Finite Group Databasebeta

← #11856266

Subgroups — #11856266

Order 1152.

2535 conjugacy classes of subgroups; only the 260 normal subgroups are shown.

normal subgroups only

This group has 2535 conjugacy classes of subgroups — too many to draw. Only its 260 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
1C23×#624C211yesmaximal, characteristic, complemented
2C2×#64146C211yesmaximal, characteristic
3C2×C3×#935C212yesmaximal, complemented
4#10591722C214yesmaximal, complemented
5#10591714C211yesmaximal, characteristic, complemented
6#10591696C212yesmaximal, complemented
7#10591374C214yesmaximal, complemented
8#82912—32nomaximal
9C24×C3⋊D3C2211yescharacteristic
10C22×C32×D4C2211yescharacteristic
11C22×#638C2212yes—
12C23×C3×Dic3C2212yes—
13C22×#624—28no—
14C22×#624—24no—
15C22×#624C2214yes—
16C2×#3757C2211yescharacteristic
17#64146—24no—
18C2×#3752C2212yes—
19C2×C3×#217C2212yes—
20C2×C3×#217C2214yescomplemented
21C3×#935—28no—
22C2×C3×Dic3⋊C4C2214yescomplemented
23C2×#3722C2214yes—
24#63579C2218yescomplemented
25C23×C4×D3—32no—
26C2×C3×#123—32no—
27C2×#937—32no—
28C2×#937—61no—
29C2×#935D312yescomplemented
30#5211—32no—
31#4989—38no—
32#4956—38no—
33#4955—32no—
34#4943—32no—
35C24×C32C2311yescharacteristic
36C24×C32C2311yescharacteristic
37C23×C3⋊D3—24no—
38C23×C3⋊D3—28no—
39C23×C3⋊D3D414yes—
40C23×C3⋊D3C2312yes—
41C2×C32×D4—22no—
42C2×C32×D4—24no—
43C22×C32×C4C2311yescharacteristic
44C2×#638—24no—
45C2×#638—28no—
46C22×C3⋊Dic3C2312yes—
47C22×C3×Dic3—24no—
48C22×C3×Dic3—24no—
49C22×C3×Dic3D418yescomplemented
50C22×C3×Dic3C2314yes—
51C22×C3×Dic3C2314yes—
52C2×#624—232no—
53C2×#624—28no—
54C2×#624—216no—
55#3757—22no—
56#3752—24no—
57C3×#217—24no—
58C3×#217—28no—
59C3×Dic3⋊C4—28no—
60#3722—28no—
61#2849—91noSylow
62C24×D3—32no—
63C24×D3—61no—
64C22×C3×D4—21no—
65C22×C3×D4C2×D312yes—
66C23×C3×C4—32no—
67C22×#67—34no—
68C22×#67—62no—
69C23×Dic3C2×D312yes—
70C22×C4×D3—38no—
71C22×C4×D3—68no—
72C22×C4×D3—68no—
73C22×C4×D3—68no—
74C3×#123—68no—
75C2×C3×C4⋊C4—34no—
76C2×C3×C22⋊C4—32no—
77C2×C3×C22⋊C4—34no—
78C2×#217—32no—
79C2×#217—61no—
80C2×#217C2×D312yes—
81C2×#217C2×D314yescomplemented
82#937—68no—
83#937—68no—
84#935—28no—
85C2×(C2×D3)⋊C4—34no—
86C2×(C2×D3)⋊C4—38no—
87C2×(C2×D3)⋊C4—62no—
88C2×Dic3⋊C4C2×D314yescomplemented
89C3⋊#95—38no—
90C3⋊#95—38no—
91C23×C32—21no—
92C23×C32—24no—
93C23×C32—22no—
94C23×C32—24no—
95C23×C32C2×D412yes—
96C23×C32C2×D414yes—
97C23×C32C2411yescharacteristic
98C22×C3⋊D3—232no—
99C22×C3⋊D3—28no—
100C22×C3⋊D3—24no—
101C22×C3⋊D3—24no—
102C22×C3⋊D3—216no—
103C22×C3⋊D3C2×D414yes—
104C22×C3⋊D3#4118yes—
105C32×D4—44no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108#638—48no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—24no—
111C2×C3×Dic3—216no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—28no—
116C2×C3×Dic3C2×D418yes—
117#624—232no—
118C24×C4—91no—
119C2×#123—32no—
120C2×#120—91no—
121#386—94no—
