Finite Group Databasebeta

← #10897636

Subgroups — #10897636

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
1C3×#2956C211yesmaximal, characteristic, complemented
2#78517C211yesmaximal, characteristic, complemented
3#72118C211yesmaximal, characteristic
4#72117C211yesmaximal, characteristic, complemented
5#72029C212yesmaximal, complemented
6#72028C212yesmaximal, complemented
7#70729C211yesmaximal, characteristic, complemented
8#70728C211yesmaximal, characteristic, complemented
9#70382C212yesmaximal, complemented
10#70378C212yesmaximal, complemented
11#69871C211yesmaximal, characteristic, complemented
12#29384—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C3×#568C2212yes—
15#5828C2212yes—
16C2×C42×D3C2211yescharacteristic
17C3×#444—22no—
18C3×#443—22no—
19C3×#430C2211yescharacteristic
20C2×C3×#104C2212yes—
21C3×C4×M16C2211yescharacteristic
22#5237C2212yes—
23#5236C2212yes—
24C2×#837C2212yes—
25#5126C2211yescharacteristic
26#5126C2211yescharacteristic
27#5126C2212yes—
28#5124C2212yes—
29#5118C2211yescharacteristic
30#5118C2211yescharacteristic
31C4×#208C2211yescharacteristic
32#4705—22no—
33#4685—22no—
34C3⋊#338C2211yescharacteristic
35#4557C2211yescharacteristic
36#4556C2212yes—
37#4556C2212yes—
38#4555C2212yes—
39#4523C2212yes—
40#2956D311yescharacteristic, complemented
41#2920—31no—
42#2102—31no—
43#2091—31no—
44#2050—31no—
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—
57C3×#131—24no—
58C3×#131—42no—
59C2×C3×M16—42no—
60C2×C3×M16C2312yes—
61C2×C3×M16C2311yescharacteristic
62C2×C3×M16C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64#946—22no—
65#938—22no—
66#937—22no—
67#935—22no—
68C4×#67—42no—
69#926C2312yes—
70C2×C4×Dic3—21no—
71C2×C4×Dic3C2311yescharacteristic
72C2×#208C2311yescharacteristic
73C2×#208C2312yes—
74C2×#208C2311yescharacteristic
75C42×D3—21no—
76C42×D3—21no—
77C42×D3—21no—
78C42×D3—21no—
79C3×#108—22no—
80C3×#104—22no—
81C3×#104—22no—
82C3×#97D412yescomplemented
83C3×C4×C8D412yescomplemented
84#837—22no—
85#837—22no—
86#837—42no—
87#836—22no—
88#833—22no—
89#822—42no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—42no—
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95#803—21no—
96C4×C3⋊C8—21no—
97#575C2×D311yescharacteristic
98#568C2×D312yes—
99#525—32no—
100C22×C42—31no—
101#444—22no—
102#443—22no—
103#432—62no—
104#431—62no—
105#430—32no—
106#430C2×D311yescharacteristic
107#423—31no—
108#423—31no—
109#421—32no—
110C2×#104—32no—
111C2×#104C2×D312yes—
112#413—32no—
113#407—31no—
114#407—31no—
115#407—32no—
116C4×M16—31no—
117C4×M16C2×D311yescharacteristic
118#344—32no—
119#344—32no—
120#339—32no—
121#339—32no—
122#338—31no—
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—
154C3×M16—44no—
155C2×C3×C8—22no—
156C2×C3×C8—44no—
157C2×C3×C8C2×D412yes—
158C2×C3×C8C2×D412yes—
159C3×C42—21no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162#217—22no—
163#217—42no—
164(C2×D3)⋊C4—42no—
165C4⋊Dic3—22no—
166Dic3⋊C4—42no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—21no—
172C4×Dic3—22no—
173C4×Dic3#4111yescharacteristic
174C4×Dic3#4111yescharacteristic
175#208—22no—
176#208—22no—
177#208—22no—
178#208—22no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—22no—
182C2×#41C22×D312yes—
183C23×C4—31no—
184C23×C4—61no—
185#137—22no—
186AGL(1,ℤ/8)—22no—
187#135—22no—
188#131—24no—
189#131—42no—
190C2×M16—32no—
191C2×M16—31no—
192C2×M16—31no—
193C2×M16—42no—
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—22no—
211#108—122no—
212C4⋊C8—62no—
213C4⋊C8—62no—
214#104—22no—
215#104—22no—
216#104—62no—
217#104—62no—
218#104—122no—
219#101—62no—
220#100—62no—
221C22⋊C8—62no—
222C22⋊C8—62no—
223C22⋊C8—122no—
224#97—61no—
225#97#6712yescomplemented
226C4×C8—61no—
227C4×C8#6712yescomplemented
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—22no—
265C3×C8—42no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—42no—
269C24—31no—
270#41—22no—
271#41—22no—
272#41—22no—
273#41—22no—
274C2×Q8#23512yes—
275C2×D4—22no—
276C2×D4—122no—
277C2×D4#23512yes—
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—62no—
287C22×C4—62no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—124no—
291C22×C4C23×D311yescharacteristic
292Q16—22no—
293SD16—24no—
294D8—22no—
295M16—22no—
296M16—22no—
297M16—22no—
298M16—22no—
299M16—44no—
300M16—62no—
301M16—62no—
302M16—62no—
303M16—62no—
304C2×C8—22no—
305C2×C8—44no—
306C2×C8—62no—
307C2×C8—61no—
308C2×C8—61no—
309C2×C8C2×#6712yes—
310C2×C8C2×#6712yes—
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—
325C42C2×#6711yescharacteristic
326C42C2×#6711yescharacteristic
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—122no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4C2×#23512yes—
379C2×C4#100412yes—
380C2×C4D3×#4111yescharacteristic
381C2×C4C22×#6711yescharacteristic
382C8—22no—
383C8—22no—
384C8—42no—
385C8—121no—
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#555612yes—
402C22#582811yescharacteristic
403C4—22no—
404C4—41no—
405C4—42no—
406C4—62no—
407C4—121no—
408C4—121no—
409C4—242no—
410C4#554012yes—
411C4#579111yescharacteristic
412C4#580511yescharacteristic
413C3#2938411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—22no—
416C2—82no—
417C2—122no—
418C2#8241811yescharacteristic
subgroupG/H
centerC4#5805
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2956C2
socleC2×C3#2598
prime radical#72118C2
hypercenter#2956D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2956

Complements

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

NH ≅ G/Ntype
1. C3×#2956C2semidirect
2. #78517C2semidirect
4. #72117C2semidirect
5. #72029C2semidirect
6. #72028C2semidirect
7. #70729C2semidirect
8. #70728C2semidirect
9. #70382C2semidirect
10. #70378C2semidirect
11. #69871C2semidirect
40. #2956D3semidirect
82. C3×#97D4semidirect
83. C3×C4×C8D4semidirect
225. #97#67semidirect
227. C4×C8#67semidirect
413. C3#29384semidirect

Maximal subgroups

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

subgroupindex
C3×#29562
#785172
#721182
#721172
#720292
#720282
#707292
#707282
#703822
#703782
#698712
#293843