Finite Group Databasebeta

← #11868086

Subgroups — #11868086

Order 1152.

1398 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#568C211yesmaximal, characteristic, complemented
2#10596293C211yesmaximal, characteristic, complemented
3C2×#64158C211yesmaximal, characteristic, complemented
4C2×#64117C211yesmaximal, characteristic, complemented
5#10592841C214yesmaximal, complemented
6#10592840C214yesmaximal, complemented
7#10592831C211yesmaximal, characteristic, complemented
8C2×#63650C211yesmaximal, characteristic, complemented
9#10592824C211yesmaximal, characteristic, complemented
10#83814—34nomaximal
11C2×C32×#41C2211yescharacteristic
12C2×#3828C2211yescharacteristic
13C22×C4⋊(C3⋊D3)C2211yescharacteristic
14C32×#131—22no—
15C32×#131C2214yescomplemented
16C2×C32×M16—22no—
17C2×C32×M16C2211yescharacteristic
18C2×C32×M16C2211yescharacteristic
19#64160C2212yescomplemented
20#64159—22no—
21#64158—22no—
22#64158C2214yes—
23#64158C2212yescomplemented
24#64151—21no—
25C2×#3755—21no—
26#64141—21no—
27C2×#3753—21no—
28C2×#3748C2211yescharacteristic
29C2×#3748C2211yescharacteristic
30#64117—22no—
31#64117—22no—
32C2×#3745C2212yescomplemented
33C2×#3744C2212yescomplemented
34#63651C2212yescomplemented
35#63650C2214yescomplemented
36#63650C2212yescomplemented
37#63646C2214yescomplemented
38C3×#568D314yescomplemented
39#5822—34no—
40C2×#949—34no—
41C2×#908—34no—
42#5142—316no—
43#5141—316no—
44#5132—34no—
45C2×#823—34no—
46#5125—34no—
47C23×C3⋊D3—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41D412yes—
51C32×#41C2312yes—
52C2×C32×Q8D411yescharacteristic
53C2×C32×D4—21no—
54C2×C32×D4D411yescharacteristic
55C22×C32×C4—21no—
56C22×C32×C4C2311yescharacteristic
57C2×#638—21no—
58#3828—21no—
59#3828—21no—
60#3828—21no—
61#3828—21no—
62C2×C4⋊(C3⋊D3)—21no—
63C2×C4⋊(C3⋊D3)—21no—
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)—21no—
67C2×C4⋊(C3⋊D3)C2312yes—
68C2×C4⋊(C3⋊D3)C2311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×#634C2311yescharacteristic
71C32×M16—21no—
72C32×M16—24no—
73C32×M16D412yes—
74C32×M16D412yes—
75C32×M16C2312yes—
76C2×C32×C8—21no—
77C2×C32×C8—24no—
78C2×C32×C8C2312yes—
79#3756—22no—
80#3755—24no—
81#3755—22no—
82#3754—22no—
83#3753—24no—
84#3753—22no—
85#3748—21no—
86#3748—21no—
87#3748C2312yes—
88#3748C2312yes—
89C2×C3⋊(C3⋊C8)—21no—
90C2×C3⋊(C3⋊C8)—21no—
91#3745—24no—
92#3744—24no—
93#3068—91noSylow
94C2×C3×#41C2×D314yes—
95C2×#235—34no—
96C22×D12—34no—
97C3×#131—28no—
98C3×#131C2×D3116yescomplemented
99C2×C3×M16—28no—
100C2×C3×M16C2×D314yes—
101C2×C3×M16C2×D314yes—
102#951—38no—
103#950—68no—
104#949—316no—
105#949—38no—
106#949—68no—
107#942—64no—
108C2×#215—64no—
109#932—64no—
110C2×#213—64no—
111C2×#208—34no—
112C2×#208—34no—
113#908—68no—
114#908—68no—
115C2×D24—38no—
116C2×#204—38no—
117#824—38no—
118#823—316no—
119#823—38no—
120#819—316no—
121C23×C32—21no—
122C23×C32#4111yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—82no—
128C32×Q8—21no—
129C32×Q8C2×D412yes—
130C32×D4—22no—
131C32×D4—22no—
132C32×D4—21no—
133C32×D4C2×D412yes—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C32×C4C2411yescharacteristic
143#638—41no—
144#638—41no—
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—22no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)—22no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—41no—
