Finite Group Databasebeta

← #11799031

Subgroups — #11799031

Order 1152.

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

normal subgroups only

This group has 371 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
1Q16×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#829C211yesmaximal, characteristic, complemented
3C2×C3×#828C211yesmaximal, characteristic
4#10590869C211yesmaximal, characteristic
5#10589133C211yesmaximal, characteristic, complemented
6#10589118C211yesmaximal, characteristic, complemented
7#10589115C211yesmaximal, characteristic
8#71498—31nomaximal
9#71478—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11Q8×C3⋊Dic3—21no—
12C2×C3×Dic12C2211yescharacteristic
13C2×C3×D24C2211yescharacteristic
14#63863—21no—
15C2×#3715—21no—
16#63662—21no—
17#63645C2211yescharacteristic
18C8×C3⋊Dic3C2211yescharacteristic
19C3×#829—22no—
20C3×#828—22no—
21C2×C3×C3⋊C16C2211yescharacteristic
22C2×C3×C3⋊C16C2211yescharacteristic
23C2×C3×Q32—31no—
24C2×C3×SD32—31no—
25#5191—31no—
26Dic3×Q16—31no—
27Dic3×Q16—31no—
28Dic3×Q16—61no—
29C2×#829D311yescharacteristic, complemented
30C2×#828D311yescharacteristic
31#5155—31no—
32#4564—31no—
33#4562—31no—
34#4509—31no—
35#4503—31no—
36Dic3⋊C16—31no—
37Dic3⋊C16—31no—
38C2×C32×Q8—21no—
39C2×C3×D12—21no—
40C2×C3×Dic6—21no—
41C32×Q16—21no—
42C32×Q16D412yes—
43C2×C32×C8C2311yescharacteristic
44C4⋊(C3⋊Dic3)—21no—
45C4⋊(C3⋊Dic3)—41no—
46C4×C3⋊Dic3—41no—
47C4×C3⋊Dic3D411yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49C3×Dic12—21no—
50C3×Dic12—21no—
51C3×D24—21no—
52C3×D24—21no—
53(C2×D3)⋊Dic3—41no—
54(C2×D3)⋊Dic3—41no—
55#3715—42no—
56C3×C3⋊C16—21no—
57C3×C3⋊C16—21no—
58#2211—91noSylow
59C2×C3×Q16—21no—
60C2×C3×Q16—31no—
61C2×C3×Q16C2×D311yescharacteristic
62C2×C3×Q16C2×D311yescharacteristic
63C2×C3×D8—31no—
64#947—61no—
65Q8×Dic3—61no—
66Q8×Dic3—61no—
67Q8×Dic3—62no—
68C2×#215—61no—
69#936—61no—
70C2×#214—61no—
71C2×Dic12C2×D311yescharacteristic
72C2×D24C2×D311yescharacteristic
73C3×Q32—62no—
74C3×SD32—62no—
75C2×C3×C16—31no—
76C2×C3×C16—31no—
77#835—61no—
78#835—61no—
79#835—62no—
80#829—22no—
81#828—22no—
82#818—31no—
83#818—31no—
84#818—61no—
85C8×Dic3—31no—
86C8×Dic3—31no—
87C8×Dic3—61no—
88C2×C3⋊C16C2×D311yescharacteristic
89C2×C3⋊C16C2×D311yescharacteristic
90C22×C3×D3—41no—
91C32×Q8—22no—
92C32×Q8—41no—
93C2×C32×C4—41no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C3⋊Dic3—41no—
96C2×C3⋊Dic3D811yescharacteristic
97C2×C3⋊Dic3D811yescharacteristic
98C2×C3×Dic3—41no—
99C3×D12—41no—
100C3×D12—41no—
101C3×Dic6—41no—
102C3×Dic6—41no—
103C32×C8C2×D411yescharacteristic
104C32×C8#4111yescharacteristic
105C3⋊(C3⋊C8)—21no—
106C2×Q32—31no—
107C2×SD32—31no—
108#495—91no—
109C4×Q16—91no—
110C4⋊C16—91no—
111#358—91no—
112#357—91no—
113C2×C3×Q8—22no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—61no—
117C2×C3×D4—61no—
118C2×D12—21no—
119C2×Dic6—21no—
120C3×Q16—22no—
121C3×Q16—21no—
122C3×Q16—21no—
123C3×Q16—22no—
124C3×Q16—61no—
125C3×Q16—61no—
126C3×Q16#6712yes—
127C3×Q16#6712yes—
128C3×D8—61no—
129C3×D8—61no—
130C2×C3×C8—21no—
131C2×C3×C8C22×D311yescharacteristic
132C2×C3×C8C22×D311yescharacteristic
133#217—121no—
134#217—121no—
135#215—122no—
136#214—122no—
137(C2×D3)⋊C4—121no—
138(C2×D3)⋊C4—121no—
139C4⋊Dic3—62no—
140C4⋊Dic3—61no—
141C4⋊Dic3—61no—
142C4⋊Dic3—121no—
143C4⋊Dic3—121no—
144C4⋊Dic3—122no—
145C4×Dic3—31no—
146C4×Dic3—31no—
147C4×Dic3—61no—
148C4×Dic3—121no—
149C4×Dic3—121no—
150C4×Dic3—122no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—61no—
154Dic12—21no—
155Dic12—21no—
156D24—21no—
157D24—21no—
158C3×C16—61no—
159C3×C16—61no—
160C3⋊C16—21no—
161C3⋊C16—21no—
162C22×C32C2×D811yescharacteristic
163C2×C3×D3—81no—
164C2×C3×D3—81no—
165C2×C3×D3—81no—
166C32×C4—42no—
