Finite Group Databasebeta

← 3−1+2×#5934 #72245443

Subgroups — 3−1+2×#5934 #72245443

Order 5184.

905 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 905 conjugacy classes of subgroups — too many to draw. Only its 56 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
13−1+2×#997C211yesmaximal, characteristic, complemented
2C32×#5934C311yesmaximal, characteristic
3C9×#5934C313yesmaximal, complemented
43−1+2×B(4,𝔽2)—31nomaximal
5C2×S4×3−1+2—43nomaximal
6C32×#997C2×C311yescharacteristic
73−1+2×H(5,𝔽2)D311yescharacteristic, complemented
8#11722589—21no—
9C9⋊#997—23no—
10C9×#997C2×C313yescomplemented
113−1+2×#120—33no—
123−1+2×#99—33no—
13C2×A4×3−1+2—43no—
14S4×3−1+2—46no—
15SL(2,𝔽3)×3−1+2—81no—
16C3×#5934—31no—
17C3×#5934C3211yescharacteristic
18C32×B(4,𝔽2)—31no—
19C9×B(4,𝔽2)—33no—
20C2×C32×S4—43no—
21C24×3−1+2—31no—
22C2×C9×S4—49no—
23#41×3−1+2—61no—
24C2×D4×3−1+2—33no—
25C2×D4×3−1+2—63no—
26C22⋊C4×3−1+2—33no—
27C22⋊C4×3−1+2—63no—
28C2×D3×3−1+2—161no—
29A4×3−1+2—43no—
30C32×H(5,𝔽2)C3×D311yescharacteristic
31C3×#997—21no—
32C3×#997—31no—
33C3×#997—61no—
34C3×#997C2×C3211yescharacteristic
35C32×#120—33no—
36C9×H(5,𝔽2)C3×D313yescomplemented
37#63703—23no—
38C32×#99—33no—
39C9×#120—39no—
40C9×#99—39no—
41C2×C32×A4—43no—
42C32×S4—46no—
43C32×SL(2,𝔽3)—81no—
44C23×3−1+2—31no—
45C23×3−1+2—63no—
46C23×3−1+2—122no—
47C23×3−1+2S413yescomplemented
48C2×#1094—83no—
49C2×C9⋊A4—89no—
50C2×C9×A4—49no—
51C9×S4—418no—
52Q8×3−1+2—21no—
53D4×3−1+2—63no—
54D4×3−1+2—126no—
55C2×C4×3−1+2—33no—
56C2×C4×3−1+2—63no—
57#6204—82no—
58C9⋊SL(2,𝔽3)—86no—
59C9×SL(2,𝔽3)—83no—
60#59343−1+211yescharacteristic, direct product factor
61C3×B(4,𝔽2)—31no—
62C3×B(4,𝔽2)—91no—
63C2×C3×3−1+2—161no—
64D3×3−1+2—162no—
65C24×C32—31no—
66C2×C3×S4—43no—
67C2×C3×S4—123no—
68C32×#41—61no—
69C2×C32×D4—33no—
70C2×C32×D4—63no—
71C24×C9—33no—
72C32×C22⋊C4—33no—
73C32×C22⋊C4—63no—
74C9×#41—63no—
75C2×D4×C9—39no—
76C2×D4×C9—69no—
77C9×C22⋊C4—39no—
78C9×C22⋊C4—69no—
79C2×C32×D3—161no—
80C32×A4—43no—
81C22×3−1+2—33no—
82C22×3−1+2—43no—
83C22×3−1+2—66no—
84C22×3−1+2—121no—
85C22×3−1+2—123no—
86C22×3−1+2—121no—
87C2×D3×C9—163no—
88#1094—83no—
89C9⋊A4—89no—
90C9×A4—49no—
91C4×3−1+2—61no—
92C4×3−1+2—123no—
93C3×H(5,𝔽2)—31no—
94C3×H(5,𝔽2)C32×D311yescharacteristic
95#997—21no—
96#997—61no—
97#997C2×3−1+211yescharacteristic, complemented
98C3×#120—33no—
99C3×#120—93no—
100C3×#99—33no—
101C3×#99—93no—
102C3×3−1+2—161noSylow
103C23×C32—31no—
104C23×C32—63no—
105C23×C32—122no—
106C23×C32C3×S413yes—
107C2×C3×A4—43no—
108C2×C3×A4—83no—
109C2×C3×A4—123no—
110C2×C3×A4—243no—
111C3×S4—46no—
112C3×S4—126no—
113C32×Q8—21no—
114C32×D4—63no—
115C32×D4—126no—
116C2×C32×C4—33no—
117C2×C32×C4—63no—
118C3×SL(2,𝔽3)—81no—
119C3×SL(2,𝔽3)—82no—
120C3×SL(2,𝔽3)—241no—
121C3×SL(2,𝔽3)—242no—
122C23×C9—33no—
123C23×C9—69no—
124C23×C9—126no—
125C23×C9C3×S419yescomplemented
126C2×C22⋊C9—89no—
127Q8×C9—23no—
128D4×C9—69no—
129D4×C9—1218no—
130C2×C4×C9—39no—
131C2×C4×C9—69no—
132Q8⋊C9—86no—
133B(4,𝔽2)—31noSylow
134C2×C33—161no—
135C32×D3—162no—
136C2×3−1+2—63no—
137C2×3−1+2—122no—
138C2×3−1+2—321no—
139C2×3−1+2—323no—
140C2×3−1+2#102011yescharacteristic
141C2×C3×C9—163no—
142D3×C9—166no—
143C24×C3—31no—
144C24×C3—91no—
145C2×S4—43no—
