Finite Group Databasebeta

← #13166637

Subgroups — #13166637

Order 1728.

960 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C22×D3×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88925C212yesmaximal, complemented
3#11723112C211yesmaximal, characteristic, complemented
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5(C2×D3)⋊#622C212yesmaximal, complemented
6#10593845—31nomaximal
7C23×C32×D3C411yescharacteristic
8C22×C3⋊(C3⋊Dic3)C411yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×D3×C3⋊Dic3—24no—
11C2×D3×C3⋊Dic3C411yescharacteristic
12C2×D3×C3⋊Dic3C411yescharacteristic
13C2×D3×C3⋊Dic3C2211yescharacteristic
14C2×D3×C3⋊Dic3C2211yescharacteristic
15C2×C3⋊#622C2212yescomplemented
16C2×C3×#622C2212yescomplemented
17#88925—24no—
18C23×C3⋊Dic3—31no—
19C22×D3×Dic3—62no—
20C2×#3792—32no—
21C22⋊#622—31no—
22C22⋊#622—32no—
23C22⋊#622D311yescharacteristic, complemented
24C23×C33C2×C411yescharacteristic
25C22×C32×D3—22no—
26C22×C32×D3—22no—
27C22×C32×D3—22no—
28C22×C32×D3C2×C411yescharacteristic
29C22×C32×D3C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)—22no—
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36D3×C3⋊Dic3—22no—
37D3×C3⋊Dic3—22no—
38D3×C3⋊Dic3—22no—
39D3×C3⋊Dic3D414yes—
40C3⋊#622—22no—
41C3×#622—22no—
42#4726—91nomaximal
43C24×C32—31no—
44C23×C3×D3—22no—
45C22×C3⋊Dic3—31no—
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—31no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—64no—
51C22×C3⋊Dic3—62no—
52C22×C3⋊Dic3—122no—
53C22×C3⋊Dic3C2×D311yescharacteristic
54C22×C3×Dic3—62no—
55C2×D3×Dic3—62no—
56C2×D3×Dic3—62no—
57C2×D3×Dic3—62no—
58C2×D3×Dic3—62no—
59C2×D3×Dic3—616no—
60#3792—64no—
61C2×#622—32no—
62C2×#622C2×D312yescomplemented
63C22×C33—21no—
64C22×C33—21no—
65C22×C33M1612yescomplemented
66C22×C33C22×C411yescharacteristic
67C2×C32×D3—28no—
68C2×C32×D3—21no—
69C2×C32×D3—21no—
70C2×C32×D3—21no—
71C2×C32×D3—21no—
72C2×C32×D3C22⋊C412yes—
73C2×C32×D3M1612yescomplemented
74C3⋊(C3⋊Dic3)—21no—
75C3⋊(C3⋊Dic3)C22⋊C412yes—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3C2×D412yes—
78C23×Dic3—182no—
79C22×C4×D3—91no—
80C2×ΓL(1,𝔽25)—92no—
81C3×C22⋊C8—91no—
82#830—91no—
83(C2×D3)⋊C8—92no—
84C23×C32—22no—
85C23×C32—31no—
86C23×C32—31no—
87C23×C32—42no—
88C23×C32—62no—
89C23×C32—62no—
90C23×C32—62no—
91C23×C32C4×D311yescharacteristic
92C22×C3×D3—28no—
93C22×C3×D3—28no—
94C22×C3×D3—28no—
95C22×C3×D3—22no—
96C22×C3×D3—22no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—34no—
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—68no—
108C2×C3⋊Dic3—128no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3C4×D311yescharacteristic
112C2×C3⋊Dic3C22×D311yescharacteristic
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—68no—
116D3×Dic3—68no—
117D3×Dic3—68no—
118D3×Dic3—68no—
119D3×Dic3—68no—
120#622—22no—
121#622—62no—
122#407—271noSylow
123C2×C33—22no—
124C2×C33C2×C22⋊C411yescharacteristic
125C2×C33C2×M1611yescharacteristic
126C2×C33C2×M1611yescharacteristic
127C32×D3—22no—
128C32×D3—22no—
129C24×C3—62no—
130C23×D3#15711yescharacteristic
131C22×C3×C4—91no—
132C22×Dic3—62no—
133C22×Dic3—91no—
134C22×Dic3—182no—
135C22×Dic3—182no—
136C22×Dic3—182no—
137C22×Dic3—182no—
138C22×Dic3—182no—
139C22×Dic3—1816no—
140C22×Dic3—362no—
141C2×C4×D3—91no—
142C2×C4×D3—91no—
143C2×C4×D3—91no—
144C2×C4×D3—91no—
145C2×C4×D3—184no—
146C2×C3×C8—92no—
147C2×C3⋊C8—92no—
148ΓL(1,𝔽25)—184no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—24no—
152C22×C32—24no—
153C22×C32—22no—
154C22×C32—24no—
155C22×C32—32no—
156C22×C32—32no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—44no—
160C22×C32—42no—
161C22×C32—68no—
162C22×C32—61no—
163C22×C32—61no—
164C22×C32—61no—
165C22×C32—61no—
166C22×C32ΓL(1,𝔽25)12yescomplemented
167C22×C32C2×C4×D311yescharacteristic
168C2×C3×D3—24no—
169C2×C3×D3—232no—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C2×C3×D3—24no—
173C2×C3×D3—24no—
174C2×C3×D3—24no—
175C3⋊Dic3—21no—
