Finite Group Databasebeta

← #13167399

Subgroups — #13167399

Order 1728.

504 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 63 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#11723697C211yesmaximal, characteristic
2#11723678C211yesmaximal, characteristic, complemented
3C3×C4×#622C211yesmaximal, characteristic, complemented
4#11722793C214yesmaximal, complemented
5#10593348—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—21no—
7C2×C3×#634—21no—
8C2×#6295—21no—
9C2×#6295C2212yes—
10C2×C3×#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#88815—21no—
13#88815C2212yes—
14#63914—31no—
15#63911—61no—
16#63873—31no—
17#63866—61no—
18C4×#622D311yescharacteristic, complemented
19#63739—34no—
20C22×C3×C3⋊D3—22no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×C4⋊(C3⋊D3)—42no—
26C3×#634—42no—
27#6295—24no—
28#6295—42no—
29C3×#622—22no—
30#4694—91nomaximal
31C2×C4⋊(C3⋊D3)—21no—
32C2×#634—21no—
33C2×C3×D12—61no—
34C2×C3×D12—61no—
35C2×C3×D12—121no—
36C2×C3×Dic6—61no—
37C2×C3×Dic6—61no—
38C2×C3×Dic6—121no—
39C2×#626—62no—
40C2×#626—62no—
41C2×#625—32no—
42C2×#625—61no—
43C2×#622C2×D312yes—
44C4×C3⋊Dic3C2×D311yescharacteristic
45C3×C4×Dic3—61no—
46C3×C4×Dic3—61no—
47#3724—32no—
48#3724—61no—
49Dic3⋊Dic3—62no—
50Dic3⋊Dic3—62no—
51C22×C33C2×D411yescharacteristic
52C2×C3×C3⋊D3—42no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3—42no—
55C33×C4D812yes—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3SD1612yes—
58C3×C3⋊Dic3#4112yes—
59#944—181no—
60#940—181no—
61#874—91no—
62#869—91no—
63C3×C4×C8—91no—
64#821—94no—
65C22×C3⋊D3—22no—
66C22×C3×D3—62no—
67C22×C3×D3—62no—
68C22×C3×D3—122no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4D1211yescharacteristic
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3D1211yescharacteristic
76C2×C3⋊Dic3C22×D311yescharacteristic
77C4⋊(C3⋊D3)—42no—
78#634—42no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—122no—
86C3×D12—122no—
87C3×D12—124no—
88C3×D12—122no—
89C3×Dic6—122no—
90C3×Dic6—124no—
91C3×Dic6—122no—
92#626—124no—
93#626—124no—
94#625—64no—
95#625—122no—
96#622—22no—
97#486—271noSylow
98C2×C33#12411yescharacteristic
99C2×C33C2×D811yescharacteristic
100C2×C33C2×SD1611yescharacteristic
101C3×C3⋊D3—82no—
102C2×C3×Q8—181no—
103C2×C3×D4—181no—
104C2×#67—182no—
105C2×#67—182no—
106C2×D12—61no—
107C2×D12—61no—
108C2×D12—92no—
109C2×D12—121no—
110C2×D12—181no—
111C2×Dic6—61no—
112C2×Dic6—61no—
113C2×Dic6—121no—
114C2×C3×C8—92no—
115C3×C42—91no—
116C4⋊Dic3—92no—
117C4⋊Dic3—181no—
118Dic3⋊C4—182no—
119Dic3⋊C4—182no—
120C4×Dic3—61no—
121C4×Dic3—61no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—41no—
125C22×C32—41no—
126C22×C32C2×D1211yescharacteristic
127C2×C3⋊D3—42no—
128C2×C3⋊D3—42no—
129C2×C3⋊D3—42no—
130C2×C3×D3—122no—
131C2×C3×D3—122no—
132C2×C3×D3—122no—
133C2×C3×D3—124no—
134C2×C3×D3—122no—
135C2×C3×D3—122no—
136C2×C3×D3—122no—
137C2×C3×D3—242no—
138C2×C3×D3—242no—
139C32×C4—22no—
140C32×C4—22no—
141C32×C4—42no—
142C32×C4—42no—
143C32×C4D2412yes—
144C3⋊Dic3—42no—
145C3⋊Dic3#20412yes—
146C3⋊Dic3#23512yes—
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—122no—
152C3×Dic3—124no—
153C3×Dic3—122no—
154C4⋊Q8—271no—
155#127—271no—
156#102—274no—
157C4×C8—91no—
158C33#48611yescharacteristic, complemented, Sylow
159C23×C3—182no—
160C22×D3—62no—
161C22×D3—62no—
162C22×D3—122no—
163C3×Q8—362no—
164C3×D4—362no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—182no—
172C2×C3×C4GO+(4,𝔽2)11yescharacteristic
173#67—364no—
174#67—364no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—122no—
182D12—122no—
183D12—124no—
184D12—122no—
185D12—184no—
186D12—362no—
187Dic6—122no—
188Dic6—124no—
189Dic6—122no—
190C3×C8—182no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32#87511yescharacteristic
204C2×C32C2×#20411yescharacteristic
205C2×C32C2×D2411yescharacteristic
206C3⋊D3—82no—
207C3×D3—242no—
208C3×D3—244no—
209C3×D3—242no—
210C2×Q8—181no—
211C2×D4—181no—
212C2×D4—272no—
213C2×D4—541no—
214C2×C8—92no—
215C4⋊C4—272no—
216C4⋊C4—541no—
217C42—91no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—362no—
223C22×C3—362no—
224C22×C3—362no—
225C22×C3C2×GO+(4,𝔽2)11yescharacteristic
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—124no—
230C2×D3—122no—
231C2×D3—122no—
232C2×D3—122no—
233C2×D3—242no—
234C2×D3—242no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—42no—
238C3×C4—42no—
239C3×C4—92no—
240C3×C4—92no—
241C3×C4—362no—
242C3×C4#377412yes—
243Dic3—62no—
244Dic3—62no—
245Dic3—62no—
246Dic3—62no—
247Dic3—122no—
248Dic3—124no—
249Dic3—122no—
250C32—21no—
251C32—21no—
252C32—41no—
253C32—41no—
254C32#469411yescharacteristic, complemented
255C23—182no—
256Q8—362no—
257D4—362no—
258D4—544no—
259D4—1082no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—182no—
263C2×C4#157⋊D311yescharacteristic
264C8—182no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—722no—
278C2×C3#6422111yescharacteristic
279C2×C3#6422711yescharacteristic
280C2×C3C2×#377411yescharacteristic
281D3—242no—
282D3—244no—
283D3—242no—
284C22—362no—
285C22—362no—
286C22—362no—
287C22C2×#157⋊D311yescharacteristic
288C4—92no—
289C4—92no—
290C4—362no—
291C4#8894112yes—
292C3—21no—
293C3—21no—
294C3—41no—
295C3—41no—
296C3#1059334811yescharacteristic, complemented
297C2—722no—
298C2#1172433411yescharacteristic
299C2#1172433911yescharacteristic
300C2C2×#8894111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical#11723697C2
hypercenterC2×C4#157⋊D3
hypocenterC33#486
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723678C2semidirect
3. C3×C4×#622C2semidirect
4. #11722793C2semidirect
18. C4×#622D3semidirect
158. C33#486semidirect
254. C32#4694semidirect
296. C3#10593348semidirect

Maximal subgroups

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

subgroupindex
#117236972
#117236782
C3×C4×#6222
#117227932
#105933483
#46949