Finite Group Databasebeta

← #11798922

Subgroups — #11798922

Order 1152.

361 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 84 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×#4872C211yesmaximal, characteristic
2#10590896C211yesmaximal, characteristic, complemented
3C2×#63548C211yesmaximal, characteristic
4C2×#63547C211yesmaximal, characteristic
5#10589358C211yesmaximal, characteristic, complemented
6C3×#4492C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊C16)C211yesmaximal, characteristic, complemented
8#70892—31nomaximal
9#69234—31nomaximal
10C2×#3746C2211yescharacteristic
11C2×C3×Dic12C2211yescharacteristic
12C2×C3×D24C2211yescharacteristic
13C3×#893—21no—
14#63866—21no—
15C2×#3719—21no—
16C32×#107C2211yescharacteristic
17C2×C3⋊(C3⋊C16)C2211yescharacteristic
18C3×C8×Dic3C2211yescharacteristic
19C2×C3×C3⋊C16C2211yescharacteristic
20C3×#809—21no—
21#63548—22no—
22#63547—22no—
23C3×C4×D8—31no—
24#5183—31no—
25C2×#828—31no—
26C2×#827—31no—
27C2×#827—31no—
28C2×#800—31no—
29#4872D311yescharacteristic
30#4699—31no—
31C3×#358—31no—
32Dic3⋊C16—31no—
33#4492—31no—
34#4492—31no—
35#4492—61no—
36#4492D311yescharacteristic, complemented
37C4⋊(C3⋊C16)—31no—
38C2×#634—21no—
39C2×C3×D12—21no—
40C2×C3×C4×D3—41no—
41C2×C3×Dic6—21no—
42C2×C32×C8C2311yescharacteristic
43C32×C4⋊C4—21no—
44#3746—21no—
45#3746—21no—
46C3×(C2×D3)⋊C4—41no—
47C3×C4×Dic3D411yescharacteristic
48C2×C3×C3⋊C8D411yescharacteristic
49C3×Dic12—21no—
50C3×Dic12—21no—
51C3×D24—21no—
52C3×D24D412yes—
53Dic3⋊Dic3—41no—
54Dic3⋊Dic3—41no—
55#3719—42no—
56C3⋊(C3⋊C16)—21no—
57C3×C3⋊C16—21no—
58#2208—91noSylow
59C2×C3×Q16—31no—
60C2×C3×D8—31no—
61C3×C4×D4—61no—
62#944—61no—
63C2×#216—61no—
64C2×Dic12—31no—
65C2×Dic12—31no—
66C2×Dic12—61no—
67C2×Dic12—61no—
68C2×Dic12C2×D311yescharacteristic
69C2×D24C2×D311yescharacteristic
70#893—21no—
71#869—61no—
72C2×C3×C16—31no—
73C3×#107—21no—
74C3×#107C2×D311yescharacteristic
75C3×#107C2×D311yescharacteristic
76C3×#102—61no—
77C3×C4×C8—31no—
78#828—62no—
79#827—62no—
80#827—62no—
81C8×Dic3C2×D311yescharacteristic
82C2×C3⋊C16—31no—
83C2×C3⋊C16—31no—
84C2×C3⋊C16—61no—
85C2×C3⋊C16C2×D311yescharacteristic
86#809—21no—
87#800—62no—
88C22×C3×D3—41no—
89C2×C32×C4—41no—
90C2×C32×C4C2×D411yescharacteristic
91C2×C3⋊Dic3—41no—
92#634—41no—
93#634—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3D811yescharacteristic
96C2×C3×Dic3D811yescharacteristic
97C3×D12—22no—
98C3×D12—41no—
99C3×C4×D3—82no—
100C3×Dic6—41no—
101C3×Dic6—41no—
102C32×C8C2×D411yescharacteristic
103C32×C8#4111yescharacteristic
104C3×C3⋊C8—21no—
105C2×SD32—91no—
106C2×SD32—91no—
107#494—91no—
108C4×D8—31no—
109C4⋊C16—91no—
110#358—31no—
111#358—91no—
112C2×C3×Q8—61no—
113C2×C3×D4—61no—
114C22×C3×C4—121no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C2×Dic6—61no—
119C2×Dic6—61no—
120C2×Dic6—121no—
121C3×Q16—61no—
122C3×Q16—61no—
123C3×D8—32no—
124C3×D8—61no—
125C2×C3×C8—21no—
126C2×C3×C8—31no—
127C2×C3×C8C22×D311yescharacteristic
128C2×C3×C8C22×D311yescharacteristic
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—121no—
133C3×C42—31no—
134#216—122no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—121no—
137C4⋊Dic3—121no—
138Dic3⋊C4—121no—
139Dic3⋊C4—121no—
140C4×Dic3#6711yescharacteristic
141C2×C3⋊C8#6711yescharacteristic
142Dic12—21no—
143Dic12—21no—
144Dic12—62no—
145Dic12—61no—
146Dic12—61no—
147Dic12—61no—
148Dic12—61no—
149Dic12—121no—
150Dic12—122no—
151D24—21no—
152D24D1212yes—
153C3×C16—61no—
154C3⋊C16—21no—
155C3⋊C16—61no—
156C3⋊C16—61no—
157C3⋊C16—121no—
158C22×C32C2×D811yescharacteristic
159C2×C3×D3—42no—
160C2×C3×D3—81no—
161C2×C3×D3—81no—
162C32×C4—81no—
163C32×C4#12111yescharacteristic
164C32×C4AGL(1,ℤ/8)11yescharacteristic
165C3⋊Dic3—81no—
166C3×Dic3—21no—
