Finite Group Databasebeta

← #11782006

Subgroups — #11782006

Order 1152.

340 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 88 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#10590337C211yesmaximal, characteristic
3#10590311C211yesmaximal, characteristic, complemented
4#10590268C211yesmaximal, characteristic
5#10589347C211yesmaximal, characteristic, complemented
6C3×#4480C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71137—31nomaximal
9#68543—31nomaximal
10#64078C2211yescharacteristic
11C3×C4×D12C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3711—21no—
14C32×C4⋊C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#821—21no—
17C3×#817—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63575—21no—
21#63575—21no—
22#63573—21no—
23#63563—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#476—31no—
26#5078—31no—
27#5012—31no—
28#5008—31no—
29#4842—31no—
30#4842D311yescharacteristic
31#4688—31no—
32#4682—31no—
33C3×#334—31no—
34#4546—31no—
35#4481—31no—
36#4481—31no—
37#4481—31no—
38#4481—61no—
39#4480D311yescharacteristic, complemented
40C2×#634—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3711—42no—
57#1695—91noSylow
58C3×C4⋊Q8—31no—
59C3×C4×D4—31no—
60C2×#208—61no—
61C2×ΓL(1,𝔽25)—61no—
62C4×D12C2×D311yescharacteristic
63#869—31no—
64#869—31no—
65#869—61no—
66#869C2×D311yescharacteristic
67C3×#106—61no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—31no—
70C3×C4⋊C8C2×D311yescharacteristic
71C3×C4⋊C8C2×D311yescharacteristic
72C3×#102—61no—
73C3×C4×C8—31no—
74#832—61no—
75#830—61no—
76#821—21no—
77#821—61no—
78(C2×D3)⋊C8—61no—
79#818—61no—
80#818—61no—
81#817—21no—
82#817—61no—
83#808—61no—
84#807—61no—
85#807—61no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4×C3⋊C8C2×D311yescharacteristic
91C22×C3×D3—21no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4#4111yescharacteristic
95C2×C3⋊Dic3—41no—
96#634—81no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—41no—
99C3×D12—21no—
100C3×D12—21no—
101C3×C4×D3—41no—
102C3×C4×D3—41no—
103C3×Dic6—81no—
104C32×C8—41no—
105C3⋊(C3⋊C8)—41no—
106C3×C3⋊C8—21no—
107C3×C3⋊C8—41no—
108C3×C3⋊C8SD1612yes—
109#500—91no—
110#476—31no—
111#476—91no—
112#436—91no—
113#334—31no—
114#333—91no—
115#333—91no—
116C2×C3×Q8—61no—
117C2×C3×D4—31no—
118C22×C3×C4—61no—
119C2×D12#6711yescharacteristic
120C2×C4×D3—21no—
121C2×Dic6—21no—
122C2×Dic6—62no—
123C2×Dic6—61no—
124C2×Dic6—61no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—31no—
128C2×C3×C8—31no—
129C2×C3×C8—41no—
130C2×C3×C8—61no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C22⋊C4—61no—
135C3×C42—21no—
136C3×C42C22×D311yescharacteristic
137C3×C42C22×D311yescharacteristic
138(C2×D3)⋊C4—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—61no—
142C4⋊Dic3—61no—
143C4⋊Dic3—61no—
144C4⋊Dic3—61no—
145C4⋊Dic3—121no—
146C4⋊Dic3—121no—
147C4⋊Dic3#6711yescharacteristic
148#208—122no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—61no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—121no—
153C2×C3⋊C8D1211yescharacteristic
154C2×C3⋊C8D1211yescharacteristic
155ΓL(1,𝔽25)—122no—
156C22×C32#12111yescharacteristic
157C2×C3×D3—41no—
158C2×C3×D3—41no—
159C2×C3×D3—41no—
160C32×C4—21no—
161C32×C4C2×SD1611yescharacteristic
162C32×C4#13711yescharacteristic
163C32×C4#13711yescharacteristic
164C32×C4#13711yescharacteristic
165C3⋊Dic3—81no—
166C3×Dic3—41no—
167C3×Dic3—81no—
168C2×M16—181no—
169C4⋊Q8—31no—
170C4⋊Q8—91no—
