Finite Group Databasebeta

← #10819920

Subgroups — #10819920

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5076C211yesmaximal, characteristic
2#82543C212yesmaximal, complemented
3#78916C212yesmaximal, complemented
4C3×#1887C211yesmaximal, characteristic
5#71763C211yesmaximal, characteristic
6#70315C212yesmaximal, complemented
7#70073C212yesmaximal, complemented
8#68765C212yesmaximal, complemented
9#68741C212yesmaximal, complemented
10#23198—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C3×#518C2211yescharacteristic
13#5798C2211yescharacteristic
14C22×#214C2212yes—
15C22×#214C2212yes—
16#5516C2212yes—
17C2×#869C2211yescharacteristic
18C3×#418C2212yes—
19C2×C3×#100C2211yescharacteristic
20C2×#833C2211yescharacteristic
21#5215C2212yes—
22C2×#824C2212yes—
23C2×#822C2212yes—
24#5122C2212yes—
25#5120C2212yes—
26#5076C418yescomplemented
27#5018—24no—
28C4×#214—24no—
29#5004C2212yes—
30#4998C2212yes—
31#4997C2212yes—
32C2×C4×C3⋊C8C2211yescharacteristic
33#4758—24no—
34#4757—24no—
35C3×#339C2212yes—
36#4523C2212yes—
37C2×#492—31no—
38#2949—32no—
39#2949—32no—
40#1977—32no—
41#1977—32no—
42#1887—31no—
43#1887D311yescharacteristic
44#1886—32no—
45#1772—32no—
46C22×C3×D4—41no—
47C22×C3×D4C2311yescharacteristic
48C22×C3×D4C2311yescharacteristic
49C22×Dic6C2312yes—
50C2×C3×M16C2312yes—
51C3×#127—41no—
52C3×#127C2×C414yes—
53C3×C4×D4—24no—
54C3×#117C2312yes—
55C2×C3×C22⋊C4—41no—
56C2×C3×C42C2311yescharacteristic
57C2×#217—41no—
58D4×Dic3—24no—
59C2×#214—22no—
60C2×#214—22no—
61C2×#214—22no—
62C2×#214—22no—
63C2×#214—24no—
64C2×#214—24no—
65C2×#214C2×C418yescomplemented
66C2×#214C2×C418yescomplemented
67#926C2312yes—
68C2×C4⋊Dic3—41no—
69C2×#208C2312yes—
70C22×C3⋊C8C2312yes—
71#887—24no—
72#877—42no—
73#869—41no—
74#869C2×C414yescomplemented
75C4×Dic6—24no—
76C3×#102—24no—
77C3×#100—41no—
78#833—41no—
79#832—24no—
80#824—42no—
81#822—24no—
82#816—24no—
83#815—22no—
84C8×Dic3—22no—
85#810—24no—
86#808D414yescomplemented
87#807D414yescomplemented
88#803—22no—
89C4×C3⋊C8—22no—
90C4×C3⋊C8C2×C414yescomplemented
91C22×SD16—32no—
92C22×SD16—32no—
93C2×C4⋊Q8—31no—
94C2×#127C2×D311yescharacteristic
95#519—32no—
96#518—31no—
97#518C2×D311yescharacteristic
98#492—38no—
99#440—64no—
100#440—64no—
101C4×SD16—64no—
102C4×SD16—64no—
103C2×#108—32no—
104#427—32no—
105#419—32no—
106#419—32no—
107#418—32no—
108#418C2×D312yes—
109C2×#101—32no—
110C2×#100—31no—
111C2×#100C2×D311yescharacteristic
112#405—32no—
113#405—32no—
114C2×C4×C8—31no—
115#339—32no—
116#339C2×D312yes—
117C24×C3—21no—
118C24×C3—21no—
119C2×C3×D4—22no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—44no—
126C2×C3×D4—84no—
127C2×C3×D4C22×C412yes—
128C2×C3×D4C22×C412yes—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—42no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6C22×C414yes—
140C3×M16—22no—
141C3×M16—22no—
142C2×C3×C8—22no—
143C3×C4⋊C4C2×D414yes—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—42no—
146C3×C42—22no—
147C3×C42—21no—
148C3×C42C22×C412yes—
149#217—22no—
150#217—42no—
151#214—28no—
152#214—28no—
153#214—28no—
154#214—28no—
155C4⋊Dic3—44no—
156C4⋊Dic3C2×D414yes—
157Dic3⋊C4—44no—
158C4×Dic3—22no—
159C4×Dic3—44no—
160#208—22no—
161#208—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8C22×C414yes—
164C2×C3⋊C8C2×D412yes—
165C2×C3⋊C8C2×D412yes—
166C2×C3⋊C8#4114yes—
167C22×Q8—32no—
168C22×D4—41no—
169C22×D4C22×D311yescharacteristic
170C22×D4C22×D311yescharacteristic
171C2×SD16—38no—
172C2×SD16—38no—
173C2×SD16—64no—
174C2×SD16—64no—
175C2×SD16—62no—
176C2×SD16—62no—
177C2×SD16—62no—
178C2×SD16—62no—
179C2×M16—32no—
180C2×M16C22×D312yes—
181C22×C8—32no—
182C4⋊Q8—34no—
183C4⋊Q8—121no—
184#127—41no—
185#127C4×D314yes—
186C4×Q8—64no—
187C4×Q8—64no—
188C4×D4—24no—
189C4×D4—64no—
190#117—32no—
191#117—122no—
192#117C22×D312yes—
