Finite Group Databasebeta

← #11792031

Subgroups — #11792031

Order 1152.

1434 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#10596301C211yesmaximal, characteristic, complemented
3C2×#64149C211yesmaximal, characteristic, complemented
4C2×#63664C211yesmaximal, characteristic, complemented
5#10592914C211yesmaximal, characteristic, complemented
6C2×#63650C211yesmaximal, characteristic, complemented
7#10589421C211yesmaximal, characteristic, complemented
8#72094—34nomaximal
9C2×C32×#41C2211yescharacteristic
10C22×C32×D4C2211yescharacteristic
11C22×#638—22no—
12C22×C4⋊(C3⋊D3)C2211yescharacteristic
13C32×AGL(1,ℤ/8)—22no—
14C32×AGL(1,ℤ/8)—22no—
15C2×C32×SD16—21no—
16C2×C32×D8—21no—
17C2×C32×M16C2211yescharacteristic
18#64156—21no—
19#64149—22no—
20#64149—21no—
21#64149—22no—
22#64146—22no—
23#64139—22no—
24#64135C2211yescharacteristic
25C2×C4×C3⋊Dic3C2211yescharacteristic
26C2×#3748C2211yescharacteristic
27#63664—22no—
28#63659—22no—
29#63650—22no—
30C2×C3×AGL(1,ℤ/8)D314yescomplemented
31#5830—34no—
32C2×#940—34no—
33C2×#837—34no—
34#5215—34no—
35C2×#823—34no—
36#4555—34no—
37C24×C32—21no—
38C23×C3⋊D3D412yes—
39C32×#41—21no—
40C32×#41—21no—
41C32×#41—21no—
42C32×#41—21no—
43C2×C32×Q8D411yescharacteristic, complemented
44C2×C32×D4—21no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4D412yescomplemented
51C2×C32×D4D411yescharacteristic, complemented
52C22×C32×C4—21no—
53C22×C32×C4C2311yescharacteristic
54C2×#638—28no—
55C2×#638—24no—
56C2×#638—48no—
57C22×C3⋊Dic3—21no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)—21no—
60C2×C4⋊(C3⋊D3)—21no—
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4⋊(C3⋊D3)—21no—
63C2×C4⋊(C3⋊D3)—21no—
64C32×SD16—24no—
65C32×D8—24no—
66C32×M16—21no—
67C32×M16D412yescomplemented
68C2×C32×C8—21no—
69#3757—21no—
70#3752—24no—
71C4⋊(C3⋊Dic3)D412yescomplemented
72C4×C3⋊Dic3—21no—
73C4×C3⋊Dic3—21no—
74C4×C3⋊Dic3D412yescomplemented
75#3748—21no—
76#3748—21no—
77C2×C3⋊(C3⋊C8)—21no—
78#2002—91noSylow
79C2×C3×#41C2×D314yes—
80C22×C3×D4C2×D314yes—
81C22×#67—68no—
82C22×D12—34no—
83C3×AGL(1,ℤ/8)—28no—
84C3×AGL(1,ℤ/8)—28no—
85C2×C3×SD16—24no—
86C2×C3×D8—24no—
87C2×C3×M16C2×D314yes—
88#947—64no—
89#940—68no—
90#940—68no—
91#940—64no—
92#937—68no—
93#930—68no—
94#926—34no—
95C2×C4×Dic3—34no—
96C2×#208—34no—
97#837—68no—
98#832—68no—
99#823—68no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—44no—
103C23×C32—41no—
104C23×C32—41no—
105C23×C32C2×D411yescharacteristic
106C22×C3⋊D3—22no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—22no—
109C22×C3⋊D3—44no—
110C32×Q8—21no—
111C32×Q8—21no—
112C32×D4—21no—
113C32×D4—22no—
114C32×D4—22no—
115C32×D4—21no—
116C32×D4—41no—
117C32×D4—41no—
118C32×D4—41no—
119C32×D4—41no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4#4111yescharacteristic
129#638—416no—
130#638—84no—
131C2×C3⋊Dic3—24no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C4⋊(C3⋊D3)—41no—
135C4⋊(C3⋊D3)—41no—
136C4⋊(C3⋊D3)—41no—
137C4⋊(C3⋊D3)—41no—
138C32×C8—22no—
139C3⋊(C3⋊C8)—41no—
