Finite Group Databasebeta

← #10944107

Subgroups — #10944107

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×C3×#519C211yesmaximal, characteristic
2C2×#5228C212yesmaximal
3C2×#5031C212yesmaximal, complemented
4#82262C211yesmaximal, characteristic
5#71632C214yesmaximal, complemented
6#71627C212yesmaximal, complemented
7#71585C212yesmaximal
8#71583C211yesmaximal, characteristic
9#32915—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C3×#519—28no—
12C2×C3×C4×Q8—22no—
13C2×C3×C4×Q8C2212yes—
14C2×C3×C4×Q8C2212yes—
15C2×C3×#117—22no—
16C2×C3×#117C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×#216C2214yes—
19C2×#924C2212yes—
20C2×#869C2211yescharacteristic
21#5228—28no—
22C2×#835C2214yes—
23C2×#830C2212yes—
24#5031—28no—
25#4981—22no—
26C2×#810C2214yes—
27#4953—22no—
28#4936C2211yescharacteristic
29#4929C2212yes—
30C2×C4⋊(C3⋊C8)C2212yes—
31#4529C2214yes—
32#4524C2212yes—
33C2×#519D311yescharacteristic
34C2×#462—32no—
35C2×#451—32no—
36#2621—31no—
37#1878—34no—
38#1873—32no—
39#1805—32no—
40#1798—31no—
41C22×C3×Q8—24no—
42C22×C3×Q8—24no—
43C22×C3×Q8—21no—
44C22×C3×Q8—22no—
45C22×C3×Q8D414yes—
46C22×C3×Q8C2312yes—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6C2312yes—
50C3×C4×Q8—216no—
51C3×C4×Q8—24no—
52C3×C4×Q8—24no—
53C3×C4×Q8—24no—
54C3×C4×Q8D418yes—
55C3×#117—216no—
56C3×#117—24no—
57C3×#117—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4D412yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4D412yes—
64C2×C3×C42—22no—
65C2×C3×C42—22no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×#217—22no—
70C2×#216—28no—
71C2×#216—28no—
72C2×#216—28no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3C2312yes—
75#924—24no—
76#924—24no—
77C2×Dic3⋊C4—22no—
78C22×C3⋊C8C2314yes—
79#869—24no—
80#835—28no—
81C3⋊#95—24no—
82C3⋊#95—22no—
83#830—24no—
84#810—28no—
85C4⋊(C3⋊C8)—24no—
86C23×Q8C2×D311yescharacteristic
87C22×Q16—34no—
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—32no—
90#519—28no—
91C2×C4×Q8—22no—
92C2×C4×Q8C2×D312yes—
93C2×C4×Q8C2×D312yes—
94C2×#117—22no—
95C2×#117C2×D311yescharacteristic
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—62no—
104#397—62no—
105#391—32no—
106#382—31no—
107#340—34no—
108#336—32no—
109C24×C3C2×D411yescharacteristic
110C2×C3×Q8—24no—
111C2×C3×Q8—216no—
112C2×C3×Q8—24no—
113C2×C3×Q8—216no—
114C2×C3×Q8—24no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8C2×D414yes—
119C2×C3×Q8C2×D414yes—
120C2×C3×Q8#4114yes—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4112yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C2×Dic6—24no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—24no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4C2×D414yes—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—24no—
147C3×C42—28no—
148C3×C42—24no—
149C3×C42—28no—
150C3×C42—22no—
151C3×C42C2×D414yes—
152#217—44no—
153#216—416no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—44no—
157Dic3⋊C4—44no—
158C2×C3⋊C8—24no—
159C2×C3⋊C8—24no—
160C2×C3⋊C8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—32no—
166C22×Q8D1214yes—
167C22×Q8C22×D312yes—
168C23×C4—21no—
169C23×C4C22×D311yescharacteristic
170C2×Q16—68no—
171C2×Q16—68no—
172C2×Q16—68no—
173C22×C8—34no—
174C4⋊Q8—64no—
175C22⋊Q8—64no—
176C22⋊Q8—64no—
177C4×Q8—216no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8#6718yes—
182#117—216no—
183#117—24no—
184#117—24no—
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—21no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4#6712yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42C22×D311yescharacteristic
199C2×C42C22×D311yescharacteristic
200C4⋊C8—64no—
201#103—68no—
202#103—68no—
203C22⋊C8—64no—
204#95—62no—
205#95—64no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—24no—
209C23×C3#12112yes—
210C23×C3C22×D411yescharacteristic
211C3×Q8—216no—
212C3×Q8—24no—
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8#12118yes—
216C2×C3×C4—216no—
217C2×C3×C4—24no—
218C2×C3×C4—21no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—44no—
229C2×C3×C4#12014yes—
230C2×C3×C4#12112yes—
231C2×C3×C4C22×D411yescharacteristic
232C2×C3×C4C22×D412yes—
233C2×C3×C4C2×#4112yes—
234C2×C3×C4C2×#4111yescharacteristic
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—44no—
240Dic6—44no—
241Dic6—44no—
242C3⋊C8—44no—
243C24C2×#6711yescharacteristic
244C2×Q8—24no—
245C2×Q8—216no—
246C2×Q8—24no—
247C2×Q8—216no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×Q8—64no—
255C2×Q8—64no—
256C2×Q8C2×D1214yes—
257C2×Q8#23514yes—
258C2×Q8C2×#6714yes—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4C2×D1212yes—
269C22×C4#23512yes—
270C22×C4C2×#6712yes—
271C22×C4C2×#6711yescharacteristic
272C22×C4C2×#6712yes—
273C22×C4C23×D311yescharacteristic
274Q16—1216no—
275C2×C8—64no—
276C2×C8—64no—
277C2×C8—64no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—124no—
285C4⋊C4—124no—
286C4⋊C4C2×#6714yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—124no—
290C42—28no—
291C42—24no—
292C42—28no—
293C42—22no—
294C42C2×#6714yes—
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—28no—
304C3×C4—22no—
305C3×C4—24no—
306C3×C4—42no—
307C3×C4—44no—
308C3×C4C2×#12112yes—
309C3×C4#52512yes—
310Dic3—82no—
311Dic3—82no—
312C23—21no—
313C23—24no—
314C23—21no—
315C23#93012yes—
316C23C22×#6711yescharacteristic
317Q8—216no—
318Q8—24no—
319Q8—24no—
320Q8—24no—
321Q8—124no—
322Q8—124no—
323Q8#93018yes—
324C2×C4—216no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—44no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—124no—
341C2×C4—122no—
342C2×C4#93012yes—
343C2×C4#95314yes—
344C2×C4C22×D1211yescharacteristic
345C2×C4C2×#23512yes—
346C2×C4C2×#23511yescharacteristic
347C2×C4C22×#6712yes—
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×#93011yescharacteristic
359C22#579111yescharacteristic
360C22#583814yes—
361C22C2×#95311yescharacteristic
362C4—28no—
363C4—22no—
364C4—24no—
365C4—42no—
366C4—44no—
367C4—242no—
368C4—242no—
369C4C2×#93012yes—
370C4#579112yes—
371C3#3291511yescharacteristic, complemented, Sylow
372C2—42no—
373C2#8255412yes—
374C2#8255512yes—
375C2#8403211yescharacteristic
376C2#8411712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#519C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#519D3
hypocenterC3#32915
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
3. C2×#5031C2semidirect
5. #71632C2semidirect
6. #71627C2semidirect
371. C3#32915semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×#52282
C2×#50312
#822622
#716322
#716272
#715852
#715832
#329153