Finite Group Databasebeta

← #18010592

Subgroups — #18010592

Order 4736.

374 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 80 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×Q8×#3866C211yesmaximal, characteristic, complemented
2C2×#11924599C211yesmaximal, characteristic, complemented
3C2×#11924598C211yesmaximal, characteristic
4#14558485C211yesmaximal, characteristic
5#14558407C211yesmaximal, characteristic, complemented
6(C2×#3866)⋊C8C211yesmaximal, characteristic, complemented
7C2×SD16×D37C211yesmaximal, characteristic
8C2×D37⋊C8C2211yescharacteristic
9Q8×#3866—22no—
10Q8×#3866—22no—
11C2×C4⋊#3866—21no—
12C2×C4⋊#3866C2211yescharacteristic
13C2×C4⋊#3866C2211yescharacteristic
14C2×C22⋊#3866—21no—
15C2×C22⋊#3866—21no—
16C2×C4×#3866—21no—
17C2×C4×#3866C2211yescharacteristic
18#11924599—22no—
19#11924598—22no—
20C2×Q8×D37C2211yescharacteristic
21C2×D4×D37C2211yescharacteristic
22C2×#10596930C411yescharacteristic, complemented
23C2×#10596940C411yescharacteristic, complemented
24C2×#10596941C411yescharacteristic, complemented
25SD16×D37—22no—
26SD16×D37—22no—
27C2×C8×D37C2211yescharacteristic
28C2×SD16×C37C411yescharacteristic, complemented
29C23×D37—21no—
30C22×#3866—21no—
31C22×#3866—21no—
32C22×#3866D411yescharacteristic
33C22×#3866D411yescharacteristic
34C2×Q8×C37C2×C411yescharacteristic
35C2×D4×C37C2×C411yescharacteristic
36C2×C22⋊D37—21no—
37Q8×D37—21no—
38Q8×D37—21no—
39Q8×D37D412yes—
40Q8×D37D412yes—
41D4×D37—21no—
42D4×D37—21no—
43D4×D37—21no—
44D4×D37—21no—
45C2×D148C2×C411yescharacteristic
46C2×C4×D37—21no—
47C2×C4×D37C2311yescharacteristic
48C2×Dic74—21no—
49C2×Dic74C2×C411yescharacteristic
50C22⋊#3866—42no—
51C22⋊#3866—42no—
52C2×#64438—21no—
53C4⋊#3866—24no—
54C4⋊#3866—21no—
55C4⋊#3866—21no—
56C4⋊#3866—21no—
57C4⋊#3866D412yes—
58C4×#3866—24no—
59C4×#3866—21no—
60C4×#3866—21no—
61D37⋊C8—21no—
62D37⋊C8—21no—
63SD16×C37—22no—
64C2×C8×C37C2×C411yescharacteristic
65#10596941—22no—
66#10596940—22no—
67C2×#64436C2×C411yescharacteristic
68#10596930—22no—
69C8×D37—21no—
70C8×D37—21no—
71C23×C37—21no—
72C22×D37—21no—
73C22×D37—41no—
74C22×D37—41no—
75C22×D37—41no—
76C22×D37—41no—
77C22×D37—41no—
78C22×D37—41no—
79C22×D37C2×D411yescharacteristic
80C2×#3866—21no—
81C2×#3866—21no—
82C2×#3866—21no—
83C2×#3866—22no—
84C2×#3866—21no—
85C2×#3866—21no—
86C2×#3866—41no—
87C2×#3866—41no—
88C2×#3866—41no—
89C2×#3866—41no—
90C2×#3866—41no—
91C2×#3866SD1612yescomplemented
92Q8×C37—21no—
93Q8×C37C22⋊C412yescomplemented
94D4×C37—21no—
95D4×C37—21no—
96C2×C4×C37—21no—
97C2×C4×C37C22×C411yescharacteristic
98C22⋊D37—41no—
99C22⋊D37—41no—
100C2×Dic37—21no—
101C2×Dic37#4111yescharacteristic
102D148—21no—
103D148—21no—
104C4×D37—22no—
105C4×D37—22no—
106C4×D37C2×D411yescharacteristic
107C4×D37C2×D411yescharacteristic
108C4×D37C2×D411yescharacteristic
109C4×D37#4111yescharacteristic
110Dic74—21no—
111Dic74—22no—
112Dic74—22no—
113Dic74C22⋊C412yescomplemented
114#64438—41no—
115C8×C37—21no—
116#64436—21no—
117C22×C37—41no—
118C22×C37—41no—
119C22×C37—41no—
120C22×C37C4×D411yescharacteristic
121C2×D37—41no—
122C2×D37—41no—
123C2×D37—41no—
124C2×D37—81no—
125C2×D37—81no—
126C2×D37—81no—
127C2×D37—81no—
128C2×D37#12011yescharacteristic
129C2×D37#12111yescharacteristic
130C2×D37C2×SD1611yescharacteristic
131C2×D37#13511yescharacteristic
132C2×D37AGL(1,ℤ/8)11yescharacteristic
133C2×D37#13711yescharacteristic
134#3866—22no—
135#3866—41no—
136#3866—42no—
137#3866—81no—
138C4×C37—22no—
139C4×C37C2×C22⋊C411yescharacteristic
140C4×C37C4×D411yescharacteristic
141Dic37—22no—
142Dic37#12111yescharacteristic
143Dic37#12311yescharacteristic
144#1880—371nomaximal, Sylow
145C2×C37—81no—
