Finite Group Databasebeta

← #10690728

Subgroups — #10690728

Order 768.

586 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2322C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3#71600C211yesmaximal, characteristic
4C2×#4528C212yesmaximal, complemented
5#67617C212yesmaximal, complemented
6#9108—31nomaximal, Sylow
7C2×C3×#123C411yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C3×#389—21no—
11C3×#388—21no—
12C3×#387—21no—
13C3×#386C411yescharacteristic
14C3×#385—21no—
15C3×#385C411yescharacteristic
16C3×#381C411yescharacteristic
17C3×C4×C22⋊C4C2211yescharacteristic
18C2×#830—21no—
19C2×#830C2212yes—
20C2×#830C2212yes—
21D4×C3⋊C8—24no—
22#4925—21no—
23#4528—21no—
24#4528—24no—
25#4525—21no—
26#2972—31no—
27#2322D311yescharacteristic
28#1459—31no—
29C2×#324—32no—
30#1282—32no—
31C22×C3×D4C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C3×#123—22no—
37C3×#123C814yescomplemented
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—22no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42D411yescharacteristic
55C22×C3⋊C8—22no—
56C22×C3⋊C8—22no—
57C22×C3⋊C8—22no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63#830—22no—
64#830—22no—
65#830—22no—
66#830—22no—
67#830D414yes—
68C4⋊(C3⋊C8)—22no—
69C4×C3⋊C8—22no—
70C2×#123Dic311yescharacteristic
71C2×C4×D4C2×D311yescharacteristic
72C22×C4⋊C4C2×D311yescharacteristic
73C8×D4—64no—
74#431—61no—
75C2×C22⋊C8—32no—
76C2×C22⋊C8—32no—
77C2×C22⋊C8—61no—
78#389—21no—
79#388—21no—
80#387—21no—
81#386Dic311yescharacteristic
82#385—21no—
83#385Dic311yescharacteristic
84#381Dic311yescharacteristic
85C4×C22⋊C4C2×D311yescharacteristic
86#324—64no—
87#324—61no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C22×C411yescharacteristic
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4C2×C812yes—
94C2×C3×D4M1612yes—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—22no—
105C22×C3×C4—81no—
106C22×C3×C4—81no—
107C22×C3×C4—81no—
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C2×C812yescomplemented
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4112yes—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—42no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—81no—
122C3×C22⋊C4—24no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—81no—
128C3×C22⋊C4C2×C812yescomplemented
129C3×C42—21no—
130C3×C42—21no—
131C3×C42—81no—
132C2×C3⋊C8—24no—
133C2×C3⋊C8—42no—
134C2×C3⋊C8—82no—
135C2×C3⋊C8—82no—
136C2×C3⋊C8—82no—
137C22×D4C2×Dic311yescharacteristic
138C23×C4C2×Dic311yescharacteristic
139C23×C4C2×Dic311yescharacteristic
140C23×C4#6711yescharacteristic
141C23×C4C22×D311yescharacteristic
142C22×C8—62no—
143C22×C8—62no—
144C22×C8—62no—
145#123—22no—
146#123C3⋊C814yescomplemented
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4#6711yescharacteristic
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4C2×Dic311yescharacteristic
159C2×C22⋊C4C2×Dic311yescharacteristic
160C2×C22⋊C4C2×Dic311yescharacteristic
161C2×C22⋊C4#6711yescharacteristic
162C2×C42—21no—
163C2×C42#6711yescharacteristic
164C4⋊C8—62no—
165C22⋊C8—34no—
166C22⋊C8—62no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169C22⋊C8—62no—
170C4×C8—62no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3C22⋊C812yes—
183C23×C3C2×C22⋊C411yescharacteristic
184C23×C3C4×D412yes—
185C23×C3C22×C811yescharacteristic
186C23×C3C2×M1611yescharacteristic
187C23×C3#13112yes—
188C3×D4—24no—
189C3×D4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—24no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4—82no—
217C2×C3×C4C22⋊C812yes—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12112yes—
220C2×C3×C4#12311yescharacteristic
221C3⋊C8—44no—
222C3⋊C8—82no—
223C24#21711yescharacteristic
224C24C22×Dic311yescharacteristic
225C2×D4—22no—
226C2×D4—22no—
227C2×D4C2×C3⋊C812yes—
228C2×D4#20812yes—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C2×C3⋊C812yescomplemented
243C22×C4#21711yescharacteristic
244C22×C4#21711yescharacteristic
245C22×C4#21711yescharacteristic
246C22×C4D3×D412yes—
247C22×C4#23712yes—
248C22×C4C2×#6711yescharacteristic
249C22×C4C2×#6711yescharacteristic
250C2×C8—64no—
251C2×C8—122no—
252C2×C8—242no—
253C2×C8—242no—
254C2×C8—242no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C22⋊C4—24no—
262C22⋊C4—21no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—81no—
267C22⋊C4C2×C3⋊C812yescomplemented
268C42—21no—
269C42—21no—
270C42—81no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—81no—
283C22×C3#38611yescharacteristic
284C22×C3C2×C22⋊C811yescharacteristic
285C22×C3#40811yescharacteristic
286C22×C3C8×D412yes—
287C22×C3#43512yes—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—22no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4—42no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—81no—
297C3×C4—81no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#83012yes—
305C23C22×C3⋊C811yescharacteristic
306C23C2×#20811yescharacteristic
307C23D4×Dic312yes—
308C23#94812yes—
309C23C2×#21711yescharacteristic
310D4—24no—
311D4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—24no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4#83012yes—
340C2×C4#93511yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93912yes—
343C8—124no—
344C8—242no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—42no—
350C2×C3#186911yescharacteristic
351C2×C3#191311yescharacteristic
352C2×C3#191511yescharacteristic
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22D4×C3⋊C812yes—
366C22#501112yes—
367C22C2×#83011yescharacteristic
368C22#522011yescharacteristic
369C22#523411yescharacteristic
370C4—21no—
371C4—22no—
372C4—21no—
373C4—22no—
374C4—42no—
375C4—41no—
376C4—42no—
377C4—41no—
378C4—81no—
379C4—81no—
380C3#910811yescharacteristic, complemented, Sylow
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—42no—
386C2#7161011yescharacteristic
387C2#7184811yescharacteristic
388C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC3×#2322C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#2322D3
hypocenterC3#9108
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2322

Complements

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

NH ≅ G/Ntype
4. C2×#4528C2semidirect
5. #67617C2semidirect
37. C3×#123C8semidirect
111. C22×C3×C4C2×C8semidirect
128. C3×C22⋊C4C2×C8semidirect
146. #123C3⋊C8semidirect
242. C22×C4C2×C3⋊C8semidirect
267. C22⋊C4C2×C3⋊C8semidirect
380. C3#9108semidirect

Maximal subgroups

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

subgroupindex
C3×#23222
#823012
#716002
C2×#45282
#676172
#91083