Finite Group Databasebeta

← #10950863

Subgroups — #10950863

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78470C211yesmaximal, characteristic, complemented
2#78469C211yesmaximal, characteristic, complemented
3C2×C3×#346C211yesmaximal, characteristic, complemented
4#71414C212yesmaximal, complemented
5#71413C212yesmaximal, complemented
6#69081C211yesmaximal, characteristic, complemented
7#69080C211yesmaximal, characteristic, complemented
8#69079C211yesmaximal, characteristic, complemented
9#69078C212yesmaximal, complemented
10#69077C212yesmaximal, complemented
11#69069C211yesmaximal, characteristic, complemented
12#33746—31nomaximal, Sylow
13#5479—21no—
14#5477—21no—
15C22×C3×C16C2212yes—
16C2×C3×C4×C8C2211yescharacteristic
17#5116C2211yescharacteristic
18#5115—21no—
19#5115C2211yescharacteristic
20#5115C2212yes—
21#5114—21no—
22C2×#800C2212yes—
23C2×D48C2212yes—
24C4×Dic12C2211yescharacteristic
25#4697—21no—
26#4696C2211yescharacteristic
27#4695C2211yescharacteristic
28#4694—21no—
29C4×D24—21no—
30C4×D24C2211yescharacteristic
31C4×D24C2212yes—
32C4×#204—21no—
33#4683C2211yescharacteristic
34#4682C2211yescharacteristic
35C3×#346C2211yescharacteristic
36C3×#346C2212yes—
37C3×#346C2211yescharacteristic
38#4509C2212yes—
39#4509C2212yes—
40#4509C2212yes—
41#4505C2212yes—
42#4504C2212yes—
43#4503C2212yes—
44#3169—31no—
45#3168—31no—
46#2255—31no—
47#2250—31no—
48#2244—31no—
49#2243—31no—
50#2217—32no—
51#2215—32no—
52#2177—32no—
53#2175—32no—
54C2×#346D311yescharacteristic, complemented
55C2×#235—41no—
56C22×C3×C8D411yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C2×C3×C42D411yescharacteristic
59#930—22no—
60#930—21no—
61#930—41no—
62#929—41no—
63C4×#67—41no—
64#924—21no—
65C2×Dic12C2311yescharacteristic
66C2×D24—21no—
67C2×D24C2311yescharacteristic
68C2×D24C2312yes—
69C2×#204—21no—
70#876—41no—
71#875—21no—
72#875—41no—
73#874—21no—
74C4×D12—21no—
75C4×D12—22no—
76C4×D12—41no—
77#872—41no—
78#870—21no—
79#869—21no—
80C4×Dic6—21no—
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—22no—
90#821—22no—
91#821—21no—
92#818—21no—
93#818C2312yes—
94#818C2311yescharacteristic
95#818C2311yescharacteristic
96#817—21no—
97#816—21no—
98#816—21no—
99#800—24no—
100D48—24no—
101#532—61no—
102#529—61no—
103C2×SD32—32no—
104C2×D16—32no—
105C22×C16C2×D312yes—
106#500—31no—
107#499—31no—
108#495—31no—
109#493—31no—
110#488—61no—
111#486—61no—
112#467—31no—
113#466—31no—
114#466—32no—
115#466—61no—
116#465—61no—
117C4×Q16—31no—
118C4×SD16—61no—
119C4×D8—31no—
120C4×D8—32no—
121C4×D8—61no—
122C2×C4×C8C2×D311yescharacteristic
123#367—32no—
124#366—32no—
125#358—32no—
126#357—32no—
127#357—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×#67—42no—
137#235—84no—
138C2×D12—22no—
139C2×D12—21no—
140C2×D12—41no—
141C2×D12—41no—
142C2×C4×D3—41no—
143C2×C4×D3—42no—
144C2×Dic6—21no—
145C2×Dic6—41no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8C2×D411yescharacteristic
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2411yescharacteristic
152C3×C42D811yescharacteristic
153C3×C42D811yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156#217—41no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—42no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—42no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—22no—
166C4⋊Dic3—41no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—41no—
172Dic12—22no—
173D24—22no—
174D24—22no—
175D24—22no—
176D24—42no—
177#204—42no—
178C3×C16—22no—
179C3×C16#4114yescomplemented
180C2×#41—121no—
181C2×Q16—31no—
182C2×SD16—61no—
183C2×D8—32no—
184C2×D8—31no—
185C2×D8—61no—
186C22×C8D1211yescharacteristic
187C22×C8C22×D311yescharacteristic
188C4⋊Q8—61no—
189#127—61no—
190#126—121no—
191#125—61no—
192#124—61no—
193#124—121no—
194#123—121no—
195C22⋊Q8—61no—
196#121—61no—
197#121—62no—
198#121—121no—
199C4×Q8—61no—
200C4×D4—61no—
201C4×D4—62no—
202C4×D4—121no—
203C4×D4—121no—
204#117—121no—
205C2×C42D1211yescharacteristic
206SD32—64no—
