Finite Group Databasebeta

← #11776360

Subgroups — #11776360

Order 1152.

625 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 110 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⋊D3×C4⋊Q8C211yesmaximal, characteristic
2#10590340C211yesmaximal, characteristic
3#10590256C211yesmaximal, characteristic
4C3×#4546C212yesmaximal, complemented
5#10589096C212yesmaximal, complemented
6#68483—32nomaximal
7C2×Q8×C3⋊D3C412yes—
8C32×C4⋊Q8C2211yescharacteristic
9#64153C412yes—
10C4⋊C4×C3⋊D3—21no—
11C4⋊C4×C3⋊D3—21no—
12#64097—21no—
13#64097—21no—
14C42×C3⋊D3C2211yescharacteristic
15#64078C2211yescharacteristic
16C2×#3712—21no—
17C3×C4⋊(C3⋊C8)C2212yes—
18C3×C4⋊(C3⋊C8)C2212yes—
19#63572—21no—
20#63557—21no—
21#63553—21no—
22D3×C4⋊Q8—32no—
23D3×C4⋊Q8—61no—
24#4837—32no—
25#4833—32no—
26C3×#333—32no—
27#4546D312yescomplemented
28#4481—32no—
29#4481—32no—
30C2×C32×Q8C2×C412yes—
31Q8×C3⋊D3—44no—
32C2×C4×C3⋊D3—22no—
33C2×C4×C3⋊D3C2×C411yescharacteristic
34C2×C4×C3⋊D3D411yescharacteristic
35C2×C4×C3⋊D3D411yescharacteristic
36C2×#634—22no—
37C2×#634C2×C412yes—
38C32×C4⋊C4—21no—
39C32×C4⋊C4—21no—
40C32×C42C2311yescharacteristic
41C4⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊Dic3)—21no—
43#3750—22no—
44#3750—22no—
45C4×C3⋊Dic3C2×C411yescharacteristic
46C4×C3⋊Dic3D411yescharacteristic
47C4×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊C8—22no—
49C2×C3×C3⋊C8—22no—
50#3712—42no—
51#1548—91noSylow
52C2×D3×Q8—34no—
53C2×D3×Q8—62no—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8C2×D312yes—
56#944—34no—
57#944—62no—
58C2×ΓL(1,𝔽25)—62no—
59D3×C4⋊C4—62no—
60D3×C4⋊C4—62no—
61D3×C4⋊C4—62no—
62D3×C4⋊C4—62no—
63#888—62no—
64#888—62no—
65#888—62no—
66#888—62no—
67C42×D3—32no—
68C42×D3—61no—
69#869—32no—
70#869—61no—
71C3×C4⋊C8—32no—
72C3×C4⋊C8—32no—
73(C2×D3)⋊C8—62no—
74Dic3⋊C8—62no—
75C8×Dic3—62no—
76C4⋊(C3⋊C8)C2×D312yes—
77C4⋊(C3⋊C8)C2×D312yes—
78C22×C3⋊D3C22⋊C411yescharacteristic
79C32×Q8—42no—
80C2×C32×C4—22no—
81C2×C32×C4C22×C411yescharacteristic
82C2×C32×C4C2×D411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3C22⋊C411yescharacteristic
86C2×C3⋊Dic3C22⋊C411yescharacteristic
87C2×C3⋊Dic3C22⋊C411yescharacteristic
88C4×C3⋊D3—22no—
89C4×C3⋊D3—22no—
90C4×C3⋊D3—44no—
91C4×C3⋊D3SD1612yes—
92C4×C3⋊D3Q1612yes—
93#634—44no—
94#634—42no—
95C3×C3⋊C8—42no—
96C3×C3⋊C8—42no—
97C2×C4⋊Q8—91no—
98#431—91no—
99#423—91no—
100#333—32no—
101#333—92no—
102C2×C3×Q8—22no—
103C2×C3×Q8C4×D314yes—
104D3×Q8—128no—
105D3×Q8—128no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3—61no—
110C2×C4×D3—61no—
111C2×C4×D3—61no—
112C2×C4×D3—64no—
113C2×C4×D3—64no—
114C2×Dic6—34no—
115C2×Dic6—62no—
116C2×Dic6—64no—
117C2×Dic6—64no—
118C2×C3×C8—62no—
119C2×C3×C8—62no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C42—21no—
125C3×C42C22×D312yes—
126C4⋊Dic3—62no—
127C4⋊Dic3—62no—
128C4⋊Dic3—62no—
129C4⋊Dic3—62no—
130Dic3⋊C4—64no—
131Dic3⋊C4—64no—
132Dic3⋊C4—64no—
133Dic3⋊C4—64no—
134C4×Dic3—32no—
135C4×Dic3—32no—
136C4×Dic3—32no—
137C4×Dic3—61no—
138C4×Dic3—61no—
139C4×Dic3—61no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142ΓL(1,𝔽25)—124no—
143C22×C32C2×C22⋊C411yescharacteristic
144C2×C3⋊D3—21no—
145C2×C3⋊D3—21no—
146C2×C3⋊D3#10312yes—
147C32×C4—21no—
148C32×C4—42no—
149C32×C4C2×SD1611yescharacteristic
