Finite Group Databasebeta

← #11781260

Subgroups — #11781260

Order 1152.

384 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×#5088C211yesmaximal, characteristic, complemented
2C3×#5084C211yesmaximal, characteristic, complemented
3#10590347C211yesmaximal, characteristic
4C3⋊C8×D12C211yesmaximal, characteristic, complemented
5#10589412C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic, complemented
8#71200—31nomaximal
9#71134—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C3×#874C2211yescharacteristic
12C3×C4×D12C2211yescharacteristic
13C2×#3713—21no—
14C2×D3×C3⋊C8—21no—
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#809—21no—
18C3×#809—21no—
19C3×#809—21no—
20C3×#808—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#486—31no—
26C3×#476—31no—
27#5088D311yescharacteristic, complemented
28#5084D311yescharacteristic, complemented
29#5072—31no—
30#5015—31no—
31D4×C3⋊C8—31no—
32C8×D12—31no—
33#4546—31no—
34#4546—31no—
35#4546—61no—
36#4544—31no—
37#4485—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C42C2311yescharacteristic
49C2×C3⋊(C3⋊C8)—21no—
50C2×C3⋊(C3⋊C8)—21no—
51C3×(C2×D3)⋊C4—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3713—42no—
57D3×C3⋊C8—42no—
58#1677—91noSylow
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×#127—31no—
63C3×C4×D4—31no—
64C2×#213—61no—
65C2×#213—61no—
66C22×C3⋊C8—61no—
67C2×D3×C8—61no—
68#874C2×D311yescharacteristic
69C4×D12C2×D311yescharacteristic
70C3×#106—61no—
71C3×C4⋊C8—31no—
72C3×#102—61no—
73C3×#102—61no—
74C3×#102—61no—
75C3×C4×C8—31no—
76#832—61no—
77#830—61no—
78(C2×D3)⋊C8—61no—
79#809—21no—
80#809—21no—
81#809—21no—
82#809—61no—
83#808—21no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3×D3—21no—
93C22×C3×D3—41no—
94C32×Q8—81no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3×Dic3—21no—
100C3×D12—21no—
101C3×D12—41no—
102C3×D12—41no—
103C3×D12—41no—
104C3×D12—41no—
105C3×D12—81no—
106C3×D12SD1612yes—
107C3×C4×D3—41no—
108C3×C4×D3—41no—
109C3⋊(C3⋊C8)—41no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—21no—
113C3×C3⋊C8—41no—
114#486—31no—
115#476—31no—
116#459—91no—
117C8×D4—91no—
118#334—91no—
119#333—91no—
120#331—91no—
121C2×C3×Q8—22no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×D4—31no—
125C2×C3×D4—61no—
126C2×C3×D4—61no—
127C2×C3×D4—61no—
128C22×C3×C4—61no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12#6711yescharacteristic
133C2×C4×D3—21no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—61no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—61no—
145C3×C42—21no—
146C3×C42C22×D311yescharacteristic
147C3×C42C22×D311yescharacteristic
148#213—122no—
149#213—122no—
150(C2×D3)⋊C4—21no—
151C4⋊Dic3#6711yescharacteristic
152C2×C3⋊C8—21no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8—121no—
158C2×C3⋊C8—121no—
159C2×C3⋊C8—122no—
160C2×C3⋊C8#6711yescharacteristic
161C2×C3⋊C8#6711yescharacteristic
162D3×C8—122no—
163C22×C32#12111yescharacteristic
164C2×C3×D3—22no—
165C2×C3×D3—41no—
166C2×C3×D3—41no—
167C2×C3×D3—81no—
168C2×C3×D3—81no—
169C2×C3×D3—81no—
170C32×C4—21no—
171C32×C4—81no—
172C32×C4C2×SD1611yescharacteristic
173C32×C4#13511yescharacteristic
174C32×C4#13511yescharacteristic
175C32×C4AGL(1,ℤ/8)11yescharacteristic
176C3×Dic3—41no—
177C2×D8—181no—
178C22×C8—181no—
179C4⋊Q8D3211yescharacteristic
180#127—31no—
181C4×D4—31no—
182#106—61no—
183C4⋊C8—31no—
184C4⋊C8—91no—
185C4⋊C8—91no—
186#102—61no—
187#102—61no—
188#102—61no—
189#102—181no—
190C22⋊C8—181no—
191C4×C8—31no—
192C23×C3—61no—
193C23×C3—121no—
194C22×D3—21no—
195C22×D3—41no—
196C3×Q8—81no—
