Finite Group Databasebeta

← #11863952

Subgroups — #11863952

Order 1152.

2153 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 2153 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3802C211yesmaximal, characteristic, complemented
2#10596299C211yesmaximal, characteristic, complemented
3C2×C3×#949C212yesmaximal, complemented
4#10591652C214yesmaximal, complemented
5#10591516C214yesmaximal, complemented
6#10591086C212yesmaximal, complemented
7#10591074C211yesmaximal, characteristic, complemented
8#83944—32nomaximal
9C23×D32—21no—
10C2×C32×#41C2211yescharacteristic, complemented
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×C3×D12C2212yes—
13C22×#625—21no—
14C2×#3802—22no—
15C2×#3802—28no—
16C2×#3802C2214yes—
17#64155C2212yescomplemented
18#64147C2212yescomplemented
19#64146—21no—
20#64144—21no—
21C4×#638—22no—
22#64135C2211yescharacteristic, complemented
23C3×#949—28no—
24C2×C3×#215C2214yescomplemented
25C2×C3×#213C2214yescomplemented
26C2×C3×#208C2212yescomplemented
27#63563C2218yescomplemented
28#63557C2214yescomplemented
29C22×D3×D4—32no—
30C2×C3×AGL(1,ℤ/8)—32no—
31#5828—32no—
32#5828—61no—
33C2×#949D312yescomplemented
34#5215—32no—
35#5169—38no—
36#5156—38no—
37#5131—32no—
38#5126—32no—
39C23×C3⋊D3D411yescharacteristic
40C23×C3×D3—22no—
41C22×D32—28no—
42C22×D32—24no—
43C22×D32—48no—
44C32×#41—24no—
45C2×C32×Q8C2311yescharacteristic, complemented
46C2×C32×D4—21no—
47C2×C32×D4C2311yescharacteristic, complemented
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×#638—22no—
51C22×C3⋊Dic3D411yescharacteristic
52C2×C4×C3⋊D3—24no—
53C2×C4×C3⋊D3D412yes—
54C2×C4×C3⋊D3C2312yes—
55C2×C3×D12—22no—
56C2×C3×D12—24no—
57C2×C3×D12D418yescomplemented
58C2×C3×D12C2314yes—
59C2×#625—28no—
60C2×#625—24no—
61#3802—216no—
62#3802—216no—
63#3757—21no—
64#3757—22no—
65#3752—22no—
66C4⋊(C3⋊Dic3)C2312yescomplemented
67#3750—22no—
68C4×C3⋊Dic3—21no—
69C3×#215—28no—
70C3×#213—28no—
71C3×#208—24no—
72C2×C3×C3⋊C8C2314yescomplemented
73#3003—91noSylow
74C24×D3—62no—
75C2×C3×#41—21no—
76C2×C3×#41C2×D312yescomplemented
77C22×C3×D4—32no—
78C22×#67—62no—
79C2×D3×D4—38no—
80C2×D3×D4—64no—
81C2×D3×D4—68no—
82C2×D3×D4—68no—
83C22×D12C2×D312yes—
84C22×C4×D3—32no—
85C22×C4×D3—61no—
86C3×AGL(1,ℤ/8)—68no—
87C2×C3×SD16—34no—
88C2×C3×D8—34no—
89C2×C3×M16—32no—
90#949—28no—
91#946—34no—
92#946—62no—
93C2×#215C2×D314yescomplemented
94#938—34no—
95#938—62no—
96#937—62no—
97#937—62no—
98#935—62no—
99#935—62no—
100C2×#213C2×D314yescomplemented
101C4×#67—64no—
102C4×#67—64no—
103#926—32no—
104#926—61no—
105C2×#208C2×D312yescomplemented
106#832—38no—
107#821—38no—
108(C2×D3)⋊C8—38no—
109C23×C32—21no—
110C23×C32C2×D411yescharacteristic
111C22×C3⋊D3—22no—
112C22×C3⋊D3—24no—
113C22×C3⋊D3C2×D412yes—
114C22×C3×D3—28no—
115C22×C3×D3—24no—
116C22×C3×D3—48no—
117C2×D32—216no—
118C2×D32—432no—
119C2×D32—48no—
120C2×D32—48no—
121C32×Q8—22no—
122C32×D4—22no—
123C32×D4—42no—
124C2×C32×C4—21no—
125C2×C32×C4—42no—
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D414yes—
128C2×C32×C4C2411yescharacteristic
129#638—44no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3C2×D412yes—
133C4×C3⋊D3—24no—
134C4×C3⋊D3—24no—
135C3×D12—28no—
136C3×D12—24no—
137C3×D12C2×D418yes—
138#625—216no—
139C3×C3⋊C8—24no—
140C23×D4—91no—
141C2×AGL(1,ℤ/8)—32no—
142#525—91no—
143#450—94no—
144#447—94no—
145#418—92no—
