Finite Group Databasebeta

← #10950862

Subgroups — #10950862

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#81907C211yesmaximal, characteristic, complemented
2#81775C211yesmaximal, characteristic, complemented
3C3×#2244C211yesmaximal, characteristic, complemented
4#71355C211yesmaximal, characteristic, complemented
5#71292C211yesmaximal, characteristic, complemented
6#69196C212yesmaximal, complemented
7#69195C212yesmaximal, complemented
8#69177C212yesmaximal, complemented
9#69176C212yesmaximal, complemented
10#69079C211yesmaximal, characteristic, complemented
11D3×#346C211yesmaximal, characteristic, complemented
12#33746—31nomaximal, Sylow
13#5716—21no—
14#5691—21no—
15C3×#499C2211yescharacteristic
16C3×#493C2211yescharacteristic
17#5157—21no—
18#5157C2212yes—
19Dic3×D8—21no—
20Dic3×D8C2212yes—
21C2×#827C2212yes—
22C2×#826C2212yes—
23#5070—21no—
24#5068—21no—
25C2×D3×C16C2212yes—
26#4887C2211yescharacteristic
27#4883C2211yescharacteristic
28#4873C2211yescharacteristic
29#4872C2211yescharacteristic
30#4864—21no—
31#4852—21no—
32#4696C2211yescharacteristic
33C4×D3×C8C2211yescharacteristic
34#4682C2211yescharacteristic
35C3×#357C2212yes—
36C3×#357C2212yes—
37C3×#346C2211yescharacteristic
38#4509C2212yes—
39#4503C2212yes—
40#4502C2212yes—
41#4490C2212yes—
42#4489C2212yes—
43#4461C2211yescharacteristic
44#3169—31no—
45#3168—31no—
46#2255—31no—
47#2250—31no—
48#2244D311yescharacteristic, complemented
49#2243—31no—
50#2217—32no—
51#2215—32no—
52#2177—32no—
53#2175—32no—
54C2×#346—31no—
55C2×#237—41no—
56C2×C3×D8—21no—
57C2×C3×D8C2312yes—
58C3×#127—21no—
59C3×#125—21no—
60#938—22no—
61#938—41no—
62#936—41no—
63D4×Dic3—22no—
64D4×Dic3—41no—
65C2×Dic12C2311yescharacteristic
66C2×D24C2311yescharacteristic
67C2×#204—21no—
68C2×D3×C8D411yescharacteristic
69C2×D3×C8C2311yescharacteristic
70#898—41no—
71#897—21no—
72#895—21no—
73#894—41no—
74#893—21no—
75#893—41no—
76#892—41no—
77#887—21no—
78#875—21no—
79C42×D3D411yescharacteristic
80#869—21no—
81C2×C3×C16C2312yes—
82C3×#107C2311yescharacteristic
83C3×#107C2311yescharacteristic
84C3×#106—21no—
85C3×C4×C8C2311yescharacteristic
86#832—22no—
87#832—22no—
88#827—24no—
89#826—24no—
90#818—21no—
91#818C2312yes—
92C8×Dic3D411yescharacteristic
93C8×Dic3C2311yescharacteristic
94C2×C3⋊C16C2312yes—
95#810—21no—
96#810—21no—
97#809—21no—
98#809—21no—
99C4×C3⋊C8D411yescharacteristic
100D3×C16—24no—
101#532—61no—
102#529—61no—
103C2×SD32—32no—
104C2×D16—32no—
105C22×C16—32no—
106#500—31no—
107#499C2×D311yescharacteristic
108#495—31no—
109#493C2×D311yescharacteristic
110#488—61no—
111#486—61no—
112#467—31no—
113#466—32no—
114#466—31no—
115#466—61no—
116#465—61no—
117C4×Q16—31no—
118C4×SD16—61no—
119C4×D8—32no—
120C4×D8—31no—
121C4×D8—61no—
122C2×C4×C8—31no—
123#367—32no—
124#366—32no—
125#358—32no—
126#357—32no—
127#357C2×D312yes—
128#357C2×D312yes—
129#346—32no—
130#346—31no—
131#346C2×D311yescharacteristic
132C2×C3×D4—22no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×#67—42no—
136C22×Dic3—42no—
137#237—84no—
138C2×D12—21no—
139C2×D12—41no—
140C2×C4×D3—41no—
141C2×C4×D3D811yescharacteristic
142C2×C4×D3D811yescharacteristic
143C2×C4×D3C2×D411yescharacteristic
144C2×Dic6—21no—
145C2×Dic6—41no—
146C3×D8—22no—
147C3×D8—22no—
148C3×D8—42no—
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2411yescharacteristic
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C42C2×D411yescharacteristic
156#217—42no—
157#217—42no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—22no—
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168C4×Dic3D811yescharacteristic
169C4×Dic3D811yescharacteristic
170C4×Dic3C2×D411yescharacteristic
171C2×C3⋊C8C2×D411yescharacteristic
172C2×C3⋊C8C2×D411yescharacteristic
173Dic12—22no—
174D24—22no—
175#204—42no—
176D3×C8—22no—
177D3×C8—22no—
178C3×C16—22no—
179C3⋊C16#4114yescomplemented
180C2×#41—121no—
181C2×Q16—31no—
182C2×SD16—61no—
183C2×D8—21no—
184C2×D8—31no—
185C2×D8C22×D312yes—
186C22×C8—31no—
187C22×C8—31no—
188C4⋊Q8—61no—
