Finite Group Databasebeta

← #13165640

Subgroups — #13165640

Order 1728.

556 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C2×#88941C211yesmaximal, characteristic, complemented
2C2×#88926C211yesmaximal, characteristic, complemented
3#11722968C212yesmaximal, complemented
4#11722772C212yesmaximal, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6#10593429—31nomaximal
7C2×#6295C2212yes—
8C2×#6292—22no—
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88941—22no—
13#88926—22no—
14C3×#3724C2212yes—
15C3×#3722—22no—
16C2×#3788—31no—
17C2×#3774—31no—
18#63912—32no—
19#63910—62no—
20#63742D311yescharacteristic, complemented
21#63739—32no—
22C22×C3⋊(C3⋊D3)D411yescharacteristic
23C22×C3×C3⋊D3—22no—
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C2×C32×Dic3—22no—
26C2×C32×Dic3D411yescharacteristic
27#6295—22no—
28#6295—22no—
29#6292—44no—
30#6289—21no—
31#6289—21no—
32C3⋊#622—21no—
33C3×#622D412yescomplemented
34#4865—91nomaximal
35C2×C4⋊(C3⋊D3)—62no—
36C2×C4×C3⋊D3—31no—
37C2×#626—62no—
38C2×#626—62no—
39C2×#626—62no—
40C2×#626—122no—
41C2×#625—32no—
42C2×#624—62no—
43C2×#622—31no—
44C2×#622C2×D311yescharacteristic
45#3788—62no—
46#3774—62no—
47C3×(C2×D3)⋊C4—62no—
48C3×(C2×D3)⋊C4—62no—
49C3×Dic3⋊C4—62no—
50#3724C2×D312yes—
51#3722—22no—
52C22×C33C2×D411yescharacteristic
53C2×C3⋊(C3⋊D3)—21no—
54C2×C3⋊(C3⋊D3)—22no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C3×C3⋊Dic3C2×D411yescharacteristic
59C3×C3⋊Dic3#4111yescharacteristic
60C32×Dic3—21no—
61C32×Dic3—42no—
62C2×D24—91no—
63C2×ΓL(1,𝔽25)—91no—
64#895—92no—
65#884—182no—
66C3×#106—91no—
67#809—92no—
68C22×C3⋊D3—22no—
69C22×C3⋊D3—31no—
70C22×C3⋊D3—62no—
71C22×C3⋊D3—122no—
72C22×C3×D3—62no—
73C22×C3×D3—62no—
74C22×C3×D3—122no—
75C2×C32×C4—31no—
76C2×C32×C4—62no—
77C2×C3⋊Dic3C22×D311yescharacteristic
78C4⋊(C3⋊D3)—124no—
79C4×C3⋊D3—61no—
80C4×C3⋊D3—61no—
81C2×C3×Dic3—22no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—62no—
86#626—122no—
87#626—122no—
88#626—124no—
89#626—124no—
90#626—128no—
91#625—62no—
92#625—62no—
93#624—122no—
94#624—122no—
95#622—61no—
96#622D1212yescomplemented
97#469—271noSylow
98C2×C33#12111yescharacteristic
99C2×C33AGL(1,ℤ/8)12yes—
100C3⋊(C3⋊D3)—41no—
101C3×C3⋊D3—82no—
102C2×#67—182no—
103C2×#67—182no—
104C2×D12—92no—
105C2×D12—182no—
106C2×D12—182no—
107C2×D12—362no—
108C2×C4×D3—91no—
109C2×C4×D3—182no—
110C2×C3×C8—91no—
111C3×C4⋊C4—92no—
112C3×C22⋊C4—182no—
113(C2×D3)⋊C4—62no—
114(C2×D3)⋊C4—62no—
115Dic3⋊C4—62no—
116C2×C3⋊C8—91no—
117D24—182no—
118ΓL(1,𝔽25)—182no—
119C22×C32—22no—
120C22×C32—42no—
121C22×C32D3×D411yescharacteristic
122C2×C3⋊D3—42no—
123C2×C3⋊D3—42no—
124C2×C3⋊D3—42no—
125C2×C3⋊D3—64no—
126C2×C3⋊D3—62no—
127C2×C3⋊D3—61no—
128C2×C3⋊D3—122no—
129C2×C3⋊D3—122no—
130C2×C3⋊D3—124no—
131C2×C3⋊D3—242no—
132C2×C3⋊D3—242no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C2×C3×D3—122no—
136C2×C3×D3—124no—
137C2×C3×D3—122no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140C2×C3×D3—242no—
141C2×C3×D3—242no—
142C32×C4—61no—
