Finite Group Databasebeta

← #13147752

Subgroups — #13147752

Order 1728.

728 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 139 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#11721716C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#11721108C214yesmaximal
5#11721106C212yesmaximal
6#11721086C214yesmaximal, complemented
7#11721082C212yesmaximal, complemented
8#10593881—31nomaximal
9#10593867—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2212yes—
14#88540C2214yes—
15#3752⋊C3C2212yes—
16C4⋊((C3⋊Dic3)⋊C3)C2214yes—
17#3750⋊C3C2212yes—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19#88450C2214yes—
20#88449C2214yes—
21#88448C2214yes—
22#88447C2214yes—
23#64081D311yescharacteristic
24#64078D311yescharacteristic, complemented
25C3×#872—31no—
26C3×#870—31no—
27C3×#870—31no—
28C3×#870—61no—
29C3×#869—31no—
30#63871—32no—
31#63861—34no—
32#63857—32no—
33#63848—34no—
34#63846—34no—
35#63845—34no—
36#63844—32no—
37#63843—32no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2314yes—
42C2×(C3⋊Dic3)⋊C3C2314yes—
43C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C4×(C3⋊D3)⋊C3—22no—
46#634⋊C3—24no—
47#5731—91no—
48#5698—91no—
49#5518—91no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×#634C2×D312yes—
52C2×C3×C4×D3—31no—
53C2×C3×Dic6—32no—
54C32×C42—21no—
55C32×C42C2×D311yescharacteristic
56C32×C42C2×D311yescharacteristic
57#3752C2×D312yes—
58C4⋊(C3⋊Dic3)C2×D314yes—
59#3750C2×D312yes—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C3×(C2×D3)⋊C4—32no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—34no—
65C3×C4⋊Dic3—62no—
66C3×Dic3⋊C4—32no—
67C3×Dic3⋊C4—34no—
68C3×Dic3⋊C4—34no—
69C3×Dic3⋊C4—64no—
70C3×C4×Dic3—31no—
71#3724—34no—
72Dic3⋊Dic3—34no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75#3722—34no—
76(C2×D3)⋊Dic3—34no—
77Dic32—34no—
78Dic32—34no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—22no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)#4112yes—
84#1084—24no—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—22no—
88C3×C4⋊Q8—91no—
89C3×#125—91no—
90C3×#117—91no—
91#946—92no—
92Q8×Dic3—92no—
93#924—92no—
94#898—94no—
95#898—94no—
96#897—92no—
97#890—94no—
98#889—94no—
99#889—94no—
100#887—92no—
101#879—94no—
102#872—31no—
103#872—91no—
104#870—31no—
105#870—31no—
106#870—61no—
107#869—31no—
108#869—91no—
109C4×Dic6—92no—
110C22×C3⋊D3C22×D311yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C22×D312yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3C22×D312yes—
119C2×C3⋊Dic3C22×D314yes—
120C2×C3⋊Dic3C22×D311yescharacteristic
121C4×C3⋊D3—22no—
122#634—24no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×C4×D3—62no—
130C3×Dic6—64no—
131#541—271noSylow
132C2×H(3,𝔽3)C2×#4111yescharacteristic
133C2×H(3,𝔽3)2−1+412yes—
134(C3⋊D3)⋊C3—41no—
135C2×C3×Q8—92no—
136C22×C3×C4—91no—
137C2×C4×D3—31no—
138C2×C4×D3—91no—
139C2×Dic6—32no—
140C2×Dic6—92no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—94no—
143C3×C4⋊C4—92no—
144C3×C4⋊C4—94no—
145C3×C22⋊C4—92no—
146C3×C42—31no—
147C3×C42—31no—
148C3×C42—61no—
149C3×C42—91no—
150C3×C42D3211yescharacteristic
151#217—94no—
152(C2×D3)⋊C4—32no—
153(C2×D3)⋊C4—92no—
154(C2×D3)⋊C4—94no—
155(C2×D3)⋊C4—94no—
156C4⋊Dic3—32no—
157C4⋊Dic3—32no—
158C4⋊Dic3—34no—
159C4⋊Dic3—62no—
160C4⋊Dic3—94no—
161C4⋊Dic3—94no—
162C4⋊Dic3—94no—
163Dic3⋊C4—34no—
164Dic3⋊C4—34no—
