Finite Group Databasebeta

← #10943968

Subgroups — #10943968

Order 768.

4032 conjugacy classes of subgroups; only the 319 normal subgroups are shown.

normal subgroups only

This group has 4032 conjugacy classes of subgroups — too many to draw. Only its 319 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#5243C214yesmaximal, complemented
3#84047C213yesmaximal, complemented
4#72142C214yesmaximal, complemented
5#72130C213yesmaximal, complemented
6#32900—31nomaximal, Sylow
7C2×C3×H(5,𝔽2)—216no—
8C2×C3×H(5,𝔽2)—26no—
9C2×C3×H(5,𝔽2)C2214yes—
10C22×C3×#41—23no—
11C23×C3×D4—23no—
12C23×C3×D4C2213yes—
13C22×#217C2211yescharacteristic, complemented
14C2×#935C2216yescomplemented
15#5247—23no—
16#5247C2213yescomplemented
17#5243—216no—
18#5222—26no—
19#5220C2216yescomplemented
20C2×#834C22112yescomplemented
21#5210—23no—
22C22×H(5,𝔽2)D311yescharacteristic
23C2×#458—34no—
24#2405—33no—
25#1790—34no—
26#1781—33no—
27C25×C3—21no—
28C25×C3D413yes—
29C25×C3C2311yescharacteristic
30C3×H(5,𝔽2)—248no—
31C3×H(5,𝔽2)—248no—
32C3×H(5,𝔽2)—224no—
33C3×H(5,𝔽2)D4116yes—
34C2×C3×#41—212no—
35C2×C3×#41—248no—
36C2×C3×#41—212no—
37C22×C3×Q8—21no—
38C22×C3×D4—23no—
39C22×C3×D4—248no—
40C22×C3×D4—212no—
41C22×C3×D4—26no—
42C22×C3×D4—212no—
43C22×C3×D4—26no—
44C22×C3×D4—224no—
45C22×C3×D4D416yes—
46C22×C3×D4C2316yes—
47C23×C3×C4—23no—
48C23×C3×C4D413yes—
49C23×Dic3C2311yescharacteristic, complemented
50C23×Dic3C2311yescharacteristic, complemented
51C2×#217—23no—
52C2×#217—23no—
53C2×#217—23no—
54C2×#217—26no—
55C2×#217—26no—
56C2×#217—23no—
57C2×#217C2316yescomplemented
58#935—224no—
59C2×Dic3⋊C4—26no—
60#834—224no—
61C3⋊#95—26no—
62C2×H(5,𝔽2)—216no—
63C2×H(5,𝔽2)—26no—
64C2×H(5,𝔽2)C2×D314yes—
65C22×#41—23no—
66C23×D4—23no—
67C23×D4C2×D313yes—
68C2×#123—36no—
69C22×C22⋊C4—31no—
70#458—616no—
71C2×#99—312no—
72#394—66no—
73#386—36no—
74#385—63no—
75#379—33no—
76#379—63no—
77C24×C3—28no—
78C24×C3—24no—
79C24×C3—23no—
80C24×C3—212no—
81C24×C3—23no—
82C24×C3—24no—
83C24×C3—42no—
84C24×C3—46no—
85C24×C3—81no—
86C24×C3—83no—
87C24×C3—82no—
88C24×C3—82no—
89C24×C3C2×D413yes—
90C24×C3C2×D416yes—
91C24×C3C2×D413yes—
92C24×C3C2411yescharacteristic
93C3×#41—296no—
94C3×#41—248no—
95C3×#41—248no—
96C2×C3×Q8—28no—
97C2×C3×Q8—24no—
98C2×C3×D4—224no—
99C2×C3×D4—248no—
100C2×C3×D4—212no—
101C2×C3×D4—296no—
102C2×C3×D4—248no—
103C2×C3×D4—248no—
104C2×C3×D4—212no—
105C2×C3×D4—248no—
106C2×C3×D4—212no—
107C2×C3×D4—212no—
108C2×C3×D4—212no—
109C2×C3×D4—212no—
110C2×C3×D4C2×D4124yes—
111C22×C3×C4—224no—
112C22×C3×C4—212no—
113C22×C3×C4—23no—
