Finite Group Databasebeta

← #11781998

Subgroups — #11781998

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×#5084C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3#10590337C211yesmaximal, characteristic
4#10590266C211yesmaximal, characteristic
5#10589412C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71374—31nomaximal
9#71137—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C3×C4×D12C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3711—21no—
14C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#809—21no—
17C3×#808—21no—
18C3×#807—21no—
19C3×#807—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63574—21no—
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#500—31no—
26C3×#476—31no—
27#5084—31no—
28#5084D311yescharacteristic
29#5078D311yescharacteristic, complemented
30#5012—31no—
31#5008—31no—
32#4688—31no—
33#4546—31no—
34#4546—31no—
35#4546—31no—
36#4546—61no—
37#4481—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—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—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4×D4—31no—
63C2×#208—61no—
64C2×ΓL(1,𝔽25)—61no—
65C4×D12C2×D311yescharacteristic
66#869C2×D311yescharacteristic
67C3×#107—61no—
68C3×#107—61no—
69C3×#106—61no—
70C3×C4⋊C8—31no—
71C3×#102—61no—
72C3×C4×C8—31no—
73#832—61no—
74#830—61no—
75(C2×D3)⋊C8—61no—
76#809—21no—
77#809—61no—
78#808—21no—
79#808—61no—
80#808—61no—
81#807—21no—
82#807—21no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—61no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4×C3⋊C8C2×D311yescharacteristic
91C22×C3×D3—21no—
92C32×Q8—81no—
93C2×C32×C4—41no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
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—
104C3⋊(C3⋊C8)—41no—
105C3⋊(C3⋊C8)—41no—
106C3×C3⋊C8—21no—
107C3×C3⋊C8—41no—
108C3×C3⋊C8SD1612yes—
109#500—31no—
110#476—31no—
111#476—91no—
112#436—91no—
113#334—91no—
114#333—91no—
115#333—91no—
116C2×C3×Q8—22no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—61no—
120C2×C3×D4—31no—
121C22×C3×C4—61no—
122C2×D12#6711yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6—21no—
125C2×C3×C8—31no—
126C2×C3×C8—31no—
127C2×C3×C8—61no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C22⋊C4—61no—
138C3×C42—21no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141(C2×D3)⋊C4—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3#6711yescharacteristic
145#208—122no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—61no—
148C2×C3⋊C8—61no—
149C2×C3⋊C8—61no—
150C2×C3⋊C8—61no—
151C2×C3⋊C8—121no—
152C2×C3⋊C8—121no—
153C2×C3⋊C8#6711yescharacteristic
154C2×C3⋊C8#6711yescharacteristic
155ΓL(1,𝔽25)—122no—
156C22×C32#12111yescharacteristic
157C2×C3×D3—41no—
158C2×C3×D3—41no—
159C2×C3×D3—41no—
160C32×C4—21no—
161C32×C4—81no—
162C32×C4C2×SD1611yescharacteristic
163C32×C4#13711yescharacteristic
164C32×C4#13711yescharacteristic
165C32×C4#13711yescharacteristic
166C3×Dic3—41no—
167C3×Dic3—81no—
168C2×M16—181no—
169C4⋊Q8—31no—
170C4⋊Q8D3211yescharacteristic
171C4×D4—31no—
172#107—61no—
173#107—61no—
174#106—61no—
175#106—181no—
176C4⋊C8—31no—
177C4⋊C8—91no—
178C4⋊C8—91no—
179#102—61no—
180#102—181no—
181C22⋊C8—181no—
182C4×C8—31no—
183C23×C3—61no—
184C22×D3—21no—
185C3×Q8—81no—
186C3×Q8—81no—
187C3×Q8—82no—
188C3×Q8—241no—
189C3×D4—61no—
190C3×D4—61no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—61no—
198C2×C3×C4—121no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4D3×D411yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4#23711yescharacteristic
204C2×C3×C4#23711yescharacteristic
205C2×C3×C4C2×#6711yescharacteristic
206C2×C3×C4C2×#6711yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—41no—
209D12—21no—
210D12—21no—
211C4×D3—41no—
212C4×D3—41no—
213Dic6—81no—
214C3×C8—32no—
215C3×C8—61no—
216C3×C8—121no—
217C3⋊C8—21no—
218C3⋊C8—41no—
219C3⋊C8—121no—
220C3⋊C8—121no—
221C3⋊C8—121no—
222C3⋊C8—121no—
223C3⋊C8—241no—
224C3⋊C8—241no—
225C3⋊C8#21512yes—
226C2×C32#46111yescharacteristic
227C2×C32#47011yescharacteristic
228C2×C32#47211yescharacteristic
229C3×D3—81no—
230C2×Q8—21no—
231C2×Q8—61no—
232C2×D4—31no—
233C22×C4—61no—
234M16—362no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—61no—
238C2×C8—181no—
239C2×C8—181no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C22⋊C4—61no—
246C42C2×D3211yescharacteristic
247C22×C3—21no—
248C22×C3—121no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3#93811yescharacteristic
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—82no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4—121no—
267C3×C4—241no—
268C3×C4D3×SD1611yescharacteristic
269C3×C4#91511yescharacteristic
270C3×C4#91811yescharacteristic
271C3×C4#91811yescharacteristic
272C3×C4C2×#21511yescharacteristic
273C3×C4#94211yescharacteristic
274C3×C4#94211yescharacteristic
275C3×C4#95111yescharacteristic
276Dic3—41no—
277Dic3—81no—
278C32#169511yescharacteristic, complemented, Sylow
279C23—61no—
280Q8—81no—
281Q8—241no—
282D4—61no—
283D4—61no—
284C2×C4—41no—
285C2×C4—61no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4#380511yescharacteristic
290C2×C4D3×#6711yescharacteristic
291C2×C4D3×#6711yescharacteristic
292C8—32no—
293C8—61no—
294C8—121no—
295C8—361no—
296C8—361no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—241no—
301C2×C3#505211yescharacteristic
302C2×C3#508311yescharacteristic
303C2×C3#516011yescharacteristic
304C2×C3#516211yescharacteristic
305C2×C3#518911yescharacteristic
306C2×C3#519911yescharacteristic
307D3—81no—
308C22—121no—
309C22—121no—
310C22—121no—
311C22#6397011yescharacteristic
312C4—21no—
313C4—81no—
314C4—121no—
315C4—241no—
316C4D3×#21511yescharacteristic
317C4#6394311yescharacteristic
318C4#6394611yescharacteristic
319C4#6394811yescharacteristic
320C3—21no—
321C3#7113711yescharacteristic, complemented
322C3#7137411yescharacteristic, complemented
323C2—241no—
324C2#1059150511yescharacteristic
325C2#1059156411yescharacteristic
326C2#1059163811yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1695
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5078C2semidirect
5. #10589412C2semidirect
6. #10589097C2semidirect
7. #10589095C2semidirect
29. #5078D3semidirect
278. C32#1695semidirect
321. C3#71137semidirect
322. C3#71374semidirect

Maximal subgroups

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

subgroupindex
C3×#50842
C3×#50782
#105903372
#105902662
#105894122
#105890972
#105890952
#713743
#711373