Finite Group Databasebeta

← #13152191

Subgroups — #13152191

Order 1728.

440 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#88691C211yesmaximal, characteristic, complemented
2C22×#6204C211yesmaximal, characteristic, complemented
3C4×#6204C211yesmaximal, characteristic, complemented
4C32×#548C311yesmaximal, characteristic
5#548⋊C9C313yesmaximal, complemented
6#88691—21no—
7C2×#6204—21no—
8C2×#6204C2211yescharacteristic
9C22⋊C4×3−1+2—41nomaximal
10C2×C32×#41C2×C311yescharacteristic
11C22×C32×Q8C2×C311yescharacteristic
12C32×#124—31no—
13C32×C22⋊Q8—31no—
14C32×#121—31no—
15C32×C4×Q8C2×C311yescharacteristic
16C32×C4×D4—31no—
17C2×#41⋊C9C2×C313yescomplemented
18C22×Q8⋊C9C2×C313yescomplemented
19C4×Q8⋊C9C2×C313yescomplemented
20C23×3−1+2—41no—
21C2×C4×3−1+2—41no—
22C2×C4×3−1+2—41no—
23#6204D411yescharacteristic, complemented
24C3×#548—31no—
25C3×#548C3211yescharacteristic
26C32×#41—21no—
27C32×#41—61no—
28C2×C32×Q8—21no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×Q8C22×C311yescharacteristic
32C2×C32×D4—31no—
33C2×C32×D4—31no—
34C22×C32×C4—31no—
35C22×C32×C4—31no—
36C32×C4⋊C4—31no—
37C32×C4⋊C4—31no—
38C32×C22⋊C4—32no—
39C32×C22⋊C4—31no—
40C32×C22⋊C4A411yescharacteristic
41C32×C42—31no—
42#41⋊C9—23no—
43C2×Q8⋊C9—23no—
44C2×Q8⋊C9C22×C313yes—
45C9×C22⋊C4—43no—
46C22×3−1+2—41no—
47C22×3−1+2—42no—
48C22×3−1+2—81no—
49C22×3−1+2—81no—
50C4×3−1+2—81no—
51C4×3−1+2—81no—
52C2×C3×#41—31no—
53C2×C3×#41C2×C3211yescharacteristic
54C22×C3×Q8—31no—
55C22×C3×Q8C2×C3211yescharacteristic
56C3×#124—33no—
57C3×#124—31no—
58C3×C22⋊Q8—33no—
59C3×C22⋊Q8—31no—
60C3×#121—31no—
61C3×#121—33no—
62C3×C4×Q8—31no—
63C3×C4×Q8C2×C3211yescharacteristic
64C3×C4×D4—33no—
65C3×C4×D4—31no—
66C23×C32—31no—
67C23×C32C2×A411yescharacteristic
68C32×Q8—31no—
69C32×Q8—61no—
70C32×Q8—61no—
71C32×Q8C3×D411yescharacteristic
72C32×D4—61no—
73C32×D4—61no—
74C32×D4—62no—
75C2×C32×C4—31no—
76C2×C32×C4—31no—
77C2×C32×C4—31no—
78C2×C32×C4—61no—
79C2×C32×C4—61no—
80C2×C32×C4—61no—
81C2×C32×C4—61no—
82C2×C32×C4C2×A411yescharacteristic
83C2×C32×C4C2×A411yescharacteristic
84C23×C9—43no—
85C2×C4×C9—43no—
86C2×C4×C9—43no—
87Q8⋊C9C3×D413yescomplemented
88#5483−1+211yescharacteristic, complemented, Sylow
89C2×3−1+2—41no—
90C2×3−1+2—41no—
91C2×3−1+2—41no—
92C2×3−1+2—82no—
93C3×#41—21no—
94C3×#41—61no—
95C3×#41—63no—
96C3×#41—61no—
97C2×C3×Q8—21no—
98C2×C3×Q8—33no—
99C2×C3×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—61no—
102C2×C3×Q8—63no—
103C2×C3×Q8—61no—
104C2×C3×Q8C22×C3211yescharacteristic
105C2×C3×D4—31no—
106C2×C3×D4—33no—
107C2×C3×D4—33no—
108C2×C3×D4—31no—
109C22×C3×C4—33no—
110C22×C3×C4—33no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—33no—
115C3×C4⋊C4—33no—
116C3×C4⋊C4—31no—
117C3×C22⋊C4—33no—
118C3×C22⋊C4—36no—
119C3×C22⋊C4—32no—
120C3×C22⋊C4—31no—
121C3×C22⋊C4—31no—
122C3×C22⋊C4C3×A411yescharacteristic
123C3×C42—33no—
124C3×C42—31no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—61no—
128C22×C32—61no—
129C22×C32—61no—
130C22×C32#41⋊C312yes—
131C22×C32C22×A411yescharacteristic
132C32×C4—21no—
133C32×C4—21no—
134C32×C4—31no—
135C32×C4—31no—
136C32×C4—61no—
137C32×C4—61no—
138C22×C9—43no—
139C22×C9—46no—
140C22×C9—83no—
141C22×C9—83no—
142C4×C9—83no—
143C4×C9—83no—
144C2×#41C2×3−1+211yescharacteristic, complemented
145C22×Q8C2×3−1+211yescharacteristic
146#124—31no—
