Finite Group Databasebeta

← #72252869

Subgroups — #72252869

Order 5184.

1054 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 122 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#72252868C211yesmaximal, characteristic, complemented
2#72252867C212yesmaximal, complemented
3C32×#64218C211yesmaximal, characteristic, complemented
4#72252860C212yesmaximal, complemented
5C2×#13049399C211yesmaximal, characteristic
6#13180373—34nomaximal
7C2×#10620150C411yescharacteristic
8C2×C3⋊(C3⋊#622)C411yescharacteristic
9C2×C32×#622C2211yescharacteristic
10#13049399C411yescharacteristic
11#13049399C411yescharacteristic
12#13049399C2211yescharacteristic
13#13049399C2211yescharacteristic
14C3⋊(C3⋊#3771)C2212yescomplemented
15C32×#3771C2212yescomplemented
16C2×#88924—34no—
17#11724242—34no—
18C3×#64218D314yescomplemented
19#11724094—38no—
20#11724092—38no—
21C22×C3⋊(C3⋊(C3⋊D3))C811yescharacteristic
22C2×C32×C3⋊Dic3C811yescharacteristic
23C2×C32×C3⋊Dic3C2×C411yescharacteristic
24#10620150C811yescharacteristic
25#10620150C811yescharacteristic
26#10620150C2×C411yescharacteristic
27#10620150C2×C411yescharacteristic
28C3⋊(C3⋊#622)C2×C411yescharacteristic
29C3⋊(C3⋊#622)C2×C411yescharacteristic
30C32×#622C2×C411yescharacteristic
31C32×#622C2311yescharacteristic
32#10596481—91no—
33C3⋊D3×M32—91nomaximal
34C2×#6289—34no—
35C2×#6289—121no—
36C2×C3⋊#622—34no—
37C2×C3×#622C2×D314yes—
38#88924—34no—
39#88924—34no—
40#88924—34no—
41#88924—34no—
42C3⋊#3771—38no—
43C3×#3771C2×D318yescomplemented
44C22×C34C2×C811yescharacteristic
45C2×C3⋊(C3⋊(C3⋊D3))—22no—
46C2×C3⋊(C3⋊(C3⋊D3))C2×C811yescharacteristic
47C2×C3⋊(C3⋊(C3⋊D3))C2×C811yescharacteristic
48C32×C3⋊Dic3C2×C811yescharacteristic
49C32×C3⋊Dic3C2×C811yescharacteristic
50C32×C3⋊Dic3C2×C811yescharacteristic
51C32×C3⋊Dic3C22×C411yescharacteristic
52C2×#3787—91no—
53#64218—91no—
54#64218C3⋊D311yescharacteristic, complemented
55#64214—92no—
56#64213—92no—
57C2×C8×C3⋊D3—91no—
58C32×M32—91no—
59#63639—91no—
60#63625—92no—
61C16×C3⋊D3—92no—
62C22×C3⋊(C3⋊D3)—34no—
63C22×C3⋊(C3⋊D3)—121no—
64C22×C3⋊(C3⋊D3)—244no—
65C2×C3×C3⋊Dic3—41no—
66C2×C3×C3⋊Dic3C4×D314yes—
67C2×C32×Dic3—34no—
68C2×C32×Dic3—121no—
69#6289—34no—
70#6289—34no—
71#6289—34no—
72#6289—34no—
73#6289—121no—
74#6289—121no—
75#6289—121no—
76#6289—121no—
77C3⋊#622—34no—
78C3⋊#622—34no—
79C3×#622C4×D314yes—
80C3×#622C22×D314yes—
81D3×M32—274no—
82C2×C34—21no—
83C2×C34C22×C811yescharacteristic
84C3⋊(C3⋊(C3⋊D3))—21no—
85C3⋊(C3⋊(C3⋊D3))M3212yescomplemented
86C2×C4×C3⋊D3—91no—
87C2×C4×C3⋊D3—91no—
88C2×#624—364no—
89C2×#622—91no—
90C2×#622C2×C3⋊D311yescharacteristic
91#3787—91no—
92#3787—91no—
93#3787—91no—
94#3787—91no—
95#3771—92no—
96#3771C2×C3⋊D312yescomplemented
97C2×C32×C8—91no—
98C2×C3⋊(C3⋊C8)—91no—
99C8×C3⋊D3—91no—
100C8×C3⋊D3—91no—
