Finite Group Databasebeta

← #10897637

Subgroups — #10897637

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84000C211yesmaximal, characteristic, complemented
2#79038C211yesmaximal, characteristic, complemented
3C3×#2050C211yesmaximal, characteristic, complemented
4#72118C211yesmaximal, characteristic
5#70526C211yesmaximal, characteristic, complemented
6#70382C212yesmaximal, complemented
7#70375C212yesmaximal, complemented
8#70136C212yesmaximal, complemented
9#70130C212yesmaximal, complemented
10#68721C211yesmaximal, characteristic, complemented
11#68716C211yesmaximal, characteristic, complemented
12#29384—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14#5828C2212yes—
15#5819C2212yes—
16C2×C42×D3C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C2×C3×#104C2212yes—
19C3×#413C2212yes—
20#5236C2212yes—
21C2×#837C2212yes—
22#5197—22no—
23#5196—22no—
24#5126C2211yescharacteristic
25#5126C2211yescharacteristic
26#5126C2212yes—
27#5125C2211yescharacteristic
28#5125C2212yes—
29Dic3×M16C2211yescharacteristic
30Dic3×M16C2211yescharacteristic
31#4924C2211yescharacteristic
32#4835—22no—
33#4833—22no—
34C3×#338C2211yescharacteristic
35#4559C2212yes—
36#4559C2212yes—
37#4557C2211yescharacteristic
38#4555C2212yes—
39#4523C2212yes—
40#2956—31no—
41#2920—31no—
42#2102—31no—
43#2091—31no—
44#2050D311yescharacteristic, complemented
45#2010—31no—
46#1932—32no—
47#1896—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C22×C4×D3—21no—
53C22×C4×D3C2311yescharacteristic
54C3×#137—22no—
55C3×AGL(1,ℤ/8)—22no—
56C3×#135—22no—
57C2×C3×M16C2312yes—
58C2×C3×M16C2311yescharacteristic
59C2×C3×M16C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61#948—24no—
62#948—42no—
63#946—22no—
64#938—22no—
65#937—22no—
66#935—22no—
67C4×#67—42no—
68#926C2312yes—
69C2×C4×Dic3—21no—
70C2×C4×Dic3C2311yescharacteristic
71C2×#208—42no—
72C2×#208C2311yescharacteristic
73C2×#208C2312yes—
74C2×#208C2311yescharacteristic
75C42×D3—21no—
76C42×D3—21no—
77C42×D3—21no—
78C42×D3—21no—
79C3×C4⋊C8—22no—
80C3×#104—22no—
81C3×#104—22no—
82C3×#101—22no—
83C3×#100—22no—
84#837—22no—
85#837—22no—
86#837—42no—
87(C2×D3)⋊C8—22no—
88(C2×D3)⋊C8—22no—
89(C2×D3)⋊C8—42no—
90#819—22no—
91#819—42no—
92#815—21no—
93#815D412yescomplemented
94C8×Dic3—21no—
95C8×Dic3D412yescomplemented
96C4⋊(C3⋊C8)—22no—
97#575C2×D311yescharacteristic
98#568—32no—
99#525—32no—
100C22×C42—31no—
101#444—62no—
102#443—62no—
103#432—62no—
104#431—62no—
105#430—31no—
106#430—32no—
107#423—31no—
108#423C2×D311yescharacteristic
109#421—32no—
110C2×#104—32no—
111C2×#104C2×D312yes—
112#413C2×D312yes—
113#407—31no—
114#407—31no—
115#407—32no—
116C4×M16—31no—
117C4×M16—31no—
118#344—32no—
119#344—32no—
120#339—32no—
121#339—32no—
122#338C2×D311yescharacteristic
123H(3,ℤ/4)—31no—
124C23×D3#4111yescharacteristic
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C2×C3×Q8#4112yes—
130C2×C3×D4—22no—
131C2×C3×D4#4112yes—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4C2411yescharacteristic
135C2×#67—42no—
136C22×Dic3—21no—
137C22×Dic3#4111yescharacteristic
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—22no—
145C2×C4×D3—22no—
146C2×C4×D3—44no—
147C3×Q16—22no—
148C3×SD16—24no—
149C3×D8—22no—
150C3×M16—22no—
151C3×M16—22no—
152C3×M16—22no—
153C3×M16—22no—
154C2×C3×C8—21no—
155C2×C3×C8—22no—
156C2×C3×C8C2×D412yes—
157C3×C42—21no—
158C3×C42#4111yescharacteristic
159C3×C42#4111yescharacteristic
160#217—22no—
161#217—42no—
162(C2×D3)⋊C4—42no—
163C4⋊Dic3—22no—
164Dic3⋊C4—42no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—22no—
171C4×Dic3C2×D411yescharacteristic
172C4×Dic3C2×D411yescharacteristic
173#208—22no—
174#208—22no—
175#208—22no—
176#208—22no—
177#208—44no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—44no—
181C2×C3⋊C8C2×D412yes—
182C2×#41C22×D312yes—
183C23×C4—31no—
184C23×C4—61no—
185#137—22no—
186AGL(1,ℤ/8)—22no—
187#135—22no—
188#131—64no—
189#131—122no—
190C2×M16—31no—
191C2×M16—32no—
192C2×M16—31no—
193C2×M16—122no—
194C2×M16C22×D312yes—
195C2×M16C22×D311yescharacteristic
196C2×M16C22×D311yescharacteristic