114C22×C3×C4—23no—
115C22×C3×C4—212no—
116C22×C3×C4C2×D416yes—
117C22×Dic3—23no—
118C22×Dic3—26no—
119C22×Dic3—23no—
120C22×Dic3—24no—
121C22×Dic3—81no—
122#217—212no—
123#217—46no—
124#217—412no—
125#217—43no—
126#217—41no—
127#217—83no—
128#217—86no—
129Dic3⋊C4—412no—
130C25—21no—
131C25#6713yes—
132C25C22×D311yescharacteristic
133H(5,𝔽2)—248no—
134H(5,𝔽2)—248no—
135H(5,𝔽2)—224no—
136H(5,𝔽2)#67116yes—
137C2×#41—212no—
138C2×#41—248no—
139C2×#41—212no—
140C22×Q8—21no—
141C22×D4—23no—
142C22×D4—248no—
143C22×D4—212no—
144C22×D4—26no—
145C22×D4—212no—
146C22×D4—26no—
147C22×D4—224no—
148C22×D4#6716yes—
149C22×D4C22×D316yes—
150C23×C4—23no—
151C23×C4—31no—
152C23×C4—31no—
153C23×C4#6713yes—
154#123—624no—
155C2×C4⋊C4—66no—
156C2×C22⋊C4—36no—
157C2×C22⋊C4—63no—
158C2×C22⋊C4—63no—
159C2×C22⋊C4—66no—
160C2×C22⋊C4—63no—
161C2×C22⋊C4—63no—
162C2×C22⋊C4—66no—
163#99—624no—
164#95—66no—
165C23×C3—212no—
166C23×C3—212no—
167C23×C3—216no—
168C23×C3—26no—
169C23×C3—26no—
170C23×C3—26no—
171C23×C3—26no—
172C23×C3—26no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—43no—
176C23×C3—43no—
177C23×C3—43no—
178C23×C3—46no—
179C23×C3—46no—
180C23×C3—46no—
181C23×C3—46no—
182C23×C3—83no—
183C23×C3—83no—
184C23×C3—88no—
185C23×C3—824no—
186C23×C3—816no—
187C23×C3—84no—
188C23×C3—84no—
189C23×C3#12013yes—
190C23×C3#12011yescharacteristic
191C23×C3#120112yes—
192C23×C3C22×D413yes—
193C23×C3C22×D414yes—
194C3×Q8—216no—
195C3×D4—248no—
196C3×D4—248no—
197C3×D4—248no—
198C2×C3×C4—212no—
199C2×C3×C4—248no—
200C2×C3×C4—212no—
201C2×C3×C4—26no—
202C2×C3×C4—26no—
203C2×C3×C4#120112yes—
204C2×Dic3—46no—
205C2×Dic3—46no—
206C2×Dic3—46no—
207C2×Dic3—46no—
208C2×Dic3—46no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C24—28no—
212C24—24no—
213C24—23no—
214C24—212no—
215C24—23no—
216C24—24no—
217C24—42no—
218C24—46no—
219C24—81no—
220C24—83no—
221C24—82no—
222C24—82no—
223C24C2×#6713yes—
224C24C2×#6713yes—
225C24C2×#6716yes—
226C24C23×D311yescharacteristic
227#41—296no—
228#41—248no—
229#41—248no—
230C2×Q8—28no—
231C2×Q8—24no—
232C2×D4—224no—
233C2×D4—248no—
234C2×D4—212no—
235C2×D4—296no—
236C2×D4—248no—
237C2×D4—248no—
238C2×D4—212no—
239C2×D4—248no—
240C2×D4—212no—
241C2×D4—212no—
242C2×D4—212no—
243C2×D4—212no—
244C2×D4C2×#67124yes—
245C22×C4—224no—
246C22×C4—212no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—212no—
250C22×C4—63no—
251C22×C4—66no—
252C22×C4—63no—
253C22×C4—64no—
254C22×C4—241no—
255C22×C4C2×#6716yes—
256C4⋊C4—1212no—
257C22⋊C4—612no—
258C22⋊C4—126no—
259C22⋊C4—123no—
260C22⋊C4—121no—
