Finite Group Databasebeta

← #10819959

Subgroups — #10819959

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#497C211yesmaximal, characteristic
2#83874C212yesmaximal, complemented
3#83843C212yesmaximal
4#72069C212yesmaximal, complemented
5#72052C212yesmaximal
6#71784C212yesmaximal, complemented
7#71724C211yesmaximal, characteristic
8#71723C211yesmaximal, characteristic
9#69829C212yesmaximal, complemented
10#23203—31nomaximal, Sylow
11C22×C3×Q16C2212yes—
12C22×C3×SD16C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15#5812C2211yescharacteristic
16#5812C2212yes—
17#5798C2211yescharacteristic
18C3×#497C418yescomplemented
19C2×C3×#97C2211yescharacteristic
20C2×#836C2211yescharacteristic
21C2×#836C2212yes—
22#5225C2212yes—
23#5225C2212yes—
24#5225C2212yes—
25C2×#833C2211yescharacteristic
26#5215C2212yes—
27#5184—24no—
28Dic3×Q16—24no—
29#5164—24no—
30Dic3×SD16—24no—
31C2×#819C2212yes—
32#5106C2212yes—
33#5101C2212yes—
34#5101C2212yes—
35#4523C2212yes—
36#4523C2212yes—
37C2×#497D311yescharacteristic
38#2950—32no—
39#2949—32no—
40#1979—32no—
41#1977—32no—
42#1888—31no—
43#1886—32no—
44#1885—31no—
45#1773—32no—
46C22×C3×Q8C2312yes—
47C22×C3×Q8C2311yescharacteristic
48C22×C3×D4C2311yescharacteristic
49C2×C3×Q16—22no—
50C2×C3×Q16—22no—
51C2×C3×Q16—24no—
52C2×C3×Q16C2×C418yescomplemented
53C2×C3×SD16—22no—
54C2×C3×SD16—22no—
55C2×C3×SD16—24no—
56C2×C3×SD16C2×C418yes—
57C22×C3×C8C2312yes—
58C3×C4⋊Q8—41no—
59C3×C4⋊Q8C2×C414yescomplemented
60C3×#124—41no—
61C3×#124C2×C414yes—
62C2×C3×C4⋊C4—41no—
63C2×C3×C22⋊C4—41no—
64C2×C3×C42C2311yescharacteristic
65C2×#217—41no—
66Q8×Dic3—24no—
67Q8×Dic3—24no—
68Q8×Dic3—24no—
69D4×Dic3—24no—
70#926—41no—
71#926—42no—
72#926C2312yes—
73#926C2312yes—
74C2×#208C2312yes—
75C2×#208C2312yes—
76C3×#97C2×C414yescomplemented
77#836—42no—
78#836—41no—
79#835—24no—
80#835—24no—
81#835—24no—
82#833—41no—
83#832—24no—
84#819—24no—
85#818D414yes—
86#817D414yes—
87#815—22no—
88#815—22no—
89C8×Dic3—22no—
90C8×Dic3—22no—
91C22×Q16C2×D312yes—
92C22×SD16C2×D312yes—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#124C2×D311yescharacteristic
95#519—32no—
96#519—31no—
97#518—31no—
98#497Dic318yescomplemented
99#441—64no—
100#440—64no—
101C4×Q16—64no—
102C4×SD16—64no—
103C2×#108—32no—
104#427—32no—
105#419—32no—
106#419—32no—
107#419—32no—
108#418—32no—
109C2×#101—32no—
110C2×#101—31no—
111C2×#100—31no—
112#405—32no—
113#405—32no—
114C2×#97C2×D311yescharacteristic
115#339—32no—
116#339—32no—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8C22×C412yes—
127C2×C3×Q8C22×C414yes—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—21no—
131C2×C3×D4C22×C412yes—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—42no—
137C3×Q16—28no—
138C3×Q16—28no—
139C3×SD16—28no—
140C3×SD16—28no—
141C2×C3×C8C22×C414yes—
142C2×C3×C8C2×D412yes—
143C2×C3×C8C2×D412yes—
144C2×C3×C8#4114yes—
145C3×C4⋊C4—44no—
146C3×C22⋊C4—44no—
147C3×C42—21no—
148C3×C42C22×C412yes—
149#217—22no—
150#217—22no—
151#217—42no—
152C4⋊Dic3—42no—
153C4⋊Dic3—44no—
154C4⋊Dic3C2×D414yes—
155C4⋊Dic3C2×D414yes—
156C4×Dic3—22no—
157C4×Dic3—22no—
158C4×Dic3—42no—
159C4×Dic3—44no—
160#208—22no—
161#208—22no—
162#208—22no—
163#208—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C22×Q8C22×D311yescharacteristic
167C22×Q8C22×D312yes—
168C22×D4C22×D311yescharacteristic
169C2×Q16—22no—
170C2×Q16—22no—
171C2×Q16—24no—
172C2×Q16C2×Dic318yescomplemented
173C2×SD16—22no—
174C2×SD16—22no—
175C2×SD16—24no—
176C2×SD16C2×Dic318yes—
177C2×M16—32no—
178C2×M16—32no—
179C22×C8C22×D312yes—
180C4⋊Q8—41no—
181C4⋊Q8C2×Dic314yescomplemented
182#124—41no—
183#124C2×Dic314yes—
184C4×Q8—64no—
185C4×Q8—64no—
186C4×Q8—64no—
