Finite Group Databasebeta

← #10897644

Subgroups — #10897644

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#78541C211yesmaximal, characteristic, complemented
2#78517C211yesmaximal, characteristic, complemented
3C3×#1841C211yesmaximal, characteristic, complemented
4#69036C212yesmaximal, complemented
5#69033C212yesmaximal, complemented
6#68756C212yesmaximal, complemented
7#68750C212yesmaximal, complemented
8#68721C211yesmaximal, characteristic, complemented
9#68719C211yesmaximal, characteristic, complemented
10#68703C211yesmaximal, characteristic
11#68701C211yesmaximal, characteristic, complemented
12#29384—31nomaximal, Sylow
13#5748C2211yescharacteristic
14#5745C2212yes—
15#5524C2212yes—
16C2×C42×D3C2211yescharacteristic
17C3×C4×M16C2211yescharacteristic
18C2×#825C2212yes—
19#5132C2212yes—
20#5126C2211yescharacteristic
21#5126C2211yescharacteristic
22#5126C2212yes—
23#5125C2211yescharacteristic
24#5125C2212yes—
25#5118C2211yescharacteristic
26#5118C2211yescharacteristic
27#5005C2212yes—
28C2×#805C2212yes—
29C4×#208C2211yescharacteristic
30#4717—22no—
31#4705—22no—
32#4700—22no—
33#4685—22no—
34C3×#344C2212yes—
35C3×#339C2212yes—
36C3×#338C2211yescharacteristic
37C3×H(3,ℤ/4)C2211yescharacteristic
38#4533C2212yes—
39#4532C2212yes—
40#2956—31no—
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#1841D311yescharacteristic, complemented
51C2×#235C2312yes—
52C22×C4×D3—21no—
53C22×C4×D3C2311yescharacteristic
54C2×C3×M16C2312yes—
55C2×C3×M16C2311yescharacteristic
56C2×C3×M16C2311yescharacteristic
57C3×#117C2312yes—
58C2×C3×C42C2311yescharacteristic
59C2×C4×Dic3—21no—
60C2×C4×Dic3C2311yescharacteristic
61C2×#208C2311yescharacteristic
62C2×#208C2312yes—
63C2×#208C2311yescharacteristic
64#909—22no—
65#908—22no—
66#907—42no—
67D3×M16—42no—
68#904—22no—
69#901—24no—
70#897—22no—
71#895—22no—
72#886—22no—
73#882—22no—
74C4×D12—42no—
75C42×D3—21no—
76C42×D3—21no—
77C42×D3—21no—
78C42×D3—21no—
79C3×#97D412yescomplemented
80C3×C4×C8D412yescomplemented
81#825—22no—
82#825—22no—
83#824—22no—
84#823—22no—
85(C2×D3)⋊C8—22no—
86(C2×D3)⋊C8—22no—
87(C2×D3)⋊C8—42no—
88#819—22no—
89#819—42no—
90Dic3⋊C8—22no—
91Dic3⋊C8—22no—
92#805—22no—
93#805—22no—
94#805—42no—
95#803—21no—
96C4×C3⋊C8—21no—
97#575—31no—
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#423—31no—
109#421—32no—
110C2×#104—32no—
111C2×#104—32no—
112#413—32no—
113#407—31no—
114#407—31no—
115#407—32no—
116C4×M16—31no—
117C4×M16C2×D311yescharacteristic
118#344—32no—
119#344C2×D312yes—
120#339—32no—
121#339C2×D312yes—
122#338C2×D311yescharacteristic
123H(3,ℤ/4)C2×D311yescharacteristic
124C23×D3#4111yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4C2411yescharacteristic
127C2×#67—22no—
128C22×Dic3—21no—
129C22×Dic3#4111yescharacteristic
130#235—22no—
131#235—22no—
132#235—22no—
133#235—22no—
134C2×D12—42no—
135C2×D12#4112yes—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—22no—
143C2×C4×D3—22no—
144C2×C4×D3—22no—
145C2×C4×D3—44no—
146C2×Dic6#4112yes—
147C3×M16—22no—
148C3×M16—22no—
149C3×M16—22no—
150C3×M16—22no—
151C2×C3×C8—22no—
152C2×C3×C8C2×D412yes—
153C2×C3×C8C2×D412yes—
154C3×C4⋊C4—22no—
155C3×C22⋊C4—22no—
156C3×C42—21no—
157C3×C42—22no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—42no—
162C4⋊Dic3—42no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3#4111yescharacteristic
169C4×Dic3#4111yescharacteristic
170#208—22no—
171#208—22no—
172#208—22no—
173#208—22no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—22no—
177Dic12—22no—
178D24—22no—
179#204—24no—
180ΓL(1,𝔽25)—44no—
181D3×C8—44no—
182C2×#41—32no—
183C23×C4—31no—
184C23×C4—61no—
185#137—62no—
186AGL(1,ℤ/8)—62no—
187#135—62no—
188#131—64no—
189#131—122no—
190C2×M16—32no—
191C2×M16—31no—
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#117C22×D312yes—
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—62no—
213C4⋊C8—62no—
