Finite Group Databasebeta

← #10848608

Subgroups — #10848608

Order 768.

1188 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85848C211yesmaximal, characteristic
2#82533C211yesmaximal, characteristic, complemented
3#78537C211yesmaximal, characteristic, complemented
4#73313C211yesmaximal, characteristic, complemented
5C3×C4×#104C211yesmaximal, characteristic, complemented
6#70386C214yesmaximal, complemented
7#69876C211yesmaximal, characteristic, complemented
8#69396C211yesmaximal, characteristic, complemented
9#68759C214yesmaximal, complemented
10#25590—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5830C2212yes—
13C2×#951C2212yes—
14#5585—22no—
15#5553—22no—
16#5550—24no—
17#5535—22no—
18#5521C2212yes—
19#5477—22no—
20C2×#875C2214yes—
21C2×#874C2211yescharacteristic
22#5470C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C3×#104C2212yes—
25C3×C4×M16C2211yescharacteristic
26C3×C43C2211yescharacteristic
27C2×#823C2214yes—
28#5120C2214yes—
29#5031—22no—
30#5019—22no—
31#5004C2212yes—
32#4939—24no—
33#4924C2211yescharacteristic
34#4703—22no—
35#4681—22no—
36C3×#339C2212yes—
37#4523C2214yes—
38#3508—31no—
39#3139—31no—
40#3039—31no—
41#2858—31no—
42#2004—34no—
43#1994—31no—
44#1965—31no—
45#1885—34no—
46C4×#104D311yescharacteristic, complemented
47C2×C3×#41C2311yescharacteristic
48C2×#235—24no—
49C22×D12D414yes—
50C22×D12C2312yes—
51C22×Dic6C2312yes—
52C2×C3×M16C2312yes—
53C3×C4×Q8D412yes—
54C3×C4×D4—22no—
55C3×C4×D4D412yes—
56C3×#117—21no—
57C3×#117C2311yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42—22no—
60C2×C3×C42C2312yes—
61C2×C3×C42C2311yescharacteristic
62#951—44no—
63#947—22no—
64C2×#216—22no—
65#940—22no—
66#937—24no—
67C2×#214—22no—
68#930—24no—
69#930—24no—
70C2×(C2×D3)⋊C4—28no—
71#926C2312yes—
72C2×C4⋊Dic3—22no—
73#924—22no—
74C2×#208C2312yes—
75#894—28no—
76#893—24no—
77#884—28no—
78D3×C22⋊C4—24no—
79#875—24no—
80#875—28no—
81#875—44no—
82#874—21no—
83#874—21no—
84#874—22no—
85C4×D12—24no—
86C4×D12—24no—
87#870D412yes—
88#869—21no—
89#869—21no—
90#869—22no—
91#869D412yes—
92C4×Dic6—22no—
93C3×#104—24no—
94C3×#97—21no—
95C3×C4×C8—21no—
96#823—44no—
97#816—28no—
98#810—24no—
99C4⋊(C3⋊C8)—22no—
100C2×#137—32no—
101#559—62no—
102#550—62no—
103#546—64no—
104#540—62no—
105#535—32no—
106#533—31no—
107#532—62no—
108C2×C4⋊Q8—31no—
109C2×#127—31no—
110C2×#124—34no—
111#518—32no—
112C4×#41C2×D311yescharacteristic
113#491—62no—
114#487—62no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104C2×D312yes—
119#419—34no—
120#419—32no—
121C2×#100—34no—
122C4×M16C2×D311yescharacteristic
123#390—64no—
124C43C2×D311yescharacteristic
125#339—34no—
126#339C2×D312yes—
127C23×D3C2×D414yes—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8C2×D411yescharacteristic
131C2×C3×D4—21no—
132C2×C3×D4C2×D411yescharacteristic
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2411yescharacteristic
138C2×#67—28no—
139C22×Dic3—22no—
140#235—88no—
141C2×D12—24no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—24no—
145C2×D12—24no—
146C2×D12—24no—
147C2×D12—48no—
148C2×C4×D3—28no—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—24no—
153C2×Dic6—24no—
154C3×M16—22no—
155C3×M16—22no—
156C2×C3×C8—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4C2×D412yes—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—22no—
161C3×C42—22no—
162C3×C42—21no—
163C3×C42—22no—
164C3×C42—44no—
165C3×C42C2×D412yes—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4114yescomplemented
169#217—22no—
170#216—44no—
171#214—44no—
172(C2×D3)⋊C4—28no—
173(C2×D3)⋊C4—28no—
174(C2×D3)⋊C4—416no—
175C4⋊Dic3—22no—
176C4⋊Dic3—42no—
177C4⋊Dic3—42no—
178C4⋊Dic3C2×D414yes—
179Dic3⋊C4—24no—
180C4×Dic3—22no—
181#208—24no—
182C2×C3⋊C8—22no—
183C2×#41—64no—
184C2×#41C22×D311yescharacteristic
185C22×Q8—32no—
186C22×D4—34no—
187C22×D4—32no—
188#137—124no—
189C2×Q16—62no—
190C2×SD16—62no—
191C2×M16—32no—
192C2×M16C22×D312yes—
193C4⋊Q8—32no—
194C4⋊Q8—62no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197#127—62no—
198#127—62no—
199#127—61no—
200#127—61no—
201#125—32no—
202#124—64no—
203#124—62no—
204#124—68no—
205#124—124no—
206#123—68no—
207C22⋊Q8—62no—