147C22⋊Q8—31no—
148#121—31no—
149C4×Q8C2×3−1+211yescharacteristic, complemented
150C4×D4—31no—
1513−1+2—41noSylow
152C23×C3—33no—
153C23×C3—31no—
154C23×C3—31no—
155C23×C3C2×C3×A411yescharacteristic
156C3×Q8—31no—
157C3×Q8—33no—
158C3×Q8—31no—
159C3×Q8—61no—
160C3×Q8—61no—
161C3×Q8—63no—
162C3×Q8—63no—
163C3×Q8C32×D411yescharacteristic
164C3×D4—61no—
165C3×D4—61no—
166C3×D4—63no—
167C3×D4—63no—
168C3×D4—66no—
169C3×D4—62no—
170C2×C3×C4—33no—
171C2×C3×C4—31no—
172C2×C3×C4—33no—
173C2×C3×C4—33no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—63no—
183C2×C3×C4—63no—
184C2×C3×C4—63no—
185C2×C3×C4—63no—
186C2×C3×C4C2×C3×A411yescharacteristic
187C2×C3×C4C2×C3×A411yescharacteristic
188C2×C32—22no—
189C2×C32—121no—
190C2×C32D4×A411yescharacteristic
191C2×C32C2×#41⋊C311yescharacteristic
192C2×C322−1+4⋊C311yescharacteristic
193C2×C9—43no—
194C2×C9—43no—
195C2×C9—43no—
196C2×C9—86no—
197#41—21no—
198#41—61no—
199C2×Q8—21no—
200C2×Q8—31no—
201C2×Q8—61no—
202C2×Q8C22×3−1+211yescharacteristic
203C2×D4—31no—
204C2×D4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C4⋊C4—31no—
208C4⋊C4—31no—
209C22⋊C4—32no—
210C22⋊C4—31no—
211C22⋊C4#109411yescharacteristic
212C42—31no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—31no—
216C22×C3—32no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—63no—
220C22×C3—63no—
221C22×C3—63no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—61no—
225C22×C3C3×#41⋊C312yes—
226C22×C3C22×C3×A411yescharacteristic
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—33no—
230C3×C4—33no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—63no—
236C3×C4—63no—
237C3×C4—61no—
238C3×C4—61no—
239C32#548⋊C311yescharacteristic
240C9—43no—
241C23—31no—
242C23C2×#109411yescharacteristic
243Q8—31no—
244Q8—61no—
245Q8—61no—
246Q8D4×3−1+211yescharacteristic
247D4—61no—
248D4—61no—
249D4—62no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4C2×#109411yescharacteristic
258C2×C4C2×#109411yescharacteristic
259C2×C3—22no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—62no—
264C2×C3—123no—
265C2×C3—121no—
266C2×C3C3×D4×A411yescharacteristic
267C2×C3C2×C3×#41⋊C311yescharacteristic
268C2×C3C3×2−1+4⋊C311yescharacteristic
269C22—21no—
270C22—21no—
271C22—61no—
272C22—61no—
273C22—61no—
274C22#8869112yescomplemented
275C22C22×#109411yescharacteristic
276C4—21no—
277C4—21no—
278C4—31no—
279C4—31no—
280C4—61no—
281C4—61no—
282C3—31no—
283C3C3×#548⋊C311yescharacteristic
284C2—22no—
285C2—121no—
286C2D4×#109411yescharacteristic
287C2C2×#8869111yescharacteristic
288C2#1172257911yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutator, cosocleC2×C3×Q8C22×C32
Fitting, prime radicalC32×#548C3
hypercenterC32×C22⋊C4A4
hypocenterQ8D4×3−1+2
LCSG ▷ C2×C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C32×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88691C2semidirect
2. C22×#6204C2semidirect
3. C4×#6204C2semidirect
5. #548⋊C9C3semidirect
17. C2×#41⋊C9C2×C3semidirect
18. C22×Q8⋊C9C2×C3semidirect
19. C4×Q8⋊C9C2×C3semidirect
23. #6204D4semidirect
87. Q8⋊C9C3×D4semidirect
88. #5483−1+2semidirect
144. C2×#41C2×3−1+2semidirect
149. C4×Q8C2×3−1+2semidirect
274. C22#88691semidirect

Maximal subgroups

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

subgroupindex
C2×#886912
C22×#62042
C4×#62042
C32×#5483
#548⋊C93
C22⋊C4×3−1+24