Finite Group Databasebeta

← #13166456

Subgroups — #13166456

Order 1728.

488 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 70 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×C3×#3788C211yesmaximal, characteristic, complemented
2C2×D3×#622C211yesmaximal, characteristic, complemented
3#11723507C211yesmaximal, characteristic
4#11723103C211yesmaximal, characteristic, complemented
5Dic3⋊#622C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593740—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6290C411yescharacteristic
11C2×D3×C3⋊Dic3C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16C3×#3788—22no—
17D3×#622—22no—
18#88857C411yescharacteristic
19C32×(C2×D3)⋊C4C411yescharacteristic
20#88803C411yescharacteristic
21Dic3×C3⋊Dic3C2211yescharacteristic
22C22×#622—31no—
23C2×#3788D311yescharacteristic, complemented
24#64090—31no—
25#63879—62no—
26C22⋊#622—31no—
27#63778—31no—
28#63777—31no—
29C4⋊#622—31no—
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C22×C32×D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C3×C4×C3⋊D3—21no—
37C3×C4×C3⋊D3—21no—
38C2×C32×Dic3C2×C411yescharacteristic
39#6290—22no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3—21no—
42C3⋊#622—21no—
43C3⋊#622—21no—
44C3×#622—21no—
45C3×#622D412yescomplemented
46#5127—91nomaximal
47C2×#638—31no—
48C22×C3⋊Dic3—31no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×C3×C4×D3—62no—
51C2×#625—62no—
52C2×D3×Dic3—62no—
53C2×#622—31no—
54C2×#622—31no—
55C2×#622—62no—
56C2×#622C2×D311yescharacteristic
57C2×#622C2×D311yescharacteristic
58#3788—22no—
59C32×C22⋊C4—31no—
60#3752—31no—
61#3750—31no—
62#3750—62no—
63#3750—122no—
64C4×C3⋊Dic3—31no—
65C3×(C2×D3)⋊C4—22no—
66(C2×D3)⋊Dic3—62no—
67Dic32—62no—
68C22×C33C22×C411yescharacteristic
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—21no—
71C2×C3×C3⋊D3—21no—
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C2×C32×D3M1612yescomplemented
75C33×C4—21no—
76C3⋊(C3⋊Dic3)—21no—
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3C2×D411yescharacteristic
79C3×C3⋊Dic3#4111yescharacteristic
80C32×Dic3—21no—
81C2×C3×M16—91no—
82C4×#67—91no—
83C2×D3×C8—91no—
84#67⋊C4—182no—
85#830—91no—
86(C2×D3)⋊C8—91no—
87#815—91no—
88Dic3⋊C8—91no—
89C23×C32—31no—
90C22×C3⋊D3C4×D311yescharacteristic
91C22×C3×D3—22no—
92C22×C3×D3—62no—
93C2×C32×C4—22no—
94C2×C32×C4—31no—
95C2×C32×C4—42no—
96C2×C32×C4C4×D311yescharacteristic
97#638—62no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4×C3⋊D3—21no—
108C4×C3⋊D3—21no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—62no—
111C3×C4×D3—122no—
112C3×C4×D3—122no—
113#625—68no—
114D3×Dic3—122no—
115D3×Dic3—122no—
116#622—21no—
117#622—61no—
118#622—61no—
119#622#6712yescomplemented
120#435—271noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33C2×M1611yescharacteristic
123C2×C33#13111yescharacteristic
124C3×C3⋊D3—41no—
125C32×D3—22no—
126C22×C3×C4—91no—
127C2×#67—91no—
128C2×#67—182no—
129C22×Dic3—182no—
130C2×C4×D3—62no—
131C2×C4×D3—91no—
132C3×M16—182no—
133C2×C3×C8—91no—
134C2×C3×C8—91no—
135C3×C22⋊C4—62no—
136#217—91no—
137(C2×D3)⋊C4—182no—
138(C2×D3)⋊C4#15711yescharacteristic
139Dic3⋊C4—91no—
140Dic3⋊C4—182no—
141Dic3⋊C4—362no—
142C4×Dic3—91no—
143C4×Dic3—182no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146D3×C8—182no—
147C22×C32—22no—
148C22×C32—32no—
149C22×C32—42no—
150C22×C32—61no—
151C22×C32—61no—
152C22×C32C2×C4×D311yescharacteristic
153C2×C3⋊D3—21no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3—21no—
156C2×C3×D3—24no—
157C2×C3×D3—42no—
158C2×C3×D3—42no—
