Finite Group Databasebeta

← #13166505

Subgroups — #13166505

Order 1728.

960 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 80 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×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#88926C212yesmaximal, complemented
3#11723103C211yesmaximal, characteristic, complemented
4C3×#63778C211yesmaximal, characteristic, complemented
5(C2×D3)⋊#622C212yesmaximal, complemented
6#10593803—31nomaximal
7C22×D3×C3⋊D3C411yescharacteristic
8C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4×D3C411yescharacteristic
11C2×#6289C2211yescharacteristic
12C2×Dic3×C3⋊D3C411yescharacteristic
13C2×D3×C3⋊Dic3C2211yescharacteristic
14C4×D3×C3⋊D3—24no—
15C2×C3⋊#622C2212yescomplemented
16C2×C3×#622C2212yescomplemented
17#88926—24no—
18C22×C4×C3⋊D3—31no—
19C2×C4×D32—62no—
20C2×#3788—32no—
21C22⋊#622—32no—
22#63778—31no—
23#63778D311yescharacteristic, complemented
24C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C22×C32×D3C2×C411yescharacteristic
27C2×D3×C3⋊D3—22no—
28C2×D3×C3⋊D3—22no—
29C2×D3×C3⋊D3—22no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C4×C3⋊(C3⋊D3)—22no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3×C4×C3⋊D3—22no—
35C2×C32×Dic3C2×C411yescharacteristic
36C32×C4×D3—22no—
37#6289D414yes—
38Dic3×C3⋊D3—22no—
39D3×C3⋊Dic3—22no—
40C3⋊#622—22no—
41C3×#622—22no—
42#4726—91nomaximal
43C23×C3⋊D3—31no—
44C22×D32—62no—
45C22×C32×C4—31no—
46C22×C3⋊Dic3—31no—
47C2×C4×C3⋊D3—31no—
48C2×C4×C3⋊D3—31no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—64no—
51C2×C4×C3⋊D3—62no—
52C2×C4×C3⋊D3—122no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×C4×D3—22no—
55C2×C3×C4×D3—62no—
56C2×#624—62no—
57C2×D3×Dic3—62no—
58C2×D3×Dic3—62no—
59C4×D32—616no—
60C2×#622—32no—
61C2×#622C2×D312yescomplemented
62#3788—64no—
63C22×C33C22×C411yescharacteristic
64C2×C3⋊(C3⋊D3)—21no—
65C2×C3⋊(C3⋊D3)—21no—
66C2×C3⋊(C3⋊D3)C22⋊C412yes—
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3M1612yescomplemented
70C2×C32×D3—21no—
71C2×C32×D3—21no—
72C2×C32×D3M1612yescomplemented
73D3×C3⋊D3—28no—
74C33×C4—21no—
75C3⋊(C3⋊Dic3)—21no—
76C3×C3⋊Dic3C2×D412yes—
77C32×Dic3C22⋊C412yes—
78C22×C4×D3—91no—
79C22×C4×D3—182no—
80C2×ΓL(1,𝔽25)—92no—
81C3×C22⋊C8—91no—
82#830—91no—
83(C2×D3)⋊C8—92no—
84C23×C32—31no—
85C22×C3⋊D3—31no—
86C22×C3⋊D3—62no—
87C22×C3⋊D3—62no—
88C22×C3⋊D3—62no—
89C22×C3⋊D3—62no—
90C22×C3⋊D3—122no—
91C22×C3⋊D3C4×D311yescharacteristic
92C22×C3×D3—22no—
93C22×C3×D3—62no—
94C2×D32—68no—
95C2×D32—68no—
96C2×D32—68no—
97C2×C32×C4—22no—
98C2×C32×C4—31no—
99C2×C32×C4—42no—
100C2×C32×C4—62no—
101C2×C32×C4C4×D311yescharacteristic
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4×C3⋊D3—22no—
108C4×C3⋊D3—34no—
109C4×C3⋊D3—62no—
110C4×C3⋊D3—62no—
111C4×C3⋊D3—68no—
112C4×C3⋊D3—128no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—62no—
115C3×C4×D3—28no—
116C3×C4×D3—68no—
117#624—68no—
118D3×Dic3—68no—
119D3×Dic3—68no—
120#622—22no—
121#622—62no—
122#407—271noSylow
123C2×C33C2×C22⋊C411yescharacteristic
124C2×C33C2×M1611yescharacteristic
125C2×C33C2×M1611yescharacteristic
126C3⋊(C3⋊D3)—22no—
127C3×C3⋊D3—22no—
128C32×D3—22no—
129C23×D3—91no—
130C23×D3—182no—
131C22×C3×C4—62no—
132C22×C3×C4—91no—
133C22×Dic3—91no—
134C22×Dic3—182no—
135C2×C4×D3—62no—
136C2×C4×D3—91no—
137C2×C4×D3—91no—
138C2×C4×D3—91no—
139C2×C4×D3—182no—
140C2×C4×D3—182no—
141C2×C4×D3—182no—
142C2×C4×D3—184no—
143C2×C4×D3—1816no—
144C2×C4×D3—362no—
145C2×C4×D3#15711yescharacteristic
146C2×C3×C8—92no—
147C2×C3⋊C8—92no—
148ΓL(1,𝔽25)—184no—
149C22×C32—22no—
150C22×C32—32no—
151C22×C32—42no—
152C22×C32—61no—
153C22×C32—61no—
154C22×C32C2×C4×D311yescharacteristic
155C2×C3⋊D3—21no—
156C2×C3⋊D3—21no—
157C2×C3⋊D3—32no—
158C2×C3⋊D3—64no—
159C2×C3⋊D3—68no—
160C2×C3⋊D3—61no—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—64no—
163C2×C3⋊D3—64no—
164C2×C3⋊D3—124no—
165C2×C3⋊D3—124no—
166C2×C3⋊D3—124no—
