Finite Group Databasebeta

← #10690968

Subgroups — #10690968

Order 768.

692 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2854C211yesmaximal, characteristic
2C2×#4936C211yesmaximal, characteristic, complemented
3C2×#4537C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#67624C212yesmaximal, complemented
6#9156—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C22×C4⋊Dic3—21no—
11C3×#390C412yescomplemented
12C3×#389C412yescomplemented
13C3×#388—22no—
14C3×#386—21no—
15C3×#380C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19#4936—24no—
20#4537—24no—
21#4524—22no—
22#2854D311yescharacteristic
23C2×#382—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—42no—
50C2×C4⋊Dic3—42no—
51C22×C3⋊C8—22no—
52C3×#95—22no—
53C3×#95—21no—
54C3⋊#95—24no—
55C3⋊#95—22no—
56C3⋊#95D414yes—
57#830—22no—
58#830—22no—
59C2×#123—21no—
60C2×#121C2×D311yescharacteristic
61C22×C4⋊C4—61no—
62C22×C42C2×D311yescharacteristic
63C2×C22⋊C8—32no—
64#390Dic312yescomplemented
65#389Dic312yescomplemented
66#388—22no—
67#386—21no—
68#382—64no—
69#380C2×D311yescharacteristic
70C2×#95—32no—
71C2×#95—61no—
72#336—62no—
73#327—64no—
74C24×C3—21no—
75C24×C3C2×D411yescharacteristic
76C2×C3×D4—22no—
77C2×C3×D4—42no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—22no—
91C22×C3×C4—21no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4C22⋊C412yes—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4D812yes—
97C22×C3×C4SD1612yes—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4112yes—
102C22×Dic3—24no—
103C22×Dic3—44no—
104C22×Dic3—44no—
105C22×Dic3—42no—
106C22×Dic3—42no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—44no—
110C3×C22⋊C4—44no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—44no—
114C3×C22⋊C4—42no—
115C3×C42—24no—
116C3×C42—24no—
117C4⋊Dic3—48no—
118C2×C3⋊C8—44no—
119C2×C3⋊C8—42no—
120C22×D4—21no—
121C22×D4C2×Dic311yescharacteristic
122C23×C4—62no—
123C23×C4#6711yescharacteristic
124C23×C4#6711yescharacteristic
125C23×C4C22×D311yescharacteristic
126C22×C8—62no—
127#123—42no—
128#123—42no—
129#121—22no—
130#121—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—64no—
133C2×C4⋊C4—122no—
134C2×C4⋊C4—122no—
135C2×C4⋊C4C2×Dic311yescharacteristic
136C2×C4⋊C4C2×Dic312yes—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—21no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C2×Dic312yes—
144C2×C42#6712yes—
145C22⋊C8—62no—
146C22⋊C8—62no—
147#95—22no—
148#95—21no—
149#95—34no—
150#95—64no—
151#95—62no—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3—41no—
155C23×C3—42no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—42no—
159C23×C3C2×C22⋊C411yescharacteristic
160C23×C3#12011yescharacteristic
161C23×C3#12311yescharacteristic
162C23×C3C2×D811yescharacteristic
163C23×C3C2×SD1611yescharacteristic
164C23×C3AGL(1,ℤ/8)11yescharacteristic
165C23×C3#13711yescharacteristic
166C3×D4—82no—
167C3×D4—81no—
168C3×D4—81no—
169C3×D4—81no—
170C3×D4—81no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
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—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4#10214yes—
193C2×C3×C4C4×D412yes—
194C2×C3×C4#12112yes—
195C2×Dic3—44no—
196C2×Dic3—44no—
197C2×Dic3—44no—
198C2×Dic3—84no—
199C2×Dic3—84no—
200C3⋊C8—82no—
201C24—21no—
202C24C2×#6711yescharacteristic
203C2×D4—22no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—64no—
222C22×C4—124no—
223C22×C4—124no—
224C22×C4—122no—
225C22×C4—122no—
226C22×C4#21312yes—
227C22×C4#21412yes—
228C22×C4#21712yes—
229C22×C4#21712yes—
230C22×C4D3×D412yes—
231C22×C4#23712yes—
232C22×C4C22×Dic311yescharacteristic
233C22×C4C2×#6711yescharacteristic
234C2×C8—124no—
235C2×C8—122no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—128no—
240C22⋊C4—44no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C42—24no—
246C42—24no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—81no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3—82no—
259C22×C3—82no—
260C22×C3#38611yescharacteristic
261C22×C3#41012yes—
262C22×C3C2×#10211yescharacteristic
263C22×C3#41711yescharacteristic
264C22×C3#42112yes—
265C22×C3#44711yescharacteristic
266C22×C3#44811yescharacteristic
267C22×C3#48011yescharacteristic
268C22×C3#48111yescharacteristic
269C3×C4—24no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—81no—
274C3×C4—82no—
275Dic3—84no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23C2×#21311yescharacteristic
284C23C2×#21411yescharacteristic
285C23#93511yescharacteristic
286C23#93711yescharacteristic
287C23#94911yescharacteristic
288C23#95111yescharacteristic
289C23C2×#21711yescharacteristic
290D4—82no—
291D4—81no—
292D4—81no—
293D4—81no—
294D4—81no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—124no—
317C2×C4—124no—
318C2×C4—124no—
319C2×C4—244no—
320C2×C4—244no—
321C2×C4#83214yes—
322C2×C4D4×Dic312yes—
323C2×C4#93912yes—
324C8—242no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—161no—
330C2×C3#151212yes—
331C2×C3#151712yes—
332C2×C3#187411yescharacteristic
333C2×C3#189711yescharacteristic
334C2×C3#189911yescharacteristic
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—81no—
348C22#503311yescharacteristic
349C22#503411yescharacteristic
350C22#503611yescharacteristic
351C22#504011yescharacteristic
352C22C2×#83211yescharacteristic
353C22#522011yescharacteristic
354C22#523211yescharacteristic
355C22#523612yes—
356C22#524012yes—
357C4—22no—
358C4—22no—
359C4—24no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—244no—
364C3#915611yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—161no—
370C2#7167711yescharacteristic
371C2#7170511yescharacteristic
372C2#7171211yescharacteristic
373C2#7175612yes—
374C2#7181412yes—
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#9156
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×#4936C2semidirect
4. #71531C2semidirect
5. #67624C2semidirect
11. C3×#390C4semidirect
12. C3×#389C4semidirect
64. #390Dic3semidirect
65. #389Dic3semidirect
364. C3#9156semidirect

Maximal subgroups

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

subgroupindex
C3×#28542
C2×#49362
C2×#45372
#715312
#676242
#91563