150C32×C4C2×Q1611yescharacteristic
151C32×C4#13712yes—
152C3⋊Dic3—21no—
153C3⋊Dic3—42no—
154C3⋊Dic3#10012yes—
155C3⋊Dic3#10312yes—
156C22×Q8—92no—
157C2×M16—181no—
158C4⋊Q8—92no—
159C4⋊Q8—91no—
160C4⋊Q8—181no—
161C4⋊Q8—181no—
162C4⋊Q8D3211yescharacteristic
163C2×C4⋊C4—181no—
164C2×C4⋊C4—181no—
165C2×C42—91no—
166C4⋊C8—32no—
167C4⋊C8—32no—
168C4⋊C8—181no—
169C22⋊C8—181no—
170C4×C8—181no—
171C22×D3—32no—
172C22×D3—61no—
173C3×Q8—44no—
174C3×Q8—44no—
175C2×C3×C4—24no—
176C2×C3×C4—24no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4C2×C4×D312yes—
181C2×C3×C4D3×D412yes—
182C2×C3×C4D3×D412yes—
183C2×Dic3—32no—
184C2×Dic3—32no—
185C2×Dic3—32no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—64no—
190C2×Dic3—64no—
191C4×D3—34no—
192C4×D3—34no—
193C4×D3—62no—
194C4×D3—62no—
195C4×D3—64no—
196C4×D3—64no—
197C4×D3—64no—
198C4×D3—62no—
199C4×D3—62no—
200C4×D3—128no—
201C4×D3—128no—
202Dic6—128no—
203Dic6—128no—
204Dic6—124no—
205Dic6—124no—
206C3×C8—122no—
207C3×C8—122no—
208C3⋊C8—42no—
209C3⋊C8—42no—
210C2×C32C2×#10011yescharacteristic
211C2×C32C2×#10311yescharacteristic
212C2×C32#41911yescharacteristic
213C3⋊D3—22no—
214C2×Q8—92no—
215C2×Q8—182no—
216C2×Q8—364no—
217C2×Q8#62412yes—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—91no—
221C22×C4—182no—
222M16—362no—
223C2×C8—62no—
224C2×C8—62no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—182no—
228C4⋊C4—182no—
229C4⋊C4—181no—
230C4⋊C4—181no—
231C42—91no—
232C42—91no—
233C42—91no—
234C42C2×D3211yescharacteristic
235C22×C3—21no—
236C22×C3D3×C22⋊C412yes—
237C2×D3—34no—
238C2×D3—62no—
239C2×D3—62no—
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—62no—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—22no—
248C3×C4—44no—
249C3×C4—44no—
250C3×C4D3×SD1612yes—
251C3×C4#91512yes—
252C3×C4D3×Q1612yes—
253C3×C4#91812yes—
254Dic3—34no—
255Dic3—34no—
256Dic3—62no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—124no—
261Dic3—124no—
262C32#154811yescharacteristic, complemented, Sylow
263C23—91no—
264Q8—42no—
265Q8—362no—
266Q8—364no—
267C2×C4—22no—
268C2×C4—91no—
269C2×C4—91no—
270C2×C4—92no—
271C2×C4—92no—
272C2×C4—91no—
273C2×C4—182no—
274C2×C4—182no—
275C2×C4—182no—
276C2×C4—364no—
277C2×C4C2×#62411yescharacteristic
278C2×C4#381111yescharacteristic
279C2×C4#381111yescharacteristic
280C8—122no—
281C8—122no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3D3×#10012yes—
286C2×C3D3×#10312yes—
287C2×C3#480212yes—
288D3—64no—
289D3—64no—
290C22—92no—
291C22—181no—
292C22—181no—
293C22#6397411yescharacteristic
294C4—21no—
295C4—42no—
296C4—92no—
297C4—92no—
298C4—181no—
299C4—362no—
300C4#6394011yescharacteristic
301C4#6394411yescharacteristic
302C4#6395112yes—
303C3—21no—
304C3#6848312yescomplemented
305C2—182no—
306C2#1059150311yescharacteristic
307C2#1059155211yescharacteristic
308C2#1059156211yescharacteristic
subgroupG/H
centerC22#63974
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32C2×C22⋊C4
prime radicalC22×C3⋊D3C22⋊C4
cosocleC32×C42C23
hypercenterC4⋊Q8D32
hypocenterC32#1548
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C3×#4546C2semidirect
5. #10589096C2semidirect
27. #4546D3semidirect
262. C32#1548semidirect
304. C3#68483semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C4⋊Q82
#105903402
#105902562
C3×#45462
#105890962
#684833