146C2×C37#38611yescharacteristic
147C2×C37C4×SD1611yescharacteristic
148C2×C37#44011yescharacteristic
149D37—81no—
150D37#44811yescharacteristic
151D37#44911yescharacteristic
152D37#46011yescharacteristic
153D37#46411yescharacteristic
154C22×SD16—371no—
155C2×C4×Q8—371no—
156C2×#103—371no—
157C2×#102—371no—
158#390—371no—
159#340—371no—
160#336—371no—
161C37#188011yescharacteristic, complemented, Sylow
162C22×Q8—371no—
163C22×D4—371no—
164C2×SD16—371no—
165C2×SD16—371no—
166C2×SD16—371no—
167C2×SD16—742no—
168C2×SD16—742no—
169C2×SD16#386611yescharacteristic, complemented
170C22×C8—371no—
171C22×C8—371no—
172C4×Q8—742no—
173C4×Q8—742no—
174C2×C4⋊C4—371no—
175C2×C4⋊C4—371no—
176C2×C4⋊C4—741no—
177C2×C22⋊C4—741no—
178C2×C22⋊C4—741no—
179C2×C42—371no—
180C2×C42—741no—
181#103—742no—
182#102—742no—
183C24—741no—
184C2×Q8—372no—
185C2×Q8—372no—
186C2×Q8—371no—
187C2×Q8—741no—
188C2×Q8—741no—
189C2×Q8—741no—
190C2×Q8C2×#386611yescharacteristic
191C2×D4—371no—
192C2×D4—741no—
193C2×D4—741no—
194C2×D4—741no—
195C2×D4—741no—
196C2×D4—741no—
197C2×D4C2×#386611yescharacteristic
198C22×C4—371no—
199C22×C4—371no—
200C22×C4—371no—
201C22×C4—741no—
202C22×C4—741no—
203C22×C4—741no—
204SD16—22no—
205SD16—742no—
206SD16—742no—
207SD16—742no—
208C2×C8—371no—
209C2×C8—741no—
210C2×C8—741no—
211C2×C8—741no—
212C2×C8—741no—
213C2×C8—741no—
214C2×C8C2×#386611yescharacteristic
215C4⋊C4—372no—
216C4⋊C4—741no—
217C4⋊C4—741no—
218C4⋊C4—741no—
219C4⋊C4—744no—
220C22⋊C4—1482no—
221C22⋊C4—1482no—
222C42—741no—
223C42—741no—
224C42—744no—
225C23—21no—
226C23—371no—
227C23—741no—
228C23—1481no—
229C23—1481no—
230C23—1481no—
231C23—1481no—
232C23—1481no—
233C23—1481no—
234Q8—21no—
235Q8—372no—
236Q8—742no—
237Q8—742no—
238Q8—741no—
239Q8C22⋊#386612yescomplemented
240D4—21no—
241D4—21no—
242D4—741no—
243D4—741no—
244D4—1481no—
245D4—1481no—
246C2×C4—21no—
247C2×C4—371no—
248C2×C4—371no—
249C2×C4—371no—
250C2×C4—371no—
251C2×C4—371no—
252C2×C4—372no—
253C2×C4—742no—
254C2×C4—742no—
255C2×C4—741no—
256C2×C4—741no—
257C2×C4—741no—
258C2×C4—742no—
259C2×C4—741no—
260C2×C4—741no—
261C2×C4—741no—
262C2×C4—1481no—
263C2×C4—1481no—
264C2×C4—1481no—
265C2×C4—1481no—
266C2×C4—1481no—
267C2×C4C22×#386611yescharacteristic
268C8—21no—
269C8—741no—
270C8—1481no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—371no—
275C22—371no—
276C22—371no—
277C22—371no—
278C22—371no—
279C22—371no—
280C22—2961no—
281C22—2961no—
282C22—2961no—
283C22—2961no—
284C22—1481no—
285C22—1481no—
286C22—1481no—
287C22D4×#386611yescharacteristic
288C4—22no—
289C4—371no—
290C4—371no—
291C4—2961no—
292C4—742no—
293C4—742no—
294C4—1481no—
295C4—1482no—
296C4C2×C22⋊#386611yescharacteristic
297C4D4×#386611yescharacteristic
298C2—81no—
299C2—371no—
300C2—371no—
301C2—371no—
302C2—371no—
303C2—2961no—
304C2#1455848111yescharacteristic
305C2SD16×#386611yescharacteristic
306C2#1455857211yescharacteristic
subgroupG/H
centerC22D4×#3866
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
FittingC2×SD16×C37C4
socleC22×C37C4×D4
prime radicalC2×D4×D37C22
cosocleC2×C4×D37C23
hypercenterC2×SD16#3866
hypocenterC37#1880
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#3866C2semidirect
2. C2×#11924599C2semidirect
5. #14558407C2semidirect
6. (C2×#3866)⋊C8C2semidirect
22. C2×#10596930C4semidirect
23. C2×#10596940C4semidirect
24. C2×#10596941C4semidirect
28. C2×SD16×C37C4semidirect
91. C2×#3866SD16semidirect
93. Q8×C37C22⋊C4semidirect
113. Dic74C22⋊C4semidirect
161. C37#1880semidirect
169. C2×SD16#3866semidirect
239. Q8C22⋊#3866semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#38662
C2×#119245992
C2×#119245982
#145584852
#145584072
(C2×#3866)⋊C82
C2×SD16×D372
#188037