122#385—94no—
123#381—92no—
124#379—91no—
125C24×C3—21no—
126C24×C3—21no—
127C24×C3C22×D312yes—
128C24×C3C22×D312yes—
129C23×D3—34no—
130C23×D3—38no—
131C23×D3—62no—
132C23×D3—64no—
133C23×D3—616no—
134C23×D3—64no—
135C23×D3—64no—
136C23×D3—128no—
137C23×D3—124no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—44no—
143C22×C3×C4—21no—
144C22×C3×C4—38no—
145C22×C3×C4—34no—
146C22×C3×C4—34no—
147C22×C3×C4—64no—
148C22×C3×C4—64no—
149C22×C3×C4C22×D312yes—
150C2×#67—68no—
151C2×#67—616no—
152C2×#67—68no—
153C2×#67—128no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—34no—
157C22×Dic3—62no—
158C22×Dic3#6718yescomplemented
159C22×Dic3C22×D314yes—
160C22×Dic3C22×D314yes—
161C2×C4×D3—616no—
162C2×C4×D3—616no—
163C2×C4×D3—664no—
164C2×C4×D3—616no—
165C3×C4⋊C4—68no—
166C3×C22⋊C4—68no—
167C3×C22⋊C4—64no—
168#217—24no—
169#217—28no—
170#217—64no—
171#217—64no—
172(C2×D3)⋊C4—616no—
173(C2×D3)⋊C4—68no—
174(C2×D3)⋊C4—68no—
175Dic3⋊C4—28no—
176C22×C32—24no—
177C22×C32—28no—
178C22×C32—41no—
179C22×C32—42no—
180C22×C32—44no—
181C22×C32#12014yes—
182C22×C32C22×D411yescharacteristic
183C22×C32C22×D412yes—
184C22×C32C2×#4114yes—
185C2×C3⋊D3—232no—
186C2×C3⋊D3—24no—
187C2×C3⋊D3—24no—
188C2×C3⋊D3—216no—
189C2×C3⋊D3#12318yescomplemented
190C32×C4—41no—
191C3⋊Dic3—42no—
192C3×Dic3—28no—
193C3×Dic3—44no—
194C25—91no—
195C22×D4—92no—
196C22×D4D3211yescharacteristic
197C23×C4—32no—
198C23×C4—94no—
199C23×C4—188no—
200C23×C4—184no—
201#123—68no—
202#120—184no—
203C2×C4⋊C4—34no—
204C2×C22⋊C4—32no—
205C2×C22⋊C4—34no—
206C2×C22⋊C4—94no—
207C2×C22⋊C4—91no—
208C2×C22⋊C4—92no—
209#95—98no—
210C23×C3—28no—
211C23×C3—22no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—28no—
215C23×C3—22no—
216C23×C3—21no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—44no—
220C23×C3—44no—
221C23×C3—42no—
222C23×C3D3×D414yes—
223C23×C3D3×D414yes—
224C23×C3C2×#6714yes—
225C23×C3C23×D312yes—
226C22×D3—38no—
227C22×D3—316no—
228C22×D3—64no—
229C22×D3—68no—
230C22×D3—632no—
231C22×D3—632no—
232C22×D3—632no—
233C22×D3—616no—
234C22×D3—64no—
235C22×D3—68no—
236C22×D3—64no—
237C22×D3—1216no—
238C22×D3—124no—
239C22×D3—124no—
240C22×D3—128no—
241C22×D3—1232no—
242C3×D4—48no—
243C3×D4—48no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—38no—
249C2×C3×C4—42no—
250C2×C3×C4—616no—
251C2×C3×C4—64no—
252C2×C3×C4—64no—
253C2×C3×C4—64no—
254C2×C3×C4—68no—
255#67—1216no—
256#67—1216no—
257C2×Dic3—216no—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—28no—
261C2×Dic3—24no—
262C2×Dic3—64no—
263C2×Dic3—64no—
264C2×Dic3—68no—
265C2×Dic3—124no—
266C2×Dic3C2×#6718yes—
267C4×D3—664no—
268C2×C32—24no—
269C2×C32—81no—
270C2×C32C2×#12011yescharacteristic
271C2×C32C2×#12312yes—
272C2×C32#52514yes—
273C3⋊D3—28no—
274C24—92no—
275C24—94no—
276C24—188no—
277C24—184no—
278C24C2×D3211yescharacteristic
279C24C2×D3211yescharacteristic
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—184no—
283C2×D4—188no—
