Finite Group Databasebeta

← #10944103

Subgroups — #10944103

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic
2C3×#2621C211yesmaximal, characteristic
3C2×#5090C212yesmaximal, complemented
4C2×#5050C212yesmaximal
5#71693C212yesmaximal
6#71583C211yesmaximal, characteristic
7#69690C214yesmaximal, complemented
8#69559C212yesmaximal, complemented
9#32915—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C2×C3×#117C2211yescharacteristic
14C22×#216C2214yes—
15C2×#926C2211yescharacteristic
16#5516—28no—
17C2×#887—22no—
18C2×#877—22no—
19C2×C4×Dic6C2212yes—
20C2×C4×Dic6C2212yes—
21C3×#398—22no—
22C3×#397—22no—
23C3×#391C2212yes—
24C3×#382C2211yescharacteristic
25C2×#835C2214yes—
26C2×#830C2212yes—
27#5090—28no—
28#5050—28no—
29C2×#810C2214yes—
30C2×C4⋊(C3⋊C8)C2212yes—
31#4529C2214yes—
32#4524C2212yes—
33C2×#519—31no—
34C2×#462—32no—
35C2×#451—32no—
36#2621D311yescharacteristic
37#1878—34no—
38#1873—32no—
39#1805—32no—
40#1798—31no—
41C22×C3×Q8C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—22no—
46C22×Dic6—24no—
47C22×Dic6—24no—
48C22×Dic6D414yes—
49C22×Dic6C2312yes—
50C3×C4⋊Q8—24no—
51C3×C22⋊Q8—24no—
52C3×C22⋊Q8—24no—
53C3×#117—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4D412yes—
60C2×C3×C42C2311yescharacteristic
61C2×C3×C42C2311yescharacteristic
62C2×#217—21no—
63C2×#216—28no—
64C2×#216—28no—
65C2×#216—28no—
66#926—24no—
67C2×C4⋊Dic3C2312yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—22no—
70C2×C4×Dic3—22no—
71C2×C4×Dic3—22no—
72C2×C4×Dic3—21no—
73C22×C3⋊C8C2314yes—
74#887—216no—
75#877—216no—
76C4×Dic6—24no—
77C4×Dic6—24no—
78C4×Dic6—24no—
79C4×Dic6D418yes—
80C3×#95—22no—
81C3×#95—24no—
82#835—28no—
83#830—24no—
84#810—28no—
85C4⋊(C3⋊C8)—24no—
86C23×Q8—31no—
87C22×Q16—34no—
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×C22⋊Q8C2×D312yes—
90#519—68no—
91C2×C4×Q8—32no—
92C2×C4×Q8—32no—
93C2×C4×Q8—62no—
94C2×#117—31no—
95C2×#117—62no—
96C2×#117C2×D311yescharacteristic
97#462—68no—
98#451—68no—
99C2×C4⋊C8—32no—
100C2×#103—34no—
101C2×#103—34no—
102C2×C22⋊C8—32no—
103#398—22no—
104#397—22no—
105#391C2×D312yes—
106#382C2×D311yescharacteristic
107#340—34no—
108#336—32no—
109C24×C3C2×D411yescharacteristic
110C2×C3×Q8—24no—
111C2×C3×Q8—22no—
112C2×C3×Q8—24no—
113C2×C3×Q8—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4112yes—
123C22×C3×C4C2411yescharacteristic
124C22×Dic3—24no—
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3—22no—
128C22×Dic3—22no—
129C2×Dic6—24no—
130C2×Dic6—24no—
131C2×Dic6—216no—
132C2×Dic6—24no—
133C2×Dic6—22no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—216no—
137C2×Dic6C2×D414yes—
138C2×Dic6C2×D414yes—
139C2×Dic6#4114yes—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—44no—
144C3×C4⋊C4—44no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—44no—
147C3×C42—22no—
148C3×C42C2×D414yes—
149#217—24no—
150#216—416no—
151C4⋊Dic3—22no—
152C4⋊Dic3C2×D414yes—
153Dic3⋊C4—28no—
154Dic3⋊C4—28no—
155C4×Dic3—28no—
156C4×Dic3—24no—
157C4×Dic3—28no—
158C2×C3⋊C8—24no—
159C2×C3⋊C8—24no—
160C2×C3⋊C8—24no—
161C22×Q8—34no—
162C22×Q8—32no—
163C22×Q8—64no—
164C22×Q8—64no—
165C22×Q8—62no—
166C22×Q8—61no—
167C22×Q8C22×D312yes—
168C23×C4—61no—
169C23×C4C22×D311yescharacteristic
170C2×Q16—68no—
171C2×Q16—68no—
172C2×Q16—68no—
173C22×C8—34no—
174C4⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C4×Q8—38no—
178C4×Q8—64no—
179C4×Q8—64no—
180C4×Q8—616no—
181C4×Q8—64no—
182#117—24no—
183#117—616no—
184#117—64no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4#6712yes—
