Finite Group Databasebeta

← #13175384

Subgroups — #13175384

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×#64078C211yesmaximal, characteristic
3C32×C4×Dic6C211yesmaximal, characteristic
4#11723536C212yesmaximal
5#11723534C212yesmaximal
6#11723399C212yesmaximal
7#11723397C212yesmaximal
8#11723394C214yesmaximal
9Q8×#634—31nomaximal
10#10593856—34nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2212yes—
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293C2214yes—
15C33×C42C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857C2212yes—
18C4×C3⋊(C3⋊Dic3)C2212yes—
19C3×C4⋊(C3⋊Dic3)C2212yes—
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C32×C4⋊Dic3C2211yescharacteristic
22C32×Dic3⋊C4C2212yes—
23C32×C4×Dic3C2212yes—
24#88806C2214yes—
25Dic3⋊(C3⋊Dic3)C2214yes—
26Dic3×C3⋊Dic3C2214yes—
27C32×C4×Q8—31no—
28Q8×C3⋊Dic3—32no—
29#64097—34no—
30#64097—32no—
31#64078—32no—
32#64078D311yescharacteristic
33C4×#634—31no—
34C4×#634—32no—
35C4×#634—34no—
36C4×#634—64no—
37C3×#869—34no—
38C3×C4×Dic6D314yes—
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⋊Dic3C2314yes—
51C3×#634—42no—
52C2×C32×Dic3C2312yes—
53C2×C32×Dic3C2312yes—
54C32×Dic6Q814yescomplemented
55#6293—44no—
56#5722—91no—
57Q8×Dic6—94no—
58C2×C32×Q8—31no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—34no—
62C2×#634—64no—
63C2×#634C2×D312yes—
64C2×C3×Dic6—38no—
65C2×C3×Dic6C2×D314yes—
66C2×#627—316no—
67C32×C4⋊C4—32no—
68C32×C4⋊C4—31no—
69C32×C42—24no—
70C32×C42—32no—
71C32×C42C2×D314yes—
72C32×C42C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—34no—
76C4⋊(C3⋊Dic3)—64no—
77C4⋊(C3⋊Dic3)C2×D312yes—
78C4⋊(C3⋊Dic3)C2×D312yes—
79#3750—32no—
80#3750—38no—
81#3750—34no—
82#3750—68no—
83C4×C3⋊Dic3—32no—
84C4×C3⋊Dic3—38no—
85C4×C3⋊Dic3—34no—
86C4×C3⋊Dic3—68no—
87C3×C4⋊Dic3—38no—
88C3×C4⋊Dic3—38no—
89C3×C4⋊Dic3C2×D314yes—
90C3×Dic3⋊C4C2×D318yes—
91C3×C4×Dic3C2×D318yes—
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—24no—
102C32×Dic3—22no—
103C32×Dic3C2×Q814yes—
104C3×C4⋊Q8—91no—
105C3×C4×Q8—34no—
106Q8×Dic3—98no—
107#944—94no—
108#888—98no—
109#888—916no—
110#888—92no—
111#888—92no—
112#887—92no—
113#887—92no—
114#869—34no—
115#869—98no—
116C4×Dic6—98no—
117C4×Dic6—94no—
118C4×Dic6—91no—
119C4×Dic6—184no—
120C4×Dic6C3⋊D311yescharacteristic
121C32×Q8—34no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—32no—
126C2×C32×C4—32no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—32no—
134C2×C3⋊Dic3—32no—
135C2×C3⋊Dic3—38no—
136C2×C3⋊Dic3—38no—
137C2×C3⋊Dic3—68no—
138C2×C3⋊Dic3—68no—
139C2×C3⋊Dic3C22×D314yes—
140#634—34no—
141#634—316no—
142#634—42no—
143#634—616no—
144#634—124no—
145C2×C3×Dic3—316no—
146C2×C3×Dic3C22×D318yes—
147C2×C3×Dic3C22×D318yes—
148C3×Dic6—128no—
149C3×Dic6Dic6116yescomplemented
150#627—1216no—
151Q82—271noSylow
152C2×C33C22×Q811yescharacteristic
153C2×C33C22×Q811yescharacteristic
154C2×C33H(5,𝔽2)11yescharacteristic
155C2×C3×Q8—34no—
156C2×C3×Q8—92no—
157C2×Dic6—38no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—916no—
161C2×Dic6—94no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×C4⋊C4—38no—
165C3×C4⋊C4—34no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—92no—
168C3×C42—24no—
169C3×C42—38no—
170C3×C42D3214yes—
171C3×C42C2×C3⋊D311yescharacteristic
172C4⋊Dic3—38no—
173C4⋊Dic3—38no—
174C4⋊Dic3—916no—
