Finite Group Databasebeta

← #10690818

Subgroups — #10690818

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2851C211yesmaximal, characteristic
2C2×#4926C211yesmaximal, characteristic
3C2×#4543C212yesmaximal, complemented
4#71529C211yesmaximal, characteristic
5#67630C212yesmaximal, complemented
6#9126—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9C3×#389C411yescharacteristic
10C3×#389C411yescharacteristic
11C3×#387—21no—
12C3×#387—21no—
13C3×#385—21no—
14C3×#385—21no—
15C3×#382C2211yescharacteristic
16C3×#380C411yescharacteristic
17C3×#380C411yescharacteristic
18C2×#830—21no—
19C2×#830—21no—
20#4926—24no—
21C2×C4⋊(C3⋊C8)—21no—
22C2×#803C2212yes—
23C2×#803C2212yes—
24#4543—24no—
25#4524—21no—
26C2×#432—31no—
27#2851D311yescharacteristic
28#1749—31no—
29C2×#330—32no—
30#1295—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C3×#125—22no—
35C3×#125—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C42—22no—
47C2×C3×C42—22no—
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C22×C3⋊C8—22no—
53C22×C3⋊C8—22no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—24no—
61#830—24no—
62C4⋊(C3⋊C8)—24no—
63#803—22no—
64#803—22no—
65#803—22no—
66#803D414yes—
67C2×#125C2×D311yescharacteristic
68C22×C42C2×D311yescharacteristic
69#432—64no—
70C2×C4⋊C8—61no—
71C2×C22⋊C8—61no—
72C2×C22⋊C8—61no—
73C2×#97—32no—
74C2×#97—32no—
75#389Dic311yescharacteristic
76#389Dic311yescharacteristic
77#387—21no—
78#387—21no—
79#385—21no—
80#385—21no—
81#382C2×D311yescharacteristic
82#380Dic311yescharacteristic
83#380Dic311yescharacteristic
84#336—61no—
85#330—64no—
86C24×C3C22⋊C411yescharacteristic
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—21no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—42no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C42—22no—
119C3×C42—21no—
120C3×C42—22no—
121C3×C42—21no—
122C3×C42C2×D412yes—
123C3×C42#4112yes—
124C2×C3⋊C8—24no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—42no—
127C2×C3⋊C8—42no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—42no—
130C23×C4C2×Dic311yescharacteristic
131C23×C4#6711yescharacteristic
132C23×C4#6711yescharacteristic
133C22×C8—62no—
134C22×C8—62no—
135#125—22no—
136#125—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C2×Dic311yescharacteristic
142C2×C4⋊C4C2×Dic311yescharacteristic
143C2×C4⋊C4C2×Dic311yescharacteristic
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42C2×Dic311yescharacteristic
150C2×C42#6711yescharacteristic
151C2×C42#6711yescharacteristic
152C2×C42C22×D311yescharacteristic
153C4⋊C8—64no—
154C22⋊C8—64no—
155C22⋊C8—64no—
156#97—34no—
157#97—62no—
158#97—62no—
159#97—62no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3#10112yes—
173C23×C3C2×C22⋊C411yescharacteristic
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4#10412yes—
205C2×C3×C4C4×D412yes—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12112yes—
208C2×C3×C4#12311yescharacteristic
209C3⋊C8—44no—
210C3⋊C8—44no—
211C24#21711yescharacteristic
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4#21711yescharacteristic
227C22×C4#21711yescharacteristic
228C22×C4#21711yescharacteristic
229C22×C4C22×Dic311yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232C2×C8—64no—
233C2×C8—64no—
234C2×C8—122no—
235C2×C8—122no—
236C2×C8—122no—
237C2×C8—122no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C42—22no—
250C42—21no—
251C42—21no—
252C42—22no—
253C42D3×D412yes—
254C42#23712yes—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3#38611yescharacteristic
263C22×C3C2×#10111yescharacteristic
264C22×C3#41311yescharacteristic
265C22×C3C2×#10411yescharacteristic
266C22×C3#42111yescharacteristic
267C22×C3#42111yescharacteristic
268C22×C3#42111yescharacteristic
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—22no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—81no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23#83612yes—
287C23C2×#21711yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4#83712yes—
319C2×C4D4×Dic312yes—
320C2×C4#93511yescharacteristic
321C2×C4#93711yescharacteristic
322C2×C4#93912yes—
323C8—124no—
324C8—124no—
325C2×C3—42no—
326C2×C3#152911yescharacteristic
327C2×C3#153111yescharacteristic
328C2×C3#153411yescharacteristic
329C2×C3#153511yescharacteristic
330C2×C3#187111yescharacteristic
331C2×C3#189711yescharacteristic
332C2×C3#190111yescharacteristic
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#522011yescharacteristic
341C22C2×#83611yescharacteristic
342C22C2×#83711yescharacteristic
343C22#523611yescharacteristic
344C22#523611yescharacteristic
345C22#523611yescharacteristic
346C22#523711yescharacteristic
347C4—22no—
348C4—21no—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C3#912611yescharacteristic, complemented, Sylow
359C2—42no—
360C2#7169811yescharacteristic
361C2#7171211yescharacteristic
362C2#7171311yescharacteristic
363C2#7173611yescharacteristic
364C2#7174711yescharacteristic
365C2#7174911yescharacteristic
366C2#7182211yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2851C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2851D3
hypocenterC3#9126
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. C2×#4543C2semidirect
5. #67630C2semidirect
358. C3#9126semidirect

Maximal subgroups

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

subgroupindex
C3×#28512
C2×#49262
C2×#45432
#715292
#676302
#91263