Finite Group Databasebeta

← #41×#1094 #13164854

Subgroups — #41×#1094 #13164854

Order 1728.

1130 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 170 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×#1094C211yesmaximal, characteristic, complemented
2D4×#1094C213yesmaximal, complemented
3C2×C4×#1094C213yesmaximal, complemented
4C22×C32×#41C311yesmaximal, characteristic
5#41×C22⋊C9C313yesmaximal, complemented
6C22×#1094C2213yes—
7#41×3−1+2—41nomaximal
8C4×#1094C2211yescharacteristic
9C4×#1094C2213yes—
10C2×C32×#41—33no—
11C2×C32×#41—33no—
12C2×C32×#41—31no—
13C2×C32×#41—31no—
14C22×C32×Q8C2×C311yescharacteristic
15C22×C32×D4C2×C313yes—
16C23×C32×C4C2×C313yes—
17Q8×C22⋊C9C2×C313yescomplemented
18D4×C22⋊C9C2×C319yescomplemented
19C2×C4×C22⋊C9C2×C319yescomplemented
20C2×#1094—23no—
21C2×#1094C2311yescharacteristic
22Q8×3−1+2—41no—
23D4×3−1+2—43no—
24C2×C4×3−1+2—43no—
25C22×C3×#41—31no—
26C22×C3×#41C3211yescharacteristic
27C24×C32C22×C313yes—
28C32×#41—36no—
29C32×#41—36no—
30C32×#41—32no—
31C32×#41—31no—
32C32×#41—33no—
33C32×#41—33no—
34C32×#41A411yescharacteristic
35C2×C32×Q8—31no—
36C2×C32×Q8—33no—
37C2×C32×D4—33no—
38C2×C32×D4—33no—
39C2×C32×D4—36no—
40C22×C32×C4—33no—
41C22×C32×C4—33no—
42C22×C32×C4—33no—
43C22×C32×C4—33no—
44C22×C32×C4C22×C311yescharacteristic
45C22×C32×C4C22×C313yes—
46C22×C22⋊C9C22×C319yes—
47C9×#41—43no—
48C4×C22⋊C9C22×C313yes—
49C4×C22⋊C9C22×C319yes—
50C22×3−1+2—43no—
51#1094#4111yescharacteristic, direct product factor
52C4×3−1+2—41no—
53C4×3−1+2—43no—
54C2×C3×#41—33no—
55C2×C3×#41—33no—
56C2×C3×#41—31no—
57C2×C3×#41—39no—
58C2×C3×#41—39no—
59C2×C3×#41—33no—
60C2×C3×#41—33no—
61C2×C3×#41—31no—
62C22×C3×Q8—31no—
63C22×C3×Q8C2×C3211yescharacteristic
64C22×C3×D4—33no—
65C22×C3×D4C2×C3213yes—
66C23×C3×C4—33no—
67C23×C3×C4C2×C3213yes—
68C23×C32—23no—
69C23×C32—33no—
70C23×C32—33no—
71C23×C32—63no—
72C23×C32C23×C311yescharacteristic
73C32×Q8—32no—
74C32×Q8—33no—
75C32×Q8C2×A411yescharacteristic
76C32×D4—36no—
77C32×D4—36no—
78C32×D4—33no—
79C32×D4C2×A413yes—
80C2×C32×C4—36no—
81C2×C32×C4—33no—
82C2×C32×C4—31no—
83C2×C32×C4—33no—
84C2×C32×C4—33no—
85C2×C32×C4—33no—
86C2×C32×C4—31no—
87C2×C32×C4—33no—
88C2×C32×C4C2×A413yes—
89C2×C22⋊C9—29no—
90C2×C22⋊C9C23×C313yes—
91Q8×C9—43no—
92D4×C9—49no—
93C2×C4×C9—49no—
94C22×#413−1+211yescharacteristic, complemented, Sylow
95C2×3−1+2—41no—
96C2×3−1+2—83no—
97C24×C3—33no—
98C24×C3C22×C3213yes—
99C3×#41—318no—
