Finite Group Databasebeta

← #11798921

Subgroups — #11798921

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
1C2×C3×#800C211yesmaximal, characteristic
2#10590896C211yesmaximal, characteristic, complemented
3Dic3×D24C211yesmaximal, characteristic
4C2×#63548C211yesmaximal, characteristic
5#10589369C211yesmaximal, characteristic, complemented
6C3×Dic3⋊C16C211yesmaximal, characteristic, complemented
7#10589118C211yesmaximal, characteristic, complemented
8#71476—31nomaximal
9#69287—31nomaximal
10C2×#3746C2211yescharacteristic
11C2×C3×Dic12C2211yescharacteristic
12C2×C3×D24C2211yescharacteristic
13Dic3×D12—21no—
14#63866—21no—
15C2×#3719—21no—
16C2×C32×C16C2211yescharacteristic
17#63645C2211yescharacteristic
18C3×C8×Dic3C2211yescharacteristic
19C2×C3×C3⋊C16C2211yescharacteristic
20C3×#800—22no—
21#63563—21no—
22#63548—22no—
23C2×C3×SD32—31no—
24#5183—31no—
25Dic3×D8—31no—
26C2×#827—31no—
27C2×#800—31no—
28C2×#800D311yescharacteristic
29#4699—31no—
30C4×D24—31no—
31C3×C4⋊C16—31no—
32#4564—31no—
33#4503—31no—
34#4503—31no—
35#4503—31no—
36#4503—61no—
37Dic3⋊C16D311yescharacteristic, complemented
38C2×#634—21no—
39C2×C3×D12—21no—
40C2×C3×Dic6—21no—
41C2×D3×Dic3—41no—
42C2×C32×C8C2311yescharacteristic
43C4⋊(C3⋊Dic3)—21no—
44#3746—21no—
45#3746—21no—
46C3×C4×Dic3D411yescharacteristic
47C2×C3×C3⋊C8D411yescharacteristic
48C3×Dic12—21no—
49C3×Dic12—21no—
50C3×D24—21no—
51C3×D24D412yes—
52Dic3⋊Dic3—41no—
53Dic3⋊Dic3—41no—
54(C2×D3)⋊Dic3—41no—
55#3719—42no—
56C32×C16—21no—
57C3×C3⋊C16—21no—
58#2208—91noSylow
59C2×C3×Q16—31no—
60C2×C3×D8—31no—
61#944—61no—
62C2×#216—61no—
63D4×Dic3—61no—
64C2×Dic12—31no—
65C2×Dic12—31no—
66C2×Dic12—61no—
67C2×Dic12—61no—
68C2×Dic12C2×D311yescharacteristic
69C2×D24C2×D311yescharacteristic
70C4×D12—61no—
71#869—61no—
72C3×SD32—62no—
73C2×C3×C16—21no—
74C2×C3×C16—31no—
75C2×C3×C16C2×D311yescharacteristic
76C2×C3×C16C2×D311yescharacteristic
77C3×C4×C8—31no—
78#832—61no—
79#827—62no—
80#821—61no—
81#818—31no—
82#818—31no—
83#818—61no—
84C8×Dic3C2×D311yescharacteristic
85C2×C3⋊C16C2×D311yescharacteristic
86#800—22no—
87#800—62no—
88C22×C3×D3—41no—
89C2×C32×C4C2×D411yescharacteristic
90C2×C3⋊Dic3—41no—
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×Dic6—41no—
100C3×Dic6—41no—
101D3×Dic3—82no—
102C32×C8C2×D411yescharacteristic
103C32×C8#4111yescharacteristic
104C3×C3⋊C8—21no—
105C2×SD32—31no—
106C2×SD32—91no—
107#494—91no—
108C4×D8—91no—
109C4⋊C16—31no—
110#358—91no—
111#358—91no—
112C2×C3×Q8—61no—
113C2×C3×D4—61no—
114C22×Dic3—121no—
115C2×D12—21no—
116C2×C4×D3—121no—
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×C42—31no—
130#217—121no—
131#216—122no—
132(C2×D3)⋊C4—121no—
133C4⋊Dic3—61no—
134C4⋊Dic3—61no—
135C4⋊Dic3—121no—
136C4⋊Dic3—121no—
137C4⋊Dic3—121no—
138Dic3⋊C4—121no—
139Dic3⋊C4—121no—
140C4×Dic3D1211yescharacteristic
141C2×C3⋊C8D1211yescharacteristic
142Dic12—21no—
143Dic12—21no—
144Dic12—62no—
145Dic12—61no—
146Dic12—61no—
147Dic12—61no—
148Dic12—61no—
149Dic12—121no—
150Dic12—122no—
151D24—21no—
152D24#6712yes—
153C3×C16—21no—
154C3×C16—21no—
155C3×C16—41no—
156C3×C16—61no—
157C3⋊C16—21no—
158C22×C32C2×D811yescharacteristic
159C2×C3×D3—42no—
160C2×C3×D3—81no—
161C2×C3×D3—81no—
162C32×C4#12111yescharacteristic