197#123—62no—
198C22⋊Q8—62no—
199#121—62no—
200#120—62no—
201C4×D4—122no—
202#117—32no—
203C2×C42—31no—
204C2×C42—61no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210#108—62no—
211#108—122no—
212C4⋊C8—22no—
213C4⋊C8—62no—
214#104—22no—
215#104—22no—
216#104—62no—
217#104—62no—
218#104—122no—
219#101—22no—
220#100—22no—
221C22⋊C8—62no—
222C22⋊C8—62no—
223C22⋊C8—122no—
224#97—32no—
225#97—61no—
226C4×C8—32no—
227C4×C8—61no—
228C23×C3—22no—
229C23×C3C2×#4111yescharacteristic
230C22×D3—21no—
231C22×D3—22no—
232C22×D3—42no—
233C3×Q8—22no—
234C3×Q8—22no—
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—42no—
238C3×D4—42no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4H(5,𝔽2)12yes—
249#67—84no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—22no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C4×D3—24no—
257C4×D3—42no—
258C4×D3—42no—
259C4×D3—42no—
260C4×D3—41no—
261C4×D3—41no—
262C4×D3—42no—
263C3×C8—22no—
264C3×C8—41no—
265C3×C8—42no—
266C3⋊C8—22no—
267C3⋊C8—42no—
268C3⋊C8—41no—
269C24—31no—
270#41—22no—
271#41—22no—
272#41—22no—
273#41—22no—
274C2×Q8#23712yes—
275C2×D4—22no—
276C2×D4—122no—
277C2×D4#23712yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—31no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—124no—
291C22×C4C23×D311yescharacteristic
292Q16—22no—
293SD16—24no—
294D8—22no—
295M16—22no—
296M16—22no—
297M16—22no—
298M16—22no—
299M16—62no—
300M16—62no—
301M16—62no—
302M16—62no—
303M16—124no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—32no—
307C2×C8—61no—
308C2×C8—62no—
309C2×C8—124no—
310C2×C8D3×D412yes—
311C4⋊C4—62no—
312C4⋊C4—122no—
313C22⋊C4—62no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C42—21no—
317C42—31no—
318C42—31no—
319C42—61no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
324C42—62no—
325C42#23711yescharacteristic
326C42#23711yescharacteristic
327C22×C3—41no—
328C22×C3—42no—
329C22×C3—82no—
330C22×C3#52511yescharacteristic
331C22×C3#55112yes—
332C2×D3—22no—
333C2×D3—24no—
334C2×D3—41no—
335C2×D3—42no—
336C2×D3—41no—
337C3×C4—22no—
338C3×C4—41no—
339C3×C4—42no—
340C3×C4#52511yescharacteristic
341C3×C4#53511yescharacteristic
342C3×C4#55312yes—
343Dic3—22no—
344Dic3—41no—
345Dic3—41no—
346Dic3—82no—
347C23—22no—
348C23—62no—
349C23—61no—
350C23—122no—
351C23D3×#4111yescharacteristic
352Q8—22no—
353Q8—22no—
354D4—22no—
355D4—22no—
356D4—42no—
357D4—42no—
358D4—244no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—64no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4C2×D3×D411yescharacteristic
379C2×C4C2×#23712yes—
380C2×C4C2×#23711yescharacteristic
381C2×C4#100912yes—
382C8—22no—
383C8—41no—
384C8—42no—
385C8—62no—
386C8—122no—
387C8—121no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—82no—
391C2×C3#259811yescharacteristic
392D3—42no—
393C22—41no—
394C22—42no—
395C22—62no—
396C22—64no—
397C22—82no—
398C22—121no—
399C22—122no—
400C22—121no—
401C22#560611yescharacteristic
402C22#565812yes—
403C4—22no—
404C4—41no—
405C4—42no—
406C4—62no—
407C4—121no—
408C4—121no—
409C4—242no—
410C4#558611yescharacteristic
411C4#561812yes—
412C4#567411yescharacteristic
413C3#2938411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—22no—
416C2—82no—
417C2—122no—
418C2#7696211yescharacteristic
subgroupG/H
centerC4#5674
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2050C2
socleC2×C3#2598
prime radical#72118C2
hypercenter#2050D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2050

Complements

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

NH ≅ G/Ntype
1. #84000C2semidirect
2. #79038C2semidirect
3. C3×#2050C2semidirect
5. #70526C2semidirect
6. #70382C2semidirect
7. #70375C2semidirect
8. #70136C2semidirect
9. #70130C2semidirect
10. #68721C2semidirect
11. #68716C2semidirect
44. #2050D3semidirect
93. #815D4semidirect
95. C8×Dic3D4semidirect
413. C3#29384semidirect

Maximal subgroups

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

subgroupindex
#840002
#790382
C3×#20502
#721182
#705262
#703822
#703752
#701362
#701302
#687212
#687162
#293843