Finite Group Databasebeta

← #10690792

Subgroups — #10690792

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#97C211yesmaximal, characteristic
2#82074C211yesmaximal, characteristic
3#71529C211yesmaximal, characteristic
4C2×#4465C212yesmaximal, complemented
5#67659C212yesmaximal, complemented
6#9121—31nomaximal, Sylow
7C23×C3×C8—21no—
8C22×C3×C42C2211yescharacteristic
9C2×#870C2211yescharacteristic
10C2×C3×#97—21no—
11C2×C3×#97—24no—
12C2×C3×#97C2212yes—
13#5212C411yescharacteristic
14#5212C411yescharacteristic
15#5210—21no—
16#5210—21no—
17C2×#830—21no—
18#4936C2211yescharacteristic
19#4933—21no—
20#4933—21no—
21#4928C411yescharacteristic
22#4928C411yescharacteristic
23C2×#803C2212yes—
24#4524—21no—
25#4465—24no—
26C22×#97D311yescharacteristic
27#2851—31no—
28#1749—31no—
29C2×#330—32no—
30#1295—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×C3×C8—24no—
35C22×C3×C8—24no—
36C22×C3×C8—24no—
37C22×C3×C8—22no—
38C2×C3×C42—22no—
39C2×C3×C42—22no—
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C22×C3⋊C8—22no—
55#870—22no—
56#870—22no—
57C3×#97—24no—
58C3×#97—22no—
59C3×#97D414yes—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—24no—
67#803—22no—
68#803—22no—
69C23×C8—21no—
70C2×#125—31no—
71C22×C42C2×D311yescharacteristic
72C2×C22⋊C8—61no—
73C2×#97—21no—
74C2×#97—24no—
75C2×#97—32no—
76C2×#97C2×D312yes—
77#389—31no—
78#389—31no—
79#387—61no—
80#387—61no—
81#385—61no—
82#385—61no—
83#382—31no—
84#380—31no—
85#380—31no—
86#336—61no—
87#330—64no—
88C24×C3C22⋊C411yescharacteristic
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×C3×C8—216no—
110C2×C3×C8—24no—
111C2×C3×C8—24no—
112C2×C3×C8—24no—
113C3×C42—22no—
114C3×C42—21no—
115C3×C42—22no—
116C3×C42—21no—
117C3×C42C2×D412yes—
118C3×C42#4112yes—
119#217—42no—
120#217—42no—
121C4⋊Dic3—22no—
122C4⋊Dic3—22no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125Dic3⋊C4—42no—
126Dic3⋊C4—42no—
127Dic3⋊C4—41no—
128Dic3⋊C4—42no—
129Dic3⋊C4—41no—
130C2×C3⋊C8—24no—
131C2×C3⋊C8—42no—
132C2×C3⋊C8—42no—
133C23×C4C4×D311yescharacteristic
134C23×C4D1211yescharacteristic
135C23×C4#6711yescharacteristic
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—22no—
140C22×C8—62no—
141#125—62no—
142#125—62no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42C4×D311yescharacteristic
156C2×C42D1211yescharacteristic
157C2×C42#6711yescharacteristic
158C2×C42C22×D311yescharacteristic
159C22⋊C8—64no—
160#97—24no—
161#97—22no—
162#97—62no—
163#97—62no—
164#97#6714yes—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3#10112yes—
176C23×C3#10412yes—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C2×C3×C4—24no—
180C2×C3×C4—24no—
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—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—44no—
196C2×C3×C4—44no—
197C2×C3×C4#10412yes—
198C2×C3×C4C4×D412yes—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12112yes—
201C2×C3×C4#12311yescharacteristic
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C3×C8—28no—
215C3⋊C8—44no—
216C24(C2×D3)⋊C411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4C2×C4×D311yescharacteristic
235C22×C4C2×D1211yescharacteristic
236C22×C4C2×#6711yescharacteristic
237C2×C8—216no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—64no—
242C2×C8—122no—
243C2×C8—122no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—121no—
251C4⋊C4—122no—
252C4⋊C4—121no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C42—22no—
256C42—21no—
257C42—22no—
258C42—21no—
259C42#23512yes—
260C42C2×#6712yes—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3#33014yescomplemented
268C22×C3#38611yescharacteristic
269C22×C3C2×#10111yescharacteristic
270C22×C3#41311yescharacteristic
271C22×C3C2×#10411yescharacteristic
272C22×C3C2×#10411yescharacteristic
273C22×C3C2×#10411yescharacteristic
274C22×C3#42111yescharacteristic
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—22no—
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—41no—
281C3×C4—41no—
282Dic3—81no—
283Dic3—81no—
284Dic3—81no—
285Dic3—81no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23#80512yes—
291C23#82412yes—
292C23#82512yes—
293C23C2×(C2×D3)⋊C411yescharacteristic
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4#80512yes—
325C2×C4C4×#6712yes—
326C2×C4#92911yescharacteristic
327C2×C4#93012yes—
328C2×C4#95311yescharacteristic
329C8—28no—
330C8—124no—
331C2×C3—24no—
332C2×C3C2×#33011yescharacteristic
333C2×C3#152411yescharacteristic
334C2×C3#153111yescharacteristic
335C2×C3#153211yescharacteristic
336C2×C3#187111yescharacteristic
337C2×C3#189611yescharacteristic
338C2×C3#190111yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22#446514yescomplemented
346C22C2×#80511yescharacteristic
347C22C2×#80511yescharacteristic
348C22#513211yescharacteristic
349C22C2×#82411yescharacteristic
350C22C2×#82511yescharacteristic
351C22#513911yescharacteristic
352C22#521311yescharacteristic
353C4—22no—
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—241no—
361C4—241no—
362C4—241no—
363C4—241no—
364C3#912111yescharacteristic, complemented, Sylow
365C2—24no—
366C2#6979711yescharacteristic
367C2#6979911yescharacteristic
368C2C2×#446511yescharacteristic
369C2#6980111yescharacteristic
370C2#7153211yescharacteristic
371C2#7198411yescharacteristic
372C2#7200311yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC22×C3×#97C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC22×#97D3
hypocenterC3#9121
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
4. C2×#4465C2semidirect
5. #67659C2semidirect
267. C22×C3#330semidirect
345. C22#4465semidirect
364. C3#9121semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#972
#820742
#715292
C2×#44652
#676592
#91213