143C32×C4—122no—
144C3⋊Dic3C2×D1211yescharacteristic
145C3⋊Dic3#23911yescharacteristic
146C3×Dic3—42no—
147C3×Dic3—42no—
148C3×Dic3—44no—
149C3×Dic3—42no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C2×D8—271no—
153C2×M16—271no—
154#121—272no—
155#106—91no—
156#102—272no—
157C33#46911yescharacteristic, complemented, Sylow
158C23×C3—182no—
159C22×D3—62no—
160C22×D3—62no—
161C22×D3—91no—
162C22×D3—122no—
163C22×D3—182no—
164C22×D3—362no—
165C2×C3×C4—62no—
166C2×C3×C4—62no—
167C2×C3×C4—62no—
168C2×C3×C4—91no—
169C2×C3×C4—122no—
170#67—362no—
171#67—362no—
172#67—364no—
173C2×Dic3—22no—
174C2×Dic3—62no—
175C2×Dic3GO+(4,𝔽2)11yescharacteristic
176D12—182no—
177D12—182no—
178D12—364no—
179D12—364no—
180D12—368no—
181C4×D3—181no—
182C4×D3—181no—
183C4×D3—362no—
184C4×D3—362no—
185C3×C8—92no—
186C3⋊C8—181no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32#89511yescharacteristic
194C2×C32#91412yes—
195C3⋊D3—82no—
196C3⋊D3—121no—
197C3⋊D3—124no—
198C3⋊D3—244no—
199C3×D3—242no—
200C3×D3—244no—
201C3×D3—242no—
202C2×D4—272no—
203C2×D4—542no—
204C22×C4—271no—
205D8—542no—
206M16—542no—
207C2×C8—91no—
208C2×C8—271no—
209C4⋊C4—92no—
210C22⋊C4—182no—
211C22×C3—22no—
212C22×C3—42no—
213C22×C3—362no—
214C22×C3—362no—
215C22×C3—362no—
216C22×C3C2×GO+(4,𝔽2)11yescharacteristic
217C2×D3—122no—
218C2×D3—122no—
219C2×D3—122no—
220C2×D3—124no—
221C2×D3—122no—
222C2×D3—122no—
223C2×D3—122no—
224C2×D3—184no—
225C2×D3—182no—
226C2×D3—181no—
227C2×D3—242no—
228C2×D3—242no—
229C2×D3—362no—
230C2×D3—362no—
231C2×D3—364no—
232C2×D3—722no—
233C2×D3—722no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—122no—
237C3×C4—122no—
238C3×C4—124no—
239C3×C4—122no—
240Dic3—21no—
241Dic3—42no—
242Dic3—62no—
243Dic3—62no—
244C32—22no—
245C32—42no—
246C32#486511yescharacteristic, complemented
247C23—182no—
248C23—271no—
249D4—542no—
250D4—542no—
251D4—1084no—
252C2×C4—31no—
253C2×C4—62no—
254C2×C4—91no—
255C2×C4—541no—
256C2×C4—541no—
257C8—92no—
258C8—541no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—42no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3—722no—
266C2×C3#6422412yes—
267C2×C3#6424211yescharacteristic
268D3—242no—
269D3—244no—
270D3—242no—
271D3—364no—
272D3—361no—
273D3—724no—
274C22—362no—
275C22—362no—
276C22—362no—
277C22—541no—
278C22—542no—
279C22D3×GO+(4,𝔽2)11yescharacteristic
280C4—61no—
281C4—91no—
282C4—91no—
283C4—122no—
284C3—22no—
285C3—42no—
286C3#1059342911yescharacteristic, complemented
287C2—722no—
288C2—1081no—
289C2#1172411611yescharacteristic
290C2#1172414112yes—
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#469
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88941C2semidirect
2. C2×#88926C2semidirect
3. #11722968C2semidirect
4. #11722772C2semidirect
5. C3×#63742C2semidirect
20. #63742D3semidirect
33. C3×#622D4semidirect
96. #622D12semidirect
157. C33#469semidirect
246. C32#4865semidirect
286. C3#10593429semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
C2×#889262
#117229682
#117227722
C3×#637422
#105934293
#48659