Finite Group Databasebeta

← #10690790

Subgroups — #10690790

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
1#82074C211yesmaximal, characteristic
2C22×#803C211yesmaximal, characteristic
3C3×#1749C211yesmaximal, characteristic
4C2×#4465C212yesmaximal, complemented
5#67659C212yesmaximal, complemented
6#9121—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C23×C3⋊C8—21no—
9C2×#870C2211yescharacteristic
10C2×C3×C22⋊C8—21no—
11C2×C3×#97C2212yes—
12#5212C411yescharacteristic
13#5212C411yescharacteristic
14#5210—21no—
15#5210—21no—
16#4936C2211yescharacteristic
17#4933—21no—
18#4933—21no—
19#4928C411yescharacteristic
20#4928C411yescharacteristic
21C2×#803—21no—
22C2×#803—24no—
23C2×#803C2212yes—
24C3×#336—21no—
25#4465—24no—
26C22×#97—31no—
27#2851—31no—
28#1749D311yescharacteristic
29C2×#330—32no—
30#1295—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×C3×C8—22no—
35C2×C3×C42—22no—
36C2×C3×C42—22no—
37C2×C3×C42C2×C411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C22×C3⋊C8—24no—
52C22×C3⋊C8—24no—
53C22×C3⋊C8—24no—
54C22×C3⋊C8—22no—
55#870—22no—
56#870—22no—
57C3×C22⋊C8—24no—
58C3×#97—22no—
59C3×#97—22no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#803—24no—
67#803—22no—
68#803D414yes—
69C23×C8—61no—
70C2×#125—31no—
71C22×C42C2×D311yescharacteristic
72C2×C22⋊C8—21no—
73C2×#97—32no—
74C2×#97—64no—
75C2×#97—61no—
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—21no—
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—24no—
110C2×C3×C8—42no—
111C2×C3×C8—42no—
112C3×C42—22no—
113C3×C42—21no—
114C3×C42—22no—
115C3×C42—21no—
116C3×C42C2×D412yes—
117C3×C42#4112yes—
118#217—42no—
119#217—42no—
120C4⋊Dic3—22no—
121C4⋊Dic3—22no—
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124Dic3⋊C4—42no—
125Dic3⋊C4—42no—
126Dic3⋊C4—41no—
127Dic3⋊C4—42no—
128Dic3⋊C4—41no—
129C2×C3⋊C8—216no—
130C2×C3⋊C8—24no—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—24no—
133C23×C4C4×D311yescharacteristic
134C23×C4D1211yescharacteristic
135C23×C4#6711yescharacteristic
136C22×C8—22no—
137C22×C8—64no—
138C22×C8—64no—
139C22×C8—64no—
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—24no—
160#97—22no—
161#97—22no—
162#97—34no—
163#97—64no—
164#97—62no—
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—44no—
215C3⋊C8—28no—
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—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—616no—
241C2×C8—64no—
242C2×C8—64no—
243C2×C8—64no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—122no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C42—22no—
256C42—21no—
257C42—21no—
258C42—22no—
259C42D3×D412yes—
260C42#23712yes—
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#82512yes—
325C2×C4#67⋊C412yes—
326C2×C4#88211yescharacteristic
327C2×C4#88611yescharacteristic
328C2×C4#93912yes—
329C8—44no—
330C8—68no—
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
347C22#495811yescharacteristic
348C22#500511yescharacteristic
349C22#513211yescharacteristic
350C22C2×#82411yescharacteristic
351C22C2×#82511yescharacteristic
352C22C2×#82511yescharacteristic
353C4—22no—
354C4—21no—
355C4—22no—
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—241no—
361C4—241no—
362C4—241no—
363C4—241no—
364C3#912111yescharacteristic, complemented, Sylow
365C2—24no—
366C2#6945311yescharacteristic
367C2C2×#446511yescharacteristic
368C2#6985811yescharacteristic
369C2#6996511yescharacteristic
370C2#6998611yescharacteristic
371C2#7080711yescharacteristic
372C2#7081311yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1749C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1749D3
hypocenterC3#9121
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

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
#820742
C22×#8032
C3×#17492
C2×#44652
#676592
#91213