193C2×C4⋊C4—121no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—121no—
196C2×C42C22×D311yescharacteristic
197#108—64no—
198#107—34no—
199#106—34no—
200#103—64no—
201#103—64no—
202#102—24no—
203#102—64no—
204#101—122no—
205#100—41no—
206#100—121no—
207#97—62no—
208#97—62no—
209C4×C8—34no—
210C4×C8—62no—
211C4×C8—62no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—82no—
219C23×C3—82no—
220C23×C3H(5,𝔽2)11yescharacteristic
221C3×D4—24no—
222C3×D4—24no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—88no—
226C2×C3×C4—24no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—162no—
230C2×C3×C4C23×C411yescharacteristic
231C2×C3×C4C22×D412yes—
232C2×C3×C4C2×#4112yes—
233C2×C3×C4H(5,𝔽2)11yescharacteristic
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—162no—
239Dic6—24no—
240Dic6—24no—
241Dic6—42no—
242Dic6—42no—
243C3×C8—42no—
244C3⋊C8—42no—
245C3⋊C8C4×D414yes—
246C3⋊C8C4×D414yes—
247C24—21no—
248C24—21no—
249C2×Q8—34no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×Q8—62no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—84no—
263C2×D4C2×C4×D312yes—
264C2×D4C2×C4×D312yes—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—62no—
268C22×C4—122no—
269C22×C4C23×D311yescharacteristic
270SD16—68no—
271SD16—68no—
272SD16—68no—
273SD16—68no—
274M16—22no—
275M16—22no—
276M16—62no—
277M16—62no—
278C2×C8—22no—
279C2×C8—34no—
280C2×C8—32no—
281C2×C8—32no—
282C2×C8—34no—
283C2×C8—62no—
284C4⋊C4—34no—
285C4⋊C4—124no—
286C4⋊C4—124no—
287C4⋊C4D3×D414yes—
288C22⋊C4—22no—
289C22⋊C4—42no—
290C22⋊C4—62no—
291C22⋊C4—122no—
292C42—22no—
293C42—21no—
294C42—62no—
295C42—124no—
296C42C2×C4×D312yes—
297C22×C3—22no—
298C22×C3—42no—
299C22×C3—42no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3—82no—
304C22×C3—161no—
305C22×C3—161no—
306C22×C3#51811yescharacteristic
307C22×C3D4211yescharacteristic
308C22×C3#55911yescharacteristic
309C3×C4—24no—
310C3×C4—44no—
311C3×C4C2×C4×D412yes—
312C3×C4#55012yes—
313Dic3—44no—
314Dic3—44no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—82no—
322C23—82no—
323C23#100411yescharacteristic
324Q8—64no—
325Q8—64no—
326Q8—122no—
327Q8—122no—
328D4—24no—
329D4—24no—
330D4—42no—
331D4—42no—
332D4—88no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—64no—
337C2×C4—64no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—162no—
341C2×C4—482no—
342C2×C4C22×C4×D311yescharacteristic
343C2×C4C2×D3×D412yes—
344C2×C4#100411yescharacteristic
345C2×C4D3×#4112yes—
346C8—34no—
347C8—34no—
348C8—42no—
349C8—122no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—82no—
353C2×C3—82no—
354C2×C3#236611yescharacteristic
355C2×C3#329112yes—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—161no—
364C22—161no—
365C22#548611yescharacteristic
366C22#559011yescharacteristic
367C22#566411yescharacteristic
368C4—24no—
369C4—44no—
370C4—124no—
371C4—124no—
372C4C4×D3×D412yes—
373C4#559912yes—
374C3#2319811yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—82no—
378C2—82no—
379C2#7467911yescharacteristic
380C2#8028112yes—
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1887C2
socleC22×C3D42
prime radicalC2×C3×#127C22
cosocleC22×C3×C4C24
hypercenter#1887D3
hypocenterC3#23198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1887

Complements

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

NH ≅ G/Ntype
2. #82543C2semidirect
3. #78916C2semidirect
6. #70315C2semidirect
7. #70073C2semidirect
8. #68765C2semidirect
9. #68741C2semidirect
26. #5076C4semidirect
65. C2×#214C2×C4semidirect
66. C2×#214C2×C4semidirect
74. #869C2×C4semidirect
86. #808D4semidirect
87. #807D4semidirect
90. C4×C3⋊C8C2×C4semidirect
374. C3#23198semidirect

Maximal subgroups

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

subgroupindex
C2×#50762
#825432
#789162
C3×#18872
#717632
#703152
#700732
#687652
#687412
#231983