Finite Group Databasebeta

← #11799032

Subgroups — #11799032

Order 1152.

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

normal subgroups only

This group has 357 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×#63656C211yesmaximal, characteristic, complemented
2C3×Dic3×Q16C211yesmaximal, characteristic, complemented
3C2×C3×#828C211yesmaximal, characteristic
4#10590864C211yesmaximal, characteristic
5Dic3⋊(C3⋊C16)C211yesmaximal, characteristic, complemented
6#10589119C211yesmaximal, characteristic, complemented
7#10589114C211yesmaximal, characteristic
8#71478—31nomaximal
9#70915—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11C2×#3746C2211yescharacteristic
12C3×Q8×Dic3—21no—
13C2×C3×D24C2211yescharacteristic
14#63860—21no—
15C2×#3716—21no—
16#63656—22no—
17C2×C3⋊(C3⋊C16)C2211yescharacteristic
18C3×#835—21no—
19C3×#828—22no—
20C3×#818C2211yescharacteristic
21C3×C8×Dic3C2211yescharacteristic
22C2×C3×C3⋊C16C2211yescharacteristic
23C2×C3×SD32—31no—
24C3×C4×Q16—31no—
25Dic3×Q16D311yescharacteristic, complemented
26C2×#829—31no—
27C2×#829—31no—
28C2×#829—61no—
29C2×#828D311yescharacteristic
30#5155—31no—
31#4696—31no—
32#4562—31no—
33#4503—31no—
34Dic3⋊C16—31no—
35#4492—31no—
36#4491—31no—
37C4⋊(C3⋊C16)—31no—
38C2×C32×Q8—21no—
39C2×#634—21no—
40C2×C3×D12—21no—
41C32×Q16—21no—
42C32×Q16D412yes—
43C2×C32×C8C2311yescharacteristic
44#3746—21no—
45#3746—21no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3—41no—
48C3×C4×Dic3—41no—
49C3×C4×Dic3D411yescharacteristic
50C2×C3×C3⋊C8D411yescharacteristic
51C3×D24—21no—
52C3×D24—21no—
53(C2×D3)⋊Dic3—41no—
54(C2×D3)⋊Dic3—41no—
55#3716—42no—
56C3⋊(C3⋊C16)—21no—
57C3×C3⋊C16—21no—
58#2211—91noSylow
59C2×C3×Q16—21no—
60C2×C3×Q16C2×D311yescharacteristic
61C2×C3×Q16C2×D311yescharacteristic
62C2×C3×D8—31no—
63C3×C4×Q8—61no—
64Q8×Dic3—21no—
65#936—61no—
66C2×#214—61no—
67C2×Dic12—31no—
68C2×Dic12—31no—
69C2×Dic12—61no—
70C2×D24C2×D311yescharacteristic
71C2×#204—61no—
72#875—61no—
73C3×SD32—62no—
74C2×C3×C16—31no—
75C3×#107—31no—
76C3×#103—61no—
77C3×C4×C8—31no—
78#835—21no—
79#829—64no—
80#829—62no—
81#829—62no—
82#828—22no—
83#818C2×D311yescharacteristic
84C8×Dic3C2×D311yescharacteristic
85C2×C3⋊C16—31no—
86C2×C3⋊C16—31no—
87C2×C3⋊C16—61no—
88C2×C3⋊C16C2×D311yescharacteristic
89C22×C3×D3—41no—
90C32×Q8—22no—
91C32×Q8—41no—
92C2×C32×C4—41no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—41no—
95#634—41no—
96#634—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3D811yescharacteristic
99C2×C3×Dic3D811yescharacteristic
100C3×D12—41no—
101C3×D12—41no—
102C32×C8C2×D411yescharacteristic
103C32×C8#4111yescharacteristic
104C3×C3⋊C8—21no—
105C2×Q32—91no—
106C2×SD32—31no—
107#495—91no—
108C4×Q16—31no—
109C4⋊C16—91no—
110#358—91no—
111#357—91no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×D4—61no—
116C2×D12—21no—
117C2×Dic6—61no—
118C2×Dic6—61no—
119C2×Dic6—121no—
120C3×Q16—21no—
121C3×Q16—21no—
122C3×Q16—22no—
123C3×Q16—41no—
124C3×Q16D1212yes—
125C3×Q16#6712yes—
126C3×D8—61no—
127C3×D8—61no—
128C2×C3×C8—21no—
129C2×C3×C8—31no—
130C2×C3×C8C22×D311yescharacteristic
131C2×C3×C8C22×D311yescharacteristic
132C3×C4⋊C4—61no—
133C3×C4⋊C4—121no—
134C3×C42—31no—
135C3×C42—121no—
136#217—121no—
137#217—121no—
138#214—122no—
139(C2×D3)⋊C4—121no—
140(C2×D3)⋊C4—121no—
141C4⋊Dic3—21no—
142C4⋊Dic3—41no—
143C4×Dic3—41no—
144C4×Dic3#6711yescharacteristic
145C2×C3⋊C8#6711yescharacteristic
146Dic12—62no—
147Dic12—61no—
148Dic12—61no—
149Dic12—61no—
150Dic12—61no—
151Dic12—121no—
152D24—21no—
153D24—21no—
154#204—122no—
155C3×C16—61no—
156C3⋊C16—21no—
157C3⋊C16—62no—
158C3⋊C16—61no—
159C3⋊C16—61no—
160C22×C32C2×D811yescharacteristic
161C2×C3×D3—81no—