146#407—91no—
147C24×C3—62no—
148C23×D3—22no—
149C23×D3—32no—
150C23×D3—61no—
151C23×D3—68no—
152C23×D3—616no—
153C23×D3—1216no—
154C3×#41—24no—
155C3×#41—24no—
156C3×#41—44no—
157C2×C3×Q8—21no—
158C2×C3×Q8C22×D312yescomplemented
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—38no—
163C2×C3×D4—34no—
164C2×C3×D4—62no—
165C2×C3×D4—64no—
166C2×C3×D4C22×D312yescomplemented
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C22×D312yes—
171C2×#67—68no—
172C2×#67—616no—
173C2×#67—64no—
174C2×#67—64no—
175C22×Dic3—32no—
176C22×Dic3—61no—
177D3×D4—616no—
178D3×D4—632no—
179D3×D4—616no—
180C2×D12—24no—
181C2×D12—22no—
182C2×D12#6718yescomplemented
183C2×D12C22×D314yes—
184C2×C4×D3—34no—
185C2×C4×D3—34no—
186C2×C4×D3—62no—
187C2×C4×D3—62no—
188C2×C4×D3—64no—
189C2×C4×D3—64no—
190C2×C4×D3—124no—
191C3×SD16—68no—
192C3×D8—68no—
193C3×M16—64no—
194C2×C3×C8—34no—
195#217—64no—
196#217—62no—
197#217—64no—
198#217—62no—
199#215—28no—
200#213—28no—
201(C2×D3)⋊C4—64no—
202(C2×D3)⋊C4—64no—
203C4⋊Dic3—34no—
204C4⋊Dic3—62no—
205Dic3⋊C4—64no—
206Dic3⋊C4—64no—
207C4×Dic3—62no—
208C4×Dic3—62no—
209#208—24no—
210C2×C3⋊C8C22×D314yescomplemented
211C22×C32—22no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32#12012yes—
215C22×C32C22×D411yescharacteristic
216C2×C3⋊D3—28no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—22no—
219C2×C3⋊D3AGL(1,ℤ/8)14yescomplemented
220C2×C3×D3—28no—
221C2×C3×D3—416no—
222C2×C3×D3—48no—
223D32—432no—
224C32×C4—41no—
225C32×C4#12012yes—
226C32×C4C22×D412yes—
227C3⋊Dic3—22no—
228C3⋊Dic3—42no—
229C25—181no—
230C2×#41D3211yescharacteristic, complemented
231C22×D4—32no—
232C22×D4—94no—
233C22×D4—182no—
234C22×D4—181no—
235C22×D4—188no—
236C23×C4—91no—
237AGL(1,ℤ/8)—68no—
238C2×SD16—34no—
239C2×D8—34no—
240C2×M16—32no—
241#123—181no—
242C22⋊Q8—92no—
243#121—92no—
244#120—181no—
245C4×D4—182no—
246#117—91no—
247#102—98no—
248C22⋊C8—94no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—21no—
252C23×C3—68no—
253C23×C3—64no—
254C23×C3—128no—
255C23×C3D3×D412yes—
256C22×D3—28no—
257C22×D3—24no—
258C22×D3—34no—
259C22×D3—48no—
260C22×D3—62no—
261C22×D3—632no—
262C22×D3—68no—
263C22×D3—64no—
264C22×D3—124no—
265C22×D3—122no—
266C22×D3—1216no—
267C22×D3—1264no—
268C22×D3—1216no—
269C3×Q8—22no—
270C3×Q8—22no—
271C3×Q8—42no—
272C3×D4—22no—
273C3×D4—22no—
274C3×D4—38no—
275C3×D4—42no—
276C3×D4—44no—
277C3×D4—44no—
278C3×D4—68no—
279C3×D4—64no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×C3×C4D3×D414yes—
288C2×C3×C4D3×D412yes—
289C2×C3×C4C2×#6714yes—
290C2×C3×C4C23×D312yes—
291#67—632no—
292#67—128no—
293#67—128no—
294C2×Dic3—34no—
295C2×Dic3—62no—
296C2×Dic3—64no—
297C2×Dic3—64no—
298C2×Dic3—64no—
299C2×Dic3—122no—
300D12—28no—
301D12—24no—
302D12C2×#6718yes—
303C4×D3—68no—
304C4×D3—64no—
305C4×D3—64no—
306C4×D3—124no—
307C4×D3—124no—
308C3×C8—64no—
309C3⋊C8—24no—
310C2×C32—22no—
311C2×C32—81no—
312C2×C32C2×#12011yescharacteristic
313C2×C32C2×AGL(1,ℤ/8)12yes—
314C3⋊D3—24no—
315C3×D3—48no—
316C24—62no—
317C24—91no—
318C24—184no—
319C24—188no—
320C24—368no—
321#41—24no—
322C2×Q8C2×D3211yescharacteristic, complemented
323C2×D4—21no—
324C2×D4—34no—
325C2×D4—38no—
326C2×D4—64no—
327C2×D4—62no—
328C2×D4—184no—
329C2×D4—1816no—
330C2×D4—1816no—
331C2×D4—188no—
332C2×D4—182no—
333C2×D4C2×D3211yescharacteristic, complemented
334C22×C4—21no—
