Finite Group Databasebeta

← #11782031

Subgroups — #11782031

Order 1152.

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

normal subgroups only

This group has 312 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×#5091C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic
3#10590310C211yesmaximal, characteristic
4#10590258C211yesmaximal, characteristic
5#10589412C211yesmaximal, characteristic
6#10589097C211yesmaximal, characteristic
7#10589095C211yesmaximal, characteristic
8#71380—31nomaximal
9#71153—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C3×#869C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#810—21no—
16C3×#807—21no—
17C3×#807—21no—
18C3×#807—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63576—21no—
22Dic3⋊(C3⋊C8)—21no—
23#63565—21no—
24#63554—21no—
25C3×#500—31no—
26C3×#477—31no—
27#5091—31no—
28#5091D311yescharacteristic
29#5078D311yescharacteristic
30#5024—31no—
31#5021—31no—
32#4679—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×Dic6—21no—
42C2×C3×Dic6D411yescharacteristic
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C42C2311yescharacteristic
46C2×C3⋊(C3⋊C8)—21no—
47C2×C3⋊(C3⋊C8)—21no—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3D411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#1696—91noSylow
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4×Q8—31no—
62#869C2×D311yescharacteristic
63C4×Dic6C2×D311yescharacteristic
64C3×#107—61no—
65C3×#107—61no—
66C3×#107—61no—
67C3×C4⋊C8—31no—
68C3×#103—61no—
69C3×C4×C8—31no—
70#835—61no—
71#815—61no—
72Dic3⋊C8—61no—
73#810—21no—
74#810—61no—
75#807—21no—
76#807—21no—
77#807—21no—
78#807—61no—
79#807—61no—
80C4⋊(C3⋊C8)—31no—
81C4⋊(C3⋊C8)—31no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—61no—
85C4⋊(C3⋊C8)—61no—
86C4⋊(C3⋊C8)—61no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88#803—61no—
89C4×C3⋊C8C2×D311yescharacteristic
90C32×Q8—81no—
91C2×C32×C4—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×Dic6—21no—
99C3×Dic6—21no—
100C3×Dic6—81no—
101C3⋊(C3⋊C8)—41no—
102C3⋊(C3⋊C8)—41no—
103C3×C3⋊C8—21no—
104C3×C3⋊C8—41no—
105C3×C3⋊C8Q1612yes—
106#500—31no—
107#477—31no—
108#477—91no—
109#446—91no—
110#334—91no—
111#333—91no—
112#333—91no—
113C2×C3×Q8—22no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—31no—
117C2×C3×Q8—61no—
118C2×Dic6—21no—
119C2×Dic6#6711yescharacteristic
120C2×C3×C8—31no—
121C2×C3×C8—31no—
122C2×C3×C8—61no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—61no—
133C3×C42—21no—
134C3×C42—61no—
135C3×C42C22×D311yescharacteristic
136C3×C42C22×D311yescharacteristic
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3#6711yescharacteristic
140Dic3⋊C4—21no—
141C4×Dic3—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—61no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8—61no—
146C2×C3⋊C8—61no—
147C2×C3⋊C8—121no—
148C2×C3⋊C8—121no—
149C2×C3⋊C8#6711yescharacteristic
150C2×C3⋊C8#6711yescharacteristic
151C22×C32#12111yescharacteristic
152C32×C4—21no—
153C32×C4—81no—
154C32×C4C2×Q1611yescharacteristic
155C32×C4#13711yescharacteristic
156C32×C4#13711yescharacteristic
157C32×C4#13711yescharacteristic
158C3×Dic3—41no—
159C3×Dic3—41no—
160C3×Dic3—81no—
161C4⋊Q8—31no—
162C4⋊Q8D3211yescharacteristic
163C4×Q8—31no—
164#107—61no—
165#107—61no—
166#107—61no—
167#107—181no—
168C4⋊C8—31no—
169C4⋊C8—91no—
170C4⋊C8—91no—
171C4⋊C8—181no—
172#103—61no—
173#103—181no—
174#97—181no—
175C4×C8—31no—
176C3×Q8—61no—
177C3×Q8—61no—
178C3×Q8—81no—
179C3×Q8—81no—
180C3×Q8—82no—
181C3×Q8—241no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—121no—
191C2×C3×C4D3×D411yescharacteristic
192C2×C3×C4D3×D411yescharacteristic
193C2×C3×C4#23711yescharacteristic
194C2×C3×C4#23711yescharacteristic
195C2×C3×C4C2×#6711yescharacteristic
196C2×C3×C4C2×#6711yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—41no—
200Dic6—21no—
201Dic6—21no—
202Dic6—81no—
203C3×C8—32no—
204C3×C8—61no—
205C3×C8—121no—
206C3⋊C8—21no—
207C3⋊C8—41no—
208C3⋊C8—121no—
209C3⋊C8—121no—
210C3⋊C8—121no—
211C3⋊C8—121no—
212C3⋊C8—241no—
213C3⋊C8—241no—
214C3⋊C8#21612yes—
215C2×C32#46211yescharacteristic
216C2×C32#47011yescharacteristic
217C2×C32#47211yescharacteristic
218C2×Q8—21no—
219C2×Q8—31no—
220C2×Q8—61no—
221C2×C8—31no—
222C2×C8—31no—
223C2×C8—61no—
224C2×C8—181no—
225C2×C8—181no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—31no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C42—61no—
233C42C2×D3211yescharacteristic
234C22×C3—21no—
235C22×C3#93811yescharacteristic
236C22×C3#93911yescharacteristic
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—41no—
244C3×C4—82no—
245C3×C4—81no—
246C3×C4—81no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—241no—
250C3×C4D3×Q1611yescharacteristic
251C3×C4#91811yescharacteristic
252C3×C4#91811yescharacteristic
253C3×C4#91811yescharacteristic
254C3×C4#94211yescharacteristic
255C3×C4#94211yescharacteristic
256C3×C4C2×#21611yescharacteristic
257C3×C4#95111yescharacteristic
258Dic3—41no—
259Dic3—41no—
260Dic3—81no—
261C32#169611yescharacteristic, complemented, Sylow
262Q8—61no—
263Q8—61no—
264Q8—81no—
265Q8—241no—
266C2×C4—41no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—121no—
270C2×C4#380511yescharacteristic
271C2×C4D3×#6711yescharacteristic
272C2×C4D3×#6711yescharacteristic
273C8—32no—
274C8—61no—
275C8—121no—
276C8—361no—
277C8—361no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3#505211yescharacteristic
282C2×C3#509011yescharacteristic
283C2×C3#516011yescharacteristic
284C2×C3#518211yescharacteristic
285C2×C3#518911yescharacteristic
286C2×C3#519911yescharacteristic
287C22#6397011yescharacteristic
288C4—21no—
289C4—81no—
290C4—121no—
291C4—121no—
292C4—241no—
293C4D3×#21611yescharacteristic
294C4#6394311yescharacteristic
295C4#6394811yescharacteristic
296C4#6394811yescharacteristic
297C3—21no—
298C3#7115311yescharacteristic, complemented
299C3#7138011yescharacteristic, complemented
300C2#1059150511yescharacteristic
301C2#1059157411yescharacteristic
302C2#1059163811yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socle, prime radicalC22×C32#121
hypercenterC4⋊Q8D32
hypocenterC32#1696
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
261. C32#1696semidirect
298. C3#71153semidirect
299. C3#71380semidirect

Maximal subgroups

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

subgroupindex
C3×#50912
C3×#50782
#105903102
#105902582
#105894122
#105890972
#105890952
#713803
#711533