Finite Group Databasebeta

← #13175385

Subgroups — #13175385

Order 1728.

1552 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic
3C32×#869C211yesmaximal, characteristic
4#11723536C212yesmaximal
5#11723535C212yesmaximal
6#11723410C212yesmaximal
7#11723406C212yesmaximal
8#11723393C214yesmaximal
9#10596193—31nomaximal
10#10593856—34nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2212yes—
14C2×#6293C2214yes—
15C33×C42C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857C2212yes—
18C4×C3⋊(C3⋊Dic3)C2212yes—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×#3750C2212yes—
21C3×C4×C3⋊Dic3C2212yes—
22C32×C4⋊Dic3C2212yes—
23C32×C4⋊Dic3C2212yes—
24#88807C2214yes—
25#88806C2214yes—
26Dic3×C3⋊Dic3C2214yes—
27C32×C4⋊Q8—31no—
28#64153—34no—
29#64097—32no—
30#64097—32no—
31#64096—32no—
32#64096—32no—
33C4×#634—31no—
34C4×#634—34no—
35C4×#634—64no—
36C4×#634D311yescharacteristic
37C3×#869D314yes—
38C3×C4×Dic6—34no—
39#63919—38no—
40#63918—38no—
41#63844—38no—
42#63843—38no—
43#63839—316no—
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)C2312yes—
48C2×C3⋊(C3⋊Dic3)C2312yes—
49#6308Q814yescomplemented
50C2×C3×C3⋊Dic3C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C3×#634Q814yescomplemented
53C2×C32×Dic3C2314yes—
54C32×Dic6—42no—
55#6293—44no—
56#5722—94no—
57Q8×Dic6—91no—
58C2×C32×Q8—32no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—34no—
62C2×#634—64no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D318yes—
66C2×#627—316no—
67C32×C4⋊C4—32no—
68C32×C4⋊C4—32no—
69C32×C42—24no—
70C32×C42C2×D314yes—
71C32×C42C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—32no—
77#3750—38no—
78#3750—34no—
79#3750—34no—
80#3750—68no—
81#3750C2×D312yes—
82C4×C3⋊Dic3—32no—
83C4×C3⋊Dic3—38no—
84C4×C3⋊Dic3—34no—
85C4×C3⋊Dic3—68no—
86C4×C3⋊Dic3C2×D312yes—
87C3×C4⋊Dic3—34no—
88C3×C4⋊Dic3C2×D318yes—
89C3×C4⋊Dic3C2×D318yes—
90C3×Dic3⋊C4—38no—
91C3×C4×Dic3—38no—
92#3724—316no—
93Dic3⋊Dic3—316no—
94Dic32—316no—
95C22×C33C2411yescharacteristic
96C33×C4—21no—
97C33×C4C2×Q812yes—
98C33×C4C2×Q812yes—
99C3⋊(C3⋊Dic3)—22no—
100C3⋊(C3⋊Dic3)C2×Q814yes—
101C3×C3⋊Dic3—22no—
102C3×C3⋊Dic3C2×Q814yes—
103C32×Dic3—24no—
104C3×C4⋊Q8—34no—
105C3×C4×Q8—91no—
106Q8×Dic3—92no—
107#944—916no—
108#888—98no—
109#888—98no—
110#888—94no—
111#888—92no—
112#887—98no—
113#887—98no—
114#869—92no—
115#869C3⋊D311yescharacteristic
116C4×Dic6—34no—
117C4×Dic6—94no—
118C4×Dic6—91no—
119C4×Dic6—92no—
120C4×Dic6—184no—
121C32×Q8—122no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—34no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—32no—
133C2×C3⋊Dic3—32no—
134C2×C3⋊Dic3—38no—
135C2×C3⋊Dic3—38no—
136C2×C3⋊Dic3—68no—
137C2×C3⋊Dic3—68no—
138C2×C3⋊Dic3C22×D312yes—
139C2×C3⋊Dic3C22×D312yes—
140#634—34no—
141#634—316no—
142#634—616no—
143#634—124no—
144#634Dic614yescomplemented
145C2×C3×Dic3—38no—
146C2×C3×Dic3—38no—
147C2×C3×Dic3C22×D3116yes—
148C3×Dic6—316no—
149C3×Dic6—48no—
150#627—1216no—
151Q82—271noSylow
152C2×C33C22×Q811yescharacteristic
153C2×C33C22×Q811yescharacteristic
154C2×C33H(5,𝔽2)11yescharacteristic
155C2×C3×Q8—38no—
156C2×C3×Q8—91no—
157C2×Dic6—34no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—916no—
161C2×Dic6—94no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D312yes—
164C3×C4⋊C4—38no—
165C3×C4⋊C4—38no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—91no—
168C3×C42—24no—
169C3×C42—92no—
170C3×C42D3214yes—
171C3×C42C2×C3⋊D311yescharacteristic
172C4⋊Dic3—34no—
173C4⋊Dic3—94no—
174C4⋊Dic3—91no—
175C4⋊Dic3—94no—
176C4⋊Dic3—184no—
177C4⋊Dic3C2×C3⋊D312yes—
178C4⋊Dic3C2×C3⋊D312yes—
179Dic3⋊C4—38no—
180Dic3⋊C4—916no—
181Dic3⋊C4—916no—
182Dic3⋊C4—98no—