167C32×C4#12111yescharacteristic
168C32×C4AGL(1,ℤ/8)11yescharacteristic
169C3⋊Dic3—21no—
170C3⋊Dic3—21no—
171C3⋊Dic3—81no—
172C3×Dic3—81no—
173C2×Q16—31no—
174C2×Q16D3211yescharacteristic
175C2×SD16—181no—
176C2×D8—31no—
177#124—181no—
178C4×Q8—181no—
179Q32—62no—
180SD32—62no—
181C2×C16—31no—
182C2×C16—31no—
183#107—91no—
184#103—181no—
185C4×C8—91no—
186C23×C3—121no—
187C22×D3—41no—
188C3×Q8—24no—
189C3×Q8—22no—
190C3×Q8—22no—
191C3×Q8—42no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—121no—
195C3×Q8—121no—
196C3×D4—121no—
197C3×D4—121no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—121no—
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×Dic3—31no—
206C2×Dic3—31no—
207C2×Dic3—31no—
208C2×Dic3—31no—
209C2×Dic3—41no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—122no—
213C2×Dic3—121no—
214C2×Dic3—121no—
215D12—41no—
216D12—41no—
217Dic6—41no—
218Dic6—41no—
219C3×C8—21no—
220C3×C8—21no—
221C3×C8#23711yescharacteristic
222C3×C8#23711yescharacteristic
223C3×C8C2×#6711yescharacteristic
224C3×C8C2×#6711yescharacteristic
225C3⋊C8—61no—
226C3⋊C8—61no—
227C3⋊C8—121no—
228C2×C32#45911yescharacteristic
229C2×C32#50811yescharacteristic
230C2×C32#51011yescharacteristic
231C3×D3—161no—
232C2×Q8—21no—
233C2×Q8—61no—
234C2×D4—61no—
235Q16—21no—
236Q16—61no—
237Q16—61no—
238Q16#62512yes—
239SD16—362no—
240D8—61no—
241D8—61no—
242C2×C8—91no—
243C2×C8C2×D3211yescharacteristic
244C4⋊C4—181no—
245C4⋊C4—361no—
246C22⋊C4—361no—
247C22⋊C4—361no—
248C42—91no—
249C42—361no—
250C16—61no—
251C16—61no—
252C22×C3—21no—
253C22×C3—241no—
254C22×C3—241no—
255C22×C3—241no—
256C22×C3D3×D811yescharacteristic
257C22×C3D3×D811yescharacteristic
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—81no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—44no—
264C3×C4—42no—
265C3×C4—42no—
266C3×C4—241no—
267C3×C4#91111yescharacteristic
268C3×C4#91111yescharacteristic
269C3×C4#93911yescharacteristic
270C3×C4#93911yescharacteristic
271Dic3—62no—
272Dic3—61no—
273Dic3—61no—
274Dic3—61no—
275Dic3—61no—
276Dic3—81no—
277Dic3—121no—
278Dic3—242no—
279Dic3—241no—
280Dic3—241no—
281C32#221111yescharacteristic, complemented, Sylow
282C23—121no—
283Q8—22no—
284Q8—41no—
285Q8—121no—
286Q8—121no—
287D4—121no—
288D4—121no—
289C2×C4—41no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—121no—
293C2×C4—361no—
294C2×C4#381111yescharacteristic
295C8—181no—
296C8#380511yescharacteristic
297C8C2×#62511yescharacteristic
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—481no—
302C2×C3#491511yescharacteristic
303C2×C3#491611yescharacteristic
304C2×C3#491811yescharacteristic
305C2×C3#491911yescharacteristic
306C2×C3#515011yescharacteristic
307C2×C3#515011yescharacteristic
308D3—161no—
309C22—241no—
310C22—241no—
311C22—241no—
312C22#6392611yescharacteristic
313C4—42no—
314C4—181no—
315C4—181no—
316C4—241no—
317C4—721no—
318C4#6392711yescharacteristic
319C4#6398011yescharacteristic
320C3—21no—
321C3#7147811yescharacteristic, complemented
322C3#7149811yescharacteristic, complemented
323C2—481no—
324C2#1059104911yescharacteristic
325C2#1059105011yescharacteristic
326C2#1059148611yescharacteristic
subgroupG/H
centerC22#63926
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC22×C32C2×D8
prime radicalC2×C3×D24C22
hypercenterC2×Q16D32
hypocenterC32#2211
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. Q16×C3⋊Dic3C2semidirect
2. C2×C3×#829C2semidirect
5. #10589133C2semidirect
6. #10589118C2semidirect
29. C2×#829D3semidirect
281. C32#2211semidirect
321. C3#71478semidirect
322. C3#71498semidirect

Maximal subgroups

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

subgroupindex
Q16×C3⋊Dic32
C2×C3×#8292
C2×C3×#8282
#105908692
#105891332
#105891182
#105891152
#714983
#714783