Finite Group Databasebeta

← #157×#204 #13166249

Subgroups — #157×#204 #13166249

Order 1728.

420 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 66 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
1D12×#157C211yesmaximal, characteristic
2Dic6×#157C211yesmaximal, characteristic, complemented
3C3⋊D3×#204C211yesmaximal, characteristic
4#11723093C211yesmaximal, characteristic, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#11722709C211yesmaximal, characteristic, complemented
7#11722707C211yesmaximal, characteristic
8SD16×#157—31nomaximal
9C2×D3×#157—21no—
10C3⋊D3×D12C2211yescharacteristic
11C3⋊D3×Dic6C2211yescharacteristic
12C4⋊(C3⋊#157)C2211yescharacteristic
13C4⋊(C3⋊#157)C2211yescharacteristic
14C3×C4×#157C2211yescharacteristic
15C3×#3787C2211yescharacteristic
16Dic3⋊#157—21no—
17(C2×D3)⋊#157—21no—
18Dic3×#157—21no—
19#88851C411yescharacteristic, complemented
20C3×C8×C3⋊D3C2211yescharacteristic
21C32×#204C411yescharacteristic, complemented
22#88795C411yescharacteristic, complemented
23#88794C411yescharacteristic, complemented
24Q8×#157—31no—
25D4×#157—31no—
26SD16×C3⋊D3—31no—
27D3×#204—62no—
28#63784—31no—
29#63782—31no—
30#63768—31no—
31C8×#157D311yescharacteristic, complemented
32C2×D3×C3⋊D3—21no—
33C2×C3⋊#157—21no—
34C2×C3⋊#157—21no—
35C2×C3×#157D411yescharacteristic
36C2×C3×#157D411yescharacteristic
37D3×#157—42no—
38C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
39#6308C2×C411yescharacteristic, complemented
40C3×C4×C3⋊D3C2311yescharacteristic
41C32×D12C2×C411yescharacteristic
42C32×Dic6C2×C411yescharacteristic, complemented
43#6293—21no—
44#6291—21no—
45Dic3×C3⋊D3—21no—
46C3×#622—21no—
47C33×C8C2×C411yescharacteristic, complemented
48C3×C3⋊(C3⋊C8)C2×C411yescharacteristic, complemented
49C4×#204—91nomaximal
50C22×#157—61no—
51Q8×C3⋊D3—31no—
52D4×C3⋊D3—31no—
53D3×D12—62no—
54D3×Dic6—62no—
55C22⋊#157—61no—
56C4⋊#157—31no—
57C4⋊#157—31no—
58C4⋊#157—61no—
59C4×#157—61no—
60C4×#157C2×D311yescharacteristic
61#3787C2×D311yescharacteristic
62C32×SD16—31no—
63#3755—31no—
64#3754—31no—
65#3744—31no—
66#3744—62no—
67#3744—122no—
68C8×C3⋊D3C2×D311yescharacteristic
69C3×#204—22no—
70C3×D3×C8—62no—
71#3717—62no—
72#3716—62no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C2×C32×D3—21no—
76D3×C3⋊D3—41no—
77D3×C3⋊D3—41no—
78C3⋊#157—41no—
79C3⋊#157—41no—
80C3×#157—21no—
81C3×#157SD1612yescomplemented
82C33×C4C22×C411yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3#4111yescharacteristic
85C32×Dic3—21no—
86D3×SD16—182no—
87C2×#204—91no—
88C4×D12—91no—
89C4×Dic6—91no—
90C3×C4×C8—91no—
91#821—91no—
92#817—91no—
93#816—91no—
94C22×C3⋊D3—61no—
95C2×D32—122no—
96C2×#157—61no—
97C2×#157—61no—
98C2×#157—122no—
99C2×#157D1211yescharacteristic
100C2×#157D1211yescharacteristic
101C32×Q8—31no—
102C32×D4—31no—
103#638—61no—
104C4⋊(C3⋊D3)—31no—
105C4⋊(C3⋊D3)—62no—
106C4⋊(C3⋊D3)—122no—
107C4×C3⋊D3—61no—
108C4×C3⋊D3C22×D311yescharacteristic
109#634—31no—
110#634—62no—
111#634—61no—
112#634—122no—
113C3×D12—22no—
114C3×C4×D3—62no—
115C3×Dic6—22no—
116#627—122no—
117#626—122no—
118D3×Dic3—122no—
119#622—21no—
120C32×C8—22no—
121C32×C8—42no—
122C32×C8C4×D311yescharacteristic, complemented
123C3⋊(C3⋊C8)C4×D311yescharacteristic, complemented
124C3×C3⋊C8—62no—
125C4×SD16—271noSylow
126C2×C33C4×D411yescharacteristic
127C3⋊(C3⋊D3)—41no—
128C3×C3⋊D3C2×SD1611yescharacteristic
129C3×C3⋊D3#13511yescharacteristic
130C32×D3—41no—
131D3×Q8—182no—
132D3×D4—182no—
133C2×D12—91no—
134C2×C4×D3—181no—
135C2×Dic6—91no—
136C3×SD16—62no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C42—91no—
140#215—182no—
141#214—182no—
142(C2×D3)⋊C4—181no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145Dic3⋊C4—181no—
146C4×Dic3—181no—
147#204—91no—
148#204—91no—
149#204—91no—
150#204—182no—
151#204—362no—
152#204#15711yescharacteristic, direct product factor
153D3×C8—62no—
154C22×C32—61no—
155C2×C3⋊D3—61no—
156C2×C3⋊D3—122no—
157C2×C3⋊D3—121no—
158C2×C3⋊D3—121no—
