Finite Group Databasebeta

← #157×#214 #13166535

Subgroups — #157×#214 #13166535

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
1C3×D4×#157C211yesmaximal, characteristic
2Dic6×#157C211yesmaximal, characteristic, complemented
3C3⋊D3×#214C211yesmaximal, characteristic
4#11723107C211yesmaximal, characteristic
5#11722714C211yesmaximal, characteristic, complemented
6#11722708C211yesmaximal, characteristic, complemented
7C3⋊C8×#157C211yesmaximal, characteristic, complemented
8SD16×#157—31nomaximal
9C22×C3×#157—21no—
10C3×D4×C3⋊D3C2211yescharacteristic
11C3⋊D3×Dic6C2211yescharacteristic
12C4⋊(C3⋊#157)C2211yescharacteristic
13#88988C2211yescharacteristic
14C3×C22⋊#157—21no—
15C3×C4⋊#157C2211yescharacteristic
16C3×C4×#157C2211yescharacteristic
17Dic3⋊#157—21no—
18Dic3×#157—21no—
19#88861C411yescharacteristic, complemented
20C32×#214C411yescharacteristic, complemented
21#88796C411yescharacteristic, complemented
22#88793C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24Q8×#157—31no—
25D4×#157D311yescharacteristic
26SD16×C3⋊D3—31no—
27D3×#214—62no—
28#63784—31no—
29#63782—31no—
30#63768—31no—
31C8×#157—31no—
32C22×C3×C3⋊D3—21no—
33C2×C3⋊#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157—42no—
36C2×C3×#157D411yescharacteristic
37C2×C3×#157D411yescharacteristic
38C33×D4C2×C411yescharacteristic
39#6308C2×C411yescharacteristic, complemented
40C3×#638—21no—
41C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C32×Dic6C2×C411yescharacteristic, complemented
44#6293—21no—
45Dic3×C3⋊D3—21no—
46C3⋊#622—21no—
47C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic, complemented
48C32×C3⋊C8C2×C411yescharacteristic, complemented
49C4×#214—91nomaximal
50C22×#157—21no—
51Q8×C3⋊D3—31no—
52D4×C3⋊D3C2×D311yescharacteristic
53C3×D3×D4—62no—
54D3×Dic6—62no—
55C22⋊#157—21no—
56C4⋊#157—31no—
57C4⋊#157—61no—
58C4⋊#157C2×D311yescharacteristic
59C4×#157—61no—
60C4×#157C2×D311yescharacteristic
61#3787—31no—
62C32×SD16—31no—
63#3755—31no—
64#3754—31no—
65#3754—62no—
66#3754—122no—
67#3744—31no—
68C8×C3⋊D3—31no—
69C3×#214—22no—
70#3716—62no—
71#3715—62no—
72D3×C3⋊C8—62no—
73C22×C33—21no—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C3⋊#157—41no—
79C3×#157—21no—
80C3×#157—41no—
81C3×#157SD1612yescomplemented
82C33×C4C22×C411yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3#4111yescharacteristic
85C32×Dic3—21no—
86C3×C4×D4—91no—
87C2×#214—91no—
88D3×SD16—182no—
89C4×Dic6—91no—
90#832—91no—
91#810—91no—
92#808—91no—
93C4×C3⋊C8—91no—
94C22×C3⋊D3—21no—
95C22×C3×D3—122no—
96C2×#157—21no—
97C2×#157—42no—
98C2×#157—61no—
99C2×#157#6711yescharacteristic
100C2×#157#6711yescharacteristic
101C32×Q8—31no—
102C32×D4—22no—
103C32×D4—42no—
104C32×D4C4×D311yescharacteristic
105#638—21no—
106C4⋊(C3⋊D3)C4×D311yescharacteristic
107C4×C3⋊D3—61no—
108C4×C3⋊D3C22×D311yescharacteristic
109#634—31no—
110#634—62no—
111#634—61no—
112#634—122no—
113C3×#67—122no—
114C3×D12—62no—
115C3×C4×D3—62no—
116C3×Dic6—22no—
117#627—122no—
118D3×Dic3—122no—
119#622—61no—
120C32×C8—31no—
121C3⋊(C3⋊C8)—31no—
122C3⋊(C3⋊C8)—62no—
123C3⋊(C3⋊C8)—122no—
124C3×C3⋊C8—22no—
125C4×SD16—271noSylow
126C2×C33—41no—
127C2×C33C4×D411yescharacteristic
128C3×C3⋊D3—41no—
129C3×C3⋊D3C2×SD1611yescharacteristic
130C3×C3⋊D3#13511yescharacteristic
131C2×C3×D4—91no—
132C22×C3×C4—181no—
133D3×Q8—182no—
134D3×D4—62no—
135C2×Dic6—91no—
136C3×SD16—62no—
137C3×C4⋊C4—91no—
138C3×C22⋊C4—181no—
139C3×C42—91no—
140#215—182no—
141#214—91no—
142#214—91no—
143#214—91no—
144#214—182no—
145#214—362no—
146#214#15711yescharacteristic, direct product factor
147C4⋊Dic3—91no—
148Dic3⋊C4—181no—
149C4×Dic3—181no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—91no—
152#204—182no—
153D3×C8—182no—
154C22×C32—21no—
155C22×C32—42no—
156C22×C32—82no—
