Finite Group Databasebeta

← #13175373

Subgroups — #13175373

Order 1728.

1556 conjugacy classes of subgroups; only the 284 normal subgroups are shown.

normal subgroups only

This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 284 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#11723911C211yesmaximal, characteristic, complemented
2C3×C42×C3⋊D3C211yesmaximal, characteristic
3C32×#870C211yesmaximal, characteristic, complemented
4#11723535C212yesmaximal, complemented
5#11723418C214yesmaximal
6#11723404C212yesmaximal
7#11723390C214yesmaximal, complemented
8#10596159—31nomaximal
9#10593874—34nomaximal
10C2×C3×C4×C3⋊D3C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6293C2214yes—
13C33×C42C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2212yes—
15#88857C2214yes—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C3×C4×C3⋊Dic3C2212yes—
18C32×C4⋊Dic3C2212yes—
19C32×Dic3⋊C4C2214yes—
20#88807C2214yes—
21#88803C2214yes—
22Dic3×C3⋊Dic3C2214yes—
23C32×#125—31no—
24#64105—34no—
25#64101—32no—
26#64097—32no—
27#64096—34no—
28C42×C3⋊D3D311yescharacteristic
29#64079—31no—
30#64079—34no—
31#64079—64no—
32C3×C42×D3—34no—
33C3×#870D314yescomplemented
34#63918—38no—
35#63851—316no—
36#63841—38no—
37#63837—316no—
38C22×C3×C3⋊D3C2311yescharacteristic
39C2×C33×C4C2312yes—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2314yes—
42C2×C3×C3⋊Dic3C2311yescharacteristic
43C2×C3×C3⋊Dic3C2312yes—
44C3×C4×C3⋊D3—24no—
45C3×C4×C3⋊D3Q814yescomplemented
46C2×C32×Dic3C2314yes—
47#6293—28no—
48#5688—94no—
49#5470—91no—
50C2×C4×C3⋊D3C2×D312yes—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×#634—34no—
53C2×C3×C4×D3—34no—
54C2×C3×C4×D3—38no—
55C2×#627—316no—
56C32×C4⋊C4—34no—
57C32×C4⋊C4—32no—
58C32×C42—24no—
59C32×C42C2×D314yes—
60C32×C42C2×D311yescharacteristic
61#3752—34no—
62C4⋊(C3⋊Dic3)—32no—
63C4⋊(C3⋊Dic3)—38no—
64C4⋊(C3⋊Dic3)—68no—
65#3750—34no—
66#3750—316no—
67#3750—34no—
68#3750—616no—
69C4×C3⋊Dic3—34no—
70C4×C3⋊Dic3C2×D312yes—
71C4×C3⋊Dic3C2×D311yescharacteristic
72C3×C4⋊Dic3C2×D318yes—
73C3×Dic3⋊C4C2×D3116yes—
74C3×C4×Dic3—38no—
75C3×C4×Dic3—34no—
76Dic3⋊Dic3—316no—
77(C2×D3)⋊Dic3—316no—
78Dic32—316no—
79C22×C33C2411yescharacteristic
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3C2×Q812yes—
82C33×C4—22no—
83C33×C4C2×Q812yes—
84C3⋊(C3⋊Dic3)—24no—
85C3×C3⋊Dic3C2×Q812yes—
86C3×C3⋊Dic3#4114yes—
87C32×Dic3—24no—
88C3×#125—34no—
89C2×C3×C42—91no—
90#926—92no—
91#924—94no—
92#896—916no—
93#892—98no—
94#888—98no—
95#887—916no—
96C42×D3—34no—
97#870—94no—
98#870—91no—
99#870—184no—
100#870C3⋊D311yescharacteristic, complemented
101#869—92no—
102C4×Dic6—94no—
103C22×C3⋊D3C22×D311yescharacteristic
104C22×C3×D3—34no—
105C2×C32×C4—24no—
106C2×C32×C4—28no—
107C2×C32×C4—34no—
108C2×C32×C4C22×D318yes—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—316no—
114C2×C3⋊Dic3—616no—
115C2×C3⋊Dic3C22×D311yescharacteristic
116C2×C3⋊Dic3C22×D312yes—
117C4×C3⋊D3—24no—
118C4×C3⋊D3Dic614yescomplemented
119#634—68no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—38no—
122C2×C3×Dic3C22×D3116yes—
123C3×C4×D3—316no—
124C3×C4×D3—616no—
125#627—632no—
126#533—271noSylow
127C2×C33C22×Q811yescharacteristic
128C2×C33C2×#4112yes—
129C3×C3⋊D3—22no—
130C22×C3×C4—91no—
131C22×C3×C4—92no—
132C2×C4×D3—34no—
133C2×C4×D3—38no—
134C2×Dic6—916no—
135C2×Dic6—94no—
136C3×C4⋊C4—316no—
137C3×C4⋊C4—38no—
138C3×C42—24no—
139C3×C42—92no—
140C3×C42—91no—
141C3×C42D3214yes—
142C3×C42C2×C3⋊D311yescharacteristic
143#217—94no—