214#104—62no—
215#104—62no—
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#97D1212yescomplemented
226C4×C8—61no—
227C4×C8D1212yescomplemented
228C23×C3C2×#4111yescharacteristic
229C22×D3—21no—
230C22×D3—22no—
231C22×D3—22no—
232C22×D3—42no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4C22×D411yescharacteristic
238C2×C3×C4C2×#4112yes—
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4H(5,𝔽2)12yes—
241#67—42no—
242#67—42no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—42no—
249D12—22no—
250D12—22no—
251D12—84no—
252C4×D3—24no—
253C4×D3—42no—
254C4×D3—42no—
255C4×D3—42no—
256C4×D3—41no—
257C4×D3—42no—
258C4×D3—41no—
259C4×D3—42no—
260C4×D3—42no—
261Dic6—22no—
262Dic6—22no—
263C3×C8—22no—
264C3×C8—22no—
265C3×C8—42no—
266C3⋊C8—41no—
267C3⋊C8—42no—
268C3⋊C8—41no—
269C24—31no—
270#41—62no—
271#41—62no—
272#41—62no—
273#41—62no—
274C2×Q8—32no—
275C2×D4—32no—
276C2×D4—62no—
277C2×D4—122no—
278C22×C4—21no—
279C22×C4—31no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—124no—
291C22×C4C23×D311yescharacteristic
292Q16—62no—
293SD16—64no—
294D8—62no—
295M16—22no—
296M16—22no—
297M16—22no—
298M16—22no—
299M16—62no—
300M16—62no—
301M16—62no—
302M16—62no—
303M16—124no—
304C2×C8—22no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—62no—
308C2×C8—124no—
309C2×C8C2×D1212yes—
310C2×C8C2×D1212yes—
311C4⋊C4—22no—
312C4⋊C4—122no—
313C22⋊C4—22no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C42—21no—
317C42—22no—
318C42—31no—
319C42—31no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
324C42—61no—
325C42C2×D1211yescharacteristic
326C42C2×D1211yescharacteristic
327C22×C3—41no—
328C22×C3#52511yescharacteristic
329C22×C3#55112yes—
330C2×D3—22no—
331C2×D3—24no—
332C2×D3—41no—
333C2×D3—42no—
334C2×D3—41no—
335C2×D3—42no—
336C2×D3—82no—
337C3×C4—22no—
338C3×C4—41no—
339C3×C4—82no—
340C3×C4#52511yescharacteristic
341C3×C4#53511yescharacteristic
342C3×C4#55312yes—
343Dic3—22no—
344Dic3—41no—
345Dic3—42no—
346Dic3—41no—
347C23—62no—
348C23—61no—
349C23—62no—
350C23—122no—
351C23D3×#4111yescharacteristic
352Q8—62no—
353Q8—62no—
354D4—62no—
355D4—62no—
356D4—122no—
357D4—122no—
358D4—244no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—64no—
364C2×C4—61no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—121no—
378C2×C4C22×D1211yescharacteristic
379C2×C4D3×#4111yescharacteristic
380C2×C4D3×#4112yes—
381C2×C4#100912yes—
382C8—22no—
383C8—22no—
384C8—42no—
385C8—121no—
386C8—122no—
387C8—121no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3#259811yescharacteristic
391D3—42no—
392D3—82no—
393C22—41no—
394C22—62no—
395C22—64no—
396C22—121no—
397C22—122no—
398C22—121no—
399C22—122no—
400C22—242no—
401C22#552411yescharacteristic
402C22#570512yes—
403C4—22no—
404C4—41no—
405C4—62no—
406C4—82no—
407C4—121no—
408C4—121no—
409C4—122no—
410C4#552411yescharacteristic
411C4#552511yescharacteristic
412C4#570912yes—
413C3#2938411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—22no—
416C2—122no—
417C2—242no—
418C2#7430711yescharacteristic
subgroupG/H
centerC4#5525
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1841C2
socleC2×C3#2598
prime radical#68703C2
hypercenter#1841D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1841

Complements

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

NH ≅ G/Ntype
1. #78541C2semidirect
2. #78517C2semidirect
3. C3×#1841C2semidirect
4. #69036C2semidirect
5. #69033C2semidirect
6. #68756C2semidirect
7. #68750C2semidirect
8. #68721C2semidirect
9. #68719C2semidirect
11. #68701C2semidirect
50. #1841D3semidirect
79. C3×#97D4semidirect
80. C3×C4×C8D4semidirect
225. #97D12semidirect
227. C4×C8D12semidirect
413. C3#29384semidirect

Maximal subgroups

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

subgroupindex
#785412
#785172
C3×#18412
#690362
#690332
#687562
#687502
#687212
#687192
#687032
#687012
#293843