261C22⋊C4—1212no—
262C22⋊C4—243no—
263C22⋊C4—246no—
264C22×C3—21no—
265C22×C3—23no—
266C22×C3—23no—
267C22×C3—21no—
268C22×C3—23no—
269C22×C3—212no—
270C22×C3—23no—
271C22×C3—23no—
272C22×C3—44no—
273C22×C3—412no—
274C22×C3—412no—
275C22×C3—424no—
276C22×C3—46no—
277C22×C3—46no—
278C22×C3—86no—
279C22×C3—86no—
280C22×C3—812no—
281C22×C3—816no—
282C22×C3#45812yescomplemented
283C22×C3C2×#12016yes—
284C22×C3C2×#12011yescharacteristic
285C3×C4—212no—
286Dic3—86no—
287Dic3—81no—
288Dic3—81no—
289C23—212no—
290C23—212no—
291C23—216no—
292C23—26no—
293C23—26no—
294C23—26no—
295C23—26no—
296C23—26no—
297C23—41no—
298C23—41no—
299C23—43no—
300C23—43no—
301C23—43no—
302C23—46no—
303C23—46no—
304C23—46no—
305C23—46no—
306C23—83no—
307C23—83no—
308C23—88no—
309C23—824no—
310C23—816no—
311C23—84no—
312C23—84no—
313C23#95313yes—
314C23#95311yescharacteristic
315C23#953112yes—
316C23C22×#6713yes—
317C23C22×#6714yes—
318Q8—216no—
319D4—248no—
320D4—248no—
321D4—248no—
322C2×C4—212no—
323C2×C4—248no—
324C2×C4—212no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—126no—
328C2×C4—126no—
329C2×C4—126no—
330C2×C4—126no—
331C2×C4—126no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4#953112yes—
335C2×C3—26no—
336C2×C3—22no—
337C2×C3—43no—
338C2×C3—412no—
339C2×C3#284611yescharacteristic
340C2×C3C2×#45811yescharacteristic
341C2×C3#302611yescharacteristic
342C22—23no—
343C22—23no—
344C22—21no—
345C22—21no—
346C22—23no—
347C22—23no—
348C22—23no—
349C22—212no—
350C22—44no—
351C22—412no—
352C22—412no—
353C22—424no—
354C22—46no—
355C22—46no—
356C22—86no—
357C22—86no—
358C22—812no—
359C22—816no—
360C22#524312yescomplemented
361C22C2×#95316yes—
362C22C2×#95311yescharacteristic
363C4—212no—
364C4—246no—
365C4—241no—
366C4—241no—
367C3#3290011yescharacteristic, complemented, Sylow
368C2—26no—
369C2—22no—
370C2—43no—
371C2—412no—
372C2C2×#524311yescharacteristic
373C2#8411011yescharacteristic
374C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC22×C3×H(5,𝔽2)C2
socleC22×C3C2×#120
hypercenterC22×H(5,𝔽2)D3
hypocenterC3#32900
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C2×#5243C2semidirect
3. #84047C2semidirect
4. #72142C2semidirect
5. #72130C2semidirect
13. C22×#217C22semidirect
14. C2×#935C22semidirect
16. #5247C22semidirect
19. #5220C22semidirect
20. C2×#834C22semidirect
49. C23×Dic3C23semidirect
50. C23×Dic3C23semidirect
57. C2×#217C23semidirect
282. C22×C3#458semidirect
360. C22#5243semidirect
367. C3#32900semidirect

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C2×#52432
#840472
#721422
#721302
#329003