Finite Group Databasebeta

← SD16×C22⋊H(3,𝔽3) #13152228

Subgroups — SD16×C22⋊H(3,𝔽3) #13152228

Order 1728.

478 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 70 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
1Q8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
3C8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C22×C32×SD16C311yesmaximal, characteristic, complemented
5C3×A4×SD16C313yesmaximal, complemented
6C22×C22⋊H(3,𝔽3)—21no—
7C4×C22⋊H(3,𝔽3)—21no—
8C4×C22⋊H(3,𝔽3)C2211yescharacteristic
9SD16×H(3,𝔽3)—41nomaximal
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C22×C32×D4C2×C311yescharacteristic
12C3×Q8×A4C2×C313yescomplemented
13C3×D4×A4C2×C313yes—
14C2×C32×SD16—31no—
15C2×C32×SD16—31no—
16C2×C32×SD16—31no—
17C2×C32×SD16—31no—
18C22×C32×C8C2×C311yescharacteristic, complemented
19C3×C8×A4C2×C313yescomplemented
20C2×C22⋊H(3,𝔽3)—41no—
21C2×C22⋊H(3,𝔽3)D411yescharacteristic
22Q8×H(3,𝔽3)—41no—
23D4×H(3,𝔽3)—41no—
24C8×H(3,𝔽3)—41no—
25C22×C3×SD16—31no—
26C22×C3×SD16C3211yescharacteristic
27A4×SD16—33no—
28C24×C32—21no—
29C22×C3×A4—23no—
30C2×C32×Q8—31no—
31C2×C32×Q8—31no—
32C2×C32×Q8—61no—
33C2×C32×D4—31no—
34C2×C32×D4—31no—
35C2×C32×D4—61no—
36C22×C32×C4—21no—
37C22×C32×C4C22×C311yescharacteristic
38C3×C4×A4—23no—
39C3×C4×A4C22×C313yes—
40C32×SD16—32no—
41C32×SD16—31no—
42C32×SD16—31no—
43C32×SD16—31no—
44C32×SD16—43no—
45C32×SD16A411yescharacteristic, complemented
46C2×C32×C8—31no—
47C2×C32×C8—31no—
48C22×H(3,𝔽3)—81no—
49C22⋊H(3,𝔽3)SD1611yescharacteristic, direct product factor
50C4×H(3,𝔽3)—41no—
51C4×H(3,𝔽3)—81no—
52C22×C3×Q8—31no—
53C22×C3×Q8C2×C3211yescharacteristic
54C22×C3×D4—31no—
55C22×C3×D4C2×C3211yescharacteristic
56Q8×A4—33no—
57D4×A4—33no—
58C2×C3×SD16—33no—
59C2×C3×SD16—33no—
60C2×C3×SD16—33no—
61C2×C3×SD16—33no—
62C2×C3×SD16—31no—
63C2×C3×SD16—31no—
64C2×C3×SD16—31no—
65C2×C3×SD16—31no—
66C22×C3×C8—31no—
67C22×C3×C8C2×C3211yescharacteristic
68C8×A4—33no—
69C23×C32—41no—
70C23×C32—61no—
71C23×C32—61no—
72C23×C32—121no—
73C23×C32C3×D411yescharacteristic
74C2×C3×A4—43no—
75C2×C3×A4C3×D413yes—
76C32×Q8—31no—
77C32×Q8—43no—
78C32×Q8—61no—
79C32×Q8—61no—
80C32×Q8C2×A411yescharacteristic, complemented
81C32×D4—31no—
82C32×D4—43no—
83C32×D4—61no—
84C32×D4—61no—
85C32×D4C2×A411yescharacteristic
86C2×C32×C4—31no—
87C2×C32×C4—31no—
88C2×C32×C4—61no—
89C2×C32×C4—61no—
90C32×C8—31no—
91C32×C8—43no—
92C32×C8C2×A411yescharacteristic, complemented
93C22×SD16H(3,𝔽3)11yescharacteristic, complemented, Sylow
94C2×H(3,𝔽3)—41no—
95C2×H(3,𝔽3)—161no—
96C24×C3—21no—
97C24×C3—61no—