175C4⋊Dic3—94no—
176C4⋊Dic3—91no—
177C4⋊Dic3—184no—
178C4⋊Dic3C2×C3⋊D311yescharacteristic
179Dic3⋊C4—916no—
180Dic3⋊C4—98no—
181Dic3⋊C4—92no—
182Dic3⋊C4—94no—
183Dic3⋊C4—94no—
184Dic3⋊C4—188no—
185Dic3⋊C4C2×C3⋊D312yes—
186C4×Dic3—916no—
187C4×Dic3—98no—
188C4×Dic3—92no—
189C4×Dic3—94no—
190C4×Dic3—188no—
191C4×Dic3C2×C3⋊D312yes—
192C22×C32—24no—
193C22×C32C23×D314yes—
194C22×C32C23×D311yescharacteristic
195C32×C4—21no—
196C32×C4—28no—
197C32×C4—28no—
198C32×C4—24no—
199C32×C4—34no—
200C32×C4—44no—
201C32×C4—62no—
202C32×C4C2×Dic618yes—
203C32×C4D3×Q818yes—
204C32×C4D3×Q812yes—
205C32×C4D3×Q812yes—
206C3⋊Dic3—24no—
207C3⋊Dic3—34no—
208C3⋊Dic3—316no—
209C3⋊Dic3—616no—
210C3⋊Dic3—62no—
211C3⋊Dic3—68no—
212C3⋊Dic3—128no—
213C3×Dic3—28no—
214C3×Dic3—616no—
215C3×Dic3C2×Dic6116yes—
216C4⋊Q8—91no—
217C4⋊Q8—272no—
218C4⋊Q8—272no—
219C4⋊Q8—274no—
220C4×Q8—31no—
221C4×Q8—272no—
222C4×Q8—272no—
223C4×Q8—271no—
224C33Q8211yescharacteristic, complemented, Sylow
225C3×Q8—316no—
226C3×Q8—362no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—38no—
231C2×C3×C4—38no—
232C2×C3×C4—94no—
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—316no—
240C2×Dic3—98no—
241C2×Dic3—92no—
242C2×Dic3—98no—
243C2×Dic3—92no—
244C2×Dic3—188no—
245C2×Dic3—188no—
246C2×Dic3C22×C3⋊D312yes—
247C2×Dic3C22×C3⋊D312yes—
248Dic6—916no—
249Dic6—94no—
250Dic6—128no—
251Dic6—1816no—
252Dic6—364no—
253Dic6—3616no—
254Dic6#63414yescomplemented
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32C22×Dic614yes—
259C2×C32#100411yescharacteristic
260C2×C32C2×D3×Q814yes—
261C2×C32C2×D3×Q811yescharacteristic
262C2×C32C2×D3×Q811yescharacteristic
263C2×C32#100914yes—
264C2×Q8—31no—
265C2×Q8—92no—
266C2×Q8—274no—
267C2×Q8—271no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—92no—
271C4⋊C4—92no—
272C4⋊C4—274no—
273C4⋊C4—274no—
274C4⋊C4—272no—
275C4⋊C4—271no—
276C42—32no—
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—316no—
288C3×C4—44no—
289C3×C4—68no—
290C3×C4—184no—
291C3×C4D3×Dic618yes—
292C3×C4#379718yes—
293C3×C4C2×#63412yes—
294C3×C4Q8×C3⋊D312yes—
295Dic3—22no—
296Dic3—616no—
297Dic3—916no—
298Dic3—94no—
299Dic3—1816no—
300Dic3—188no—
301Dic3—182no—
302Dic3—368no—
303Dic3C2×#63414yes—
304C32—24no—
305C32Q8×Dic614yescomplemented
306C32#572211yescharacteristic, complemented
307Q8—34no—
308Q8—274no—
309Q8—362no—
310Q8—1084no—
311C2×C4—32no—
312C2×C4—32no—
313C2×C4—94no—
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×C3C22×#63411yescharacteristic
326C2×C3C2×Q8×C3⋊D311yescharacteristic
327C2×C3#6436611yescharacteristic
328C22C22×D3×C3⋊D311yescharacteristic
329C4—21no—
330C4—34no—
331C4—62no—
332C4—184no—
333C4—274no—
334C4—542no—
335C4D3×#63412yes—
336C4#8901912yes—
337C3—24no—
338C3#1059385614yescomplemented
339C3Q8×#63411yescharacteristic, complemented
340C2C2×D3×#63411yescharacteristic
341C2C2×#8901911yescharacteristic
342C2#1172458011yescharacteristic
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
54. C32×Dic6Q8semidirect
149. C3×Dic6Dic6semidirect
224. C33Q82semidirect
254. Dic6#634semidirect
305. C32Q8×Dic6semidirect
306. C32#5722semidirect
338. C3#10593856semidirect
339. C3Q8×#634semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×#640782
C32×C4×Dic62
#117235362
#117235342
#117233992
#117233972
#117233942
Q8×#6343
#105938563