Finite Group Databasebeta

← #72228415

Subgroups — #72228415

Order 2916.

562 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#11758104C311yesmaximal, characteristic, complemented
2C22×#6909C311yesmaximal, characteristic, complemented
3C3×#66366C311yesmaximal, characteristic, complemented
4#11757856C311yesmaximal, characteristic, complemented
5#11757851C311yesmaximal, characteristic, complemented
6#11757850C311yesmaximal, characteristic, complemented
7#11757850C311yesmaximal, characteristic, complemented
8#11757841C311yesmaximal, characteristic, complemented
9#11757841C311yesmaximal, characteristic, complemented
10C3×#66363C311yesmaximal, characteristic, complemented
11(C3×A4)⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
12#11757838C311yesmaximal, characteristic, complemented
13#11757837C311yesmaximal, characteristic, complemented
14#10624749—41nomaximal, Sylow
15C2×#6909—31no—
16C33×A4C3211yescharacteristic
17C22×C3×3−1+2C3211yescharacteristic
18C22×C3×3−1+2C3211yescharacteristic
19C22×C3×H(3,𝔽3)C3211yescharacteristic
20C22×C3×H(3,𝔽3)C3211yescharacteristic
21C3×C22⋊H(3,𝔽3)—33no—
22C3×C22⋊H(3,𝔽3)C3211yescharacteristic
23C3×C22⋊H(3,𝔽3)C3211yescharacteristic
24C3×C22⋊H(3,𝔽3)C3211yescharacteristic
25C3×#1094C3211yescharacteristic
26C3×#1094C3211yescharacteristic
27C3×#1094C3211yescharacteristic
28C3×#1094C3211yescharacteristic
29C3×#1094C3211yescharacteristic
30#66440—33no—
31#66440—33no—
32A4×3−1+2—33no—
33#66436—33no—
34C22×AΣL(1,𝔽27)—33no—
35#66366—33no—
36#66366—33no—
37#66365—33no—
38#66365—33no—
39#66363—33no—
40#66363—33no—
41#66363—33no—
42#66362—33no—
43#66361—33no—
44(C3×A4)⋊C9—33no—
453−1+4—41no—
46#6911—41no—
47#6910—41no—
48#6910—41no—
49#6910—41no—
50#6909—41no—
51#6909—41no—
52#6909—41no—
53#6909A411yescharacteristic, complemented
54C3×AΣL(1,𝔽27)—41no—
55C3×AΣL(1,𝔽27)—41no—
56#6891—41no—
57C32⋊H(3,𝔽3)—41no—
58C2×C3×3−1+2—31no—
59C2×C3×3−1+2—31no—
60C2×C3×H(3,𝔽3)—31no—
61C2×C3×H(3,𝔽3)—31no—
62C2×AΣL(1,𝔽27)—39no—
63C22×C33—31no—
64C22×C33—31no—
65C22×C33C3311yescharacteristic
66C32×A4—31no—
67C32×A4—31no—
68C32×A4—31no—
69C32×A4—31no—
70C32×A4—31no—
71C32×A4—31no—
72C32×A4—93no—
73C32×A4H(3,𝔽3)13yes—
74C22×3−1+2—31no—
75C22×3−1+2—31no—
76C22×3−1+2—31no—
77C22×3−1+2—31no—
78C22×3−1+2—31no—
79C22×3−1+2H(3,𝔽3)13yescomplemented
80C22×H(3,𝔽3)—31no—
81C22×H(3,𝔽3)—31no—
82C22×H(3,𝔽3)—31no—
83C22×H(3,𝔽3)—31no—
84C22×H(3,𝔽3)—31no—
85C22×H(3,𝔽3)—31no—
86C22×C3×C9—31no—
87C22×C3×C9—31no—
88C22⋊H(3,𝔽3)—31no—
89C22⋊H(3,𝔽3)—31no—
90C22⋊H(3,𝔽3)—31no—
91C22⋊H(3,𝔽3)—31no—
92C22⋊H(3,𝔽3)—31no—