189#127—21no—
190#126—121no—
191#125—21no—
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×C42—31no—
206SD32—64no—
207D16—64no—
208C2×C16—32no—
209C2×C16—64no—
210C2×C16C22×D312yes—
211#107—32no—
212#107—61no—
213#107C22×D311yescharacteristic
214#107C22×D311yescharacteristic
215#106—21no—
216#103—61no—
217#103—61no—
218#102—61no—
219#102—61no—
220#102—62no—
221#102—62no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8C22×D311yescharacteristic
226C23×C3—42no—
227C22×D3—41no—
228C22×D3C2×D811yescharacteristic
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—82no—
232C3×D4—82no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4C2×D811yescharacteristic
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4C22×D411yescharacteristic
238#67—84no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—84no—
242C2×Dic3C2×D811yescharacteristic
243C2×Dic3C2×D811yescharacteristic
244C2×Dic3C2×D811yescharacteristic
245D12—41no—
246D12—41no—
247D12—82no—
248C4×D3—22no—
249C4×D3—21no—
250C4×D3—21no—
251C4×D3—41no—
252C4×D3—82no—
253Dic6—41no—
254Dic6—41no—
255Dic6—82no—
256C3×C8#13512yes—
257C3×C8C2×#4112yes—
258C3⋊C8—21no—
259C3⋊C8—21no—
260#41—244no—
261C2×Q8—61no—
262C2×Q8—121no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—61no—
267C2×D4—121no—
268C2×D4—122no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—121no—
273C22×C4—122no—
274Q16—62no—
275SD16—122no—
276D8—22no—
277D8—22no—
278D8—42no—
279D8—62no—
280C2×C8—31no—
281C2×C8—31no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8D3×D411yescharacteristic
285C2×C8C23×D311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—62no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C22⋊C4—122no—
297C22⋊C4—122no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C42—31no—
302C42—31no—
303C42—31no—
304C42—121no—
305C42D3×D411yescharacteristic
306C16—22no—
307C16—34no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3C22×D811yescharacteristic
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3#50912yes—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4#53211yescharacteristic
323C3×C4C2×#13511yescharacteristic
324Dic3—21no—
325Dic3—21no—
326Dic3—21no—
327Dic3—82no—
328Dic3—81no—
329C23—31no—
330C23—42no—
331C23—121no—
332Q8—121no—
333Q8—121no—
334Q8—242no—
335D4—42no—
336D4—42no—
337D4—82no—
338D4—82no—
339D4—121no—
340D4—121no—
341D4—242no—
342D4—244no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—62no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—242no—
355C2×C4—244no—
356C2×C4D3×D811yescharacteristic
357C2×C4D3×D811yescharacteristic
358C2×C4C2×D3×D411yescharacteristic
359C8—61no—
360C8—61no—
361C8#91612yes—
362C8D3×#4112yes—
363C2×C3—162no—
364C2×C3#316911yescharacteristic
365C2×C3C2×#50911yescharacteristic
366C2×C3#341411yescharacteristic
367D3—22no—
368D3—161no—
369C22—32no—
370C22—61no—
371C22—61no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—241no—
376C22—241no—
377C22—241no—
378C22C2×D3×D811yescharacteristic
379C4—21no—
380C4—21no—
381C4—61no—
382C4—61no—
383C4—61no—
384C4—81no—
385C4—81no—
386C4—242no—
387C4—241no—
388C4#566811yescharacteristic
389C4C2×#91611yescharacteristic
390C3#3374611yescharacteristic, complemented, Sylow
391C2—62no—
392C2—162no—
393C2—481no—
394C2#8155711yescharacteristic
395C2D3×#50911yescharacteristic
396C2#8203511yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2244C2
socleC22×C3C22×D8
hypercenter#2244D3
hypocenterC3#33746
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #81907C2semidirect
2. #81775C2semidirect
3. C3×#2244C2semidirect
4. #71355C2semidirect
5. #71292C2semidirect
6. #69196C2semidirect
7. #69195C2semidirect
8. #69177C2semidirect
9. #69176C2semidirect
10. #69079C2semidirect
11. D3×#346C2semidirect
48. #2244D3semidirect
179. C3⋊C16#41semidirect
390. C3#33746semidirect

Maximal subgroups

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

subgroupindex
#819072
#817752
C3×#22442
#713552
#712922
#691962
#691952
#691772
#691762
#690792
D3×#3462
#337463