Finite Group Databasebeta

← #13167076

Subgroups — #13167076

Order 1728.

460 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88989C211yesmaximal, characteristic
2#11723673C211yesmaximal, characteristic
3C3×#63900C211yesmaximal, characteristic
4#11722779C212yesmaximal, complemented
5#11722778C212yesmaximal, complemented
6#10593394—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#624—21no—
9C2×C3⋊#622C2212yes—
10#88989C414yescomplemented
11C3×C4×C3⋊Dic3—21no—
12C32×C4⋊Dic3—21no—
13#88815C2211yescharacteristic
14#88815C2211yescharacteristic
15#88814—21no—
16C3×#3724C2212yes—
17C2×#3788—31no—
18C3×D3×C4⋊C4—61no—
19#63900D311yescharacteristic
20#63874—31no—
21#63855—61no—
22#63743—32no—
23#63742—32no—
24C22×C3×C3⋊D3D411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—21no—
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4×C3⋊D3C2×C412yes—
30C3×C4×C3⋊D3Q812yes—
31C2×C32×Dic3—22no—
32C3×#624—42no—
33C3⋊#622C2×C414yescomplemented
34#4964—91nomaximal
35C2×C4×C3⋊D3C2×D311yescharacteristic
36C2×C3×C4×D3—61no—
37C2×C3×C4×D3—61no—
38C2×C3×C4×D3—62no—
39C2×#624—21no—
40C2×#622—32no—
41#3788—34no—
42C32×C4⋊C4—61no—
43C4×C3⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—21no—
46C3×C4⋊Dic3—41no—
47C3×Dic3⋊C4—62no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—121no—
51#3724—31no—
52#3724—31no—
53#3724C2×D312yes—
54Dic3⋊Dic3—61no—
55Dic3⋊Dic3—61no—
56#3722—61no—
57(C2×D3)⋊Dic3—61no—
58(C2×D3)⋊Dic3—61no—
59C22×C33C2×D411yescharacteristic
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3—21no—
62C2×C3×C3⋊D3C4⋊C412yes—
63C33×C4C4⋊C412yes—
64C3×C3⋊Dic3—41no—
65C3×C3⋊Dic3—41no—
66C3×C3⋊Dic3C22×C411yescharacteristic
67C3×C3⋊Dic3C2×Q811yescharacteristic
68C32×Dic3—42no—
69C2×C3×C4⋊C4—91no—
70#926—91no—
71C2×#208—91no—
72D3×C4⋊C4—61no—
73#872—181no—
74#808—92no—
75#807—92no—
76C22×C3⋊D3#6711yescharacteristic
77C22×C3×D3—61no—
78C22×C3×D3—61no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—62no—
84C2×C32×C4#6711yescharacteristic
85C2×C3⋊Dic3—21no—
86C2×C3⋊Dic3—21no—
87C2×C3⋊Dic3C22×D311yescharacteristic
88C4×C3⋊D3Dic612yes—
89C4×C3⋊D3C4×D312yes—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—42no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—121no—
100C2×C3×Dic3—121no—
101C3×C4×D3—62no—
102C3×C4×D3—62no—
103C3×C4×D3—62no—
104C3×C4×D3—62no—
105C3×C4×D3—124no—
106#624—42no—
107#622—34no—
108#428—271noSylow
109C2×C33C2×C4⋊C411yescharacteristic
110C2×C33AGL(1,ℤ/8)11yescharacteristic
111C2×C33#13711yescharacteristic
112C3×C3⋊D3—22no—
113C22×C3×C4—91no—
114C22×C3×C4—181no—
115C2×C4×D3—61no—
116C2×C4×D3—61no—
117C2×C4×D3—62no—
118C3×C4⋊C4—61no—
119C3×C4⋊C4—61no—
120C3×C4⋊C4—92no—
121C3×C4⋊C4—121no—
122C3×C42—181no—
123#217—181no—
124(C2×D3)⋊C4—181no—
125(C2×D3)⋊C4—181no—
126C4⋊Dic3—21no—
127C4⋊Dic3—91no—
128C4⋊Dic3—91no—
129Dic3⋊C4—62no—
130Dic3⋊C4—181no—
131Dic3⋊C4—181no—
132C4×Dic3—61no—
133C4×Dic3—61no—
134C4×Dic3—121no—
135#208—94no—
136C2×C3⋊C8—92no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—41no—
140C22×C32—41no—
141C22×C32C2×#6711yescharacteristic
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3⋊D3Dic3⋊C412yes—
145C2×C3×D3—62no—
