124no—
238C2×C8—122no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—616no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C42—24no—
249C42—24no—
250C42—128no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—82no—
263C22×C3—82no—
264C22×C3#38611yescharacteristic
265C22×C3#41012yes—
266C22×C3C2×#10211yescharacteristic
267C22×C3#41711yescharacteristic
268C22×C3C2×#10412yes—
269C22×C3#44711yescharacteristic
270C22×C3#44811yescharacteristic
271C22×C3#48011yescharacteristic
272C22×C3#48111yescharacteristic
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—81no—
278C3×C4—82no—
279Dic3—48no—
280C23—22no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23C2×#21311yescharacteristic
288C23C2×#21411yescharacteristic
289C23#93511yescharacteristic
290C23#93711yescharacteristic
291C23#94911yescharacteristic
292C23#95111yescharacteristic
293C23C2×#21711yescharacteristic
294D4—82no—
295D4—81no—
296D4—81no—
297D4—81no—
298D4—81no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—68no—
320C2×C4—124no—
321C2×C4—128no—
322C2×C4—128no—
323C2×C4—124no—
324C2×C4#83214yes—
325C2×C4#83714yes—
326C2×C4D4×Dic312yes—
327C2×C4#93912yes—
328C8—242no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—161no—
334C2×C3#150212yes—
335C2×C3#150912yes—
336C2×C3#187411yescharacteristic
337C2×C3#189611yescharacteristic
338C2×C3#189911yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—82no—
350C22—82no—
351C22—81no—
352C22#503311yescharacteristic
353C22#503411yescharacteristic
354C22#503611yescharacteristic
355C22#504011yescharacteristic
356C22C2×#83211yescharacteristic
357C22#522011yescharacteristic
358C22#523211yescharacteristic
359C22C2×#83712yes—
360C22#524012yes—
361C4—22no—
362C4—22no—
363C4—24no—
364C4—24no—
365C4—82no—
366C4—81no—
367C4—128no—
368C3#914811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—21no—
372C2—21no—
373C2—161no—
374C2#7167711yescharacteristic
375C2#7170511yescharacteristic
376C2#7171111yescharacteristic
377C2#7175512yes—
378C2#7181112yes—
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2854C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2854D3
hypocenterC3#9148
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic3C2semidirect
4. #71531C2semidirect
5. #67624C2semidirect
12. C3×#390C4semidirect
13. C3×#389C4semidirect
66. #390Dic3semidirect
67. #389Dic3semidirect
368. C3#9148semidirect

Maximal subgroups

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

subgroupindex
C3×#28542
C2×C4×C4⋊Dic32
C2×#45372
#715312
#676242
#91483