98C22×A4—63no—
99C2×C3×Q8—33no—
100C2×C3×Q8—33no—
101C2×C3×Q8—31no—
102C2×C3×Q8—31no—
103C2×C3×Q8—61no—
104C2×C3×Q8—63no—
105C2×C3×D4—33no—
106C2×C3×D4—33no—
107C2×C3×D4—31no—
108C2×C3×D4—31no—
109C2×C3×D4—63no—
110C2×C3×D4—61no—
111C22×C3×C4—21no—
112C22×C3×C4—31no—
113C22×C3×C4—61no—
114C22×C3×C4C22×C3211yescharacteristic
115C4×A4—33no—
116C4×A4—63no—
117C3×SD16—36no—
118C3×SD16—32no—
119C3×SD16—33no—
120C3×SD16—33no—
121C3×SD16—33no—
122C3×SD16—31no—
123C3×SD16—31no—
124C3×SD16—31no—
125C3×SD16—31no—
126C3×SD16—123no—
127C3×SD16C3×A411yescharacteristic
128C2×C3×C8—33no—
129C2×C3×C8—33no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C22×C32—21no—
133C22×C32—31no—
134C22×C32—31no—
135C22×C32—61no—
136C22×C32—83no—
137C22×C32—121no—
138C22×C32—121no—
139C22×C32—121no—
140C22×C32—121no—
141C22×C32C3×SD1611yescharacteristic, complemented
142C3×A4C3×SD1613yescomplemented
143C32×C4—21no—
144C32×C4—31no—
145C32×C4—43no—
146C32×C4—61no—
147C32×C4—83no—
148C32×C4C22×A411yescharacteristic
149C22×Q8C2×H(3,𝔽3)11yescharacteristic, complemented
150C22×D4C2×H(3,𝔽3)11yescharacteristic
151C2×SD16—31no—
152C2×SD16—31no—
153C2×SD16—31no—
154C2×SD16—31no—
155C22×C8C2×H(3,𝔽3)11yescharacteristic, complemented
156H(3,𝔽3)—41noSylow
157C23×C3—31no—
158C23×C3—41no—
159C23×C3—61no—
160C23×C3—63no—
161C23×C3—63no—
162C23×C3—61no—
163C23×C3—121no—
164C23×C3—121no—
165C23×C3—123no—
166C23×C3C32×D411yescharacteristic
167C2×A4—33no—
168C2×A4—123no—
169C3×Q8—33no—
170C3×Q8—31no—
171C3×Q8—31no—
172C3×Q8—61no—
173C3×Q8—61no—
174C3×Q8—63no—
175C3×Q8—63no—
176C3×Q8—123no—
177C3×Q8C2×C3×A411yescharacteristic
178C3×D4—33no—
179C3×D4—31no—
180C3×D4—31no—
181C3×D4—61no—
182C3×D4—61no—
183C3×D4—63no—
184C3×D4—63no—
185C3×D4—123no—
186C3×D4C2×C3×A411yescharacteristic
187C2×C3×C4—33no—
188C2×C3×C4—33no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—61no—
192C2×C3×C4—63no—
193C2×C3×C4—63no—
194C2×C3×C4—61no—
195C3×C8—33no—
196C3×C8—31no—
197C3×C8—31no—
198C3×C8—123no—
199C3×C8C2×C3×A411yescharacteristic
200C2×C32—31no—
201C2×C32—31no—
202C2×C32—41no—
203C2×C32—43no—
204C2×C32—121no—
205C2×C32—163no—
206C2×C32D4×A411yescharacteristic
207C24—21no—
208C2×Q8—31no—
209C2×Q8—31no—
210C2×Q8—61no—
211C2×D4—31no—
212C2×D4—31no—
213C2×D4—61no—
214C22×C4—21no—
215C22×C4C22×H(3,𝔽3)11yescharacteristic
216SD16—32no—
217SD16—31no—
218SD16—31no—
219SD16—31no—
220SD16C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
221C2×C8—31no—
222C2×C8—31no—
223C22×C3—21no—
224C22×C3—31no—
225C22×C3—33no—
226C22×C3—33no—
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—61no—
