Finite Group Databasebeta

← #10897642

Subgroups — #10897642

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×#2920C211yesmaximal, characteristic, complemented
2#78541C211yesmaximal, characteristic, complemented
3#71532C212yesmaximal, complemented
4#70531C211yesmaximal, characteristic, complemented
5#70530C212yesmaximal, complemented
6#70526C211yesmaximal, characteristic, complemented
7#69993C211yesmaximal, characteristic
8#69992C212yesmaximal, complemented
9#69987C211yesmaximal, characteristic, complemented
10#69874C212yesmaximal, complemented
11#69871C211yesmaximal, characteristic, complemented
12#29384—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14#5791C2212yes—
15#5748C2211yescharacteristic
16#5745C2212yes—
17C3×#432—22no—
18C3×#431—22no—
19C3×#423C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#407C2211yescharacteristic
22C3×#407C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24C3×C4×M16C2211yescharacteristic
25#5207C2212yes—
26#5139C2212yes—
27#5132C2212yes—
28#5125C2211yescharacteristic
29#5124C2212yes—
30C2×#805C2212yes—
31C2×#805C2212yes—
32#4717—22no—
33#4700—22no—
34C3⋊#338C2211yescharacteristic
35#4559C2212yes—
36#4557C2211yescharacteristic
37#4556C2212yes—
38#4523C2212yes—
39#4523C2212yes—
40#2956—31no—
41#2920D311yescharacteristic, complemented
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—
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C2×#235C2312yes—
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C2×C3×M16C2311yescharacteristic
57C2×C3×M16C2311yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42—21no—
60C2×C3×C42—21no—
61C2×C3×C42—21no—
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65#953—22no—
66#930—22no—
67#929—22no—
68C4×#67—42no—
69#926C2312yes—
70#924—22no—
71C2×#208C2312yes—
72C2×#208C2312yes—
73#909—22no—
74#908—22no—
75#907—42no—
76D3×M16—42no—
77#904—22no—
78#901—24no—
79C3×C4⋊C8—22no—
80C3×C4⋊C8—22no—
81C3×C22⋊C8—22no—
82C3×C22⋊C8—22no—
83C3×#97—21no—
84C3×#97D412yescomplemented
85C3×C4×C8—21no—
86C3×C4×C8D412yescomplemented
87#830—42no—
88#825—42no—
89#824—22no—
90#823—22no—
91#822—42no—
92#819—22no—
93#805—22no—
94#805—22no—
95#805—22no—
96#805—22no—
97#575—31no—
98#568—32no—
99#525—32no—
100C22×C42C2×D311yescharacteristic
101#444—62no—
102#443—62no—
103#432—22no—
104#431—22no—
105#430—31no—
106#430—32no—
107#423C2×D311yescharacteristic
108#423C2×D311yescharacteristic
109#421—32no—
110C2×#104—32no—
111C2×#104—32no—
112#413—32no—
113#407—32no—
114#407C2×D311yescharacteristic
115#407C2×D311yescharacteristic
116C4×M16C2×D311yescharacteristic
117C4×M16C2×D311yescharacteristic
118#344—32no—
119#344—32no—
120#339—32no—
121#339—32no—
122#338—31no—
123H(3,ℤ/4)—31no—
124C24×C3#4111yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—44no—
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C2×#67—22no—
139C2×#67—42no—
140#235—22no—
141#235—22no—
142#235—22no—
143#235—22no—
144C2×D12#4112yes—
145C2×C4×D3—22no—
146C2×Dic6#4112yes—
147C3×M16—22no—
148C3×M16—22no—
149C3×M16—22no—
150C3×M16—22no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8C2×D412yes—
154C2×C3×C8C2×D412yes—
155C3×C42—21no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42—21no—
159C3×C42—21no—
160C3×C42—21no—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164C3×C42#4111yescharacteristic
165#217—22no—
166#217—42no—
167(C2×D3)⋊C4—42no—
168C4⋊Dic3—22no—
169Dic3⋊C4—42no—
170C4×Dic3—22no—
171#208—22no—
172#208—22no—
173#208—22no—
174#208—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177Dic12—22no—
178D24—22no—
179#204—24no—
180ΓL(1,𝔽25)—44no—
181D3×C8—44no—
182C2×#41—32no—
183C23×C4—21no—
184C23×C4C22×D311yescharacteristic