284C22×C4—34no—
285C22×C4—38no—
286C22×C4—34no—
287C22×C4—64no—
288C22×C4—64no—
289C22×C4—92no—
290C22×C4—1816no—
291C22×C4—188no—
292C22×C4—1832no—
293C22×C4C2×D3211yescharacteristic
294C4⋊C4—68no—
295C22⋊C4—68no—
296C22⋊C4—64no—
297C22⋊C4—184no—
298C22⋊C4—188no—
299C22⋊C4—182no—
300C22×C3—28no—
301C22×C3—28no—
302C22×C3—28no—
303C22×C3—24no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—24no—
307C22×C3—48no—
308C22×C3—44no—
309C22×C3—44no—
310C22×C3—42no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—42no—
314C22×C3—44no—
315C22×C3—84no—
316C22×C3#93718yes—
317C22×C3C2×D3×D412yes—
318C22×C3C2×D3×D412yes—
319C22×C3D3×#4118yes—
320C22×C3C22×#6712yes—
321C2×D3—316no—
322C2×D3—68no—
323C2×D3—664no—
324C2×D3—68no—
325C2×D3—616no—
326C2×D3—68no—
327C2×D3—616no—
328C2×D3—1216no—
329C2×D3—124no—
330C2×D3—124no—
331C2×D3—1232no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—68no—
335C3×C4—124no—
336Dic3—28no—
337Dic3—44no—
338Dic3—124no—
339Dic3—124no—
340C32#284911yescharacteristic, complemented, Sylow
341C23—24no—
342C23—24no—
343C23—22no—
344C23—21no—
345C23—98no—
346C23—94no—
347C23—1816no—
348C23—1832no—
349C23—184no—
350C23—184no—
351C23—188no—
352C23D3×#6714yes—
353C23#381112yes—
354C23C22×D3211yescharacteristic
355D4—44no—
356D4—368no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—38no—
360C2×C4—64no—
361C2×C4—64no—
362C2×C4—616no—
363C2×C4—68no—
364C2×C4—64no—
365C2×C4—1832no—
366C2×C4—184no—
367C2×C4—182no—
368C2×C3—28no—
369C2×C3—24no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—44no—
373C2×C3—82no—
374C2×C3—82no—
375C2×C3D3×#12314yes—
376C2×C3#565614yes—
377C2×C3C2×#93712yes—
378C2×C3#582814yes—
379D3—616no—
380D3—128no—
381C22—28no—
382C22—24no—
383C22—44no—
384C22—42no—
385C22—41no—
386C22—98no—
387C22—1832no—
388C22—1816no—
389C22—184no—
390C22—184no—
391C22#6398314yes—
392C22#6431214yes—
393C22C2×D3×#6712yes—
394C22C2×#381111yescharacteristic
395C4—41no—
396C4—68no—
397C4—124no—
398C4—362no—
399C3—21no—
400C3#8291212yescomplemented
401C2—24no—
402C2—81no—
403C2—188no—
404C2#1059515914yes—
405C2#1059516912yes—
406C2C2×#6398311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC22×C32×D4C22
prime radicalC2×#64146C2
hypercenterC22×D4D32
hypocenterC32#2849
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C23×#624C2semidirect
3. C2×C3×#935C2semidirect
4. #10591722C2semidirect
5. #10591714C2semidirect
6. #10591696C2semidirect
7. #10591374C2semidirect
20. C2×C3×#217C22semidirect
22. C2×C3×Dic3⋊C4C22semidirect
24. #63579C22semidirect
29. C2×#935D3semidirect
49. C22×C3×Dic3D4semidirect
81. C2×#217C2×D3semidirect
88. C2×Dic3⋊C4C2×D3semidirect
158. C22×Dic3#67semidirect
189. C2×C3⋊D3#123semidirect
340. C32#2849semidirect
400. C3#82912semidirect

Maximal subgroups

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

subgroupindex
C23×#6242
C2×#641462
C2×C3×#9352
#105917222
#105917142
#105916962
#105913742
#829123