208#121—68no—
209#121—64no—
210#121—64no—
211#120—64no—
212C4×Q8—62no—
213C4×Q8D1212yes—
214C4×D4—22no—
215C4×D4—64no—
216C4×D4—64no—
217C4×D4—64no—
218C4×D4D1212yes—
219#117—21no—
220#117—32no—
221#117C22×D311yescharacteristic
222C2×C4⋊C4—62no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—68no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42C22×D312yes—
228C2×C42C22×D311yescharacteristic
229C4⋊C8—62no—
230#104—24no—
231#103—68no—
232#103—64no—
233#100—124no—
234#97—21no—
235C4×C8—21no—
236C23×C3—21no—
237C23×C3C22×D411yescharacteristic
238C22×D3—24no—
239C22×D3—28no—
240C22×D3—48no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—42no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—44no—
251C2×C3×C4—42no—
252C2×C3×C4—44no—
253C2×C3×C4#12012yes—
254C2×C3×C4#12012yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C4H(5,𝔽2)12yes—
259#67—88no—
260C2×Dic3—24no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263D12—42no—
264D12—42no—
265D12—44no—
266D12—48no—
267D12—84no—
268C4×D3—88no—
269Dic6—44no—
270Dic6—42no—
271Dic6—42no—
272Dic6—44no—
273C3×C8—42no—
274C3⋊C8—42no—
275C24—34no—
276#41—22no—
277#41—22no—
278#41—248no—
279C2×Q8—62no—
280C2×Q8—62no—
281C2×Q8—64no—
282C2×Q8—62no—
283C2×Q8—64no—
284C2×Q8C2×D1211yescharacteristic
285C2×D4—21no—
286C2×D4—68no—
287C2×D4—64no—
288C2×D4—64no—
289C2×D4—62no—
290C2×D4—64no—
291C2×D4—62no—
292C2×D4—64no—
293C2×D4—128no—
294C2×D4C2×D1211yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—68no—
299C22×C4—62no—
300C22×C4D3×D412yes—
301C22×C4C23×D311yescharacteristic
302Q16—124no—
303SD16—124no—
304M16—22no—
305M16—22no—
306M16—64no—
307C2×C8—22no—
308C2×C8—62no—
309C4⋊C4—22no—
310C4⋊C4—34no—
311C4⋊C4—62no—
312C4⋊C4—64no—
313C4⋊C4—122no—
314C4⋊C4—122no—
315C4⋊C4C2×D1212yes—
316C22⋊C4—21no—
317C22⋊C4—22no—
318C22⋊C4—68no—
319C22⋊C4—68no—
320C22⋊C4—62no—
321C22⋊C4—1216no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42—44no—
326C42—62no—
327C42C2×D1212yes—
328C42#23514yescomplemented
329C42D3×D411yescharacteristic
330C42D3×D411yescharacteristic
331C22×C3—22no—
332C22×C3—41no—
333C22×C3—42no—
334C22×C3C2×#12011yescharacteristic
335C22×C3#54612yes—
336C2×D3—44no—
337C2×D3—48no—
338C2×D3—88no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—41no—
342C3×C4—42no—
343C3×C4#45514yes—
344C3×C4#53211yescharacteristic
345C3×C4#53511yescharacteristic
346C3×C4#55012yes—
347Dic3—44no—
348Dic3—82no—
349C23—21no—
350C23—64no—
351C23—68no—
352C23—128no—
353C23C2×D3×D411yescharacteristic
354Q8—21no—
355Q8—21no—
356Q8—124no—
357Q8—122no—
358Q8—122no—
359Q8—124no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—122no—
364D4—122no—
365D4—124no—
366D4—128no—
367D4—244no—
368D4—248no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—64no—
377C2×C4—122no—
378C2×C4—124no—
379C2×C4—248no—
380C2×C4#88212yes—
381C2×C4#88212yes—
382C2×C4C22×D1211yescharacteristic
383C2×C4C2×#23512yes—
384C2×C4C2×D3×D411yescharacteristic
385C2×C4#100912yes—
386C8—42no—
387C8—122no—
388C2×C3—22no—
389C2×C3—81no—
390C2×C3#243111yescharacteristic
391C2×C3C2×#45512yes—
392D3—84no—
393C22—22no—
394C22—41no—
395C22—42no—
396C22—124no—
397C22—128no—
398C22—248no—
399C22C2×#88211yescharacteristic
400C22#555012yes—
401C4—22no—
402C4—44no—
403C4—41no—
404C4—42no—
405C4—124no—
406C4—242no—
407C4#482514yes—
408C4#547711yescharacteristic
409C4#552511yescharacteristic
410C4#555312yes—
411C3#2559011yescharacteristic, complemented, Sylow
412C2—22no—
413C2—81no—
414C2—244no—
415C2#7358611yescharacteristic
416C2C2×#482512yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×#120
prime radical#85848C2
hypercenterC4×#104D3
hypocenterC3#25590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. #82533C2semidirect
3. #78537C2semidirect
4. #73313C2semidirect
5. C3×C4×#104C2semidirect
6. #70386C2semidirect
7. #69876C2semidirect
8. #69396C2semidirect
9. #68759C2semidirect
46. C4×#104D3semidirect
168. C3×C42#41semidirect
328. C42#235semidirect
411. C3#25590semidirect

Maximal subgroups

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

subgroupindex
#858482
#825332
#785372
#733132
C3×C4×#1042
#703862
#698762
#693962
#687592
#255903