140C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
141#535—91no—
142C2×#127—91no—
143C2×#104—91no—
144#418—91no—
145C2×#100—91no—
146#339—91no—
147C24×C3—24no—
148C23×D3—38no—
149C3×#41—24no—
150C3×#41—24no—
151C3×#41—24no—
152C3×#41—24no—
153C2×C3×Q8#6714yescomplemented
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4#6714yescomplemented
161C2×C3×D4#6718yescomplemented
162C22×C3×C4—24no—
163C22×C3×C4C22×D314yes—
164C2×#67—632no—
165C2×#67—616no—
166C2×#67—1232no—
167C22×Dic3—64no—
168C2×D12—64no—
169C2×D12—64no—
170C2×D12—64no—
171C2×D12—64no—
172C2×D12—64no—
173C2×D12—64no—
174C3×SD16—216no—
175C3×D8—216no—
176C3×M16—24no—
177C3×M16#6718yescomplemented
178C2×C3×C8—24no—
179#217—64no—
180(C2×D3)⋊C4—616no—
181C4⋊Dic3—38no—
182C4×Dic3—38no—
183C4×Dic3—64no—
184C4×Dic3—64no—
185#208—64no—
186#208—64no—
187C2×C3⋊C8—64no—
188C22×C32—22no—
189C22×C32—44no—
190C22×C32—41no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32—82no—
194C22×C32—82no—
195C22×C32#12011yescharacteristic
196C22×C32#12011yescharacteristic
197C22×C32#12111yescharacteristic
198C2×C3⋊D3—44no—
199C2×C3⋊D3—42no—
200C2×C3⋊D3—44no—
201C2×C3⋊D3—82no—
202C32×C4—41no—
203C32×C4#12011yescharacteristic
204C32×C4#12111yescharacteristic
205C32×C4#12111yescharacteristic
206C32×C4#12711yescharacteristic
207C3⋊Dic3—24no—
208C3⋊Dic3—42no—
209C2×#41C2×C3⋊D311yescharacteristic
210C22×D4—91no—
211C22×D4—182no—
212C22×D4C2×C3⋊D311yescharacteristic
213AGL(1,ℤ/8)—22no—
214AGL(1,ℤ/8)—22no—
215C2×SD16—21no—
216C2×D8—21no—
217C2×M16—91no—
218C2×M16C2×C3⋊D311yescharacteristic
219#127—182no—
220#127—182no—
221#127—181no—
222#124—181no—
223#121—182no—
224#120—182no—
225#117—91no—
226C2×C42—91no—
227#104—182no—
228#102—182no—
229#100—182no—
230C23×C3—24no—
231C23×C3—28no—
232C23×C3—416no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3D3×D414yes—
236C22×D3—68no—
237C22×D3—68no—
238C22×D3—68no—
239C22×D3—1216no—
240C3×Q8—24no—
241C3×Q8—24no—
242C3×D4—28no—
243C3×D4—24no—
244C3×D4—24no—
245C3×D4—28no—
246C3×D4—44no—
247C3×D4—44no—
248C3×D4—44no—
249C3×D4—44no—
250C2×C3×C4—24no—
251C2×C3×C4—44no—
252C2×C3×C4—44no—
253C2×C3×C4D3×D414yes—
254C2×C3×C4D3×D414yes—
255C2×C3×C4#23714yes—
256C2×C3×C4C2×#6714yes—
257C2×C3×C4C2×#6714yes—
258C2×C3×C4C2×#6714yes—
259#67—1264no—
260#67—2416no—
261C2×Dic3—616no—
262C2×Dic3—68no—
263C2×Dic3—68no—
264D12—124no—
265D12—124no—
266D12—124no—
267D12—124no—
268C3×C8—28no—
269C3⋊C8—124no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—81no—
273C2×C32—82no—
274C2×C32#39211yescharacteristic
275C2×C32#45312yes—
276C3⋊D3—82no—
277C24—21no—
278C24—92no—
279#41—21no—
280#41—21no—
281#41—21no—
282#41—21no—
283C2×Q8#63811yescharacteristic, complemented
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—188no—
291C2×D4—184no—
292C2×D4—181no—
293C2×D4—181no—
294C2×D4—181no—
295C2×D4—181no—
296C2×D4—181no—
297C2×D4—181no—
298C2×D4—368no—
299C2×D4#63811yescharacteristic, complemented
300C2×D4#63812yescomplemented
301C22×C4—21no—