171C4×D4—31no—
172#107—181no—
173#107—181no—
174#106—61no—
175#106—181no—
176C4⋊C8—31no—
177C4⋊C8—91no—
178C4⋊C8D3211yescharacteristic
179#102—61no—
180#102—181no—
181C22⋊C8—181no—
182C4×C8—31no—
183C23×C3—61no—
184C22×D3—21no—
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×C4C2×D1211yescharacteristic
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4#23711yescharacteristic
199C2×C3×C4#23911yescharacteristic
200C2×C3×C4C2×#6711yescharacteristic
201C2×Dic3—21no—
202C2×Dic3—41no—
203C2×Dic3—122no—
204C2×Dic3—121no—
205C2×Dic3—121no—
206D12—21no—
207D12—21no—
208C4×D3—41no—
209C4×D3—41no—
210Dic6—81no—
211Dic6—241no—
212Dic6—241no—
213Dic6—242no—
214C3×C8—32no—
215C3×C8—41no—
216C3×C8—41no—
217C3×C8—61no—
218C3×C8—81no—
219C3×C8—121no—
220C3⋊C8—21no—
221C3⋊C8—41no—
222C3⋊C8—121no—
223C3⋊C8—121no—
224C3⋊C8—241no—
225C3⋊C8#20412yes—
226C2×C32#46111yescharacteristic
227C2×C32#47011yescharacteristic
228C2×C32#47211yescharacteristic
229C3×D3—81no—
230C2×Q8—61no—
231C2×Q8—181no—
232C2×D4—31no—
233C22×C4—61no—
234M16—362no—
235C2×C8—21no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—61no—
239C2×C8—181no—
240C4⋊C4—31no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—181no—
244C4⋊C4—181no—
245C22⋊C4—61no—
246C42C2×D3211yescharacteristic
247C22×C3—21no—
248C22×C3—121no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3#89511yescharacteristic
252C22×C3#93911yescharacteristic
253C2×D3—41no—
254C2×D3—41no—
255C2×D3—41no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—121no—
264C3×C4—241no—
265C3×C4C2×#20411yescharacteristic
266C3×C4#90911yescharacteristic
267C3×C4D3×SD1611yescharacteristic
268C3×C4#91511yescharacteristic
269C3×C4#91511yescharacteristic
270C3×C4#91511yescharacteristic
271C3×C4#94211yescharacteristic
272C3×C4#95111yescharacteristic
273Dic3—41no—
274Dic3—81no—
275Dic3—242no—
276Dic3—241no—
277Dic3—241no—
278C32#169511yescharacteristic, complemented, Sylow
279C23—61no—
280Q8—241no—
281Q8—721no—
282D4—61no—
283D4—61no—
284C2×C4—61no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—361no—
289C2×C4#379611yescharacteristic
290C2×C4D3×D1211yescharacteristic
291C2×C4D3×#6711yescharacteristic
292C8—32no—
293C8—41no—
294C8—61no—
295C8—121no—
296C8—361no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—241no—
301C2×C3#484111yescharacteristic
302C2×C3#486711yescharacteristic
303C2×C3#489711yescharacteristic
304C2×C3#505211yescharacteristic
305C2×C3#516211yescharacteristic
306C2×C3#519911yescharacteristic
307D3—81no—
308C22—121no—
309C22—121no—
310C22—121no—
311C22#6388611yescharacteristic
312C4—21no—
313C4—121no—
314C4—241no—
315C4—721no—
316C4D3×#20411yescharacteristic
317C4#6380011yescharacteristic
318C4#6392911yescharacteristic
319C4#6394611yescharacteristic
320C3—21no—
321C3#6854311yescharacteristic, complemented
322C3#7113711yescharacteristic, complemented
323C2—241no—
324C2#1059058711yescharacteristic
325C2#1059075211yescharacteristic
326C2#1059098411yescharacteristic
subgroupG/H
centerC22#63886
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. #10590311C2semidirect
5. #10589347C2semidirect
6. C3×#4480C2semidirect
7. #10589095C2semidirect
39. #4480D3semidirect
278. C32#1695semidirect
321. C3#68543semidirect
322. C3#71137semidirect

Maximal subgroups

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

subgroupindex
C3×#48422
#105903372
#105903112
#105902682
#105893472
C3×#44802
#105890952
#711373
#685433