101C8×C3⋊D3—91no—
102C8×C3⋊D3—91no—
103C32×C16—92no—
104C3⋊(C3⋊C16)—92no—
105C22×C33—41no—
106C22×C33—84no—
107C22×C33D3×C814yes—
108C2×C3⋊(C3⋊D3)—34no—
109C2×C3⋊(C3⋊D3)—34no—
110C2×C3⋊(C3⋊D3)—68no—
111C2×C3⋊(C3⋊D3)—121no—
112C2×C3⋊(C3⋊D3)—121no—
113C2×C3⋊(C3⋊D3)—124no—
114C2×C3⋊(C3⋊D3)—244no—
115C2×C3⋊(C3⋊D3)—244no—
116C2×C3⋊(C3⋊D3)—2416no—
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C3×C3⋊Dic3D3×C814yes—
120C3×C3⋊Dic3C2×C4×D314yes—
121C32×Dic3—34no—
122C32×Dic3—34no—
123C32×Dic3—121no—
124C32×Dic3—121no—
125C2×D3×C8—274no—
126C3×M32—94no—
127#812—274no—
128#798—278no—
129D3×C16—278no—
130C34C2×M3211yescharacteristic, complemented, Sylow
131C22×C3⋊D3—91no—
132C22×C3⋊D3—91no—
133C22×C3⋊D3—364no—
134C22×C3⋊D3—724no—
135C22×C3⋊D3—724no—
136C22×C3⋊D3—726no—
137C2×C32×C4—91no—
138C2×C32×C4—91no—
139C2×C3⋊Dic3C4×C3⋊D311yescharacteristic
140C2×C3⋊Dic3AGL(1,𝔽9)11yescharacteristic
141C4×C3⋊D3—91no—
142C4×C3⋊D3—91no—
143C4×C3⋊D3—91no—
144C4×C3⋊D3—91no—
145C4×C3⋊D3—91no—
146C4×C3⋊D3—91no—
147C4×C3⋊D3—91no—
148C4×C3⋊D3—91no—
149C2×C3×Dic3—124no—
150C2×C3×Dic3—124no—
151#624—364no—
152#624—364no—
153#624—364no—
154#624—364no—
155#622—91no—
156#622—91no—
157#622C4×C3⋊D311yescharacteristic
158#622C22×C3⋊D311yescharacteristic
159C32×C8—91no—
160C32×C8—91no—
161C3⋊(C3⋊C8)—91no—
162C3⋊(C3⋊C8)—91no—
163C2×M32—811noSylow
164C2×C33—24no—
165C2×C33—42no—
166C2×C33—41no—
167C2×C33—88no—
168C2×C33—84no—
169C2×C33C2×D3×C814yes—
170C3⋊(C3⋊D3)—38no—
171C3⋊(C3⋊D3)—64no—
172C3⋊(C3⋊D3)—122no—
173C3⋊(C3⋊D3)—122no—
174C3⋊(C3⋊D3)—248no—
175C3⋊(C3⋊D3)—248no—
176C2×C4×D3—274no—
177C2×C4×D3—1081no—
178C2×C3×C8—94no—
179C2×C3⋊C8—274no—
180D3×C8—274no—
181D3×C8—274no—
182D3×C8—274no—
183D3×C8—274no—
184C3×C16—98no—
185C3⋊C16—278no—
186C22×C32—44no—
187C22×C32—84no—
188C22×C32—86no—
189C22×C32—84no—
190C22×C32C8×C3⋊D311yescharacteristic
191C22×C32C2×AGL(1,𝔽9)11yescharacteristic
192C2×C3⋊D3—91no—
193C2×C3⋊D3—91no—
194C2×C3⋊D3—91no—
195C2×C3⋊D3—91no—
196C2×C3⋊D3—182no—
197C2×C3⋊D3—182no—
198C2×C3⋊D3—364no—
199C2×C3⋊D3—364no—
200C2×C3⋊D3—3616no—
201C2×C3⋊D3—724no—
202C2×C3⋊D3—724no—
203C2×C3⋊D3—7216no—
204C2×C3⋊D3—7216no—
205C2×C3⋊D3—724no—
206C2×C3⋊D3—724no—
207C2×C3⋊D3—726no—
208C2×C3⋊D3—726no—
209C2×C3⋊D3—7224no—
210C32×C4—91no—
211C32×C4—91no—
212C32×C4—91no—
213C32×C4—91no—
214C3⋊Dic3C8×C3⋊D311yescharacteristic
215C3⋊Dic3C2×C4×C3⋊D311yescharacteristic
216C3⋊Dic3C2×AGL(1,𝔽9)11yescharacteristic
217C3⋊Dic3C2×AGL(1,𝔽9)11yescharacteristic
218C3×Dic3—124no—
219C3×Dic3—124no—
220C3×Dic3—124no—
221C3×Dic3—124no—
222C22×C8—811no—
223M32—91no—
224M32—812no—
225M32—811no—
226C2×C16—812no—
227C33—41no—
228C33—84no—
229C33D3×M3214yescomplemented
230C22×D3—274no—
231C22×D3—1081no—
232C22×D3—2164no—
233C2×C3×C4—94no—
234C2×C3×C4—361no—
235C2×Dic3—34no—
236C2×Dic3—121no—