167C2×C3⋊D3ΓL(1,𝔽25)12yescomplemented
168C2×C3×D3—24no—
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174D32—632no—
175C32×C4—21no—
176C32×C4—24no—
177C32×C4—32no—
178C32×C4—44no—
179C3⋊Dic3—61no—
180C3⋊Dic3—64no—
181C3⋊Dic3—124no—
182C3⋊Dic3D3×D412yes—
183C3×Dic3—24no—
184C3×Dic3—64no—
185C23×C4—271no—
186C2×M16—272no—
187C22⋊C8—91no—
188C22⋊C8—272no—
189C22⋊C8—271no—
190C33#40711yescharacteristic, complemented, Sylow
191C23×C3—62no—
192C23×C3—91no—
193C22×D3—62no—
194C22×D3—91no—
195C22×D3—182no—
196C22×D3—182no—
197C22×D3—182no—
198C22×D3—188no—
199C22×D3—188no—
200C22×D3—188no—
201C22×D3—182no—
202C22×D3—362no—
203C22×D3C2×#15711yescharacteristic
204C2×C3×C4—22no—
205C2×C3×C4—42no—
206C2×C3×C4—62no—
207C2×C3×C4—68no—
208C2×C3×C4—91no—
209C2×C3×C4—182no—
210C2×C3×C4C2×#15711yescharacteristic
211C2×Dic3—62no—
212C2×Dic3—91no—
213C2×Dic3—182no—
214C2×Dic3—188no—
215C2×Dic3—182no—
216C2×Dic3—362no—
217C2×Dic3C2×#15711yescharacteristic
218C4×D3—22no—
219C4×D3—68no—
220C4×D3—94no—
221C4×D3—188no—
222C4×D3—182no—
223C4×D3—188no—
224C4×D3—188no—
225C4×D3—182no—
226C4×D3—368no—
227C3×C8—182no—
228C3⋊C8—182no—
229C2×C32—22no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—42no—
233C2×C32—42no—
234C2×C32—42no—
235C2×C32—62no—
236C2×C32D3×C22⋊C411yescharacteristic
237C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
238C2×C32D3×M1611yescharacteristic
239C3⋊D3—22no—
240C3⋊D3—62no—
241C3⋊D3—68no—
242C3⋊D3—128no—
243C3×D3—28no—
244C3×D3—68no—
245C24—271no—
246C22×C4—31no—
247C22×C4—91no—
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—22no—
257C22×C3—42no—
258C22×C3—64no—
259C22×C3—64no—
260C22×C3—64no—
261C22×C3—92no—
262C22×C3—181no—
263C22×C3—181no—
264C22×C3C22×#15711yescharacteristic
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—64no—
268C2×D3—64no—
269C2×D3—64no—
270C2×D3—92no—
271C2×D3—184no—
272C2×D3—188no—
273C2×D3—1832no—
274C2×D3—184no—
275C2×D3—184no—
276C2×D3—181no—
277C2×D3—181no—
278C2×D3—364no—
279C2×D3—364no—
280C2×D3—364no—
281C2×D3#378812yescomplemented
282C3×C4—24no—
283C3×C4—21no—
284C3×C4—44no—
285C3×C4—64no—
286C3×C4—92no—
287Dic3—64no—
288Dic3—184no—
289Dic3—181no—
290Dic3—364no—
291Dic3C22⋊#15712yes—
292C32—22no—
293C32—42no—
294C32#472611yescharacteristic, complemented
295C23—31no—
296C23—91no—
297C23—271no—
298C23—542no—
299C23—542no—
300C23—542no—
301C2×C4—31no—
302C2×C4—62no—
303C2×C4—91no—
304C2×C4—182no—
305C2×C4—274no—
306C2×C4—271no—
307C2×C4—542no—
308C2×C4—542no—
309C2×C4—542no—
310C2×C4D3×#15711yescharacteristic
311C8—182no—
312C8—542no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—22no—
316C2×C3—42no—
317C2×C3—42no—
318C2×C3—42no—
319C2×C3—68no—
320C2×C3—182no—
321C2×C3C2×#378811yescharacteristic
322C2×C3#6428311yescharacteristic
323C2×C3C2×C22⋊#15711yescharacteristic
324D3—22no—
325D3—68no—
326D3—188no—
327D3—182no—
328D3—368no—
329C22—32no—
330C22—61no—
331C22—61no—
332C22—92no—
333C22—181no—
334C22—181no—
335C22—272no—
336C22—548no—
337C22—541no—
338C22—541no—
339C22C2×D3×#15711yescharacteristic
340C4—21no—
341C4—32no—
342C4—92no—
343C4—541no—
344C3—22no—
345C3—42no—
346C3#1059380311yescharacteristic, complemented
347C2—62no—
348C2—182no—
349C2—542no—
350C2D3×#378811yescharacteristic
351C2#1172425411yescharacteristic
352C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×D3×C3⋊D3C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88926C2semidirect
3. #11723103C2semidirect
4. C3×#63778C2semidirect
5. (C2×D3)⋊#622C2semidirect
15. C2×C3⋊#622C22semidirect
16. C2×C3×#622C22semidirect
23. #63778D3semidirect
61. C2×#622C2×D3semidirect
69. C2×C3×C3⋊D3M16semidirect
72. C2×C32×D3M16semidirect
167. C2×C3⋊D3ΓL(1,𝔽25)semidirect
190. C33#407semidirect
281. C2×D3#3788semidirect
294. C32#4726semidirect
346. C3#10593803semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C2×#889262
#117231032
C3×#637782
(C2×D3)⋊#6222
#105938033
#47269