Finite Group Databasebeta

← #10819918

Subgroups — #10819918

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5085C211yesmaximal, characteristic
2#82548C212yesmaximal, complemented
3#78947C212yesmaximal, complemented
4C3×#1886C211yesmaximal, characteristic, complemented
5#71784C211yesmaximal, characteristic, complemented
6#70335C212yesmaximal, complemented
7#70107C212yesmaximal, complemented
8#68760C212yesmaximal
9#68741C212yesmaximal, complemented
10#23198—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#519C2211yescharacteristic
13#5812C2211yescharacteristic
14C22×#215C2212yes—
15C22×#215C2212yes—
16#5521C2212yes—
17C2×#874C2211yescharacteristic
18C3×#419C2212yes—
19C2×C3×#101C2211yescharacteristic
20C2×#836C2211yescharacteristic
21#5225C2212yes—
22#5131C2212yes—
23C2×#823C2212yes—
24C2×#822C2212yes—
25#5122C2212yes—
26#5085C418yescomplemented
27#5026—24no—
28C4×#215—24no—
29#5001C2212yes—
30#4998C2212yes—
31#4997C2212yes—
32C2×C4×C3⋊C8C2211yescharacteristic
33#4817—24no—
34#4788—24no—
35C3×#339C2212yes—
36#4523C2212yes—
37C2×#492—31no—
38#2949—32no—
39#2949—32no—
40#1977—32no—
41#1977—32no—
42#1887—32no—
43#1886—31no—
44#1886D311yescharacteristic, complemented
45#1772—32no—
46C22×C3×Q8C2311yescharacteristic
47C22×C3×Q8C2311yescharacteristic
48C22×D12—41no—
49C22×D12C2312yes—
50C2×C3×M16C2312yes—
51C3×C4⋊Q8—41no—
52C3×C4⋊Q8C2×C414yescomplemented
53C3×C4×Q8—24no—
54C3×#117—41no—
55C3×#117C2312yes—
56C2×C3×C4⋊C4—41no—
57C2×C3×C42C2311yescharacteristic
58Q8×Dic3—24no—
59C2×#215—22no—
60C2×#215—22no—
61C2×#215—22no—
62C2×#215—22no—
63C2×#215—24no—
64C2×#215—24no—
65C2×#215C2×C418yescomplemented
66C2×#215C2×C418yescomplemented
67#926—41no—
68#926C2312yes—
69C2×#208C2312yes—
70C22×C3⋊C8C2312yes—
71#893—24no—
72D3×C22⋊C4—42no—
73#874—41no—
74#874C2×C414yes—
75C4×D12—24no—
76C3×#103—24no—
77C3×#101—41no—
78#836—41no—
79#835—24no—
80#823—42no—
81#822—24no—
82#821—24no—
83#815—22no—
84C8×Dic3—22no—
85#809—24no—
86#808D414yescomplemented
87#807D414yescomplemented
88#803—22no—
89C4×C3⋊C8—22no—
90C4×C3⋊C8C2×C414yescomplemented
91C22×SD16—32no—
92C22×SD16—32no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#127—31no—
95#519—31no—
96#519C2×D311yescharacteristic
97#518—32no—
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#419C2×D312yes—
107#418—32no—
108#418—32no—
109C2×#101—31no—
110C2×#101C2×D311yescharacteristic
111C2×#100—32no—
112#405—32no—
113#405—32no—
114C2×C4×C8—31no—
115#339—32no—
116#339C2×D312yes—
117C23×D3—22no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8C22×C412yes—
125C2×C3×Q8C22×C412yes—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4C2411yescharacteristic
130C2×D12—22no—
131C2×D12—22no—
132C2×D12—22no—
133C2×D12—22no—
134C2×D12—22no—
135C2×D12—44no—
136C2×D12—84no—
137C2×D12C22×C414yes—
138C2×C4×D3—44no—
139C3×M16—22no—
140C3×M16—22no—
141C2×C3×C8—22no—
142C3×C4⋊C4—44no—
143C3×C4⋊C4—42no—
144C3×C4⋊C4C2×D414yes—
145C3×C22⋊C4—22no—
146C3×C42—22no—
147C3×C42—21no—
148C3×C42—42no—
149C3×C42C22×C412yes—
150#217—22no—
151#215—28no—
152#215—28no—
153#215—28no—
154#215—28no—
155(C2×D3)⋊C4—44no—
156C4⋊Dic3—42no—
157C4⋊Dic3C2×D414yes—
158C4×Dic3—22no—
159C4×Dic3—42no—
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×Q8C22×D311yescharacteristic
168C22×Q8C22×D311yescharacteristic
169C22×D4—32no—
170C22×D4—121no—
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—41no—
183C4⋊Q8C4×D314yescomplemented
184#127—34no—
185#127—121no—
186C4×Q8—24no—
187C4×Q8—64no—
188C4×D4—64no—
189C4×D4—64no—
190#117—32no—
191#117—41no—
192#117—121no—
193#117C22×D312yes—
194C2×C4⋊C4—41no—
195C2×C22⋊C4—122no—
