Finite Group Databasebeta

← #11781308

Subgroups — #11781308

Order 1152.

347 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 86 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×#4848C211yesmaximal, characteristic
2#10590310C211yesmaximal, characteristic, complemented
3#10590277C211yesmaximal, characteristic
4#10590250C211yesmaximal, characteristic, complemented
5#10589346C211yesmaximal, characteristic, complemented
6C3×#4481C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71022—31nomaximal
9#68510—31nomaximal
10C2×C3×#204—21no—
11C3×#874C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#821—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19#63576—21no—
20Dic3⋊(C3⋊C8)—21no—
21#63565—21no—
22#63562—21no—
23#63562—21no—
24Dic3×C3⋊C8—21no—
25C3×#460—31no—
26#5091—31no—
27#5088—31no—
28#5070—31no—
29Q8×C3⋊C8—31no—
30#5021—31no—
31#4848D311yescharacteristic
32#4830—31no—
33C3×#333—31no—
34#4544—31no—
35#4485—31no—
36#4481D311yescharacteristic, complemented
37#4480—31no—
38#4480—31no—
39#4480—61no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×Dic6—21no—
44C2×C3×Dic6D411yescharacteristic
45C2×C32×C8—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C3×Dic3⋊C4—21no—
54C3×C4×Dic3—21no—
55C2×C3×C3⋊C8—21no—
56C3×#204—42no—
57#1678—91noSylow
58C2×C3×SD16—61no—
59C3×C4⋊Q8—31no—
60C3×#127—31no—
61C3×C4×Q8—31no—
62C2×#204—21no—
63#874C2×D311yescharacteristic
64#869C2×D311yescharacteristic
65C4×Dic6C2×D311yescharacteristic
66C3×C4⋊C8—21no—
67C3×C4⋊C8—31no—
68C3×C4⋊C8C2×D311yescharacteristic
69C3×C4⋊C8C2×D311yescharacteristic
70C3×#102—61no—
71#835—61no—
72#832—61no—
73#832—61no—
74#821—21no—
75Dic3⋊C8—61no—
76C8×Dic3—61no—
77#810—61no—
78#809—61no—
79#809—61no—
80#807—61no—
81#807—61no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—61no—
85C4⋊(C3⋊C8)—61no—
86C4⋊(C3⋊C8)C2×D311yescharacteristic
87C4×C3⋊C8—31no—
88C4×C3⋊C8—31no—
89C4×C3⋊C8—61no—
90C4×C3⋊C8—61no—
91C22×C3×D3—41no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4#4111yescharacteristic
95C2×C3×Dic3—21no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—41no—
98C3×D12—41no—
99C3×D12—41no—
100C3×D12—41no—
101C3×D12—41no—
102C3×D12—81no—
103C3×Dic6—21no—
104C3×Dic6—81no—
105C3×Dic6SD1612yes—
106C32×C8—41no—
107C3⋊(C3⋊C8)—21no—
108C3⋊(C3⋊C8)—21no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—41no—
111#486—91no—
112#477—91no—
113#460—31no—
114C8×Q8—91no—
115#334—91no—
116#333—31no—
117#331—91no—
118C2×C3×Q8—31no—
119C2×C3×Q8—61no—
120C2×C3×D4—61no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×Dic6—21no—
127C2×Dic6#6711yescharacteristic
128C3×SD16—122no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8—61no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C42—21no—
138C3×C42—61no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3#6711yescharacteristic
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—121no—
156#204—42no—
157C22×C32#12111yescharacteristic
158C2×C3×D3—81no—
159C2×C3×D3—81no—
160C2×C3×D3—81no—
161C32×C4—21no—
162C32×C4C2×SD1611yescharacteristic
163C32×C4#13511yescharacteristic
164C32×C4#13511yescharacteristic
165C32×C4#13711yescharacteristic
166C3×Dic3—22no—
167C3×Dic3—41no—
168C3×Dic3—81no—
169C2×SD16—61no—
170C4⋊Q8—31no—
171#127—31no—
172C4×Q8—31no—
173#107—181no—
174C4⋊C8—31no—
175C4⋊C8—91no—
176C4⋊C8—181no—
177C4⋊C8D3211yescharacteristic
178#103—181no—
179#102—61no—
180#102—181no—
181#102—181no—
182C4×C8—91no—