207D16—64no—
208C2×C16—24no—
209C2×C16C22×D312yes—
210C2×C16C22×D312yes—
211#107—31no—
212#107—31no—
213#107—32no—
214#107—61no—
215#106—61no—
216#103—61no—
217#103—61no—
218#102—61no—
219#102—62no—
220#102—62no—
221#102—61no—
222C4×C8D1211yescharacteristic
223C4×C8D1211yescharacteristic
224C4×C8C22×D311yescharacteristic
225C4×C8C22×D311yescharacteristic
226C23×C3C2×D811yescharacteristic
227C22×D3—41no—
228C22×D3—42no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4C2×D811yescharacteristic
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4C22×D411yescharacteristic
239#67—84no—
240#67—82no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245D12—41no—
246D12—42no—
247D12—41no—
248D12—42no—
249D12—82no—
250D12—82no—
251C4×D3—82no—
252C4×D3—84no—
253Dic6—41no—
254Dic6—41no—
255Dic6—82no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8#13512yes—
259C3×C8C2×#4112yes—
260#41—244no—
261C2×Q8—61no—
262C2×Q8—121no—
263C2×D4—61no—
264C2×D4—62no—
265C2×D4—121no—
266C2×D4—122no—
267C2×D4—121no—
268C2×D4—121no—
269C22×C4—121no—
270C22×C4—122no—
271C22×C4D2411yescharacteristic
272C22×C4D2411yescharacteristic
273C22×C4C2×D1211yescharacteristic
274Q16—62no—
275SD16—122no—
276D8—62no—
277D8—62no—
278D8—62no—
279D8—122no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8C2×D1211yescharacteristic
283C2×C8C2×D1211yescharacteristic
284C2×C8C2×D1211yescharacteristic
285C2×C8C23×D311yescharacteristic
286C4⋊C4—62no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—122no—
298C22⋊C4—121no—
299C22⋊C4—122no—
300C22⋊C4—121no—
301C42—121no—
302C42D2411yescharacteristic
303C42D2411yescharacteristic
304C42C2×D1211yescharacteristic
305C42C2×D1211yescharacteristic
306C16—22no—
307C16#23514yescomplemented
308C22×C3—21no—
309C22×C3—21no—
310C22×C3#50912yes—
311C22×C3C22×D811yescharacteristic
312C2×D3—82no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—82no—
316C2×D3—81no—
317C2×D3—82no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4#53211yescharacteristic
324C3×C4C2×#13511yescharacteristic
325Dic3—82no—
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329C23—121no—
330C23—122no—
331C23C2×D2411yescharacteristic
332Q8—121no—
333Q8—121no—
334Q8—242no—
335D4—121no—
336D4—122no—
337D4—122no—
338D4—121no—
339D4—242no—
340D4—242no—
341D4—244no—
342D4—242no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—244no—
352C2×C4—242no—
353C2×C4C2×D2411yescharacteristic
354C2×C4C2×D2411yescharacteristic
355C2×C4C2×D2411yescharacteristic
356C2×C4C2×D2411yescharacteristic
357C2×C4C2×D2411yescharacteristic
358C2×C4C22×D1211yescharacteristic
359C8—21no—
360C8—21no—
361C8#90412yes—
362C8C2×#23512yes—
363C2×C3—22no—
364C2×C3#316911yescharacteristic
365C2×C3C2×#50911yescharacteristic
366C2×C3#341411yescharacteristic
367D3—162no—
368D3—161no—
369C22—21no—
370C22—21no—
371C22—242no—
372C22—241no—
373C22—241no—
374C22—242no—
375C22—242no—
376C22—241no—
377C22#490812yes—
378C22C22×D2411yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—242no—
385C4—241no—
386C4—241no—
387C4—241no—
388C4#547711yescharacteristic
389C4C2×#90411yescharacteristic
390C3#3374611yescharacteristic, complemented, Sylow
391C2—22no—
392C2—482no—
393C2—481no—
394C2#7847011yescharacteristic
395C2C2×#490811yescharacteristic
396C2#8200911yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#346C2
socleC22×C3C22×D8
hypercenterC2×#346D3
hypocenterC3#33746
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. #78470C2semidirect
2. #78469C2semidirect
3. C2×C3×#346C2semidirect
4. #71414C2semidirect
5. #71413C2semidirect
6. #69081C2semidirect
7. #69080C2semidirect
8. #69079C2semidirect
9. #69078C2semidirect
10. #69077C2semidirect
11. #69069C2semidirect
54. C2×#346D3semidirect
179. C3×C16#41semidirect
307. C16#235semidirect
390. C3#33746semidirect

Maximal subgroups

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

subgroupindex
#784702
#784692
C2×C3×#3462
#714142
#714132
#690812
#690802
#690792
#690782
#690772
#690692
#337463