151C4⋊(C3⋊D3)—41no—
152C4×C3⋊D3—41no—
153C4×C3⋊D3—41no—
154#634—21no—
155#634—21no—
156C32×C8—22no—
157C32×C8C2×D414yes—
158C3⋊(C3⋊C8)—22no—
159C3⋊(C3⋊C8)—22no—
160#575—91no—
161C2×AGL(1,ℤ/8)—91no—
162C2×AGL(1,ℤ/8)—91no—
163#568C3⋊D311yescharacteristic, complemented
164#472—94no—
165#471—94no—
166#430—91no—
167#413—91no—
168C2×#100—91no—
169C23×D3—64no—
170C3×#41—24no—
171C3×#41—24no—
172C3×#41D1218yes—
173C3×#41C22×D318yes—
174C2×C3×Q8D1214yes—
175C2×C3×D4—24no—
176C2×C3×D4D1214yes—
177C22×C3×C4—24no—
178C22×C3×C4C22×D314yes—
179C2×#67—64no—
180#235—64no—
181#235—64no—
182#235—64no—
183#235—64no—
184C2×D12—38no—
185C2×D12—34no—
186C2×D12—64no—
187C2×D12—64no—
188C2×D12—64no—
189C2×D12—64no—
190C2×D12—64no—
191C2×C4×D3—64no—
192C2×Dic6—34no—
193C3×M16—24no—
194C3×M16—216no—
195C3×M16#6718yes—
196C3×M16#6718yes—
197C3×M16C22×D318yes—
198C2×C3×C8—24no—
199C2×C3×C8—216no—
200C2×C3×C8C22×D318yes—
201#216—68no—
202#215—616no—
203#215—68no—
204#214—68no—
205#213—616no—
206#213—68no—
207#208—38no—
208#208—38no—
209#208—64no—
210#208—64no—
211C2×C3⋊C8—64no—
212C2×C3⋊C8—64no—
213D24—616no—
214#204—616no—
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—81no—
218C22×C32#12111yescharacteristic
219C22×C32#12111yescharacteristic
220C22×C32C2×#4111yescharacteristic
221C2×C3⋊D3—41no—
222C2×C3⋊D3—42no—
223C2×C3⋊D3—82no—
224C2×C3⋊D3—82no—
225C2×C3⋊D3—81no—
226C2×C3⋊D3—82no—
227C32×C4—22no—
228C32×C4#12111yescharacteristic
229C32×C4#12111yescharacteristic
230C32×C4C22×D411yescharacteristic
231C32×C4C22×D411yescharacteristic
232C3⋊Dic3—41no—
233C2×#41—91no—
234C2×#41C2×C3⋊D311yescharacteristic
235C22×D4—91no—
236#137—92no—
237#137—181no—
238AGL(1,ℤ/8)—94no—
239AGL(1,ℤ/8)—92no—
240AGL(1,ℤ/8)—182no—
241AGL(1,ℤ/8)—181no—
242AGL(1,ℤ/8)—182no—
243AGL(1,ℤ/8)—182no—
244#135—182no—
245C2×SD16—92no—
246C2×SD16—181no—
247C2×D8—92no—
248C2×D8—181no—
249#131—22no—
250#131C2×C3⋊D314yescomplemented
251C2×M16—22no—
252C2×M16—91no—
253C2×M16—91no—
254C2×M16C2×C3⋊D311yescharacteristic
255C2×M16C2×C3⋊D311yescharacteristic
256#108—94no—
257#101—92no—
258#100—94no—
259#100—92no—
260C23×C3—24no—
261C23×C3#23514yes—
262C22×D3—64no—
263C22×D3—68no—
264C22×D3—124no—
265C22×D3—124no—
266C22×D3—248no—
267C3×Q8—24no—
268C3×Q8C2×D1218yes—
269C3×D4—28no—
270C3×D4—28no—
271C3×D4—24no—
272C3×D4C2×D1218yes—
273C2×C3×C4—24no—
274C2×C3×C4—28no—
275C2×C3×C4—28no—
276C2×C3×C4C2×D1214yes—
277C2×C3×C4C2×D1214yes—
278C2×C3×C4#23514yes—
279C2×C3×C4C2×#6714yes—
280C2×C3×C4C2×#6714yes—
281C2×C3×C4C23×D314yes—
282#67—124no—
283#67—124no—
284C2×Dic3—64no—
285D12—68no—
286D12—64no—
287D12—68no—
288D12—64no—
289D12—124no—
290D12—124no—
291C4×D3—124no—
292C4×D3—124no—
293Dic6—64no—
294Dic6—64no—
295C3×C8—28no—
296C3×C8C2×#67116yes—
297C3⋊C8—68no—
298C3⋊C8—68no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—42no—
303C2×C32C2×#12111yescharacteristic
304C3⋊D3—81no—
305C3⋊D3—82no—
306C24—181no—
307#41—21no—
308#41—21no—
309#41—181no—
310#41—181no—
311#41—181no—
312#41—181no—
313#41C4⋊(C3⋊D3)12yes—
314#41C22×C3⋊D312yes—
315C2×Q8—91no—
316C2×Q8C4⋊(C3⋊D3)11yescharacteristic
317C2×D4—21no—
318C2×D4—91no—
319C2×D4—92no—
320C2×D4—181no—
321C2×D4—181no—
322C2×D4—181no—
323C2×D4—181no—