196C2×C42C22×D311yescharacteristic
197#108—64no—
198#107—34no—
199#106—34no—
200#103—24no—
201#103—64no—
202#102—64no—
203#102—64no—
204#101—41no—
205#101—121no—
206#100—122no—
207#97—62no—
208#97—62no—
209C4×C8—34no—
210C4×C8—62no—
211C4×C8—62no—
212C23×C3H(5,𝔽2)11yescharacteristic
213C22×D3—24no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—84no—
217C3×Q8—24no—
218C3×Q8—24no—
219C3×Q8—42no—
220C3×Q8—42no—
221C2×C3×C4—24no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4C23×C411yescharacteristic
231C2×C3×C4C22×D412yes—
232C2×C3×C4C2×#4112yes—
233C2×C3×C4H(5,𝔽2)11yescharacteristic
234C2×Dic3—24no—
235D12—24no—
236D12—24no—
237D12—42no—
238D12—42no—
239D12—88no—
240C4×D3—164no—
241C3×C8—42no—
242C3⋊C8—42no—
243C3⋊C8C4×D414yes—
244C3⋊C8C4×D414yes—
245C24—62no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8C2×C4×D312yes—
253C2×Q8C2×C4×D312yes—
254C2×D4—34no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—124no—
261C2×D4—244no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—124no—
266C22×C4C23×D311yescharacteristic
267SD16—68no—
268SD16—68no—
269SD16—68no—
270SD16—68no—
271M16—22no—
272M16—22no—
273M16—62no—
274M16—62no—
275C2×C8—22no—
276C2×C8—34no—
277C2×C8—32no—
278C2×C8—32no—
279C2×C8—34no—
280C2×C8—62no—
281C4⋊C4—34no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—122no—
285C4⋊C4D3×D414yes—
286C22⋊C4—22no—
287C22⋊C4—62no—
288C22⋊C4—124no—
289C42—22no—
290C42—21no—
291C42—42no—
292C42—62no—
293C42—122no—
294C42C2×C4×D312yes—
295C22×C3—22no—
296C22×C3#51811yescharacteristic
297C22×C3D4211yescharacteristic
298C22×C3#55911yescharacteristic
299C2×D3—44no—
300C2×D3—84no—
301C2×D3—84no—
302C2×D3—162no—
303C3×C4—24no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—44no—
307C3×C4C2×C4×D412yes—
308C3×C4#55012yes—
309Dic3—44no—
310C23—64no—
311C23—122no—
312C23—122no—
313C23—244no—
314C23#100411yescharacteristic
315Q8—24no—
316Q8—24no—
317Q8—42no—
318Q8—42no—
319D4—64no—
320D4—64no—
321D4—122no—
322D4—122no—
323D4—248no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—64no—
334C2×C4—484no—
335C2×C4C22×C4×D311yescharacteristic
336C2×C4C2×D3×D412yes—
337C2×C4#100411yescharacteristic
338C2×C4D3×#4112yes—
339C8—34no—
340C8—34no—
341C8—42no—
342C8—122no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3#236611yescharacteristic
346C2×C3#329112yes—
347D3—84no—
348C22—22no—
349C22—124no—
350C22—244no—
351C22—244no—
352C22—482no—
353C22#548611yescharacteristic
354C22#559011yescharacteristic
355C22#566411yescharacteristic
356C4—24no—
357C4—44no—
358C4—42no—
359C4—42no—
360C4—124no—
361C4C4×D3×D412yes—
362C4#559912yes—
363C3#2319811yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2—244no—
367C2#7467911yescharacteristic
368C2#8030212yes—
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1886C2
socleC22×C3D42
prime radicalC2×#874C22
cosocleC22×C3×C4C24
hypercenter#1886D3
hypocenterC3#23198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1886

Complements

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

NH ≅ G/Ntype
2. #82548C2semidirect
3. #78947C2semidirect
4. C3×#1886C2semidirect
5. #71784C2semidirect
6. #70335C2semidirect
7. #70107C2semidirect
9. #68741C2semidirect
26. #5085C4semidirect
44. #1886D3semidirect
52. C3×C4⋊Q8C2×C4semidirect
65. C2×#215C2×C4semidirect
66. C2×#215C2×C4semidirect
86. #808D4semidirect
87. #807D4semidirect
90. C4×C3⋊C8C2×C4semidirect
183. C4⋊Q8C4×D3semidirect
363. C3#23198semidirect

Maximal subgroups

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

subgroupindex
C2×#50852
#825482
#789472
C3×#18862
#717842
#703352
#701072
#687602
#687412
#231983