187C4×D4—64no—
188#117—32no—
189#117—32no—
190#117—122no—
191#117—121no—
192C2×C4⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—121no—
195C2×C42C22×D311yescharacteristic
196#108—64no—
197#107—34no—
198#106—34no—
199#103—64no—
200#103—64no—
201#103—64no—
202#102—64no—
203#101—122no—
204#101—121no—
205#100—121no—
206#97—62no—
207#97—62no—
208#97C2×Dic314yescomplemented
209C4×C8—62no—
210C4×C8—62no—
211C23×C3—22no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—82no—
215C23×C3H(5,𝔽2)11yescharacteristic
216C3×Q8—24no—
217C3×Q8—24no—
218C3×Q8—24no—
219C3×Q8—42no—
220C3×Q8—42no—
221C3×Q8—42no—
222C3×D4—24no—
223C3×D4—42no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—24no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4C23×C411yescharacteristic
233C2×C3×C4C22×D412yes—
234C2×C3×C4C2×#4112yes—
235C2×C3×C4H(5,𝔽2)11yescharacteristic
236C2×Dic3—24no—
237C2×Dic3—24no—
238C2×Dic3—162no—
239C3×C8C4×D414yes—
240C3×C8C4×D414yes—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C24—21no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8C22×Dic312yes—
253C2×Q8C22×Dic314yes—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4C22×Dic312yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—122no—
262C22×C4C23×D311yescharacteristic
263Q16—28no—
264Q16—28no—
265SD16—28no—
266SD16—28no—
267M16—62no—
268M16—62no—
269M16—62no—
270M16—62no—
271C2×C8—62no—
272C2×C8—62no—
273C2×C8D3×D412yes—
274C2×C8D3×D412yes—
275C2×C8#23714yes—
276C2×C8C22×Dic314yes—
277C4⋊C4—34no—
278C4⋊C4—34no—
279C4⋊C4—44no—
280C4⋊C4—122no—
281C4⋊C4—124no—
282C22⋊C4—44no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—122no—
286C42—21no—
287C42—62no—
288C42—62no—
289C42—122no—
290C42—124no—
291C42C22×Dic312yes—
292C22×C3—22no—
293C22×C3—42no—
294C22×C3—82no—
295C22×C3—82no—
296C22×C3—161no—
297C22×C3#51811yescharacteristic
298C22×C3D4211yescharacteristic
299C22×C3#55911yescharacteristic
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—44no—
303C3×C4C2×C4×D412yes—
304C3×C4#55012yes—
305Dic3—44no—
306Dic3—44no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—82no—
311C23#100411yescharacteristic
312Q8—24no—
313Q8—24no—
314Q8—24no—
315Q8—42no—
316Q8—42no—
317Q8—42no—
318D4—24no—
319D4—42no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—64no—
329C2×C4—64no—
330C2×C4—482no—
331C2×C4C2×D3×D412yes—
332C2×C4C2×#23712yes—
333C2×C4#100411yescharacteristic
334C2×C4C23×Dic311yescharacteristic
335C8—122no—
336C8—122no—
337C8D4×Dic314yes—
338C8D4×Dic314yes—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—82no—
342C2×C3#236611yescharacteristic
343C2×C3#328912yes—
344C22—22no—
345C22—42no—
346C22—82no—
347C22—82no—
348C22—161no—
349C22#571311yescharacteristic
350C22#571711yescharacteristic
351C22#579811yescharacteristic
352C4—42no—
353C4—42no—
354C4—44no—
355C4—124no—
356C4—124no—
357C4#571812yes—
358C4C2×D4×Dic312yes—
359C3#2320311yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2—82no—
363C2#8188512yes—
364C2#8317311yescharacteristic
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#497C2
socleC22×C3D42
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenterC2×#497D3
hypocenterC3#23203
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. #83874C2semidirect
4. #72069C2semidirect
6. #71784C2semidirect
9. #69829C2semidirect
18. C3×#497C4semidirect
52. C2×C3×Q16C2×C4semidirect
59. C3×C4⋊Q8C2×C4semidirect
76. C3×#97C2×C4semidirect
98. #497Dic3semidirect
172. C2×Q16C2×Dic3semidirect
181. C4⋊Q8C2×Dic3semidirect
208. #97C2×Dic3semidirect
359. C3#23203semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4972
#838742
#838432
#720692
#720522
#717842
#717242
#717232
#698292
#232033