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—121no—
150C2×C3×D3—121no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—42no—
154C32×C4—42no—
155C32×C4—122no—
156C32×C4Dic3⋊C412yes—
157C3⋊Dic3—41no—
158C3⋊Dic3—41no—
159C3⋊Dic3C2×Dic611yescharacteristic
160C3⋊Dic3C2×C4×D311yescharacteristic
161C3×Dic3—42no—
162C3×Dic3—42no—
163C3×Dic3—61no—
164C3×Dic3—61no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—82no—
168C3×Dic3—121no—
169C3×Dic3—121no—
170C3×Dic3—122no—
171C3×Dic3—122no—
172C3×Dic3—121no—
173C3×Dic3—121no—
174C2×M16—271no—
175#117—271no—
176C2×C4⋊C4—91no—
177#107—272no—
178#106—272no—
179C33#42811yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C22×D3—61no—
182C22×D3—61no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—122no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4—362no—
196C2×C3×C4GO+(4,𝔽2)11yescharacteristic
197C2×Dic3—22no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—121no—
205C2×Dic3—121no—
206C4×D3—62no—
207C4×D3—62no—
208C4×D3—62no—
209C4×D3—62no—
210C4×D3—124no—
211C3⋊C8—94no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32C2×Dic3⋊C411yescharacteristic
225C2×C32#93211yescharacteristic
226C2×C32#94211yescharacteristic
227C3⋊D3—22no—
228C3×D3—64no—
229C3×D3—122no—
230C22×C4—91no—
231C22×C4—181no—
232M16—274no—
233C2×C8—272no—
234C4⋊C4—61no—
235C4⋊C4—92no—
236C4⋊C4—271no—
237C4⋊C4—271no—
238C22⋊C4—541no—
239C42—181no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—92no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3C2×GO+(4,𝔽2)11yescharacteristic
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—121no—
253C2×D3—121no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—42no—
257C3×C4—42no—
258C3×C4—91no—
259C3×C4—91no—
260C3×C4—122no—
261C3×C4—122no—
262C3×C4—242no—
263C3×C4—361no—
264C3×C4—361no—
265C3×C4#377312yes—
266Dic3—42no—
267Dic3—61no—
268Dic3—61no—
269Dic3—61no—
270Dic3—61no—
271Dic3—121no—
272Dic3—121no—
273Dic3—122no—
274Dic3—122no—
275Dic3—121no—
276Dic3—121no—
277C32—21no—
278C32—21no—
279C32—41no—
280C32—41no—
281C32#496411yescharacteristic, complemented
282C23—91no—
283C2×C4—62no—
284C2×C4—91no—
285C2×C4—92no—
286C2×C4—92no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—362no—
290C2×C4D32⋊D311yescharacteristic
291C8—274no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—182no—
305C2×C3#6422411yescharacteristic
306C2×C3#6422611yescharacteristic
307C2×C3C2×#377311yescharacteristic
308D3—64no—
309D3—122no—
310C22—92no—
311C22—181no—
312C22—181no—
313C22C2×D32⋊D311yescharacteristic
314C4—91no—
315C4—91no—
316C4—122no—
317C4—361no—
318C4—361no—
319C4#8893412yes—
320C3—21no—
321C3—21no—
322C3—41no—
323C3—41no—
324C3#1059339411yescharacteristic, complemented
325C2—182no—
326C2#1172430411yescharacteristic
327C2#1172430711yescharacteristic
328C2C2×#8893411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#428
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722779C2semidirect
5. #11722778C2semidirect
10. #88989C4semidirect
33. C3⋊#622C2×C4semidirect
179. C33#428semidirect
281. C32#4964semidirect
324. C3#10593394semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
#117236732
C3×#639002
#117227792
#117227782
#105933943
#49649