Finite Group Databasebeta

← #13165620

Subgroups — #13165620

Order 1728.

564 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 54 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×#88924C211yesmaximal, characteristic, complemented
3#11722968C212yesmaximal, complemented
4#11722772C212yesmaximal, complemented
5C3×#63743C211yesmaximal, characteristic, complemented
6#10593423—31nomaximal
7C2×#6295C2212yes—
8C2×#6292—22no—
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88941—22no—
13#88924—22no—
14C3×#3724C2212yes—
15C3×#3722—22no—
16C2×#3787—31no—
17C2×#3774—31no—
18#63912—32no—
19#63910—62no—
20#63743D311yescharacteristic, 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#4883—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#3787—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)—21no—
55C2×C3⋊(C3⋊D3)D812yescomplemented
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3—42no—
59C3×C3⋊Dic3C2×D411yescharacteristic
60C3×C3⋊Dic3#4111yescharacteristic
61C32×Dic3—21no—
62C32×Dic3—42no—
63C2×D24—91no—
64C2×D3×C8—91no—
65#895—92no—
66#884—182no—
67C3×#107—91no—
68#809—92no—
69C22×C3⋊D3—22no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3—62no—
72C22×C3⋊D3—122no—
73C22×C3×D3—62no—
74C22×C3×D3—62no—
75C22×C3×D3—122no—
76C2×C32×C4—31no—
77C2×C32×C4—62no—
78C2×C3⋊Dic3C22×D311yescharacteristic
79C4⋊(C3⋊D3)—124no—
80C4×C3⋊D3—61no—
81C4×C3⋊D3—61no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—62no—
87#626—122no—
88#626—122no—
89#626—124no—
90#626—124no—
91#626—128no—
92#625—62no—
93#625—62no—
94#624—122no—
95#624—122no—
96#622—61no—
97#622D1212yescomplemented
98#466—271noSylow
99C2×C33#12111yescharacteristic
100C2×C33C2×D811yescharacteristic
101C2×C33#13511yescharacteristic
102C3⋊(C3⋊D3)—22no—
103C3×C3⋊D3—82no—
104C2×#67—182no—
105C2×#67—182no—
106C2×D12—92no—
107C2×D12—182no—
108C2×D12—182no—
109C2×D12—362no—
110C2×C4×D3—91no—
111C2×C4×D3—182no—
112C2×C3×C8—91no—
113C3×C4⋊C4—92no—
114C3×C22⋊C4—182no—
115(C2×D3)⋊C4—62no—
116(C2×D3)⋊C4—62no—
117Dic3⋊C4—62no—
118C2×C3⋊C8—91no—
119D24—182no—
120D3×C8—182no—
121C22×C32—22no—
122C22×C32—42no—
123C22×C32D3×D411yescharacteristic
124C2×C3⋊D3—32no—
125C2×C3⋊D3—42no—
126C2×C3⋊D3—42no—
127C2×C3⋊D3—42no—
128C2×C3⋊D3—64no—
129C2×C3⋊D3—61no—
130C2×C3⋊D3—61no—
131C2×C3⋊D3—124no—
132C2×C3⋊D3—122no—
133C2×C3⋊D3—122no—
134C2×C3⋊D3—242no—
135C2×C3⋊D3—242no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C2×C3×D3—122no—
139C2×C3×D3—124no—
140C2×C3×D3—122no—
141C2×C3×D3—122no—
142C2×C3×D3—122no—
143C2×C3×D3—242no—
144C2×C3×D3—242no—
145C32×C4—61no—
146C32×C4—122no—
147C3⋊Dic3C2×D1211yescharacteristic
148C3⋊Dic3#23911yescharacteristic
149C3×Dic3—42no—
150C3×Dic3—42no—
151C3×Dic3—44no—
152C3×Dic3—42no—
153C3×Dic3—62no—
154C3×Dic3—62no—
155C2×D8—271no—
156C22×C8—271no—
157#121—272no—
158#107—91no—
159#102—272no—