100C3×#41—318no—
101C3×#41—36no—
102C3×#41—36no—
103C3×#41—36no—
104C3×#41—32no—
105C3×#41—33no—
106C3×#41—39no—
107C3×#41—39no—
108C3×#41—31no—
109C3×#41—33no—
110C3×#41—33no—
111C3×#41—31no—
112C3×#41C3×A411yescharacteristic
113C2×C3×Q8—33no—
114C2×C3×Q8—39no—
115C2×C3×Q8—33no—
116C2×C3×Q8—31no—
117C2×C3×D4—39no—
118C2×C3×D4—39no—
119C2×C3×D4—318no—
120C2×C3×D4—33no—
121C2×C3×D4—33no—
122C2×C3×D4—36no—
123C22×C3×C4—39no—
124C22×C3×C4—39no—
125C22×C3×C4—39no—
126C22×C3×C4—39no—
127C22×C3×C4—33no—
128C22×C3×C4—33no—
129C22×C3×C4—33no—
130C22×C3×C4—33no—
131C22×C3×C4—31no—
132C22×C3×C4—33no—
133C22×C3×C4C22×C3211yescharacteristic
134C22×C3×C4C22×C3213yes—
135C22×C32—31no—
136C22×C32—31no—
137C22×C32—33no—
138C22×C32—63no—
139C22×C32—63no—
140C22×C32—63no—
141C22×C32—63no—
142C22×C32C3×#4111yescharacteristic
143C22×C32C22×A413yes—
144C32×C4—31no—
145C32×C4—33no—
146C32×C4C22×A413yes—
147C32×C4C22×A411yescharacteristic
148C22×C9—49no—
149C22⋊C9C3×#4113yescomplemented
150C4×C9—49no—
151C4×C9—43no—
152C2×#41—33no—
153C2×#41—33no—
154C2×#41—31no—
155C2×#41—31no—
156C22×Q8C2×3−1+211yescharacteristic, complemented
157C22×D4C2×3−1+213yescomplemented
158C23×C4C2×3−1+213yescomplemented
1593−1+2—41noSylow
160C23×C3—23no—
161C23×C3—39no—
162C23×C3—39no—
163C23×C3—33no—
164C23×C3—33no—
165C23×C3—31no—
166C23×C3—63no—
167C23×C3—63no—
168C23×C3—69no—
169C23×C3C23×C3211yescharacteristic
170C3×Q8—36no—
171C3×Q8—32no—
172C3×Q8—39no—
173C3×Q8—33no—
174C3×Q8—31no—
175C3×Q8C2×C3×A411yescharacteristic
176C3×D4—318no—
177C3×D4—36no—
178C3×D4—318no—
179C3×D4—39no—
180C3×D4—36no—
181C3×D4—33no—
182C3×D4—33no—
183C3×D4C2×C3×A413yes—
184C2×C3×C4—33no—
185C2×C3×C4—33no—
186C2×C3×C4—39no—
187C2×C3×C4—39no—
188C2×C3×C4—318no—
189C2×C3×C4—36no—
190C2×C3×C4—39no—
191C2×C3×C4—39no—
192C2×C3×C4—39no—
193C2×C3×C4—33no—
194C2×C3×C4—33no—
195C2×C3×C4—33no—
196C2×C3×C4—31no—
197C2×C3×C4—33no—
198C2×C3×C4—31no—
199C2×C3×C4—33no—
200C2×C3×C4—33no—
201C2×C3×C4C2×C3×A413yes—
202C2×C32—23no—
203C2×C32—31no—
204C2×C32—31no—
205C2×C32—63no—
206C2×C32C23×A411yescharacteristic
207C2×C9—43no—
208C2×C9—89no—
209C24C22×3−1+213yes—
210#41—36no—
211#41—36no—
212#41—32no—
213#41—31no—
214#41—33no—
215#41—33no—
216#41#109411yescharacteristic, direct product factor
217C2×Q8—33no—
218C2×Q8—31no—
219C2×D4—33no—
220C2×D4—33no—
221C2×D4—36no—
222C22×C4—33no—
223C22×C4—33no—
224C22×C4—33no—
225C22×C4—33no—
226C22×C4C22×3−1+211yescharacteristic