93C22⋊H(3,𝔽3)—31no—
94C22⋊H(3,𝔽3)—31no—
95C22⋊H(3,𝔽3)—31no—
96C22⋊H(3,𝔽3)—31no—
97C22⋊H(3,𝔽3)—99no—
98#1094—31no—
99#1094—31no—
100#1094—31no—
101#1094—31no—
102#1094—31no—
103#1094—31no—
104#1094—31no—
105#1094—31no—
106#1094—31no—
107#1094—31no—
108#1094—31no—
109#1094—31no—
110#1094—31no—
111#1094—31no—
112#1094H(3,𝔽3)13yescomplemented
113C3×C22⋊C9—31no—
114C3×C22⋊C9—31no—
115C3×C22⋊C9—31no—
116C3×C22⋊C9—31no—
117C3×C22⋊C9H(3,𝔽3)13yescomplemented
118C9⋊A4—318no—
119C9×A4—39no—
120C34—41no—
121#737—123no—
122#737—123no—
123#737—123no—
124C3×3−1+2—41no—
125C3×3−1+2—41no—
126C3×3−1+2—41no—
127C3×3−1+2—41no—
128C3×3−1+2—41no—
129C3×3−1+2—123no—
130C3×3−1+2C3×A411yescharacteristic
131C3×3−1+2C3×A411yescharacteristic
132C3×H(3,𝔽3)—41no—
133C3×H(3,𝔽3)—41no—
134C3×H(3,𝔽3)—41no—
135C3×H(3,𝔽3)—123no—
136C3×H(3,𝔽3)C3×A411yescharacteristic
137C3×H(3,𝔽3)C3×A411yescharacteristic
138#732—123no—
139#731—123no—
140#731—123no—
141#731—123no—
142AΣL(1,𝔽27)—33no—
143AΣL(1,𝔽27)—123no—
144AΣL(1,𝔽27)—123no—
145AΣL(1,𝔽27)—123no—
146AΣL(1,𝔽27)—123no—
147AΣL(1,𝔽27)—123no—
148C32⋊C9—123no—
149C2×C33—31no—
150C2×C33—33no—
151C2×C33—91no—
152C2×3−1+2—33no—
153C2×3−1+2—33no—
154C2×3−1+2—91no—
155C2×3−1+2—91no—
156C2×3−1+2—91no—
157C2×3−1+2—91no—
158C2×H(3,𝔽3)—33no—
159C2×H(3,𝔽3)—91no—
160C2×H(3,𝔽3)—91no—
161C2×H(3,𝔽3)—91no—
162C2×H(3,𝔽3)—91no—
163C2×H(3,𝔽3)—91no—
164C2×C3×C9—91no—
165C2×C3×C9—91no—
166C22×C32—91no—
167C22×C32—93no—
168C22×C32—91no—
169C22×C32—91no—
170C22×C32—271no—
171C22×C32C3×H(3,𝔽3)11yescharacteristic
172C22×C32C3×H(3,𝔽3)11yescharacteristic
173C22×C32C3×H(3,𝔽3)11yescharacteristic
174C22×C32C3×H(3,𝔽3)11yescharacteristic
175C3×A4—33no—
176C3×A4—33no—
177C3×A4—33no—
178C3×A4—92no—
179C3×A4—92no—
180C3×A4—91no—
181C3×A4—93no—
182C3×A4—93no—
183C3×A4—93no—
184C3×A4—93no—
185C3×A4—91no—
186C3×A4—91no—
187C3×A4—271no—
188C22×C9—33no—
189C22×C9—91no—
190C22⋊C9—33no—
191C22⋊C9—33no—
192C22⋊C9—91no—
193C22⋊C9—91no—
194C22⋊C9—91no—
195C33—31no—
196C33—31no—
197C33—43no—
198C33—121no—
199C33—121no—
200C33—121no—
201C33—121no—
202C33—121no—
203C33—121no—
204C33—363no—
205C33C32×A411yescharacteristic
2063−1+2—31no—
2073−1+2—31no—
2083−1+2—31no—
2093−1+2—31no—
2103−1+2—31no—
2113−1+2—43no—
2123−1+2—1218no—
2133−1+2—121no—
2143−1+2—121no—
2153−1+2—121no—
2163−1+2—121no—
2173−1+2—121no—
2183−1+2—121no—
2193−1+2—121no—
2203−1+2—121no—
2213−1+2—121no—
2223−1+2—121no—
2233−1+2—121no—
2243−1+2—121no—
2253−1+2—121no—
2263−1+2—121no—
2273−1+2C22⋊H(3,𝔽3)13yescomplemented
228H(3,𝔽3)—31no—
229H(3,𝔽3)—31no—
230H(3,𝔽3)—31no—
231H(3,𝔽3)—31no—
