Finite Group Databasebeta

← #13175370

Subgroups — #13175370

Order 1728.

1872 conjugacy classes of subgroups; only the 260 normal subgroups are shown.

normal subgroups only

This group has 1872 conjugacy classes of subgroups — too many to draw. Only its 260 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#11723915C211yesmaximal, characteristic, complemented
2C3×#64081C211yesmaximal, characteristic, complemented
3C32×#870C211yesmaximal, characteristic, complemented
4#11723526C212yesmaximal, complemented
5#11723498C214yesmaximal, complemented
6#11723494C212yesmaximal, complemented
7#11723486C214yesmaximal, complemented
8#10596168—31nomaximal
9#10593989—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
11C2×C4⋊(C3⋊(C3⋊D3))C2214yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6292C2214yes—
14C2×#6289C2214yes—
15C33×C42C2211yescharacteristic
16C3×#3752C2212yes—
17C3×#3750C2212yes—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2212yes—
20C32×Dic3⋊C4C2214yes—
21#88804C2214yes—
22#88804C2214yes—
23C32×#125—31no—
24#64104—32no—
25#64104—34no—
26#64103—34no—
27#64102—32no—
28#64083—31no—
29#64083—34no—
30#64083—64no—
31#64081D311yescharacteristic, complemented
32C3×#872—34no—
33C3×#870D314yescomplemented
34#63910—316no—
35#63909—38no—
36#63897—316no—
37#63894—38no—
38C22×C3⋊(C3⋊D3)C2314yes—
39C22×C3×C3⋊D3C2311yescharacteristic
40C2×C33×C4C2312yes—
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—24no—
43C4⋊(C3⋊(C3⋊D3))—44no—
44C2×C3×C3⋊Dic3C2312yes—
45C2×C3×C3⋊Dic3C2311yescharacteristic
46C3×C4×C3⋊D3—22no—
47C2×C32×Dic3C2314yes—
48#6292—44no—
49#6289—44no—
50#5697—94no—
51#5535—91no—
52C2×C4⋊(C3⋊D3)—32no—
53C2×C4⋊(C3⋊D3)—38no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—316no—
57C2×C4⋊(C3⋊D3)—68no—
58C2×C4⋊(C3⋊D3)—616no—
59C2×C4×C3⋊D3—34no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×C4×D3—34no—
62C2×#626—316no—
63C2×#624—316no—
64C32×C4⋊C4—32no—
65C32×C4⋊C4—34no—
66C32×C42—24no—
67C32×C42C2×D314yes—
68C32×C42C2×D311yescharacteristic
69#3752—34no—
70#3752—34no—
71#3752C2×D312yes—
72#3750C2×D312yes—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C3×(C2×D3)⋊C4—38no—
75C3×C4⋊Dic3C2×D318yes—
76C3×Dic3⋊C4—38no—
77C3×Dic3⋊C4C2×D3116yes—
78C3×C4×Dic3—34no—
79#3722—316no—
80#3722—316no—
81C22×C33C2411yescharacteristic
82C2×C3⋊(C3⋊D3)—24no—
83C2×C3⋊(C3⋊D3)—24no—
84C2×C3⋊(C3⋊D3)—24no—
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—22no—
87C33×C4—22no—
88C33×C4#4112yes—
89C3×C3⋊Dic3—21no—
90C3×C3⋊Dic3—22no—
91C32×Dic3—24no—
92C3×#125—34no—
93C3×#117—91no—
94#930—92no—
95#895—916no—
96#895—98no—
97#894—916no—
98#894—94no—
99#893—98no—
100#884—94no—
101#874—94no—
102#874—91no—
103#874—184no—
104C4×D12—92no—
105#872—34no—
106#870C3⋊D311yescharacteristic, complemented
107C22×C3⋊D3—34no—
108C22×C3⋊D3—316no—
109C22×C3⋊D3—616no—
110C22×C3⋊D3C22×D311yescharacteristic
111C22×C3×D3—34no—
112C2×C32×C4—28no—
113C2×C32×C4—24no—
114C2×C32×C4—34no—
115C2×C32×C4C22×D318yes—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3C22×D312yes—
120C2×C3⋊Dic3C22×D311yescharacteristic
121C4⋊(C3⋊D3)—64no—
122C4⋊(C3⋊D3)—616no—
123C4⋊(C3⋊D3)—1216no—
124C4⋊(C3⋊D3)—124no—
125C4⋊(C3⋊D3)—1216no—
126C4⋊(C3⋊D3)—1232no—
127C4⋊(C3⋊D3)—124no—
128C4×C3⋊D3—22no—
129C4×C3⋊D3—124no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—38no—
132C2×C3×Dic3C22×D3116yes—
133C3×C4×D3—68no—
134#626—1216no—
135#624—1216no—
136#540—271noSylow
137C2×C33C2×#4111yescharacteristic
138C2×C33H(5,𝔽2)12yes—
139C3⋊(C3⋊D3)—44no—
140C3×C3⋊D3—41no—
141C22×C3×C4—91no—
142C2×#67—94no—
143C2×D12—98no—
144C2×D12—92no—
145C2×D12—916no—
146C2×D12—916no—
147C2×D12—94no—
148C2×D12—188no—
149C2×D12—1816no—
150C2×C4×D3—34no—
151C2×C4×D3—916no—
152C2×C4×D3—94no—
153C3×C4⋊C4—38no—
154C3×C4⋊C4—316no—
155C3×C4⋊C4—92no—
156C3×C22⋊C4—92no—
157C3×C42—24no—
158C3×C42—91no—
159C3×C42D3214yes—