159C2×C3⋊D3—242no—
160C2×C3⋊D3C2×D1211yescharacteristic
161C2×C3×D3—42no—
162C2×C3×D3—62no—
163D32—242no—
164D32—242no—
165#157—21no—
166#157—121no—
167#157—121no—
168#157#20412yesdirect product factor
169C32×C4—22no—
170C32×C4—42no—
171C32×C4—61no—
172C32×C4C2×C4×D311yescharacteristic
173C3⋊Dic3—61no—
174C3⋊Dic3—122no—
175C3⋊Dic3—242no—
176C3⋊Dic3#23511yescharacteristic
177C3×Dic3—42no—
178C3×Dic3—62no—
179C2×SD16—271no—
180C4×Q8—271no—
181C4×D4—271no—
182#106—271no—
183#103—271no—
184#102—271no—
185C4×C8—91no—
186C33C4×SD1611yescharacteristic, complemented, Sylow
187C22×D3—181no—
188C22×D3—362no—
189C3×Q8—62no—
190C3×D4—62no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194#67—362no—
195C2×Dic3—181no—
196C2×Dic3—181no—
197C2×Dic3—181no—
198D12—91no—
199D12—182no—
200D12—181no—
201D12—362no—
202D12C2×#15711yescharacteristic
203C4×D3—62no—
204C4×D3—362no—
205C4×D3—362no—
206Dic6—91no—
207Dic6—182no—
208Dic6—181no—
209Dic6—362no—
210Dic6—362no—
211Dic6C2×#15711yescharacteristic, complemented
212C3×C8—22no—
213C3×C8—42no—
214C3×C8—91no—
215C3×C8—181no—
216C3×C8C2×#15711yescharacteristic, complemented
217C3⋊C8—62no—
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—121no—
221C2×C32C4×D1211yescharacteristic
222C3⋊D3—121no—
223C3⋊D3—242no—
224C3⋊D3—482no—
225C3⋊D3C2×#20411yescharacteristic
226C3⋊D3#90411yescharacteristic
227C3×D3—62no—
228C3×D3—62no—
229C3×D3—82no—
230C2×Q8—271no—
231C2×D4—271no—
232C22×C4—541no—
233SD16—31no—
234SD16—271no—
235SD16—271no—
236SD16—271no—
237C2×C8—91no—
238C2×C8—91no—
239C4⋊C4—271no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C22⋊C4—541no—
243C42—91no—
244C42—541no—
245C22×C3—91no—
246C22×C3—122no—
247C2×D3—21no—
248C2×D3—62no—
249C2×D3—181no—
250C2×D3—362no—
251C2×D3—361no—
252C2×D3—361no—
253C2×D3—722no—
254C2×D3—722no—
255C2×D3—722no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—92no—
259C3×C4—91no—
260C3×C4—122no—
261C3×C4—181no—
262C3×C4C22×#15711yescharacteristic
263Dic3—21no—
264Dic3—62no—
265Dic3—181no—
266Dic3—362no—
267Dic3—361no—
268Dic3—361no—
269Dic3—722no—
270C32—22no—
271C32—42no—
272C32C4×#20411yescharacteristic, complemented
273C23—541no—
274Q8—31no—
275Q8—271no—
276Q8—541no—
277D4—31no—
278D4—271no—
279D4—541no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—541no—
284C2×C4—541no—
285C2×C4—541no—
286C2×C4—1082no—
287C8—91no—
288C8—181no—
289C8D3×#15711yescharacteristic, complemented
290C2×C3—22no—
291C2×C3—42no—
292C2×C3—91no—
293C2×C3—91no—
294C2×C3—242no—
295C2×C3D4×#15711yescharacteristic
296D3—41no—
297D3—62no—
298D3—62no—
299D3—361no—
300D3—722no—
301D3—1442no—
302C22—61no—
303C22—91no—
304C22—541no—
305C22—1081no—
306C22—1081no—
307C4—61no—
308C4—92no—
309C4—91no—
310C4—181no—
311C4—541no—
312C4—1081no—
313C4—1081no—
314C4C2×D3×#15711yescharacteristic
315C3—22no—
316C3—42no—
317C3SD16×#15711yescharacteristic, complemented
318C2—91no—
319C2—91no—
320C2—121no—
321C2—1081no—
322C2D12×#15711yescharacteristic
subgroupG/H
centerC2D12×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radicalC3⋊D3×D12C22
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. Dic6×#157C2semidirect
4. #11723093C2semidirect
5. C3×C8×#157C2semidirect
6. #11722709C2semidirect
19. #88851C4semidirect
21. C32×#204C4semidirect
22. #88795C4semidirect
23. #88794C4semidirect
31. C8×#157D3semidirect
39. #6308C2×C4semidirect
42. C32×Dic6C2×C4semidirect
47. C33×C8C2×C4semidirect
48. C3×C3⋊(C3⋊C8)C2×C4semidirect
81. C3×#157SD16semidirect
122. C32×C8C4×D3semidirect
123. C3⋊(C3⋊C8)C4×D3semidirect
152. #204#157direct (N × H)
168. #157#204direct (N × H)
186. C33C4×SD16semidirect
211. Dic6C2×#157semidirect
216. C3×C8C2×#157semidirect
272. C32C4×#204semidirect
289. C8D3×#157semidirect
317. C3SD16×#157semidirect

Maximal subgroups

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

subgroupindex
D12×#1572
Dic6×#1572
C3⋊D3×#2042
#117230932
C3×C8×#1572
#117227092
#117227072
SD16×#1573
C4×#2049