232H(3,𝔽3)—31no—
233H(3,𝔽3)—31no—
234H(3,𝔽3)—121no—
235H(3,𝔽3)—121no—
236H(3,𝔽3)—121no—
237H(3,𝔽3)—121no—
238H(3,𝔽3)—121no—
239H(3,𝔽3)—121no—
240H(3,𝔽3)—121no—
241H(3,𝔽3)—121no—
242H(3,𝔽3)—121no—
243H(3,𝔽3)—369no—
244C3×C9—31no—
245C3×C9—31no—
246C3×C9—43no—
247C3×C9—129no—
248C3×C9—121no—
249C3×C9—121no—
250C3×C9—121no—
251C3×C9—121no—
252C2×C32—31no—
253C2×C32—31no—
254C2×C32—31no—
255C2×C32—31no—
256C2×C32—93no—
257C2×C32—93no—
258C2×C32—271no—
259C2×C32—273no—
260C2×C32—273no—
261C2×C9—93no—
262C2×C9—93no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—271no—
268C22×C3—271no—
269C22×C3C32⋊H(3,𝔽3)11yescharacteristic
270A4—93no—
271A4—93no—
272A4—271no—
273A4—271no—
274C32—91no—
275C32—93no—
276C32—91no—
277C32—91no—
278C32—123no—
279C32—123no—
280C32—123no—
281C32—271no—
282C32—363no—
283C32—363no—
284C32—363no—
285C32—362no—
286C32—362no—
287C32—363no—
288C32—361no—
289C32—361no—
290C32—361no—
291C32—1081no—
292C32A4×H(3,𝔽3)11yescharacteristic
293C32A4×H(3,𝔽3)11yescharacteristic
294C32A4×H(3,𝔽3)11yescharacteristic
295C32C3×C22⋊H(3,𝔽3)11yescharacteristic
296C9—33no—
297C9—91no—
298C9—123no—
299C9—123no—
300C9—361no—
301C9—361no—
302C9—361no—
303C2×C3—31no—
304C2×C3—91no—
305C2×C3—91no—
306C2×C3—91no—
307C2×C3—91no—
308C2×C3—273no—
309C2×C3—273no—
310C22#1062474911yescharacteristic, complemented, Sylow
311C3—31no—
312C3—31no—
313C3—31no—
314C3—31no—
315C3—271no—
316C3—271no—
317C3—363no—
318C3—363no—
319C3—1081no—
320C3—1081no—
321C3(C3×A4)⋊H(3,𝔽3)11yescharacteristic
322C2—31no—
subgroupG/H
centerC3(C3×A4)⋊H(3,𝔽3)
commutator, cosocleC22×C33C33
FrattiniC33C32×A4
FittingC22×#6909C3
socleC22×C3C32⋊H(3,𝔽3)
hypercenter#6909A4
hypocenterC22#10624749
LCSG ▷ C22×C33 ▷ C22×C3 ▷ C22
UCSC1 ◁ C3 ◁ C33 ◁ #6909

Complements

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

NH ≅ G/Ntype
1. #11758104C3semidirect
2. C22×#6909C3semidirect
3. C3×#66366C3semidirect
4. #11757856C3semidirect
5. #11757851C3semidirect
6. #11757850C3semidirect
7. #11757850C3semidirect
8. #11757841C3semidirect
9. #11757841C3semidirect
10. C3×#66363C3semidirect
11. (C3×A4)⋊H(3,𝔽3)C3semidirect
12. #11757838C3semidirect
13. #11757837C3semidirect
53. #6909A4semidirect
79. C22×3−1+2H(3,𝔽3)semidirect
112. #1094H(3,𝔽3)semidirect
117. C3×C22⋊C9H(3,𝔽3)semidirect
227. 3−1+2C22⋊H(3,𝔽3)semidirect
310. C22#10624749semidirect

Maximal subgroups

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

subgroupindex
#117581043
C22×#69093
C3×#663663
#117578563
#117578513
#117578503
#117578413
C3×#663633
(C3×A4)⋊H(3,𝔽3)3
#117578383
#117578373
#106247494