160C33#46611yescharacteristic, complemented, Sylow
161C23×C3—182no—
162C22×D3—62no—
163C22×D3—62no—
164C22×D3—91no—
165C22×D3—122no—
166C22×D3—182no—
167C22×D3—362no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—91no—
172C2×C3×C4—122no—
173#67—362no—
174#67—362no—
175#67—364no—
176C2×Dic3—22no—
177C2×Dic3—62no—
178C2×Dic3GO+(4,𝔽2)11yescharacteristic
179D12—182no—
180D12—182no—
181D12—364no—
182D12—364no—
183D12—368no—
184C4×D3—181no—
185C4×D3—181no—
186C4×D3—362no—
187C4×D3—362no—
188C3×C8—92no—
189C3⋊C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—42no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32#89511yescharacteristic
197C2×C32D3×D811yescharacteristic
198C2×C32#91911yescharacteristic
199C3⋊D3—62no—
200C3⋊D3—82no—
201C3⋊D3—124no—
202C3⋊D3—244no—
203C3×D3—242no—
204C3×D3—244no—
205C3×D3—242no—
206C2×D4—272no—
207C2×D4—542no—
208C22×C4—271no—
209D8—542no—
210C2×C8—91no—
211C2×C8—271no—
212C2×C8—542no—
213C4⋊C4—92no—
214C22⋊C4—182no—
215C22×C3—22no—
216C22×C3—42no—
217C22×C3—362no—
218C22×C3—362no—
219C22×C3—362no—
220C22×C3C2×GO+(4,𝔽2)11yescharacteristic
221C2×D3—92no—
222C2×D3—122no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—124no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—184no—
230C2×D3—181no—
231C2×D3—181no—
232C2×D3—242no—
233C2×D3—242no—
234C2×D3—364no—
235C2×D3—362no—
236C2×D3—362no—
237C2×D3—722no—
238C2×D3—722no—
239C3×C4—91no—
240C3×C4—91no—
241C3×C4—122no—
242C3×C4—122no—
243C3×C4—124no—
244C3×C4—122no—
245Dic3—21no—
246Dic3—42no—
247Dic3—62no—
248Dic3—62no—
249C32—22no—
250C32—42no—
251C32#488311yescharacteristic, complemented
252C23—182no—
253C23—271no—
254D4—542no—
255D4—542no—
256D4—1084no—
257C2×C4—31no—
258C2×C4—62no—
259C2×C4—91no—
260C2×C4—541no—
261C2×C4—541no—
262C8—92no—
263C8—541no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3—722no—
271C2×C3#6422311yescharacteristic
272C2×C3#6422511yescharacteristic
273C2×C3#6424211yescharacteristic
274D3—182no—
275D3—242no—
276D3—244no—
277D3—242no—
278D3—364no—
279D3—724no—
280C22—272no—
281C22—362no—
282C22—362no—
283C22—362no—
284C22—541no—
285C22—541no—
286C22D3×GO+(4,𝔽2)11yescharacteristic
287C4—61no—
288C4—91no—
289C4—91no—
290C4—122no—
291C3—22no—
292C3—42no—
293C3#1059342311yescharacteristic, complemented
294C2—542no—
295C2—722no—
296C2#1172411611yescharacteristic
297C2#1172412711yescharacteristic
298C2#1172413711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#466
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×#88924C2semidirect
3. #11722968C2semidirect
4. #11722772C2semidirect
5. C3×#63743C2semidirect
20. #63743D3semidirect
33. C3×#622D4semidirect
55. C2×C3⋊(C3⋊D3)D8semidirect
97. #622D12semidirect
160. C33#466semidirect
251. C32#4883semidirect
293. C3#10593423semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
C2×#889242
#117229682
#117227722
C3×#637432
#105934233
#48839