176C3⋊Dic3—32no—
177C3⋊Dic3—64no—
178C3⋊Dic3—61no—
179C3⋊Dic3—64no—
180C3⋊Dic3—124no—
181C3⋊Dic3—124no—
182C3⋊Dic3D3×D412yes—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C23×C4—271no—
186C2×M16—272no—
187C22⋊C8—91no—
188C22⋊C8—272no—
189C22⋊C8—271no—
190C33#40711yescharacteristic, complemented, Sylow
191C23×C3—22no—
192C23×C3—42no—
193C23×C3—62no—
194C23×C3—62no—
195C23×C3—68no—
196C23×C3—68no—
197C23×C3—68no—
198C23×C3C2×#15711yescharacteristic
199C22×D3—22no—
200C22×D3—22no—
201C22×D3—22no—
202C22×D3C2×#15711yescharacteristic
203C22×D3C2×#15711yescharacteristic
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—182no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—68no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—188no—
213C2×Dic3—182no—
214C2×Dic3—182no—
215C2×Dic3—188no—
216C2×Dic3—188no—
217C2×Dic3—188no—
218C2×Dic3—182no—
219C2×Dic3—188no—
220C2×Dic3—362no—
221C2×Dic3—362no—
222C2×Dic3—368no—
223C4×D3—94no—
224C4×D3—182no—
225C4×D3—182no—
226C4×D3—182no—
227C3×C8—182no—
228C3⋊C8—182no—
229C2×C32—22no—
230C2×C32—28no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—48no—
235C2×C32—42no—
236C2×C32—42no—
237C2×C32—42no—
238C2×C32—62no—
239C2×C32—62no—
240C2×C32D3×C22⋊C411yescharacteristic
241C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
242C2×C32D3×M1611yescharacteristic
243C3×D3—28no—
244C3×D3—28no—
245C24—31no—
246C22×C4—91no—
247C22×C4—271no—
248C22×C4—271no—
249C22×C4—271no—
250C22×C4—271no—
251C22×C4—271no—
252C22×C4—544no—
253M16—544no—
254C2×C8—92no—
255C2×C8—272no—
256C22×C3—24no—
257C22×C3—24no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—24no—
262C22×C3—44no—
263C22×C3—44no—
264C22×C3—42no—
265C22×C3—44no—
266C22×C3—64no—
267C22×C3—64no—
268C22×C3—64no—
269C22×C3—632no—
270C22×C3—64no—
271C22×C3—64no—
272C22×C3—64no—
273C22×C3#379212yescomplemented
274C22×C3C22×#15711yescharacteristic
275C2×D3—28no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3#379212yescomplemented
281C2×D3C22⋊#15712yes—
282C3×C4—92no—
283C3×C4—181no—
284Dic3—64no—
285Dic3—64no—
286Dic3—92no—
287Dic3—184no—
288Dic3—184no—
289Dic3—181no—
290Dic3—364no—
291Dic3—364no—
292C32—22no—
293C32—42no—
294C32#472611yescharacteristic, complemented
295C23—31no—
296C23—31no—
297C23—62no—
298C23—62no—
299C23—62no—
300C23D3×#15711yescharacteristic
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—182no—
304C2×C4—274no—
305C2×C4—271no—
306C2×C4—271no—
307C2×C4—542no—
308C2×C4—542no—
309C2×C4—542no—
310C2×C4—542no—
311C8—182no—
312C8—542no—
313C2×C3—28no—
314C2×C3—22no—
315C2×C3—22no—
316C2×C3—22no—
317C2×C3—22no—
318C2×C3—48no—
319C2×C3—42no—
320C2×C3—42no—
321C2×C3—42no—
322C2×C3—68no—
323C2×C3—68no—
324C2×C3C2×#379211yescharacteristic
325C2×C3C2×#379211yescharacteristic
326C2×C3C2×C22⋊#15711yescharacteristic
327D3—22no—
328D3—22no—
329C22—21no—
330C22—21no—
331C22—32no—
332C22—32no—
333C22—68no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—61no—
338C22#8892512yescomplemented
339C22C2×D3×#15711yescharacteristic
340C4—92no—
341C4—181no—
342C4—272no—
343C4—541no—
344C3—22no—
345C3—42no—
346C3#1059384511yescharacteristic, complemented
347C2—22no—
348C2—62no—
349C2—62no—
350C2D3×#379211yescharacteristic
351C2C2×#8892511yescharacteristic
352C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC23×C32×D3C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88925C2semidirect
3. #11723112C2semidirect
4. C3×C22⋊#622C2semidirect
5. (C2×D3)⋊#622C2semidirect
15. C2×C3⋊#622C22semidirect
16. C2×C3×#622C22semidirect
23. C22⋊#622D3semidirect
62. C2×#622C2×D3semidirect
65. C22×C33M16semidirect
73. C2×C32×D3M16semidirect
166. C22×C32ΓL(1,𝔽25)semidirect
190. C33#407semidirect
273. C22×C3#3792semidirect
280. C2×D3#3792semidirect
294. C32#4726semidirect
338. C22#88925semidirect
346. C3#10593845semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊Dic32
C2×#889252
#117231122
C3×C22⋊#6222
(C2×D3)⋊#6222
#105938453
#47269