237C4×D3—274no—
238C4×D3—274no—
239C4×D3—274no—
240C4×D3—274no—
241C4×D3—1081no—
242C4×D3—1081no—
243C4×D3—1081no—
244C4×D3—1081no—
245C3×C8—94no—
246C3×C8—94no—
247C3⋊C8—274no—
248C3⋊C8—274no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—48no—
252C2×C32—44no—
253C2×C32—88no—
254C2×C32—88no—
255C2×C32—84no—
256C2×C32—812no—
257C2×C32—86no—
258C2×C32—84no—
259C2×C32C2×C8×C3⋊D311yescharacteristic
260C2×C32C22×AGL(1,𝔽9)11yescharacteristic
261C3⋊D3—92no—
262C3⋊D3—92no—
263C3⋊D3—181no—
264C3⋊D3—181no—
265C3⋊D3—368no—
266C3⋊D3—368no—
267C3⋊D3—728no—
268C3⋊D3—728no—
269C3⋊D3—728no—
270C3⋊D3—728no—
271C3⋊D3—7212no—
272C3⋊D3—7212no—
273C22×C4—811no—
274C2×C8—91no—
275C2×C8—811no—
276C2×C8—811no—
277C2×C8—811no—
278C2×C8—811no—
279C2×C8—811no—
280C16—92no—
281C16—812no—
282C22×C3—41no—
283C22×C3—84no—
284C22×C3D3×AGL(1,𝔽9)14yes—
285C2×D3—274no—
286C2×D3—274no—
287C2×D3—548no—
288C2×D3—1081no—
289C2×D3—1081no—
290C2×D3—1084no—
291C2×D3—2164no—
292C2×D3—2164no—
293C2×D3—21616no—
294C3×C4—94no—
295C3×C4—94no—
296C3×C4—361no—
297C3×C4—361no—
298Dic3—34no—
299Dic3—34no—
300Dic3—121no—
301Dic3—121no—
302C32—44no—
303C32—84no—
304C32—86no—
305C32—84no—
306C32C3⋊D3×M3211yescharacteristic, complemented
307C32#1059648111yescharacteristic, complemented
308C23—811no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—811no—
312C2×C4—811no—
313C2×C4—811no—
314C2×C4—811no—
315C8—91no—
316C8—91no—
317C8—811no—
318C8—811no—
319C2×C3—24no—
320C2×C3—42no—
321C2×C3—41no—
322C2×C3—88no—
323C2×C3—84no—
324C2×C3C2×D3×AGL(1,𝔽9)14yes—
325D3—278no—
326D3—544no—
327D3—1082no—
328D3—1082no—
329D3—2168no—
330D3—2168no—
331C22—811no—
332C22—811no—
333C22—1622no—
334C22C3⋊D3×AGL(1,𝔽9)11yescharacteristic
335C4—91no—
336C4—91no—
337C4—91no—
338C4—91no—
339C3—41no—
340C3—84no—
341C3#1318037314yescomplemented
342C2—21no—
343C2—812no—
344C2—1621no—
345C2C2×C3⋊D3×AGL(1,𝔽9)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×AGL(1,𝔽9)
commutator, socleC2×C34C22×C8
FittingC22×C34C2×C8
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C8
cosocleC32×#622C23
hypercenterC22C3⋊D3×AGL(1,𝔽9)
hypocenterC34C2×M32
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #72252868C2semidirect
2. #72252867C2semidirect
3. C32×#64218C2semidirect
4. #72252860C2semidirect
14. C3⋊(C3⋊#3771)C22semidirect
15. C32×#3771C22semidirect
18. C3×#64218D3semidirect
43. C3×#3771C2×D3semidirect
54. #64218C3⋊D3semidirect
85. C3⋊(C3⋊(C3⋊D3))M32semidirect
96. #3771C2×C3⋊D3semidirect
130. C34C2×M32semidirect
229. C33D3×M32semidirect
306. C32C3⋊D3×M32semidirect
307. C32#10596481semidirect
341. C3#13180373semidirect

Maximal subgroups

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

subgroupindex
#722528682
#722528672
C32×#642182
#722528602
C2×#130493992
#131803733
C3⋊D3×M329