230C22×C3—63no—
231C22×C3—61no—
232C22×C3—121no—
233C22×C3—121no—
234C22×C3—121no—
235C22×C3—121no—
236C22×C3—123no—
237C22×C3—123no—
238C22×C3—123no—
239C22×C3—123no—
240C22×C3—243no—
241C22×C3C32×SD1611yescharacteristic
242A4—33no—
243C3×C4—21no—
244C3×C4—33no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—61no—
248C3×C4—63no—
249C3×C4—61no—
250C3×C4—123no—
251C3×C4—243no—
252C3×C4C22×C3×A411yescharacteristic
253C32—43no—
254C32A4×SD1611yescharacteristic, complemented
255C23—41no—
256C23—61no—
257C23—61no—
258C23—121no—
259C23D4×H(3,𝔽3)11yescharacteristic
260Q8—31no—
261Q8—61no—
262Q8—61no—
263Q8C2×C22⋊H(3,𝔽3)11yescharacteristic, complemented
264D4—31no—
265D4—61no—
266D4—61no—
267D4C2×C22⋊H(3,𝔽3)11yescharacteristic
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—61no—
271C2×C4—61no—
272C8—31no—
273C8C2×C22⋊H(3,𝔽3)11yescharacteristic, complemented
274C2×C3—33no—
275C2×C3—33no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—41no—
280C2×C3—121no—
281C2×C3—123no—
282C2×C3—121no—
283C2×C3—123no—
284C2×C3—483no—
285C2×C3C3×D4×A411yescharacteristic
286C22—21no—
287C22—31no—
288C22—31no—
289C22—61no—
290C22—121no—
291C22—121no—
292C22—121no—
293C22—121no—
294C22SD16×H(3,𝔽3)11yescharacteristic, complemented
295C4—21no—
296C4—31no—
297C4—61no—
298C4C22×C22⋊H(3,𝔽3)11yescharacteristic
299C3—31no—
300C3—123no—
301C3C3×A4×SD1611yescharacteristic
302C2—31no—
303C2—31no—
304C2—41no—
305C2—121no—
306C2D4×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×D4×A4
commutator, cosocleC22×C3×C4C22×C32
FrattiniC3×C4C22×C3×A4
FittingC22×C32×SD16C3
socleC23×C3C32×D4
prime radicalD4×C22⋊H(3,𝔽3)C2
hypercenterC32×SD16A4
hypocenterC22SD16×H(3,𝔽3)
LCSG ▷ C22×C3×C4 ▷ C23 ▷ C22
UCSC1 ◁ C2×C3 ◁ C32×C4 ◁ C32×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊H(3,𝔽3)C2semidirect
3. C8×C22⋊H(3,𝔽3)C2semidirect
4. C22×C32×SD16C3semidirect
5. C3×A4×SD16C3semidirect
10. C22×C32×Q8C2×C3semidirect
12. C3×Q8×A4C2×C3semidirect
18. C22×C32×C8C2×C3semidirect
19. C3×C8×A4C2×C3semidirect
45. C32×SD16A4semidirect
49. C22⋊H(3,𝔽3)SD16direct (N × H)
80. C32×Q8C2×A4semidirect
92. C32×C8C2×A4semidirect
93. C22×SD16H(3,𝔽3)semidirect
141. C22×C32C3×SD16semidirect
142. C3×A4C3×SD16semidirect
149. C22×Q8C2×H(3,𝔽3)semidirect
155. C22×C8C2×H(3,𝔽3)semidirect
220. SD16C22⋊H(3,𝔽3)direct (N × H)
254. C32A4×SD16semidirect
263. Q8C2×C22⋊H(3,𝔽3)semidirect
273. C8C2×C22⋊H(3,𝔽3)semidirect
294. C22SD16×H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
D4×C22⋊H(3,𝔽3)2
C8×C22⋊H(3,𝔽3)2
C22×C32×SD163
C3×A4×SD163
SD16×H(3,𝔽3)4