157C2×C3⋊D3—21no—
158C2×C3⋊D3—41no—
159C2×C3⋊D3—41no—
160C2×C3⋊D3C2×#6711yescharacteristic
161C2×C3×D3—62no—
162C2×C3×D3—122no—
163C2×C3×D3—242no—
164C2×C3×D3—242no—
165#157—21no—
166#157—41no—
167#157—121no—
168#157#21412yesdirect 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—91no—
182#106—271no—
183#103—271no—
184#102—271no—
185C4×C8—271no—
186C33C4×SD1611yescharacteristic, complemented, Sylow
187C23×C3—181no—
188C22×D3—122no—
189C3×Q8—62no—
190C3×D4—22no—
191C3×D4—42no—
192C3×D4—91no—
193C3×D4—181no—
194C3×D4C2×#15711yescharacteristic
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—181no—
199C2×C3×C4—362no—
200#67—122no—
201C2×Dic3—181no—
202C2×Dic3—181no—
203D12—62no—
204C4×D3—62no—
205C4×D3—362no—
206Dic6—91no—
207Dic6—182no—
208Dic6—181no—
209Dic6—362no—
210Dic6—362no—
211Dic6C2×#15711yescharacteristic, complemented
212C3×C8—62no—
213C3⋊C8—91no—
214C3⋊C8—182no—
215C3⋊C8—181no—
216C3⋊C8—362no—
217C3⋊C8C2×#15711yescharacteristic, complemented
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—41no—
221C2×C32—82no—
222C2×C32—162no—
223C2×C32C4×#6711yescharacteristic
224C3⋊D3—41no—
225C3⋊D3C2×#21411yescharacteristic
226C3⋊D3#95011yescharacteristic
227C3×D3—62no—
228C3×D3—62no—
229C3×D3—242no—
230C2×Q8—271no—
231C2×D4—91no—
232C22×C4—181no—
233SD16—31no—
234SD16—271no—
235SD16—271no—
236SD16—271no—
237C2×C8—271no—
238C2×C8—271no—
239C4⋊C4—91no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C22⋊C4—181no—
243C42—91no—
244C42—541no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—82no—
248C22×C3—91no—
249C22×C3—181no—
250C22×C3—361no—
251C22×C3—361no—
252C2×D3—62no—
253C2×D3—122no—
254C2×D3—242no—
255C2×D3—242no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—92no—
259C3×C4—91no—
260C3×C4—122no—
261C3×C4—181no—
262C3×C4—361no—
263C3×C4C22×#15711yescharacteristic
264Dic3—21no—
265Dic3—62no—
266Dic3—181no—
267Dic3—362no—
268Dic3—361no—
269Dic3—722no—
270C32—22no—
271C32—42no—
272C32C4×#21411yescharacteristic, complemented
273C23—181no—
274Q8—31no—
275Q8—271no—
276Q8—541no—
277D4—91no—
278D4—181no—
279D4D3×#15711yescharacteristic
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—181no—
284C2×C4—362no—
285C2×C4—541no—
286C2×C4—541no—
287C8—31no—
288C8—271no—
289C8—541no—
290C2×C3—22no—
291C2×C3—41no—
292C2×C3—42no—
293C2×C3—82no—
294C2×C3—91no—
295C2×C3—91no—
296C2×C3—162no—
297C2×C3—361no—
298C2×C3D4×#15711yescharacteristic
299D3—62no—
300D3—62no—
301D3—242no—
302C22—21no—
303C22—91no—
304C22—181no—
305C22—361no—
306C22—361no—
307C4—61no—
308C4—92no—
309C4—91no—
310C4—181no—
311C4—361no—
312C4—541no—
313C4—1081no—
314C4C2×D3×#15711yescharacteristic
315C3—22no—
316C3—42no—
317C3SD16×#15711yescharacteristic, complemented
318C2—41no—
319C2—91no—
320C2—91no—
321C2—361no—
322C2#67×#15711yescharacteristic
subgroupG/H
centerC2#67×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×D4C2×C4
socleC2×C33C4×D4
prime radicalC3×D4×C3⋊D3C22
cosocleC3×C4×C3⋊D3C23
hypercenterD4D3×#157
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. Dic6×#157C2semidirect
5. #11722714C2semidirect
6. #11722708C2semidirect
7. C3⋊C8×#157C2semidirect
19. #88861C4semidirect
20. C32×#214C4semidirect
21. #88796C4semidirect
22. #88793C4semidirect
39. #6308C2×C4semidirect
43. C32×Dic6C2×C4semidirect
47. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
48. C32×C3⋊C8C2×C4semidirect
81. C3×#157SD16semidirect
146. #214#157direct (N × H)
168. #157#214direct (N × H)
186. C33C4×SD16semidirect
211. Dic6C2×#157semidirect
217. C3⋊C8C2×#157semidirect
272. C32C4×#214semidirect
317. C3SD16×#157semidirect

Maximal subgroups

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

subgroupindex
C3×D4×#1572
Dic6×#1572
C3⋊D3×#2142
#117231072
#117227142
#117227082
C3⋊C8×#1572
SD16×#1573
C4×#2149