227C22×C4C22×3−1+213yes—
228C22×C3—31no—
229C22×C3—33no—
230C22×C3—33no—
231C22×C3—31no—
232C22×C3—39no—
233C22×C3—33no—
234C22×C3—33no—
235C22×C3—31no—
236C22×C3—63no—
237C22×C3—63no—
238C22×C3—63no—
239C22×C3—63no—
240C22×C3—69no—
241C22×C3—69no—
242C22×C3—69no—
243C22×C3—69no—
244C22×C3C32×#4111yescharacteristic
245C22×C3C22×C3×A413yes—
246C3×C4—33no—
247C3×C4—39no—
248C3×C4—31no—
249C3×C4—33no—
250C3×C4—31no—
251C3×C4—33no—
252C3×C4C22×C3×A413yes—
253C3×C4C22×C3×A411yescharacteristic
254C32A4×#4111yescharacteristic
255C9—43no—
256C23—23no—
257C23—33no—
258C23—33no—
259C23—63no—
260C23C23×3−1+211yescharacteristic
261Q8—32no—
262Q8—33no—
263Q8C2×#109411yescharacteristic, complemented
264D4—36no—
265D4—36no—
266D4—33no—
267D4C2×#109413yescomplemented
268C2×C4—36no—
269C2×C4—33no—
270C2×C4—31no—
271C2×C4—33no—
272C2×C4—33no—
273C2×C4—33no—
274C2×C4—31no—
275C2×C4—33no—
276C2×C4C2×#109413yescomplemented
277C2×C3—23no—
278C2×C3—33no—
279C2×C3—33no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—63no—
284C2×C3—69no—
285C2×C3—63no—
286C2×C3C23×C3×A411yescharacteristic
287C22—31no—
288C22—31no—
289C22—33no—
290C22—63no—
291C22—63no—
292C22—63no—
293C22—63no—
294C22#41×3−1+211yescharacteristic, complemented
295C22C22×#109413yes—
296C4—31no—
297C4—33no—
298C4C22×#109411yescharacteristic
299C4C22×#109413yes—
300C3—31no—
301C3C3×A4×#4111yescharacteristic
302C2—23no—
303C2—31no—
304C2—31no—
305C2—63no—
306C2C23×#109411yescharacteristic
subgroupG/H
centerC3×C4C22×C3×A4
commutator, socle, cosocleC23×C3C23×C32
FrattiniC2×C3C23×C3×A4
Fitting, prime radicalC22×C32×#41C3
hypercenterC32×#41A4
hypocenterC22#41×3−1+2
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C3×C4 ◁ C32×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#1094C2semidirect
2. D4×#1094C2semidirect
3. C2×C4×#1094C2semidirect
5. #41×C22⋊C9C3semidirect
17. Q8×C22⋊C9C2×C3semidirect
18. D4×C22⋊C9C2×C3semidirect
19. C2×C4×C22⋊C9C2×C3semidirect
51. #1094#41direct (N × H)
94. C22×#413−1+2semidirect
149. C22⋊C9C3×#41semidirect
156. C22×Q8C2×3−1+2semidirect
157. C22×D4C2×3−1+2semidirect
158. C23×C4C2×3−1+2semidirect
216. #41#1094direct (N × H)
263. Q8C2×#1094semidirect
267. D4C2×#1094semidirect
276. C2×C4C2×#1094semidirect
294. C22#41×3−1+2semidirect

Maximal subgroups

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

subgroupindex
Q8×#10942
D4×#10942
C2×C4×#10942
C22×C32×#413
#41×C22⋊C93
#41×3−1+24