335C22×C4—91no—
336C22×C4—92no—
337C22×C4—92no—
338C22×C4—184no—
339C22×C4C2×D3211yescharacteristic
340SD16—68no—
341D8—68no—
342M16—64no—
343C2×C8—34no—
344C4⋊C4—92no—
345C4⋊C4—182no—
346C22⋊C4—182no—
347C22⋊C4—182no—
348C22⋊C4—181no—
349C42—181no—
350C22×C3—24no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—42no—
354C22×C3—44no—
355C22×C3—42no—
356C22×C3—44no—
357C22×C3—42no—
358C22×C3—68no—
359C22×C3—1216no—
360C22×C3—128no—
361C22×C3#93714yes—
362C22×C3C2×D3×D412yes—
363C2×D3—28no—
364C2×D3—38no—
365C2×D3—416no—
366C2×D3—48no—
367C2×D3—64no—
368C2×D3—616no—
369C2×D3—64no—
370C2×D3—64no—
371C2×D3—122no—
372C2×D3—122no—
373C2×D3—128no—
374C2×D3—1264no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4#93714yes—
380C3×C4C2×D3×D412yes—
381C3×C4C22×#6712yes—
382Dic3—64no—
383Dic3—122no—
384Dic3—124no—
385Dic3—124no—
386C32#300311yescharacteristic, complemented, Sylow
387C23—21no—
388C23—64no—
389C23—68no—
390C23—92no—
391C23—128no—
392C23—184no—
393C23—1816no—
394C23—182no—
395C23—368no—
396C23—3632no—
397C23—368no—
398C23#381111yescharacteristic
399Q8—22no—
400D4—22no—
401D4—38no—
402D4—42no—
403D4—64no—
404D4—68no—
405D4—1816no—
406D4—364no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—92no—
410C2×C4—182no—
411C2×C4—184no—
412C2×C4—184no—
413C2×C4—182no—
414C2×C4D3×#6714yes—
415C2×C4#381111yescharacteristic
416C2×C4C22×D3211yescharacteristic
417C8—64no—
418C2×C3—24no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—42no—
422C2×C3—82no—
423C2×C3—82no—
424C2×C3—128no—
425C2×C3D3×AGL(1,ℤ/8)14yes—
426C2×C3C2×#93712yes—
427D3—48no—
428D3—68no—
429D3—124no—
430C22—22no—
431C22—42no—
432C22—41no—
433C22—68no—
434C22—94no—
435C22—1216no—
436C22—128no—
437C22—188no—
438C22—182no—
439C22—182no—
440C22—3632no—
441C22#6398312yes—
442C22C2×#381111yescharacteristic
443C4—41no—
444C4—182no—
445C4—362no—
446C4#6398312yes—
447C4C2×D3×#6712yes—
448C3—21no—
449C3#8394412yescomplemented
450C2—22no—
451C2—81no—
452C2—128no—
453C2—184no—
454C2#1059545812yes—
455C2C2×#6398311yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×#41C22
socleC22×C32C22×D4
hypercenterC2×#41D32
hypocenterC32#3003
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C22×#3802C2semidirect
2. #10596299C2semidirect
3. C2×C3×#949C2semidirect
4. #10591652C2semidirect
5. #10591516C2semidirect
6. #10591086C2semidirect
7. #10591074C2semidirect
10. C2×C32×#41C22semidirect
17. #64155C22semidirect
18. #64147C22semidirect
22. #64135C22semidirect
24. C2×C3×#215C22semidirect
25. C2×C3×#213C22semidirect
26. C2×C3×#208C22semidirect
27. #63563C22semidirect
28. #63557C22semidirect
33. C2×#949D3semidirect
45. C2×C32×Q8C23semidirect
47. C2×C32×D4C23semidirect
57. C2×C3×D12D4semidirect
66. C4⋊(C3⋊Dic3)C23semidirect
72. C2×C3×C3⋊C8C23semidirect
76. C2×C3×#41C2×D3semidirect
93. C2×#215C2×D3semidirect
100. C2×#213C2×D3semidirect
105. C2×#208C2×D3semidirect
158. C2×C3×Q8C22×D3semidirect
166. C2×C3×D4C22×D3semidirect
182. C2×D12#67semidirect
210. C2×C3⋊C8C22×D3semidirect
219. C2×C3⋊D3AGL(1,ℤ/8)semidirect
230. C2×#41D32semidirect
322. C2×Q8C2×D32semidirect
333. C2×D4C2×D32semidirect
386. C32#3003semidirect
449. C3#83944semidirect

Maximal subgroups

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

subgroupindex
C22×#38022
#105962992
C2×C3×#9492
#105916522
#105915162
#105910862
#105910742
#839443