160C3×C42C2×C3⋊D311yescharacteristic
161(C2×D3)⋊C4—38no—
162(C2×D3)⋊C4—916no—
163(C2×D3)⋊C4—916no—
164(C2×D3)⋊C4—94no—
165(C2×D3)⋊C4—94no—
166C4⋊Dic3C2×C3⋊D312yes—
167Dic3⋊C4—38no—
168Dic3⋊C4C2×C3⋊D314yes—
169C4×Dic3—34no—
170C22×C32—24no—
171C22×C32C23×D314yes—
172C22×C32C23×D311yescharacteristic
173C2×C3⋊D3—21no—
174C2×C3⋊D3—22no—
175C2×C3⋊D3—64no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—616no—
179C2×C3⋊D3—616no—
180C2×C3⋊D3—616no—
181C2×C3⋊D3—1216no—
182C2×C3⋊D3—1216no—
183C2×C3⋊D3—1216no—
184C2×C3×D3—64no—
185C2×C3×D3—68no—
186C32×C4—216no—
187C32×C4—28no—
188C32×C4—22no—
189C32×C4—28no—
190C32×C4—64no—
191C32×C4#23512yes—
192C32×C4#23918yes—
193C3⋊Dic3—21no—
194C3⋊Dic3—22no—
195C3×Dic3—216no—
196C3×Dic3—64no—
197C3×Dic3—68no—
198#127—271no—
199#125—31no—
200#123—274no—
201#121—274no—
202#121—272no—
203C4×D4—272no—
204#117—91no—
205C33#54011yescharacteristic, complemented, Sylow
206C23×C3—91no—
207C22×D3—34no—
208C22×D3—916no—
209C22×D3—94no—
210C22×D3—1816no—
211C2×C3×C4—28no—
212C2×C3×C4—24no—
213C2×C3×C4—316no—
214C2×C3×C4—91no—
215C2×C3×C4—92no—
216C2×C3×C4—182no—
217C2×C3×C4C2×D3218yes—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D312yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221#67—364no—
222C2×Dic3—38no—
223C2×Dic3—34no—
224C2×Dic3C22×C3⋊D314yes—
225D12—1816no—
226D12—184no—
227D12—3616no—
228D12—3616no—
229D12—364no—
230D12—3632no—
231D12—3616no—
232C4×D3—68no—
233C4×D3—364no—
234C4×D3—3616no—
235C2×C32—24no—
236C2×C32—28no—
237C2×C32C2×#23511yescharacteristic
238C2×C32C2×#23914yes—
239C2×C32#100918yes—
240C2×C32#100912yes—
241C3⋊D3—41no—
242C3⋊D3—124no—
243C3⋊D3—1216no—
244C3⋊D3—2416no—
245C3×D3—124no—
246C2×D4—274no—
247C2×D4—272no—
248C2×D4—274no—
249C22×C4—91no—
250C22×C4—274no—
251C4⋊C4—32no—
252C4⋊C4—34no—
253C4⋊C4—92no—
254C22⋊C4—92no—
255C22⋊C4—274no—
256C22⋊C4—274no—
257C42—91no—
258C42D3×C3⋊D311yescharacteristic
259C22×C3—24no—
260C22×C3—181no—
261C22×C3—182no—
262C22×C3C22×D3214yes—
263C22×C3C23×C3⋊D311yescharacteristic
264C2×D3—64no—
265C2×D3—68no—
266C2×D3—1816no—
267C2×D3—1816no—
268C2×D3—1816no—
269C2×D3—184no—
270C2×D3—184no—
271C2×D3—184no—
272C2×D3—3616no—
273C2×D3—3616no—
274C2×D3—3616no—
275C3×C4—216no—
276C3×C4—28no—
277C3×C4—22no—
278C3×C4—28no—
279C3×C4—616no—
280C3×C4—181no—
281C3×C4—182no—
282C3×C4#379918yes—
283C3×C4#383212yes—
284Dic3—24no—
285Dic3—64no—
286Dic3—68no—
287C32—24no—
288C32#553511yescharacteristic, complemented
289C32#569714yescomplemented
290C23—91no—
291C23—274no—
292D4—544no—
293D4—1084no—
294D4—1084no—
295C2×C4—34no—
296C2×C4—92no—
297C2×C4—91no—
298C2×C4—182no—
299C2×C4—1084no—
300C2×C4C2×D3×C3⋊D312yes—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C2×C3—24no—
303C2×C3—28no—
304C2×C3—361no—
305C2×C3C2×#379914yes—
306C2×C3#6430818yes—
307C2×C3C2×#383211yescharacteristic
308C2×C3#6436612yes—
309D3—124no—
310D3—364no—
311D3—3616no—
312D3—7216no—
313C22—181no—
314C22—182no—
315C22—544no—
316C22—544no—
317C22—544no—
318C22C22×D3×C3⋊D311yescharacteristic
319C4—22no—
320C4—64no—
321C4—181no—
322C4—182no—
323C4#8901812yes—
324C3—24no—
325C3#1059398914yescomplemented
326C3#1059616811yescharacteristic, complemented
327C2—361no—
328C2—1084no—
329C2C2×#8901811yescharacteristic
330C2#1172458112yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723915C2semidirect
2. C3×#64081C2semidirect
3. C32×#870C2semidirect
4. #11723526C2semidirect
5. #11723498C2semidirect
6. #11723494C2semidirect
7. #11723486C2semidirect
31. #64081D3semidirect
33. C3×#870D3semidirect
106. #870C3⋊D3semidirect
205. C33#540semidirect
288. C32#5535semidirect
289. C32#5697semidirect
325. C3#10593989semidirect
326. C3#10596168semidirect

Maximal subgroups

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

subgroupindex
#117239152
C3×#640812
C32×#8702
#117235262
#117234982
#117234942
#117234862
#105961683
#105939893