Finite Group Databasebeta

← #13166995

Subgroups — #13166995

Order 1728.

450 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 59 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#11723678C211yesmaximal, characteristic, complemented
2C3×#63902C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic, complemented
4#11722778C212yesmaximal, complemented
5#11722773C212yesmaximal
6#10593357—31nomaximal
7C2×C3×#634—21no—
8C2×C3×#625C2211yescharacteristic
9C2×C3×D3×Dic3—21no—
10C2×C3⋊#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×(C2×D3)⋊C4—21no—
13#88815—21no—
14#88815C2212yes—
15C3×#3724C2211yescharacteristic
16C3×#67⋊C4—61no—
17#63902D311yescharacteristic
18#63873—31no—
19#63866—61no—
20#63779—31no—
21#63742—32no—
22#63739—32no—
23C22×C32×D3—21no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×#634—42no—
29C2×C32×Dic3—21no—
30C3×#625—21no—
31C3×#625Q812yescomplemented
32C3×D3×Dic3—42no—
33C3⋊#622—22no—
34#5008—91nomaximal
35C2×#634—21no—
36C2×C3×#67—61no—
37C22×C3×Dic3—61no—
38C2×C3×C4×D3—61no—
39C2×C3×Dic6—61no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—121no—
42C2×#625C2×D311yescharacteristic
43C2×D3×Dic3—21no—
44C2×#622—32no—
45C32×C22⋊C4—61no—
46C4×C3⋊Dic3C2×D311yescharacteristic
47C3×(C2×D3)⋊C4—21no—
48C3×(C2×D3)⋊C4—21no—
49C3×(C2×D3)⋊C4—41no—
50C3×Dic3⋊C4—61no—
51C3×C4×Dic3—61no—
52C3×C4×Dic3—61no—
53#3724—32no—
54#3724—61no—
55#3724C2×D311yescharacteristic
56Dic3⋊Dic3—62no—
57Dic3⋊Dic3—62no—
58C22×C33C2×D411yescharacteristic
59C2×C32×D3—22no—
60C2×C32×D3—41no—
61C2×C32×D3—41no—
62C33×C4—21no—
63C3×C3⋊Dic3—42no—
64C3×C3⋊Dic3SD1612yes—
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C3×C3⋊Dic3#4111yescharacteristic
67C32×Dic3—41no—
68C3×C4×D4—91no—
69#944—181no—
70#67⋊C4—61no—
71#869—91no—
72#832—92no—
73#808—92no—
74C4⋊(C3⋊C8)—91no—
75C23×C32—61no—
76C22×C3×D3—21no—
77C22×C3×D3—21no—
78C22×C3×D3—41no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—61no—
84C2×C32×C4#6711yescharacteristic
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3#6711yescharacteristic
87C2×C3⋊Dic3C22×D311yescharacteristic
88#634—42no—
89C3×#67—62no—
90C3×#67—62no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C2×C3×Dic3—122no—
102C3×C4×D3—122no—
103C3×Dic6—122no—
104C3×Dic6—122no—
105C3×Dic6—124no—
106#625—21no—
107#625Dic612yescomplemented
108D3×Dic3—42no—
109#622—62no—
110#476—271noSylow
111C2×C33C22⋊Q811yescharacteristic
112C2×C33C2×SD1611yescharacteristic
113C2×C33AGL(1,ℤ/8)11yescharacteristic
114C32×D3—42no—
115C2×C3×Q8—181no—
116C2×C3×D4—91no—
117C22×C3×C4—181no—
118C2×#67—61no—
119C22×Dic3—61no—
120C2×C4×D3—61no—
121C2×Dic6—61no—
122C2×Dic6—61no—
123C2×Dic6—121no—
124C3×C4⋊C4—91no—
125C3×C22⋊C4—61no—
126C3×C22⋊C4—61no—
127C3×C22⋊C4—121no—
128C3×C42—91no—
129(C2×D3)⋊C4—21no—
130C4⋊Dic3—92no—
131C4⋊Dic3—181no—
132Dic3⋊C4—61no—
133Dic3⋊C4—182no—
134Dic3⋊C4—182no—
135C4×Dic3—61no—
136C4×Dic3—61no—
137C2×C3⋊C8—92no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—41no—
141C22×C32—41no—
142C22×C32—62no—
143C22×C32—121no—
144C22×C32—121no—
145C22×C32C2×#6711yescharacteristic
146C2×C3×D3—22no—
147C2×C3×D3—22no—
148C2×C3×D3—42no—
149C2×C3×D3—41no—
150C2×C3×D3—41no—
151C2×C3×D3—41no—
152C2×C3×D3—41no—
153C2×C3×D3—81no—
