Finite Group Databasebeta

← #10950874

Subgroups — #10950874

Order 768.

512 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
1#78475C211yesmaximal, characteristic, complemented
2#78469C211yesmaximal, characteristic
3C2×C3×#346C211yesmaximal, characteristic, complemented
4#71416C212yesmaximal, complemented
5#71414C212yesmaximal
6#69084C211yesmaximal, characteristic, complemented
7#69083C211yesmaximal, characteristic, complemented
8#69082C212yesmaximal, complemented
9#69079C211yesmaximal, characteristic, complemented
10#69077C212yesmaximal, complemented
11#69069C211yesmaximal, characteristic, complemented
12#33747—31nomaximal, Sylow
13#5479—21no—
14#5470—21no—
15C22×C3×C16C2212yes—
16C2×C3×C4×C8C2211yescharacteristic
17#5116—21no—
18#5116C2211yescharacteristic
19#5116C2212yes—
20#5115C2211yescharacteristic
21#5114—21no—
22C2×Dic24C2212yes—
23C2×#800C2212yes—
24#4699C2211yescharacteristic
25C4×Dic12—21no—
26C4×Dic12C2211yescharacteristic
27C4×Dic12C2212yes—
28#4697—21no—
29#4696C2211yescharacteristic
30C4×D24C2211yescharacteristic
31C4×#204—21no—
32#4683C2211yescharacteristic
33#4682C2211yescharacteristic
34#4681—21no—
35C3×#346C2211yescharacteristic
36C3×#346C2212yes—
37C3×#346C2211yescharacteristic
38#4509C2212yes—
39#4505C2212yes—
40#4504C2212yes—
41#4503C2212yes—
42#4503C2212yes—
43#4503C2212yes—
44#3170—31no—
45#3168—31no—
46#2255—31no—
47#2251—31no—
48#2245—31no—
49#2243—31no—
50#2217—32no—
51#2216—32no—
52#2176—32no—
53#2175—32no—
54C2×#346D311yescharacteristic, complemented
55C22×C3×C8D411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C2×C3×C42D411yescharacteristic
58#930—21no—
59#929—41no—
60C4×#67—41no—
61#926—41no—
62#924—22no—
63#924—21no—
64#924—41no—
65C2×Dic12—21no—
66C2×Dic12C2312yes—
67C2×Dic12C2311yescharacteristic
68C2×D24C2311yescharacteristic
69C2×#204—21no—
70#876—41no—
71#875—21no—
72C4×D12—21no—
73#872—41no—
74#870—21no—
75#870—41no—
76#869—21no—
77#869—21no—
78C4×Dic6—21no—
79C4×Dic6—22no—
80C4×Dic6—41no—
81C2×C3×C16—24no—
82C2×C3×C16C2312yes—
83C2×C3×C16C2312yes—
84C3×C4×C8D411yescharacteristic
85C3×C4×C8D411yescharacteristic
86C3×C4×C8C2311yescharacteristic
87C3×C4×C8C2311yescharacteristic
88#821—21no—
89#821—21no—
90#818—21no—
91#818C2312yes—
92#818C2311yescharacteristic
93#818C2311yescharacteristic
94#817—21no—
95#816—22no—
96#816—22no—
97#816—21no—
98#816—21no—
99Dic24—24no—
100#800—24no—
101#533—61no—
102#529—61no—
103C2×Q32—32no—
104C2×SD32—32no—
105C22×C16C2×D312yes—
106#500—31no—
107#499—31no—
108#495—31no—
109#494—31no—
110#488—61no—
111#487—61no—
112#467—31no—
113#467—32no—
114#467—61no—
115#466—31no—
116#465—61no—
117C4×Q16—31no—
118C4×Q16—32no—
119C4×Q16—61no—
120C4×SD16—61no—
121C4×D8—31no—
122C2×C4×C8C2×D311yescharacteristic
123#367—32no—
124#366—32no—
125#358—32no—
126#358—32no—
127#358—32no—
128#357—32no—
129#346C2×D312yes—
130#346C2×D311yescharacteristic
131#346C2×D311yescharacteristic
132C22×C3×C4D811yescharacteristic
133C22×C3×C4D811yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C2×#67—41no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—22no—
139C2×Dic6—21no—
140C2×Dic6—41no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2×D411yescharacteristic
146C2×C3×C8C2411yescharacteristic
147C3×C42D811yescharacteristic
148C3×C42D811yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#217—42no—
152#217—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—22no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163Dic3⋊C4—42no—
164Dic3⋊C4—41no—
165Dic3⋊C4—42no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169C4×Dic3—42no—
170C4×Dic3—41no—
171Dic12—22no—
172Dic12—22no—
173Dic12—22no—
174Dic12—42no—
175D24—22no—
176#204—42no—
177C3×C16—22no—
178C3×C16#4114yes—
179C2×Q16—32no—
180C2×Q16—31no—
181C2×Q16—61no—
182C2×SD16—61no—
183C2×D8—31no—
184C22×C8D1211yescharacteristic
185C22×C8C22×D311yescharacteristic
186C4⋊Q8—61no—
187C4⋊Q8—61no—
188#126—121no—
189#125—61no—
190#125—121no—
191#124—61no—
192#123—121no—
193C22⋊Q8—61no—
