Finite Group Databasebeta

← #11782004

Subgroups — #11782004

Order 1152.

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

normal subgroups only

This group has 335 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×#4842C211yesmaximal, characteristic
2#10590339C211yesmaximal, characteristic
3#10590312C211yesmaximal, characteristic, complemented
4#10590266C211yesmaximal, characteristic
5#10589346C211yesmaximal, characteristic, complemented
6C3×#4481C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71137—31nomaximal
9#68501—31nomaximal
10C3×C4×D12C2211yescharacteristic
11C3×#869C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3711—21no—
14C32×C4⋊C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#821—21no—
18C3×#817—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20#63576—21no—
21#63576—21no—
22#63574—21no—
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#476—31no—
26#5084—31no—
27#5078—31no—
28#5078—31no—
29#5012—31no—
30#5008—31no—
31#4842D311yescharacteristic
32#4838—31no—
33C3×#333—31no—
34#4546—31no—
35#4481—31no—
36#4481D311yescharacteristic, complemented
37#4480—31no—
38#4480—31no—
39#4480—61no—
40C2×C3×D12D411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46C2×C3⋊(C3⋊C8)—21no—
47C2×C3⋊(C3⋊C8)D411yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3D411yescharacteristic
55C2×C3×C3⋊C8—21no—
56#3711—42no—
57#1695—91noSylow
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8—31no—
60C3×C4×D4—31no—
61C2×#208—61no—
62C2×ΓL(1,𝔽25)—61no—
63C4×D12C2×D311yescharacteristic
64#869C2×D311yescharacteristic
65#869C2×D311yescharacteristic
66C3×#106—61no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—31no—
69C3×C4⋊C8C2×D311yescharacteristic
70C3×C4⋊C8C2×D311yescharacteristic
71C3×#102—61no—
72#832—61no—
73#830—61no—
74#821—21no—
75(C2×D3)⋊C8—61no—
76#817—21no—
77#809—61no—
78#808—61no—
79#808—61no—
80#807—61no—
81#807—61no—
82#807—61no—
83#807—61no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—61no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88C4×C3⋊C8—31no—
89C4×C3⋊C8—31no—
90C4×C3⋊C8—61no—
91C22×C3×D3—21no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4#4111yescharacteristic
95C2×C3×Dic3—21no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C3×D12—21no—
99C3×D12—21no—
100C3×C4×D3—41no—
101C3×C4×D3—41no—
102C3×Dic6—81no—
103C3×Dic6—81no—
104C32×C8—41no—
105C3⋊(C3⋊C8)—21no—
106C3⋊(C3⋊C8)—41no—
107C3⋊(C3⋊C8)SD1612yes—
108C3×C3⋊C8—41no—
109#500—91no—
110#476—31no—
111#476—91no—
112#436—91no—
113#334—91no—
114#333—31no—
115#333—91no—
116C2×C3×Q8—61no—
117C2×C3×Q8—61no—
118C2×C3×D4—31no—
119C22×C3×C4—61no—
120C2×D12#6711yescharacteristic
121C2×C4×D3—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—41no—
127C2×C3×C8—61no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—61no—
133C3×C22⋊C4—61no—
134C3×C42—21no—
135C3×C42C22×D311yescharacteristic
136C3×C42C22×D311yescharacteristic
137(C2×D3)⋊C4—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3#6711yescharacteristic
143#208—122no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—31no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—61no—
150C2×C3⋊C8—61no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—121no—
154ΓL(1,𝔽25)—122no—
155C22×C32#12111yescharacteristic
156C2×C3×D3—41no—
157C2×C3×D3—41no—
158C2×C3×D3—41no—
159C32×C4—21no—
160C32×C4C2×SD1611yescharacteristic
161C32×C4#13711yescharacteristic
162C32×C4#13711yescharacteristic
163C32×C4#13711yescharacteristic
164C3×Dic3—41no—
165C3×Dic3—81no—
166C3×Dic3—81no—