167C3×Dic3—81no—
168C3×Dic3SD3212yescomplemented
169C2×Q16—31no—
170C2×Q16—91no—
171C2×Q16—181no—
172C2×D8—31no—
173C4⋊Q8—181no—
174C4×D4—61no—
175SD32—182no—
176SD32—182no—
177C2×C16—31no—
178C2×C16—91no—
179#107D3211yescharacteristic
180#102—61no—
181C4×C8—31no—
182C23×C3—121no—
183C22×D3—41no—
184C3×Q8—121no—
185C3×Q8—121no—
186C3×D4—62no—
187C3×D4—121no—
188C2×C3×C4—21no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4—121no—
195C2×C3×C4—242no—
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4C2×#6711yescharacteristic
198C2×Dic3—41no—
199C2×Dic3—121no—
200C2×Dic3—121no—
201C2×Dic3—241no—
202C2×Dic3#21311yescharacteristic
203C2×Dic3#21311yescharacteristic
204D12—22no—
205D12—41no—
206C4×D3—82no—
207Dic6—41no—
208Dic6—41no—
209Dic6—122no—
210Dic6—121no—
211Dic6—121no—
212Dic6—121no—
213Dic6—121no—
214Dic6—241no—
215C3×C8—21no—
216C3×C8—21no—
217C3×C8—61no—
218C3×C8C2×D1211yescharacteristic
219C3×C8#23511yescharacteristic
220C3×C8#23511yescharacteristic
221C3×C8D3×D411yescharacteristic
222C3⋊C8—21no—
223C2×C32#45911yescharacteristic
224C2×C32C2×SD3211yescharacteristic
225C2×C32#51011yescharacteristic
226C3×D3—82no—
227C2×Q8—61no—
228C2×Q8—181no—
229C2×D4—61no—
230C22×C4—121no—
231Q16—61no—
232Q16—61no—
233Q16—181no—
234Q16—181no—
235Q16—362no—
236D8—32no—
237D8—61no—
238C2×C8—31no—
239C2×C8C2×D3211yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—361no—
242C4⋊C4—361no—
243C22⋊C4—121no—
244C42—31no—
245C16—61no—
246C16—181no—
247C22×C3—21no—
248C22×C3—122no—
249C22×C3—241no—
250C22×C3—241no—
251C22×C3D3×D811yescharacteristic
252C22×C3C2×#21311yescharacteristic
253C2×D3—42no—
254C2×D3—81no—
255C2×D3—81no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—32no—
259C3×C4—61no—
260C3×C4—81no—
261C3×C4—81no—
262C3×C4—161no—
263C3×C4—241no—
264C3×C4#88411yescharacteristic
265C3×C4#91411yescharacteristic
266C3×C4#93011yescharacteristic
267C3×C4#94911yescharacteristic
268Dic3—21no—
269Dic3—81no—
270Dic3—242no—
271Dic3—241no—
272Dic3—241no—
273Dic3#82712yescomplemented
274C32#220811yescharacteristic, complemented, Sylow
275C23—121no—
276Q8—121no—
277Q8—121no—
278Q8—361no—
279Q8—361no—
280D4—62no—
281D4—121no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—41no—
285C2×C4—121no—
286C2×C4—242no—
287C2×C4—361no—
288C2×C4D3×#6711yescharacteristic
289C8—61no—
290C8#379811yescharacteristic
291C8D3×D1211yescharacteristic
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—242no—
296C2×C3#478611yescharacteristic
297C2×C3D3×SD3211yescharacteristic
298C2×C3#491811yescharacteristic
299C2×C3#501511yescharacteristic
300C2×C3C2×#82711yescharacteristic
301C2×C3#519511yescharacteristic
302D3—82no—
303C22—122no—
304C22—241no—
305C22—241no—
306C22D3×#21311yescharacteristic
307C4—32no—
308C4—61no—
309C4—81no—
310C4—241no—
311C4—721no—
312C4#6390611yescharacteristic
313C4#6393911yescharacteristic
314C3—21no—
315C3#6923411yescharacteristic, complemented
316C3#7089211yescharacteristic, complemented
317C2—242no—
318C2#1059054711yescharacteristic
319C2D3×#82711yescharacteristic
320C2#1059104211yescharacteristic
subgroupG/H
centerC22D3×#213
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC32×#107C22
socleC22×C32C2×D8
prime radicalC2×C3×D24C22
hypercenter#107D32
hypocenterC32#2208
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #10590896C2semidirect
5. #10589358C2semidirect
6. C3×#4492C2semidirect
7. Dic3⋊(C3⋊C16)C2semidirect
36. #4492D3semidirect
168. C3×Dic3SD32semidirect
273. Dic3#827semidirect
274. C32#2208semidirect
315. C3#69234semidirect
316. C3#70892semidirect

Maximal subgroups

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

subgroupindex
C3×#48722
#105908962
C2×#635482
C2×#635472
#105893582
C3×#44922
Dic3⋊(C3⋊C16)2
#708923
#692343