163C32×C4AGL(1,ℤ/8)11yescharacteristic
164C3⋊Dic3—81no—
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—181no—
175SD32—62no—
176SD32—182no—
177C2×C16—31no—
178C2×C16D3211yescharacteristic
179#107—91no—
180#102—181no—
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—121no—
192C2×C3×C4C2×D1211yescharacteristic
193C2×C3×C4D3×D411yescharacteristic
194C2×Dic3—41no—
195C2×Dic3—121no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198C2×Dic3—121no—
199C2×Dic3—241no—
200C2×Dic3—241no—
201C2×Dic3—242no—
202C2×Dic3D2411yescharacteristic
203C2×Dic3D2411yescharacteristic
204D12—22no—
205D12—41no—
206C4×D3—242no—
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×C8D3×D411yescharacteristic
219C3×C8#23711yescharacteristic
220C3×C8#23911yescharacteristic
221C3×C8C2×#6711yescharacteristic
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—361no—
231Q16—61no—
232Q16—61no—
233Q16—181no—
234Q16—181no—
235Q16—362no—
236D8—32no—
237D8—61no—
238C2×C8—31no—
239C2×C8C2×D3211yescharacteristic
240C4⋊C4—181no—
241C4⋊C4—361no—
242C4⋊C4—361no—
243C22⋊C4—361no—
244C42—31no—
245C16—21no—
246C16—61no—
247C22×C3—21no—
248C22×C3—122no—
249C22×C3—241no—
250C22×C3—241no—
251C22×C3C2×D2411yescharacteristic
252C22×C3D3×D811yescharacteristic
253C2×D3—42no—
254C2×D3—81no—
255C2×D3—81no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—32no—
259C3×C4—61no—
260C3×C4—241no—
261C3×C4#89511yescharacteristic
262C3×C4#90811yescharacteristic
263C3×C4#91111yescharacteristic
264C3×C4#93911yescharacteristic
265Dic3—21no—
266Dic3—81no—
267Dic3—242no—
268Dic3—241no—
269Dic3—241no—
270Dic3—241no—
271Dic3—241no—
272Dic3—481no—
273Dic3#80012yescomplemented
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—121no—
285C2×C4—361no—
286C2×C4—361no—
287C2×C4—722no—
288C2×C4D3×D1211yescharacteristic
289C8—61no—
290C8#379611yescharacteristic
291C8D3×#6711yescharacteristic
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—242no—
296C2×C3#484311yescharacteristic
297C2×C3C2×#80011yescharacteristic
298C2×C3#490911yescharacteristic
299C2×C3D3×SD3211yescharacteristic
300C2×C3#491511yescharacteristic
301C2×C3#515011yescharacteristic
302D3—82no—
303C22—122no—
304C22—241no—
305C22—241no—
306C22D3×D2411yescharacteristic
307C4—32no—
308C4—61no—
309C4—241no—
310C4—721no—
311C4—721no—
312C4#6379511yescharacteristic
313C4#6388611yescharacteristic
314C3—21no—
315C3#6928711yescharacteristic, complemented
316C3#7147611yescharacteristic, complemented
317C2—242no—
318C2D3×#80011yescharacteristic
319C2#1059023911yescharacteristic
320C2#1059075311yescharacteristic
subgroupG/H
centerC22D3×D24
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×C16C22
socleC22×C32C2×D8
prime radicalC2×C3×D24C22
hypercenterC2×C16D32
hypocenterC32#2208
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16

Complements

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

NH ≅ G/Ntype
2. #10590896C2semidirect
5. #10589369C2semidirect
6. C3×Dic3⋊C16C2semidirect
7. #10589118C2semidirect
37. Dic3⋊C16D3semidirect
168. C3×Dic3SD32semidirect
273. Dic3#800semidirect
274. C32#2208semidirect
315. C3#69287semidirect
316. C3#71476semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8002
#105908962
Dic3×D242
C2×#635482
#105893692
C3×Dic3⋊C162
#105891182
#714763
#692873