183C4×C8—181no—
184C23×C3—121no—
185C22×D3—41no—
186C3×Q8—32no—
187C3×Q8—61no—
188C3×Q8—241no—
189C3×D4—121no—
190C3×D4—121no—
191C3×D4—121no—
192C3×D4—121no—
193C3×D4—241no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—121no—
200C2×C3×C4#23511yescharacteristic
201C2×C3×C4D3×D411yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4#23711yescharacteristic
205C2×C3×C4C2×#6711yescharacteristic
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—41no—
209D12—41no—
210D12—41no—
211D12—41no—
212D12—41no—
213D12—81no—
214Dic6—21no—
215Dic6—81no—
216Dic6#21412yes—
217C3×C8—41no—
218C3×C8—41no—
219C3×C8—81no—
220C3×C8—121no—
221C3⋊C8—41no—
222C3⋊C8—62no—
223C3⋊C8—61no—
224C3⋊C8—61no—
225C3⋊C8—61no—
226C3⋊C8—61no—
227C3⋊C8—121no—
228C3⋊C8—121no—
229C3⋊C8—121no—
230C3⋊C8—241no—
231C2×C32#46411yescharacteristic
232C2×C32#46511yescharacteristic
233C2×C32#47211yescharacteristic
234C3×D3—161no—
235C2×Q8—31no—
236C2×Q8—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240SD16—122no—
241C2×C8—21no—
242C2×C8—61no—
243C2×C8—91no—
244C2×C8—91no—
245C2×C8—181no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C42—61no—
251C42C2×D3211yescharacteristic
252C22×C3—21no—
253C22×C3—241no—
254C22×C3—241no—
255C22×C3—241no—
256C22×C3#88411yescharacteristic
257C22×C3#93911yescharacteristic
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—81no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—62no—
269C3×C4—121no—
270C3×C4—241no—
271C3×C4D3×SD1611yescharacteristic
272C3×C4#91511yescharacteristic
273C3×C4#91611yescharacteristic
274C3×C4#91611yescharacteristic
275C3×C4#91811yescharacteristic
276C3×C4#91911yescharacteristic
277C3×C4C2×#21411yescharacteristic
278C3×C4#95011yescharacteristic
279Dic3—22no—
280Dic3—41no—
281Dic3—81no—
282C32#167811yescharacteristic, complemented, Sylow
283C23—121no—
284Q8—32no—
285Q8—61no—
286Q8—241no—
287D4—121no—
288D4—121no—
289D4—121no—
290D4—121no—
291D4—241no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—121no—
295C2×C4#379511yescharacteristic
296C2×C4#380211yescharacteristic
297C2×C4D3×#6711yescharacteristic
298C8—41no—
299C8—121no—
300C8—181no—
301C8—181no—
302C8—361no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—481no—
307C2×C3#474911yescharacteristic
308C2×C3#479811yescharacteristic
309C2×C3#480711yescharacteristic
310C2×C3#504111yescharacteristic
311C2×C3#516611yescharacteristic
312C2×C3#519911yescharacteristic
313D3—161no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22#6388511yescharacteristic
318C4—21no—
319C4—62no—
320C4—121no—
321C4—241no—
322C4#6380911yescharacteristic
323C4#6381111yescharacteristic
324C4D3×#21411yescharacteristic
325C4#6394911yescharacteristic
326C3—21no—
327C3#6851011yescharacteristic, complemented
328C3#7102211yescharacteristic, complemented
329C2—481no—
330C2#1059051411yescharacteristic
331C2#1059079611yescharacteristic
332C2#1059098211yescharacteristic
subgroupG/H
centerC22#63885
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC4⋊C8D32
hypocenterC32#1678
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10590310C2semidirect
4. #10590250C2semidirect
5. #10589346C2semidirect
6. C3×#4481C2semidirect
36. #4481D3semidirect
282. C32#1678semidirect
327. C3#68510semidirect
328. C3#71022semidirect

Maximal subgroups

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

subgroupindex
C3×#48482
#105903102
#105902772
#105902502
#105893462
C3×#44812
#105890902
#710223
#685103