144(C2×D3)⋊C4—916no—
145C4⋊Dic3—98no—
146C4⋊Dic3—92no—
147C4⋊Dic3—94no—
148C4⋊Dic3—188no—
149C4⋊Dic3C2×C3⋊D312yes—
150Dic3⋊C4—916no—
151Dic3⋊C4—916no—
152Dic3⋊C4—94no—
153Dic3⋊C4—1816no—
154Dic3⋊C4C2×C3⋊D314yes—
155C4×Dic3—38no—
156C4×Dic3—34no—
157C4×Dic3—916no—
158C4×Dic3—94no—
159C22×C32—24no—
160C22×C32C23×D314yes—
161C22×C32C23×D311yescharacteristic
162C2×C3⋊D3—22no—
163C2×C3⋊D3C2×Dic612yes—
164C2×C3×D3—38no—
165C2×C3×D3—68no—
166C32×C4—22no—
167C32×C4—28no—
168C32×C4—28no—
169C32×C4—48no—
170C32×C4—64no—
171C32×C4C2×Dic612yes—
172C32×C4D3×Q818yes—
173C3⋊Dic3—64no—
174C3⋊Dic3—616no—
175C3⋊Dic3—1216no—
176C3⋊Dic3C2×Dic612yes—
177C3⋊Dic3#23514yes—
178C3×Dic3—216no—
179C3×Dic3—38no—
180C3×Dic3—316no—
181C4⋊Q8—272no—
182#125—31no—
183#125—271no—
184C22⋊Q8—274no—
185C4×Q8—274no—
186#117—272no—
187C2×C42—91no—
188C33#53311yescharacteristic, complemented, Sylow
189C23×C3—91no—
190C22×D3—34no—
191C2×C3×C4—24no—
192C2×C3×C4—28no—
193C2×C3×C4—316no—
194C2×C3×C4—91no—
195C2×C3×C4—92no—
196C2×C3×C4—94no—
197C2×C3×C4—184no—
198C2×C3×C4C2×D3218yes—
199C2×C3×C4C2×D3214yes—
200C2×C3×C4C22×C3⋊D312yes—
201C2×C3×C4C22×C3⋊D311yescharacteristic
202C2×Dic3—34no—
203C2×Dic3—38no—
204C2×Dic3—916no—
205C2×Dic3—94no—
206C2×Dic3—1816no—
207C2×Dic3C22×C3⋊D314yes—
208C4×D3—316no—
209C4×D3—616no—
210Dic6—188no—
211Dic6—1832no—
212C2×C32—24no—
213C2×C32—28no—
214C2×C32C22×Dic611yescharacteristic
215C2×C32C2×#23512yes—
216C2×C32C2×D3×Q814yes—
217C2×C32D3×#4118yes—
218C3⋊D3—22no—
219C3×D3—68no—
220C2×Q8—274no—
221C22×C4—91no—
222C22×C4—92no—
223C4⋊C4—32no—
224C4⋊C4—34no—
225C4⋊C4—274no—
226C4⋊C4—272no—
227C4⋊C4—274no—
228C22⋊C4—274no—
229C42—92no—
230C42—91no—
231C42—274no—
232C42D3×C3⋊D311yescharacteristic
233C22×C3—24no—
234C22×C3—92no—
235C22×C3—182no—
236C22×C3C22×D3214yes—
237C22×C3C23×C3⋊D311yescharacteristic
238C2×D3—38no—
239C2×D3—68no—
240C3×C4—28no—
241C3×C4—22no—
242C3×C4—28no—
243C3×C4—48no—
244C3×C4—616no—
245C3×C4—92no—
246C3×C4—94no—
247C3×C4D3×Dic618yes—
248C3×C4Q8×C3⋊D312yes—
249Dic3—24no—
250Dic3—38no—
251Dic3—316no—
252Dic3—1816no—
253Dic3—184no—
254Dic3—3616no—
255C32—24no—
256C32#547011yescharacteristic, complemented
257C32#568814yescomplemented
258C23—91no—
259Q8—548no—
260C2×C4—34no—
261C2×C4—91no—
262C2×C4—92no—
263C2×C4—94no—
264C2×C4—184no—
265C2×C4—274no—
266C2×C4C2×D3×C3⋊D312yes—
267C2×C4C2×D3×C3⋊D311yescharacteristic
268C2×C3—24no—
269C2×C3—28no—
270C2×C3—182no—
271C2×C3C2×D3×Dic614yes—
272C2×C3D3×#23518yes—
273C2×C3C2×Q8×C3⋊D311yescharacteristic
274C2×C3#41×C3⋊D312yes—
275D3—68no—
276C22—92no—
277C22—182no—
278C22C22×D3×C3⋊D311yescharacteristic
279C4—22no—
280C4—64no—
281C4—92no—
282C4—94no—
283C4—544no—
284C4C3⋊D3×Dic612yes—
285C3—24no—
286C3#1059387414yescomplemented
287C3#1059615911yescharacteristic, complemented
288C2—182no—
289C2C2×C3⋊D3×Dic611yescharacteristic
290C2C3⋊D3×#23512yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC42D3×C3⋊D3
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723911C2semidirect
3. C32×#870C2semidirect
4. #11723535C2semidirect
7. #11723390C2semidirect
33. C3×#870D3semidirect
45. C3×C4×C3⋊D3Q8semidirect
100. #870C3⋊D3semidirect
118. C4×C3⋊D3Dic6semidirect
188. C33#533semidirect
256. C32#5470semidirect
257. C32#5688semidirect
286. C3#10593874semidirect
287. C3#10596159semidirect

Maximal subgroups

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

subgroupindex
#117239112
C3×C42×C3⋊D32
C32×#8702
#117235352
#117234182
#117234042
#117233902
#105961593
#105938743