162C2×C3×D3—81no—
163C2×C3×D3—81no—
164C32×C4—42no—
165C32×C4#12111yescharacteristic
166C32×C4AGL(1,ℤ/8)11yescharacteristic
167C3⋊Dic3—81no—
168C3×Dic3—21no—
169C3×Dic3—21no—
170C3×Dic3—81no—
171C2×Q16—91no—
172C2×Q16D3211yescharacteristic
173C2×SD16—181no—
174C2×D8—31no—
175#124—181no—
176C4×Q8—61no—
177Q32—182no—
178SD32—62no—
179C2×C16—31no—
180C2×C16—91no—
181#107—31no—
182#103—61no—
183C4×C8—31no—
184C23×C3—121no—
185C22×D3—41no—
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—42no—
189C3×Q8—41no—
190C3×Q8—41no—
191C3×Q8—81no—
192C3×D4—121no—
193C3×D4—121no—
194C2×C3×C4—21no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—81no—
200C2×C3×C4—121no—
201C2×C3×C4D3×D411yescharacteristic
202C2×C3×C4C2×#6711yescharacteristic
203C2×Dic3—41no—
204C2×Dic3—121no—
205C2×Dic3—121no—
206C2×Dic3—241no—
207C2×Dic3#21311yescharacteristic
208C2×Dic3#21311yescharacteristic
209D12—41no—
210D12—41no—
211Dic6—122no—
212Dic6—121no—
213Dic6—121no—
214Dic6—121no—
215Dic6—121no—
216Dic6—241no—
217C3×C8—21no—
218C3×C8—21no—
219C3×C8—61no—
220C3×C8C2×D1211yescharacteristic
221C3×C8#23511yescharacteristic
222C3×C8#23711yescharacteristic
223C3×C8C2×#6711yescharacteristic
224C3⋊C8—21no—
225C2×C32#45911yescharacteristic
226C2×C32#50811yescharacteristic
227C2×C32#51011yescharacteristic
228C3×D3—161no—
229C2×Q8—21no—
230C2×Q8—181no—
231C2×D4—61no—
232Q16—21no—
233Q16—181no—
234Q16—181no—
235Q16#62612yes—
236SD16—362no—
237D8—61no—
238D8—61no—
239C2×C8—31no—
240C2×C8C2×D3211yescharacteristic
241C4⋊C4—61no—
242C4⋊C4—121no—
243C22⋊C4—361no—
244C22⋊C4—361no—
245C42—31no—
246C42—121no—
247C16—61no—
248C16—181no—
249C22×C3—21no—
250C22×C3—241no—
251C22×C3—241no—
252C22×C3—241no—
253C22×C3D3×D811yescharacteristic
254C22×C3C2×#21311yescharacteristic
255C2×D3—81no—
256C2×D3—81no—
257C2×D3—81no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—42no—
261C3×C4—42no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—82no—
265C3×C4—241no—
266C3×C4#91111yescharacteristic
267C3×C4#93011yescharacteristic
268C3×C4#93911yescharacteristic
269C3×C4#94911yescharacteristic
270Dic3—21no—
271Dic3—21no—
272Dic3—81no—
273Dic3—242no—
274Dic3—241no—
275Dic3—241no—
276C32#221111yescharacteristic, complemented, Sylow
277C23—121no—
278Q8—22no—
279Q8—41no—
280Q8—361no—
281Q8—361no—
282D4—121no—
283D4—121no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—41no—
287C2×C4—121no—
288C2×C4—361no—
289C2×C4D3×#6711yescharacteristic
290C8—61no—
291C8#380411yescharacteristic
292C8C2×#62611yescharacteristic
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—481no—
297C2×C3#491511yescharacteristic
298C2×C3#491611yescharacteristic
299C2×C3#501511yescharacteristic
300C2×C3#515011yescharacteristic
301C2×C3#517911yescharacteristic
302C2×C3#519411yescharacteristic
303D3—161no—
304C22—241no—
305C22—241no—
306C22—241no—
307C22D3×#21311yescharacteristic
308C4—42no—
309C4—61no—
310C4—61no—
311C4—241no—
312C4—721no—
313C4#6392511yescharacteristic
314C4#6397811yescharacteristic
315C3—21no—
316C3#7091511yescharacteristic, complemented
317C3#7147811yescharacteristic, complemented
318C2—481no—
319C2#1059104511yescharacteristic
320C2#1059104611yescharacteristic
321C2#1059148111yescharacteristic
subgroupG/H
centerC22D3×#213
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. C2×#63656C2semidirect
2. C3×Dic3×Q16C2semidirect
5. Dic3⋊(C3⋊C16)C2semidirect
6. #10589119C2semidirect
25. Dic3×Q16D3semidirect
276. C32#2211semidirect
316. C3#70915semidirect
317. C3#71478semidirect

Maximal subgroups

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

subgroupindex
C2×#636562
C3×Dic3×Q162
C2×C3×#8282
#105908642
Dic3⋊(C3⋊C16)2
#105891192
#105891142
#714783
#709153