194C22⋊Q8—62no—
195C22⋊Q8—121no—
196#121—61no—
197C4×Q8—61no—
198C4×Q8—62no—
199C4×Q8—121no—
200C4×D4—61no—
201C4×D4—121no—
202#117—121no—
203#117—121no—
204C2×C42D1211yescharacteristic
205Q32—64no—
206SD32—64no—
207C2×C16—24no—
208C2×C16C22×D312yes—
209C2×C16C22×D312yes—
210#107—32no—
211#107—31no—
212#107—31no—
213#107—61no—
214#106—61no—
215#103—61no—
216#103—61no—
217#103—62no—
218#103—62no—
219#102—61no—
220#102—61no—
221C4×C8D1211yescharacteristic
222C4×C8D1211yescharacteristic
223C4×C8C22×D311yescharacteristic
224C4×C8C22×D311yescharacteristic
225C23×C3C2×D811yescharacteristic
226C22×D3—41no—
227C2×C3×C4—22no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4C2×D811yescharacteristic
232C2×C3×C4C2×D811yescharacteristic
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4C2×D811yescharacteristic
236C2×C3×C4C22×D411yescharacteristic
237#67—82no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243D12—41no—
244D12—41no—
245C4×D3—82no—
246Dic6—42no—
247Dic6—41no—
248Dic6—41no—
249Dic6—42no—
250Dic6—82no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8#13512yes—
254C3×C8C2×#4112yes—
255C2×Q8—61no—
256C2×Q8—62no—
257C2×Q8—121no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—121no—
261C22×C4D2411yescharacteristic
262C22×C4D2411yescharacteristic
263C22×C4C2×D1211yescharacteristic
264Q16—62no—
265Q16—62no—
266Q16—62no—
267Q16—122no—
268SD16—122no—
269D8—62no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8C2×D1211yescharacteristic
273C2×C8C2×D1211yescharacteristic
274C2×C8C2×D1211yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—62no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—122no—
285C4⋊C4—121no—
286C4⋊C4—122no—
287C4⋊C4—121no—
288C22⋊C4—122no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—121no—
295C42—122no—
296C42D2411yescharacteristic
297C42D2411yescharacteristic
298C42C2×D1211yescharacteristic
299C42C2×D1211yescharacteristic
300C16—22no—
301C16#23514yes—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3#51012yes—
305C22×C3C22×D811yescharacteristic
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4#53211yescharacteristic
315C3×C4C2×#13511yescharacteristic
316Dic3—81no—
317Dic3—82no—
318Dic3—81no—
319Dic3—81no—
320Dic3—82no—
321C23—121no—
322C23C2×D2411yescharacteristic
323Q8—122no—
324Q8—121no—
325Q8—121no—
326Q8—122no—
327Q8—242no—
328D4—121no—
329D4—121no—
330D4—242no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—242no—
341C2×C4C2×D2411yescharacteristic
342C2×C4C2×D2411yescharacteristic
343C2×C4C2×D2411yescharacteristic
344C2×C4C2×D2411yescharacteristic
345C2×C4C2×D2411yescharacteristic
346C2×C4C22×D1211yescharacteristic
347C8—21no—
348C8—21no—
349C8#90412yes—
350C8C2×#23512yes—
351C2×C3—22no—
352C2×C3#316911yescharacteristic
353C2×C3C2×#51011yescharacteristic
354C2×C3#341411yescharacteristic
355D3—161no—
356C22—21no—
357C22—21no—
358C22—241no—
359C22—241no—
360C22—241no—
361C22#490912yes—
362C22C22×D2411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—241no—
369C4—242no—
370C4—241no—
371C4—241no—
372C4—242no—
373C4#547711yescharacteristic
374C4C2×#90411yescharacteristic
375C3#3374711yescharacteristic, complemented, Sylow
376C2—22no—
377C2—481no—
378C2#7847011yescharacteristic
379C2#8200911yescharacteristic
380C2C2×#490911yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#346C2
socleC22×C3C22×D8
prime radical#5115C22
hypercenterC2×#346D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×#346

Complements

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

NH ≅ G/Ntype
1. #78475C2semidirect
3. C2×C3×#346C2semidirect
4. #71416C2semidirect
6. #69084C2semidirect
7. #69083C2semidirect
8. #69082C2semidirect
9. #69079C2semidirect
10. #69077C2semidirect
11. #69069C2semidirect
54. C2×#346D3semidirect
375. C3#33747semidirect

Maximal subgroups

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

subgroupindex
#784752
#784692
C2×C3×#3462
#714162
#714142
#690842
#690832
#690822
#690792
#690772
#690692
#337473