183Dic3⋊C4—92no—
184Dic3⋊C4—94no—
185Dic3⋊C4—188no—
186C4×Dic3—38no—
187C4×Dic3—916no—
188C4×Dic3—98no—
189C4×Dic3—92no—
190C4×Dic3—94no—
191C4×Dic3—188no—
192C22×C32—24no—
193C22×C32C23×D314yes—
194C22×C32C23×D311yescharacteristic
195C32×C4—21no—
196C32×C4—28no—
197C32×C4—28no—
198C32×C4—24no—
199C32×C4—44no—
200C32×C4—64no—
201C32×C4C2×Dic612yes—
202C32×C4D3×Q818yes—
203C32×C4D3×Q818yes—
204C32×C4D3×Q812yes—
205C3⋊Dic3—22no—
206C3⋊Dic3—34no—
207C3⋊Dic3—316no—
208C3⋊Dic3—616no—
209C3⋊Dic3—62no—
210C3⋊Dic3—68no—
211C3⋊Dic3—128no—
212C3⋊Dic3C2×Dic614yes—
213C3×Dic3—216no—
214C3×Dic3—316no—
215C3×Dic3—68no—
216C4⋊Q8—31no—
217C4⋊Q8—272no—
218C4⋊Q8—272no—
219C4⋊Q8—274no—
220C4×Q8—91no—
221C4×Q8—272no—
222C4×Q8—272no—
223C4×Q8—271no—
224C33Q8211yescharacteristic, complemented, Sylow
225C3×Q8—94no—
226C3×Q8—128no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—316no—
231C2×C3×C4—92no—
232C2×C3×C4—92no—
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C2×D3214yes—
235C2×C3×C4C2×D3214yes—
236C2×C3×C4C22×C3⋊D311yescharacteristic
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—38no—
240C2×Dic3—38no—
241C2×Dic3—98no—
242C2×Dic3—92no—
243C2×Dic3—98no—
244C2×Dic3—92no—
245C2×Dic3—188no—
246C2×Dic3—188no—
247C2×Dic3C22×C3⋊D314yes—
248Dic6—316no—
249Dic6—42no—
250Dic6—916no—
251Dic6—94no—
252Dic6—1816no—
253Dic6—364no—
254Dic6—3616no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32C22×Dic611yescharacteristic
259C2×C32#100414yes—
260C2×C32C2×D3×Q814yes—
261C2×C32C2×D3×Q814yes—
262C2×C32C2×D3×Q811yescharacteristic
263C2×C32#100911yescharacteristic
264C2×Q8—32no—
265C2×Q8—91no—
266C2×Q8—274no—
267C2×Q8—271no—
268C4⋊C4—32no—
269C4⋊C4—32no—
270C4⋊C4—92no—
271C4⋊C4—91no—
272C4⋊C4—274no—
273C4⋊C4—274no—
274C4⋊C4—272no—
275C4⋊C4—271no—
276C42—92no—
277C42—274no—
278C42—272no—
279C42D3×C3⋊D311yescharacteristic
280C22×C3—24no—
281C22×C3C22×D3214yes—
282C22×C3C23×C3⋊D311yescharacteristic
283C3×C4—24no—
284C3×C4—21no—
285C3×C4—28no—
286C3×C4—28no—
287C3×C4—44no—
288C3×C4—616no—
289C3×C4—94no—
290C3×C4—182no—
291C3×C4D3×Dic618yes—
292C3×C4#379718yes—
293C3×C4Q8×C3⋊D312yes—
294C3×C4Q8×C3⋊D312yes—
295Dic3—24no—
296Dic3—316no—
297Dic3—68no—
298Dic3—916no—
299Dic3—94no—
300Dic3—1816no—
301Dic3—188no—
302Dic3—182no—
303Dic3—368no—
304C32—24no—
305C32Q8×Dic611yescharacteristic, complemented
306C32#572214yescomplemented
307Q8—94no—
308Q8—122no—
309Q8—274no—
310Q8—1084no—
311C2×C4—34no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—272no—
315C2×C4—272no—
316C2×C4C2×D3×C3⋊D311yescharacteristic
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3C2×D3×Dic614yes—
323C2×C3C2×#379714yes—
324C2×C3#6430714yes—
325C2×C3#6436111yescharacteristic
326C2×C3C2×Q8×C3⋊D311yescharacteristic
327C2×C3C2×Q8×C3⋊D311yescharacteristic
328C22C22×D3×C3⋊D311yescharacteristic
329C4—21no—
330C4—64no—
331C4—94no—
332C4—182no—
333C4—274no—
334C4—542no—
335C4C3⋊D3×Dic612yes—
336C4#8901912yes—
337C3—24no—
338C3#1059385614yescomplemented
339C3#1059619311yescharacteristic, complemented
340C2C2×C3⋊D3×Dic611yescharacteristic
341C2C2×#8901911yescharacteristic
342C2#1172457911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, prime radical, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33Q82
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
49. #6308Q8semidirect
52. C3×#634Q8semidirect
144. #634Dic6semidirect
224. C33Q82semidirect
305. C32Q8×Dic6semidirect
306. C32#5722semidirect
338. C3#10593856semidirect
339. C3#10596193semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×C4×#6342
C32×#8692
#117235362
#117235352
#117234102
#117234062
#117233932
#105961933
#105938563