165Dic3⋊C4—32no—
166Dic3⋊C4—64no—
167Dic3⋊C4—92no—
168Dic3⋊C4—94no—
169Dic3⋊C4—94no—
170Dic3⋊C4—94no—
171Dic3⋊C4—94no—
172C4×Dic3—31no—
173C4×Dic3—91no—
174C4×Dic3—94no—
175C4×Dic3—94no—
176C4×Dic3—94no—
177C22×C32—21no—
178C22×C32C23×D311yescharacteristic
179C22×C32C23×D311yescharacteristic
180C2×C3⋊D3—21no—
181C2×C3⋊D3—22no—
182C2×C3×D3—61no—
183C2×C3×D3—62no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—42no—
188C32×C4#23512yes—
189C32×C4#23712yes—
190C3⋊Dic3—21no—
191C3⋊Dic3—24no—
192C3⋊Dic3—22no—
193C3×Dic3—64no—
194C3×Dic3—61no—
195C3×Dic3—62no—
196C3×Dic3—64no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199C4⋊Q8—91no—
200#126—274no—
201#125—91no—
202#125—274no—
203C22⋊Q8—272no—
204C4×Q8—272no—
205#117—91no—
206H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
207C23×C3—91no—
208C22×D3—31no—
209C22×D3—91no—
210C3×Q8—184no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—61no—
216C2×C3×C4—62no—
217C2×C3×C4—92no—
218C2×C3×C4—94no—
219C2×C3×C4—91no—
220C2×C3×C4—94no—
221C2×C3×C4—182no—
222C2×C3×C4C2×D3212yes—
223C2×C3×C4C2×D3211yescharacteristic
224C2×Dic3—34no—
225C2×Dic3—34no—
226C2×Dic3—32no—
227C2×Dic3—34no—
228C2×Dic3—31no—
229C2×Dic3—64no—
230C2×Dic3—92no—
231C2×Dic3—94no—
232C2×Dic3—91no—
233C4×D3—62no—
234C4×D3—182no—
235Dic6—64no—
236Dic6—184no—
237C2×C32—21no—
238C2×C32—22no—
239C2×C32C2×#23511yescharacteristic
240C2×C32C2×#23711yescharacteristic
241C2×C32#100712yes—
242C2×C32#101012yes—
243C3⋊D3—41no—
244C3×D3—121no—
245C2×Q8—92no—
246C22×C4—91no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C4⋊C4—94no—
250C4⋊C4—94no—
251C4⋊C4—274no—
252C4⋊C4—274no—
253C22⋊C4—92no—
254C22⋊C4—274no—
255C42—91no—
256C42—274no—
257C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—61no—
261C22×C3—181no—
262C22×C3—182no—
263C22×C3C22×D3211yescharacteristic
264C2×D3—61no—
265C2×D3—62no—
266C2×D3—181no—
267C2×D3—182no—
268C3×C4—22no—
269C3×C4—32no—
270C3×C4—32no—
271C3×C4—62no—
272C3×C4—62no—
273C3×C4—62no—
274C3×C4—122no—
275C3×C4—181no—
276C3×C4—184no—
277C3×C4—182no—
278C3×C4—184no—
279C3×C4#379512yes—
280Dic3—64no—
281Dic3—64no—
282Dic3—64no—
283Dic3—61no—
284Dic3—62no—
285Dic3—124no—
286Dic3—184no—
287Dic3—181no—
288Dic3—182no—
289C32—21no—
290C32#551811yescharacteristic, complemented
291C32#573111yescharacteristic, complemented
292C23—91no—
293Q8—184no—
294C2×C4—92no—
295C2×C4—94no—
296C2×C4—94no—
297C2×C4—91no—
298C2×C4—182no—
299C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
300C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—32no—
304C2×C3—32no—
305C2×C3—61no—
306C2×C3—62no—
307C2×C3—361no—
308C2×C3C2×#379511yescharacteristic
309C2×C3#6429712yes—
310D3—121no—
311D3—361no—
312C22—181no—
313C22—182no—
314C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C4—22no—
316C4—181no—
317C4—184no—
318C4—184no—
319C4—182no—
320C4#8864812yes—
321C3—31no—
322C3—31no—
323C3—61no—
324C3#1059386211yescharacteristic
325C2—361no—
326C2C2×#8864811yescharacteristic
327C2#1172248212yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
3. #64078⋊C3C2semidirect
6. #11721086C2semidirect
7. #11721082C2semidirect
24. #64078D3semidirect
206. H(3,𝔽3)#541semidirect
290. C32#5518semidirect
291. C32#5731semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64081⋊C32
#64078⋊C32
#117211082
#117211062
#117210862
#117210822
#105938813
#105938673