191C2×C4⋊C4C22×D312yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4#6712yes—
195C2×C42—61no—
196C2×C42—62no—
197C2×C42—62no—
198C2×C42C22×D311yescharacteristic
199C2×C42C22×D311yescharacteristic
200C4⋊C8—64no—
201#103—68no—
202#103—68no—
203C22⋊C8—64no—
204#95—22no—
205#95—24no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—24no—
209C23×C3#12112yes—
210C23×C3C22×D411yescharacteristic
211C3×Q8—44no—
212C3×Q8—44no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—44no—
225C2×C3×C4#12014yes—
226C2×C3×C4#12112yes—
227C2×C3×C4C22×D411yescharacteristic
228C2×C3×C4C22×D412yes—
229C2×C3×C4C2×#4111yescharacteristic
230C2×C3×C4C2×#4112yes—
231C2×Dic3—216no—
232C2×Dic3—24no—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—44no—
237Dic6—216no—
238Dic6—24no—
239Dic6—24no—
240Dic6—24no—
241Dic6#12118yes—
242C3⋊C8—44no—
243C24C2×#6711yescharacteristic
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—34no—
249C2×Q8—34no—
250C2×Q8—34no—
251C2×Q8—616no—
252C2×Q8—64no—
253C2×Q8—64no—
254C2×Q8—616no—
255C2×Q8—64no—
256C2×Q8—62no—
257C2×Q8—62no—
258C2×Q8—62no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—64no—
264C22×C4—64no—
265C22×C4—64no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4D3×D412yes—
269C22×C4#23712yes—
270C22×C4C2×#6712yes—
271C22×C4C2×#6712yes—
272C22×C4C2×#6711yescharacteristic
273C22×C4C23×D311yescharacteristic
274Q16—1216no—
275C2×C8—64no—
276C2×C8—64no—
277C2×C8—64no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—34no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—68no—
285C4⋊C4—68no—
286C4⋊C4—62no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—64no—
290C42—22no—
291C42—64no—
292C42—68no—
293C42—68no—
294C42D3×D414yes—
295C22×C3—22no—
296C22×C3—24no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3C2×#12011yescharacteristic
300C22×C3C2×#12111yescharacteristic
301C22×C3#52511yescharacteristic
302C22×C3#57814yes—
303C3×C4—22no—
304C3×C4—24no—
305C3×C4—42no—
306C3×C4—82no—
307C3×C4—82no—
308C3×C4C2×#12112yes—
309C3×C4#52512yes—
310Dic3—28no—
311Dic3—44no—
312C23—21no—
313C23—21no—
314C23—24no—
315C23#93812yes—
316C23C22×#6711yescharacteristic
317Q8—38no—
318Q8—44no—
319Q8—44no—
320Q8—616no—
321Q8—64no—
322Q8—64no—
323Q8—64no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—616no—
337C2×C4—64no—
338C2×C4—64no—
339C2×C4—64no—
340C2×C4—64no—
341C2×C4—124no—
342C2×C4#93714yes—
343C2×C4#93812yes—
344C2×C4C2×D3×D411yescharacteristic
345C2×C4C2×D3×D412yes—
346C2×C4C2×#23711yescharacteristic
347C2×C4D3×#4112yes—
348C8—124no—
349C2×C3—42no—
350C2×C3#284711yescharacteristic
351C2×C3#301112yes—
352C2×C3#304412yes—
353C2×C3#312712yes—
354C22—22no—
355C22—24no—
356C22—42no—
357C22—42no—
358C22C2×#93711yescharacteristic
359C22C2×#93811yescharacteristic
360C22#582811yescharacteristic
361C22#583814yes—
362C4—22no—
363C4—24no—
364C4—42no—
365C4—68no—
366C4—82no—
367C4—82no—
368C4—124no—
369C4D3×#12112yes—
370C4#560612yes—
371C3#3291511yescharacteristic, complemented, Sylow
372C2—42no—
373C2#8272311yescharacteristic
374C2#8341112yes—
375C2#8341212yes—
376C2#8345112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2621C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2621D3
hypocenterC3#32915
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. C2×#5090C2semidirect
7. #69690C2semidirect
8. #69559C2semidirect
371. C3#32915semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
C3×#26212
C2×#50902
C2×#50502
#716932
#715832
#696902
#695592
#329153