159C2×C3×D3—64no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C32×C4—21no—
163C32×C4—42no—
164C32×C4—61no—
165C32×C4—82no—
166C3⋊Dic3—61no—
167C3⋊Dic3—64no—
168C3⋊Dic3—61no—
169C3⋊Dic3—124no—
170C3⋊Dic3—122no—
171C3⋊Dic3—242no—
172C3⋊Dic3#23511yescharacteristic
173C3⋊Dic3C2×#6711yescharacteristic
174C3×Dic3—42no—
175C3×Dic3—62no—
176C3×Dic3—62no—
177C2×M16—91no—
178C22×C8—271no—
179C4×D4—271no—
180C4⋊C8—271no—
181C22⋊C8—271no—
182C22⋊C8—271no—
183#97—271no—
184C33#43511yescharacteristic, complemented, Sylow
185C23×C3—62no—
186C23×C3—91no—
187C22×D3—62no—
188C22×D3C2×#15711yescharacteristic
189C2×C3×C4—22no—
190C2×C3×C4—42no—
191C2×C3×C4—62no—
192C2×C3×C4—91no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C2×#15711yescharacteristic
196#67—188no—
197#67—182no—
198C2×Dic3—62no—
199C2×Dic3—91no—
200C2×Dic3—91no—
201C2×Dic3—182no—
202C2×Dic3—182no—
203C2×Dic3—362no—
204C2×Dic3—362no—
205C2×Dic3—362no—
206C2×Dic3—362no—
207C2×Dic3C2×#15711yescharacteristic
208C4×D3—122no—
209C4×D3—122no—
210C4×D3—181no—
211C4×D3—181no—
212C3×C8—92no—
213C3×C8—181no—
214C3⋊C8—181no—
215C3⋊C8—181no—
216C2×C32—22no—
217C2×C32—22no—
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—42no—
221C2×C32—42no—
222C2×C32—62no—
223C2×C32D3×M1611yescharacteristic
224C2×C32#90711yescharacteristic
225C2×C32C4×#6711yescharacteristic
226C3⋊D3—41no—
227C3×D3—44no—
228C3×D3—124no—
229C2×D4—271no—
230C22×C4—91no—
231C22×C4—271no—
232M16—182no—
233C2×C8—91no—
234C2×C8—91no—
235C2×C8—271no—
236C2×C8—271no—
237C2×C8—542no—
238C4⋊C4—271no—
239C22⋊C4—31no—
240C22⋊C4—271no—
241C42—271no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—64no—
245C22×C3—122no—
246C22×C3—122no—
247C22×C3—181no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C22×#15711yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—64no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3#378812yescomplemented
257C3×C4—21no—
258C3×C4—42no—
259C3×C4—82no—
260C3×C4—91no—
261C3×C4—91no—
262C3×C4—122no—
263Dic3—21no—
264Dic3—62no—
265Dic3—62no—
266Dic3—184no—
267Dic3—181no—
268Dic3—181no—
269Dic3—362no—
270Dic3—364no—
271Dic3—722no—
272C32—22no—
273C32—42no—
274C32#512711yescharacteristic, complemented
275C23—31no—
276C23—91no—
277D4—542no—
278C2×C4—31no—
279C2×C4—91no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4—541no—
285C2×C4—541no—
286C2×C4D3×#15711yescharacteristic
287C8—92no—
288C8—181no—
289C8—541no—
290C8—541no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—124no—
298C2×C3—361no—
299C2×C3C2×#378811yescharacteristic
300C2×C3#6428711yescharacteristic
301C2×C3D4×#15711yescharacteristic
302D3—22no—
303D3—124no—
304C22—32no—
305C22—61no—
306C22—61no—
307C22—181no—
308C22—181no—
309C22—181no—
310C22C2×D3×#15711yescharacteristic
311C4—21no—
312C4—61no—
313C4—91no—
314C4—91no—
315C4—541no—
316C4—541no—
317C3—22no—
318C3—42no—
319C3#1059374011yescharacteristic, complemented
320C2—62no—
321C2—361no—
322C2D3×#378811yescharacteristic
323C2#1172425111yescharacteristic
324C2#67×#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#6290C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3788C2semidirect
2. C2×D3×#622C2semidirect
4. #11723103C2semidirect
5. Dic3⋊#622C2semidirect
6. (C2×D3)⋊#622C2semidirect
7. #11722724C2semidirect
23. C2×#3788D3semidirect
45. C3×#622D4semidirect
74. C2×C32×D3M16semidirect
119. #622#67semidirect
184. C33#435semidirect
256. C2×D3#3788semidirect
274. C32#5127semidirect
319. C3#10593740semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×D3×#6222
#117235072
#117231032
Dic3⋊#6222
(C2×D3)⋊#6222
#117227242
#105937403
#51279