185#137—62no—
186AGL(1,ℤ/8)—62no—
187#135—62no—
188#131—64no—
189#131—122no—
190C2×M16—32no—
191C2×M16—32no—
192C2×M16—122no—
193C2×M16C22×D311yescharacteristic
194C2×M16C22×D311yescharacteristic
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—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22×D311yescharacteristic
209C2×C42C22×D311yescharacteristic
210#108—62no—
211#108—122no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214#104—62no—
215#104—62no—
216#104—62no—
217#104—62no—
218#104—122no—
219#101—62no—
220#100—62no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—122no—
224#97—21no—
225#97#6712yescomplemented
226C4×C8—21no—
227C4×C8#6712yescomplemented
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3C2×#4111yescharacteristic
232C22×D3—22no—
233C2×C3×C4—24no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4C2×#4112yes—
251C2×C3×C4H(5,𝔽2)12yes—
252#67—42no—
253#67—42no—
254#67—84no—
255C2×Dic3—22no—
256C2×Dic3—42no—
257D12—22no—
258D12—22no—
259C4×D3—42no—
260C4×D3—42no—
261Dic6—22no—
262Dic6—22no—
263C3×C8—22no—
264C3×C8—22no—
265C3×C8—41no—
266C3×C8—41no—
267C3⋊C8—42no—
268C3⋊C8—42no—
269C24#23511yescharacteristic
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—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—44no—
289C22×C4—62no—
290C22×C4#23511yescharacteristic
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—21no—
305C2×C8—21no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—124no—
309C2×C8C2×#6712yes—
310C2×C8C2×#6712yes—
311C4⋊C4—62no—
312C4⋊C4—122no—
313C22⋊C4—62no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—62no—
323C42#23511yescharacteristic
324C42#23511yescharacteristic
325C42C2×#6711yescharacteristic
326C42C2×#6711yescharacteristic
327C22×C3—24no—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3#52511yescharacteristic
334C22×C3#55112yes—
335C2×D3—42no—
336C2×D3—82no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4#52511yescharacteristic
343C3×C4#53511yescharacteristic
344C3×C4#55312yes—
345Dic3—42no—
346Dic3—82no—
347C23—22no—
348C23—21no—
349C23—42no—
350C23—62no—
351C23C2×#23511yescharacteristic
352Q8—62no—
353Q8—62no—
354D4—62no—
355D4—62no—
356D4—122no—
357D4—122no—
358D4—244no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—62no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4C2×#23511yescharacteristic
379C2×C4D3×#4112yes—
380C2×C4#100912yes—
381C2×C4C22×#6711yescharacteristic
382C8—22no—
383C8—22no—
384C8—41no—
385C8—41no—
386C8—122no—
387C8—122no—
388C2×C3—22no—
389C2×C3—21no—
390C2×C3—42no—
391C2×C3#259811yescharacteristic
392D3—82no—
393C22—22no—
394C22—24no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—122no—
400C22—242no—
401C22#552712yes—
402C22#579111yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—122no—
409C4—242no—
410C4#553012yes—
411C4#582811yescharacteristic
412C4#583011yescharacteristic
413C3#2938411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—22no—
416C2—42no—
417C2—242no—
418C2#8228411yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2920C2
socleC2×C3#2598
prime radical#69993C2
hypercenter#2920D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2920

Complements

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

NH ≅ G/Ntype
1. C3×#2920C2semidirect
2. #78541C2semidirect
3. #71532C2semidirect
4. #70531C2semidirect
5. #70530C2semidirect
6. #70526C2semidirect
8. #69992C2semidirect
9. #69987C2semidirect
10. #69874C2semidirect
11. #69871C2semidirect
41. #2920D3semidirect
84. C3×#97D4semidirect
86. 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×#29202
#785412
#715322
#705312
#705302
#705262
#699932
#699922
#699872
#698742
#698712
#293843