Finite Group Databasebeta

← #13173419

Subgroups — #13173419

Order 1728.

752 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 93 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×#3830C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic, complemented
3C22×C3⋊#622C211yesmaximal, characteristic, complemented
4#11723112C212yesmaximal, complemented
5#11723103C212yesmaximal, complemented
6#10593822—31nomaximal
7C2×C33×D4C411yescharacteristic
8C2×C3×#638C411yescharacteristic
9C2×C3×#638C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C3×#3830—22no—
13C3×#3830—22no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C411yescharacteristic
16#88993—22no—
17C2×C3⋊#622—22no—
18C2×C3⋊#622C2212yes—
19C2×C3⋊#622C2212yes—
20C2×#3830D311yescharacteristic
21C2×C3×#237—62no—
22C2×#3792—31no—
23C22×#622—31no—
24C22⋊#622—32no—
25#63778—32no—
26C23×C33C2×C411yescharacteristic
27C23×C33C2×C411yescharacteristic
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C33×D4—22no—
30C2×C33×C4C2×C411yescharacteristic
31C3×#638—22no—
32C3×#638—22no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3D412yes—
38C2×C3×C3⋊Dic3D412yes—
39C2×C3×C3⋊Dic3C2311yescharacteristic
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42C3×#634—22no—
43C3⋊#622—22no—
44C3⋊#622—22no—
45#5234—91nomaximal
46C2×C32×D4—22no—
47C2×C32×D4—42no—
48C2×C32×D4Dic311yescharacteristic
49C2×#638Dic311yescharacteristic
50C2×#638Dic311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D311yescharacteristic
53#3830—22no—
54#3830—22no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634Dic311yescharacteristic
57C2×C3×#67—62no—
58C2×C3×#67—62no—
59C22×C3×Dic3—62no—
60C22×C3×Dic3—62no—
61C3×#237—68no—
62C3×#237—68no—
63C2×C3×C4×D3—62no—
64C2×C3×Dic6—62no—
65#3792—62no—
66C2×#622—32no—
67C2×#622—32no—
68C2×#622—62no—
69C22×C33—22no—
70C22×C33—21no—
71C22×C33—22no—
72C22×C33C22⋊C412yes—
73C22×C33C22×C411yescharacteristic
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3—22no—
76C33×C4—21no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3C22⋊C412yes—
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3C2×D411yescharacteristic
81C2×C3×#41—91no—
82C2×#237—62no—
83C2×#208—91no—
84C22×C3⋊C8—91no—
85#830—92no—
86#830—92no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—42no—
90C23×C32—42no—
91C23×C32C2×Dic311yescharacteristic
92C23×C32C2×Dic311yescharacteristic
93C22×C3⋊D3C2×Dic311yescharacteristic
94C22×C3×D3—62no—
95C32×D4—22no—
96C32×D4—28no—
97C32×D4—48no—
98C2×C32×C4—22no—
99C2×C32×C4—42no—
100C2×C32×C4C2×Dic311yescharacteristic
101#638—22no—
102#638—22no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3C2×Dic311yescharacteristic
106C2×C3⋊Dic3C2×Dic311yescharacteristic
107C2×C3⋊Dic3#6712yes—
108C2×C3⋊Dic3#6712yes—
109C2×C3⋊Dic3C22×D311yescharacteristic
110C4×C3⋊D3—21no—
111C4×C3⋊D3—21no—
112#634—22no—
113C3×#67—68no—
114C3×#67—68no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C3×C4×D3—64no—
123C3×C4×D3—64no—
124C3×Dic6—68no—
125#622—62no—
126#622—62no—
127#408—271noSylow
128C2×C33—22no—
129C2×C33—41no—
130C2×C33C2×C22⋊C411yescharacteristic
131C2×C33#13112yes—
132C3×C3⋊D3—41no—
133C3×#41—182no—
134C3×#41—182no—
135C2×C3×Q8—91no—
136C2×C3×D4—22no—
137C2×C3×D4—42no—
138C2×C3×D4—91no—
139C2×C3×D4—91no—
140C2×C3×D4#15711yescharacteristic
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C22×C3×C4—91no—
144C2×#67—62no—
145C2×#67—62no—
146C22×Dic3—62no—
147C22×Dic3—62no—
148#237—68no—
149#237—68no—
150C2×C4×D3—62no—
151C2×Dic6—62no—
152#208—182no—
153C2×C3⋊C8—92no—
154C2×C3⋊C8—92no—
155C2×C3⋊C8—182no—
156C22×C32—21no—
157C22×C32—24no—
158C22×C32—22no—