324C2×D4—181no—
325C2×D4—181no—
326C2×D4C4⋊(C3⋊D3)11yescharacteristic
327C22×C4—21no—
328C22×C4—181no—
329C22×C4C22×C3⋊D311yescharacteristic
330Q16—182no—
331SD16—184no—
332SD16—182no—
333SD16—182no—
334SD16—184no—
335D8—184no—
336D8—182no—
337D8—184no—
338M16—21no—
339M16—24no—
340M16—92no—
341M16—92no—
342M16—181no—
343M16—181no—
344M16#63812yes—
345M16#63812yes—
346M16C22×C3⋊D312yes—
347C2×C8—21no—
348C2×C8—24no—
349C2×C8—181no—
350C2×C8—181no—
351C2×C8C22×C3⋊D312yes—
352C22×C3—28no—
353C22×C3—44no—
354C22×C3—84no—
355C22×C3#93014yes—
356C22×C3#93014yes—
357C22×C3C2×#23514yes—
358C2×D3—124no—
359C2×D3—128no—
360C2×D3—248no—
361C2×D3—248no—
362C2×D3—248no—
363C2×D3—244no—
364C3×C4—28no—
365C3×C4#93014yes—
366C3×C4#93014yes—
367C3×C4C22×D1214yes—
368C3×C4C22×#6714yes—
369Dic3—124no—
370C32#306811yescharacteristic, complemented, Sylow
371C23—21no—
372C23—181no—
373C23—182no—
374C23—361no—
375C23—361no—
376C23—722no—
377C23#382811yescharacteristic
378Q8—21no—
379Q8—181no—
380Q8—181no—
381Q8C2×C4⋊(C3⋊D3)12yes—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—182no—
386D4—181no—
387D4—181no—
388D4—182no—
389D4—361no—
390D4—361no—
391D4—361no—
392D4—361no—
393D4C2×C4⋊(C3⋊D3)12yes—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—181no—
398C2×C4—361no—
399C2×C4—361no—
400C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
401C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
402C2×C4#382811yescharacteristic
403C2×C4C2×#63811yescharacteristic
404C2×C4C2×#63811yescharacteristic
405C2×C4C23×C3⋊D311yescharacteristic
406C8—22no—
407C8—182no—
408C8—182no—
409C8C2×#63814yes—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—48no—
414C2×C3C2×#93014yes—
415D3—248no—
416D3—244no—
417C22—22no—
418C22—41no—
419C22—81no—
420C22—361no—
421C22—362no—
422C22—722no—
423C22—721no—
424C22—722no—
425C22—722no—
426C22#6413911yescharacteristic
427C22#6413911yescharacteristic
428C22C2×#382811yescharacteristic
429C4—22no—
430C4—361no—
431C4#6413911yescharacteristic
432C4#6413911yescharacteristic
433C4C22×C4⋊(C3⋊D3)11yescharacteristic
434C4C22×#63811yescharacteristic
435C3#8381414yescomplemented
436C2—21no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—722no—
441C2—721no—
442C2C2×#6413911yescharacteristic
subgroupG/H
centerC2C2×#64139
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#568C2
socleC2×C32C2×#121
hypercenter#568C3⋊D3
hypocenterC32#3068
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
1. C32×#568C2semidirect
2. #10596293C2semidirect
3. C2×#64158C2semidirect
4. C2×#64117C2semidirect
5. #10592841C2semidirect
6. #10592840C2semidirect
7. #10592831C2semidirect
8. C2×#63650C2semidirect
9. #10592824C2semidirect
15. C32×#131C22semidirect
19. #64160C22semidirect
23. #64158C22semidirect
32. C2×#3745C22semidirect
33. C2×#3744C22semidirect
34. #63651C22semidirect
35. #63650C22semidirect
36. #63650C22semidirect
37. #63646C22semidirect
38. C3×#568D3semidirect
98. C3×#131C2×D3semidirect
163. #568C3⋊D3semidirect
250. #131C2×C3⋊D3semidirect
370. C32#3068semidirect
435. C3#83814semidirect

Maximal subgroups

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

subgroupindex
C32×#5682
#105962932
C2×#641582
C2×#641172
#105928412
#105928402
#105928312
C2×#636502
#105928242
#838143