167C2×M16—181no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170C4×D4—31no—
171#107—181no—
172#107—181no—
173#106—61no—
174#106—181no—
175C4⋊C8—31no—
176C4⋊C8—91no—
177C4⋊C8D3211yescharacteristic
178#102—61no—
179#102—181no—
180C22⋊C8—181no—
181C4×C8—91no—
182C23×C3—61no—
183C22×D3—21no—
184C3×Q8—241no—
185C3×Q8—241no—
186C3×D4—61no—
187C3×D4—61no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—61no—
192C2×C3×C4—121no—
193C2×C3×C4—121no—
194C2×C3×C4—121no—
195C2×C3×C4—121no—
196C2×C3×C4#23511yescharacteristic
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4D3×D411yescharacteristic
199C2×C3×C4D3×D411yescharacteristic
200C2×C3×C4#23711yescharacteristic
201C2×C3×C4C2×#6711yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205D12—21no—
206D12—21no—
207C4×D3—41no—
208C4×D3—41no—
209Dic6—81no—
210Dic6—81no—
211C3×C8—41no—
212C3×C8—41no—
213C3×C8—81no—
214C3×C8—121no—
215C3⋊C8—32no—
216C3⋊C8—32no—
217C3⋊C8—41no—
218C3⋊C8—62no—
219C3⋊C8—61no—
220C3⋊C8—61no—
221C3⋊C8—121no—
222C3⋊C8—121no—
223C3⋊C8—121no—
224C3⋊C8—241no—
225C2×C32#46111yescharacteristic
226C2×C32#47011yescharacteristic
227C2×C32#47211yescharacteristic
228C3×D3—81no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×D4—31no—
232C22×C4—61no—
233M16—362no—
234C2×C8—21no—
235C2×C8—61no—
236C2×C8—91no—
237C2×C8—91no—
238C2×C8—181no—
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C22⋊C4—61no—
245C42C2×D3211yescharacteristic
246C22×C3—21no—
247C22×C3—121no—
248C22×C3—121no—
249C22×C3—121no—
250C22×C3#88411yescharacteristic
251C22×C3#93911yescharacteristic
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—41no—
262C3×C4—121no—
263C3×C4—241no—
264C3×C4—241no—
265C3×C4D3×SD1611yescharacteristic
266C3×C4D3×SD1611yescharacteristic
267C3×C4#91511yescharacteristic
268C3×C4#91511yescharacteristic
269C3×C4#91811yescharacteristic
270C3×C4#91811yescharacteristic
271C3×C4#94211yescharacteristic
272C3×C4#95111yescharacteristic
273Dic3—41no—
274Dic3—81no—
275Dic3—81no—
276C32#169511yescharacteristic, complemented, Sylow
277C23—61no—
278Q8—241no—
279Q8—241no—
280D4—61no—
281D4—61no—
282C2×C4—61no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4#379511yescharacteristic
288C2×C4#380211yescharacteristic
289C2×C4D3×#6711yescharacteristic
290C8—41no—
291C8—92no—
292C8—121no—
293C8—181no—
294C8—361no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—241no—
299C2×C3#474911yescharacteristic
300C2×C3#481611yescharacteristic
301C2×C3#481811yescharacteristic
302C2×C3#505211yescharacteristic
303C2×C3#516211yescharacteristic
304C2×C3#519911yescharacteristic
305D3—81no—
306C22—121no—
307C22—121no—
308C22—121no—
309C22#6388511yescharacteristic
310C4—21no—
311C4—121no—
312C4—241no—
313C4—241no—
314C4#6379611yescharacteristic
315C4#6381111yescharacteristic
316C4#6392911yescharacteristic
317C4#6394811yescharacteristic
318C3—21no—
319C3#6850111yescharacteristic, complemented
320C3#7113711yescharacteristic, complemented
321C2—241no—
322C2#1059052011yescharacteristic
323C2#1059074811yescharacteristic
324C2#1059098211yescharacteristic
subgroupG/H
centerC22#63885
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊C8D32
hypocenterC32#1695
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
3. #10590312C2semidirect
5. #10589346C2semidirect
6. C3×#4481C2semidirect
7. #10589095C2semidirect
36. #4481D3semidirect
276. C32#1695semidirect
319. C3#68501semidirect
320. C3#71137semidirect

Maximal subgroups

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

subgroupindex
C3×#48422
#105903392
#105903122
#105902662
#105893462
C3×#44812
#105890952
#711373
#685013