146C3×#41—61no—
147C3×#41—181no—
148C2×C3×D4—33no—
149C2×C3×D4—63no—
150C2×C3×D4—93no—
151C2×C3×D4—183no—
152C3×C22⋊C4—33no—
153C3×C22⋊C4—63no—
154C3×C22⋊C4—93no—
155C3×C22⋊C4—183no—
156C22×C32—33no—
157C22×C32—43no—
158C22×C32—66no—
159C22×C32—121no—
160C22×C32—123no—
161C22×C32—121no—
162C2×C3×D3—161no—
163C2×C3×D3—481no—
164C3×A4—43no—
165C3×A4—83no—
166C3×A4—123no—
167C3×A4—243no—
168C32×C4—61no—
169C32×C4—123no—
170C22×C9—39no—
171C22×C9—49no—
172C22×C9—618no—
173C22×C9—123no—
174C22×C9—129no—
175C22×C9—123no—
176C22⋊C9—89no—
177C4×C9—63no—
178C4×C9—129no—
179H(5,𝔽2)D3×3−1+211yescharacteristic, complemented
180#120—33no—
181#99—33no—
182C33—161no—
1833−1+2—321no—
1843−1+2—323no—
1853−1+2#593411yescharacteristic, direct product factor
186C3×C9—163no—
187C23×C3—31no—
188C23×C3—33no—
189C23×C3—63no—
190C23×C3—91no—
191C23×C3—122no—
192C23×C3—183no—
193C23×C3—362no—
194C23×C3C32×S413yes—
195C2×A4—43no—
196C2×A4—83no—
197C2×A4—243no—
198S4—46no—
199C3×Q8—21no—
200C3×Q8—61no—
201C3×D4—63no—
202C3×D4—126no—
203C3×D4—183no—
204C3×D4—366no—
205C2×C3×C4—33no—
206C2×C3×C4—63no—
207C2×C3×C4—93no—
208C2×C3×C4—183no—
209SL(2,𝔽3)—81no—
210SL(2,𝔽3)—82no—
211SL(2,𝔽3)—242no—
212C2×C32—63no—
213C2×C32—122no—
214C2×C32—161no—
215C2×C32—321no—
216C2×C32—481no—
217C2×C32—961no—
218C2×C32C3×#102011yescharacteristic
219C3×D3—162no—
220C3×D3—482no—
221C2×C9—69no—
222C2×C9—126no—
223C2×C9—323no—
224C2×C9C3×#102013yes—
225C24—31no—
226#41—61no—
227C2×D4—33no—
228C2×D4—63no—
229C22⋊C4—33no—
230C22⋊C4—63no—
231C22×C3—33no—
232C22×C3—43no—
233C22×C3—66no—
234C22×C3—93no—
235C22×C3—123no—
236C22×C3—121no—
237C22×C3—123no—
238C22×C3—121no—
239C22×C3—186no—
240C22×C3—361no—
241C22×C3—361no—
242C22×C3—363no—
243C2×D3—161no—
244A4—43no—
245A4—83no—
246A4—243no—
247C3×C4—61no—
248C3×C4—123no—
249C3×C4—181no—
250C3×C4—363no—
251C32—161no—
252C32—321no—
253C32—481no—
254C32—961no—
255C32C3×#593411yescharacteristic
256C9—323no—
257C9C3×#593413yescomplemented
258C23—31no—
259C23—63no—
260C23—122no—
261C23S4×3−1+213yescomplemented
262Q8—21no—
263D4—63no—
264D4—126no—
265C2×C4—33no—
266C2×C4—63no—
267C2×C3—31no—
268C2×C3—63no—
269C2×C3—122no—
270C2×C3—161no—
271C2×C3—183no—
272C2×C3—321no—
273C2×C3—362no—
274C2×C3—961no—
275C2×C3C32×#102011yescharacteristic
276D3—162no—
277C22—33no—
278C22—43no—
279C22—66no—
280C22—121no—
281C22—123no—
282C22—121no—
283C4—61no—
284C4—123no—
285C3—31no—
286C3—161no—
287C3—321no—
288C3—961no—
289C3C32×#593411yescharacteristic
290C2—63no—
291C2—122no—
292C23−1+2×#102011yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C32×#1020
commutator, cosocleC3×#997C2×C32
Fitting3−1+2×H(5,𝔽2)D3
prime radicalC32×#5934C3
hypercenterC2×3−1+2#1020
hypocenter#997C2×3−1+2
LCSG ▷ C3×#997 ▷ #997
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#997C2semidirect
3. C9×#5934C3semidirect
7. 3−1+2×H(5,𝔽2)D3semidirect
10. C9×#997C2×C3semidirect
36. C9×H(5,𝔽2)C3×D3semidirect
47. C23×3−1+2S4semidirect
60. #59343−1+2direct (N × H)
97. #997C2×3−1+2semidirect
125. C23×C9C3×S4semidirect
179. H(5,𝔽2)D3×3−1+2semidirect
185. 3−1+2#5934direct (N × H)
257. C9C3×#5934semidirect
261. C23S4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#9972
C32×#59343
C9×#59343
3−1+2×B(4,𝔽2)3
C2×S4×3−1+24