197C3×Q8—81no—
198C3×Q8—82no—
199C3×D4—32no—
200C3×D4—61no—
201C3×D4—121no—
202C3×D4—121no—
203C3×D4—121no—
204C3×D4—121no—
205C3×D4—241no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—61no—
213C2×C3×C4—121no—
214C2×C3×C4—121no—
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4#23711yescharacteristic
218C2×C3×C4#23711yescharacteristic
219C2×C3×C4C2×#6711yescharacteristic
220C2×C3×C4C2×#6711yescharacteristic
221C2×Dic3—21no—
222D12—21no—
223D12—41no—
224D12—41no—
225D12—41no—
226D12—41no—
227D12—81no—
228D12#21512yes—
229C4×D3—41no—
230C4×D3—41no—
231C3×C8—61no—
232C3×C8—61no—
233C3×C8—121no—
234C3⋊C8—21no—
235C3⋊C8—21no—
236C3⋊C8—41no—
237C3⋊C8—121no—
238C3⋊C8—121no—
239C3⋊C8—121no—
240C3⋊C8—121no—
241C3⋊C8—241no—
242C3⋊C8—241no—
243C2×C32#46311yescharacteristic
244C2×C32#46511yescharacteristic
245C2×C32#47211yescharacteristic
246C3×D3—42no—
247C3×D3—161no—
248C2×Q8—21no—
249C2×D4—31no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C22×C4—61no—
254D8—362no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—61no—
258C2×C8—181no—
259C2×C8—181no—
260C2×C8—362no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—31no—
264C22⋊C4—61no—
265C42C2×D3211yescharacteristic
266C22×C3—21no—
267C22×C3—62no—
268C22×C3—121no—
269C22×C3—121no—
270C22×C3—241no—
271C22×C3—241no—
272C22×C3—241no—
273C22×C3#93811yescharacteristic
274C22×C3#93911yescharacteristic
275C2×D3—22no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4—121no—
292C3×C4D3×SD1611yescharacteristic
293C3×C4#91411yescharacteristic
294C3×C4#91611yescharacteristic
295C3×C4#91611yescharacteristic
296C3×C4C2×#21511yescharacteristic
297C3×C4#94911yescharacteristic
298C3×C4#95011yescharacteristic
299C3×C4#95011yescharacteristic
300Dic3—41no—
301C32#167711yescharacteristic, complemented, Sylow
302C23—61no—
303C23—121no—
304Q8—81no—
305D4—32no—
306D4—61no—
307D4—121no—
308D4—121no—
309D4—121no—
310D4—121no—
311D4—241no—
312C2×C4—41no—
313C2×C4—61no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4#380511yescharacteristic
317C2×C4D3×#6711yescharacteristic
318C2×C4D3×#6711yescharacteristic
319C8—61no—
320C8—61no—
321C8—121no—
322C8—361no—
323C8—361no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—122no—
328C2×C3—481no—
329C2×C3#504811yescharacteristic
330C2×C3#505711yescharacteristic
331C2×C3#516011yescharacteristic
332C2×C3#516511yescharacteristic
333C2×C3#517111yescharacteristic
334C2×C3#519911yescharacteristic
335D3—42no—
336D3—161no—
337C22—62no—
338C22—121no—
339C22—121no—
340C22—241no—
341C22—241no—
342C22—241no—
343C22#6397011yescharacteristic
344C4—21no—
345C4—81no—
346C4—121no—
347C4D3×#21511yescharacteristic
348C4#6393911yescharacteristic
349C4#6394711yescharacteristic
350C4#6394711yescharacteristic
351C3—21no—
352C3#7113411yescharacteristic, complemented
353C3#7120011yescharacteristic, complemented
354C2—122no—
355C2—481no—
356C2#1059150511yescharacteristic
357C2#1059156711yescharacteristic
358C2#1059163411yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radical#10590347C2
hypercenterC4⋊Q8D32
hypocenterC32#1677
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5088C2semidirect
2. C3×#5084C2semidirect
4. C3⋊C8×D12C2semidirect
5. #10589412C2semidirect
6. #10589097C2semidirect
7. #10589090C2semidirect
27. #5088D3semidirect
28. #5084D3semidirect
301. C32#1677semidirect
352. C3#71134semidirect
353. C3#71200semidirect

Maximal subgroups

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

subgroupindex
C3×#50882
C3×#50842
#105903472
C3⋊C8×D122
#105894122
#105890972
#105890902
#712003
#711343