154C2×C3×D3—81no—
155C32×C4—22no—
156C32×C4—21no—
157C32×C4—41no—
158C32×C4—42no—
159C32×C4—81no—
160C32×C4—121no—
161C3⋊Dic3—42no—
162C3⋊Dic3#21412yes—
163C3⋊Dic3C2×Dic611yescharacteristic
164C3⋊Dic3#23511yescharacteristic
165C3×Dic3—41no—
166C3×Dic3—41no—
167C3×Dic3—62no—
168C3×Dic3—62no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—81no—
174C3×Dic3—122no—
175C3×Dic3—124no—
176C3×Dic3—122no—
177C4⋊Q8—271no—
178C4×D4—91no—
179#106—272no—
180C4⋊C8—271no—
181#102—272no—
182C33#47611yescharacteristic, complemented, Sylow
183C23×C3—61no—
184C23×C3—61no—
185C23×C3—121no—
186C22×D3—21no—
187C3×Q8—362no—
188C3×D4—92no—
189C3×D4—181no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—121no—
199C2×C3×C4—182no—
200C2×C3×C4—362no—
201C2×C3×C4GO+(4,𝔽2)11yescharacteristic
202#67—62no—
203#67—62no—
204C2×Dic3—21no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—122no—
212C2×Dic3—122no—
213C4×D3—122no—
214Dic6—122no—
215Dic6—122no—
216Dic6—124no—
217C3⋊C8—182no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—122no—
231C2×C32#92411yescharacteristic
232C2×C32C2×#21411yescharacteristic
233C2×C32#94911yescharacteristic
234C3×D3—42no—
235C3×D3—42no—
236C3×D3—82no—
237C2×Q8—181no—
238C2×D4—91no—
239C22×C4—181no—
240C2×C8—272no—
241C4⋊C4—91no—
242C4⋊C4—272no—
243C4⋊C4—541no—
244C22⋊C4—61no—
245C42—91no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—62no—
251C22×C3—62no—
252C22×C3—122no—
253C22×C3—121no—
254C22×C3—121no—
255C22×C3—121no—
256C22×C3—121no—
257C22×C3—241no—
258C22×C3—241no—
259C22×C3C2×GO+(4,𝔽2)11yescharacteristic
260C2×D3—22no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—42no—
267C3×C4—81no—
268C3×C4—91no—
269C3×C4—91no—
270C3×C4—92no—
271C3×C4—121no—
272C3×C4—121no—
273C3×C4—241no—
274C3×C4—362no—
275Dic3—41no—
276Dic3—62no—
277Dic3—61no—
278Dic3—61no—
279Dic3—62no—
280Dic3—61no—
281Dic3—61no—
282Dic3—122no—
283Dic3—124no—
284Dic3—122no—
285C32—21no—
286C32—21no—
287C32—41no—
288C32—41no—
289C32#500811yescharacteristic, complemented
290C23—61no—
291Q8—362no—
292D4—92no—
293D4—181no—
294C2×C4—61no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—182no—
298C2×C4—362no—
299C2×C4D32⋊D311yescharacteristic
300C8—542no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—122no—
314C2×C3—122no—
315C2×C3—242no—
316C2×C3D32⋊Q811yescharacteristic
317C2×C3#6422711yescharacteristic
318C2×C3#6423611yescharacteristic
319D3—42no—
320C22—62no—
321C22—121no—
322C22—121no—
323C22C2×D32⋊D311yescharacteristic
324C4—21no—
325C4—91no—
326C4—91no—
327C4—92no—
328C4—121no—
329C4—362no—
330C3—21no—
331C3—21no—
332C3—41no—
333C3—41no—
334C3#1059335711yescharacteristic, complemented
335C2—122no—
336C2#1172430011yescharacteristic
337C2#1172431011yescharacteristic
338C2#1172432211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C4D32⋊D3
hypocenterC33#476
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
1. #11723678C2semidirect
3. #11723104C2semidirect
4. #11722778C2semidirect
31. C3×#625Q8semidirect
107. #625Dic6semidirect
182. C33#476semidirect
289. C32#5008semidirect
334. C3#10593357semidirect

Maximal subgroups

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

subgroupindex
#117236782
C3×#639022
#117231042
#117227782
#117227732
#105933573
#50089