302C22×C4—181no—
303C22×C4C22×C3⋊D311yescharacteristic
304SD16—24no—
305D8—24no—
306M16—21no—
307M16—181no—
308M16—181no—
309M16#63812yescomplemented
310C2×C8—21no—
311C2×C8—181no—
312C4⋊C4—92no—
313C22⋊C4—184no—
314C22⋊C4—181no—
315C42—92no—
316C42—181no—
317C42—181no—
318C22×C3—28no—
319C22×C3—416no—
320C22×C3—48no—
321C22×C3—44no—
322C22×C3—48no—
323C22×C3—88no—
324C22×C3—88no—
325C22×C3#93714yes—
326C22×C3#93714yes—
327C22×C3#93814yes—
328C2×D3—1216no—
329C2×D3—1216no—
330C2×D3—128no—
331C2×D3—248no—
332C3×C4—44no—
333C3×C4#93914yes—
334C3×C4#93914yes—
335C3×C4#94014yes—
336C3×C4#95314yes—
337Dic3—616no—
338Dic3—128no—
339C32#200211yescharacteristic, complemented, Sylow
340C23—22no—
341C23—21no—
342C23—44no—
343C23—41no—
344C23—41no—
345C23—182no—
346C23—182no—
347C23—182no—
348C23—364no—
349C23D4×C3⋊D311yescharacteristic
350Q8—21no—
351Q8—21no—
352D4—22no—
353D4—21no—
354D4—21no—
355D4—22no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—3616no—
361D4—361no—
362D4—361no—
363D4—361no—
364D4—361no—
365D4—724no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—184no—
370C2×C4—182no—
371C2×C4—182no—
372C2×C4D4×C3⋊D311yescharacteristic
373C2×C4D4×C3⋊D311yescharacteristic
374C2×C4#383011yescharacteristic
375C2×C4C2×#63811yescharacteristic
376C2×C4C2×#63811yescharacteristic
377C2×C4C2×#63811yescharacteristic
378C8—22no—
379C8—361no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—88no—
383C2×C3—84no—
384C2×C3#519818yes—
385C2×C3#522314yes—
386D3—248no—
387C22—22no—
388C22—44no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—82no—
393C22—82no—
394C22—364no—
395C22—364no—
396C22—362no—
397C22—722no—
398C22#6414611yescharacteristic
399C22#6414611yescharacteristic
400C22#6414711yescharacteristic
401C4—41no—
402C4—184no—
403C4—362no—
404C4#6414811yescharacteristic
405C4#6414811yescharacteristic
406C4#6414911yescharacteristic
407C4#6416211yescharacteristic
408C3#7209414yescomplemented
409C2—21no—
410C2—21no—
411C2—82no—
412C2—81no—
413C2—722no—
414C2#1059289712yes—
415C2#1059292211yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC22×C32#120
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#2002
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C32×AGL(1,ℤ/8)C2semidirect
2. #10596301C2semidirect
3. C2×#64149C2semidirect
4. C2×#63664C2semidirect
5. #10592914C2semidirect
6. C2×#63650C2semidirect
7. #10589421C2semidirect
30. C2×C3×AGL(1,ℤ/8)D3semidirect
43. C2×C32×Q8D4semidirect
50. C2×C32×D4D4semidirect
51. C2×C32×D4D4semidirect
67. C32×M16D4semidirect
71. C4⋊(C3⋊Dic3)D4semidirect
74. C4×C3⋊Dic3D4semidirect
140. C2×AGL(1,ℤ/8)C3⋊D3semidirect
153. C2×C3×Q8#67semidirect
160. C2×C3×D4#67semidirect
161. C2×C3×D4#67semidirect
177. C3×M16#67semidirect
283. C2×Q8#638semidirect
299. C2×D4#638semidirect
300. C2×D4#638semidirect
309. M16#638semidirect
339. C32#2002semidirect
408. C3#72094semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
#105963012
C2×#641492
C2×#636642
#105929142
C2×#636502
#105894212
#720943