159C22×C32—24no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—44no—
163C22×C32—42no—
164C22×C32—44no—
165C22×C32—44no—
166C22×C32—44no—
167C22×C32—84no—
168C22×C32—84no—
169C22×C32#21712yes—
170C22×C32C22×Dic311yescharacteristic
171C2×C3⋊D3—21no—
172C2×C3⋊D3—22no—
173C2×C3×D3—64no—
174C2×C3×D3—124no—
175C32×C4—21no—
176C32×C4—24no—
177C32×C4—44no—
178C3⋊Dic3—21no—
179C3⋊Dic3#21712yes—
180C3⋊Dic3C2×#6711yescharacteristic
181C3⋊Dic3C2×#6711yescharacteristic
182C3×Dic3—64no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—64no—
186C2×#41—91no—
187C2×M16—271no—
188C22×C8—271no—
189C22⋊C8—272no—
190C22⋊C8—272no—
191C33#40811yescharacteristic, complemented, Sylow
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—91no—
197C23×C3C2×#15711yescharacteristic
198C23×C3C2×#15711yescharacteristic
199C22×D3—62no—
200C3×Q8—182no—
201C3×D4—28no—
202C3×D4—22no—
203C3×D4—48no—
204C3×D4—182no—
205C3×D4—182no—
206C2×C3×C4—22no—
207C2×C3×C4—42no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4—92no—
211C2×C3×C4—92no—
212C2×C3×C4—91no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4C2×#15711yescharacteristic
218#67—68no—
219#67—68no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—64no—
223C2×Dic3—64no—
224C2×Dic3—62no—
225C2×Dic3—64no—
226C2×Dic3—64no—
227C4×D3—64no—
228C4×D3—64no—
229Dic6—68no—
230C3⋊C8—182no—
231C3⋊C8—182no—
232C2×C32—24no—
233C2×C32—22no—
234C2×C32—22no—
235C2×C32—42no—
236C2×C32—44no—
237C2×C32—44no—
238C2×C32—41no—
239C2×C32—44no—
240C2×C32—84no—
241C2×C32—84no—
242C2×C32#94812yes—
243C2×C32C2×#21711yescharacteristic
244C3⋊D3—41no—
245C3×D3—124no—
246#41—182no—
247#41—182no—
248C2×Q8—91no—
249C2×D4—91no—
250C2×D4—91no—
251C2×D4C3⋊#15711yescharacteristic
252C22×C4—91no—
253C22×C4—91no—
254C22×C4—91no—
255M16—542no—
256C2×C8—272no—
257C2×C8—272no—
258C2×C8—542no—
259C22×C3—22no—
260C22×C3—24no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—24no—
265C22×C3—44no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3—42no—
269C22×C3—44no—
270C22×C3—84no—
271C22×C3—84no—
272C22×C3—182no—
273C22×C3—181no—
274C22×C3C22⋊#15712yes—
275C22×C3C22×#15711yescharacteristic
276C2×D3—64no—
277C2×D3—124no—
278C3×C4—24no—
279C3×C4—21no—
280C3×C4—44no—
281C3×C4—91no—
282C3×C4—91no—
283C3×C4—92no—
284C3×C4—181no—
285Dic3—64no—
286Dic3—62no—
287Dic3—62no—
288Dic3—64no—
289C32—22no—
290C32—42no—
291C32#523411yescharacteristic, complemented
292C23—91no—
293C23C2×C3⋊#15711yescharacteristic
294C23C2×C3⋊#15711yescharacteristic
295Q8—182no—
296D4—22no—
297D4—182no—
298D4—182no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—92no—
302C2×C4—92no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4C2×C3⋊#15711yescharacteristic
309C8—542no—
310C8—542no—
311C2×C3—22no—
312C2×C3—24no—
313C2×C3—22no—
314C2×C3—44no—
315C2×C3—44no—
316C2×C3—41no—
317C2×C3—44no—
318C2×C3—42no—
319C2×C3—84no—
320C2×C3—84no—
321C2×C3—361no—
322C2×C3#6428712yes—
323C2×C3C2×C22⋊#15711yescharacteristic
324D3—124no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—182no—
329C22—181no—
330C22#8899412yes—
331C22C22×C3⋊#15711yescharacteristic
332C4—21no—
333C4—91no—
334C4—91no—
335C4—92no—
336C4—181no—
337C3—22no—
338C3—42no—
339C3#1059382211yescharacteristic, complemented
340C2—22no—
341C2—41no—
342C2—361no—
343C2#1172451512yes—
344C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC2×C33×D4C4
prime radicalC2×C3×#3830C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. C22×C3⋊#622C2semidirect
4. #11723112C2semidirect
5. #11723103C2semidirect
191. C33#408semidirect
291. C32#5234semidirect
339. C3#10593822semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#889932
C22×C3⋊#6222
#117231122
#117231032
#105938223
#52349