Finite Group Databasebeta

← #13166556

Subgroups — #13166556

Order 1728.

416 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 62 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×Q8×#157C211yesmaximal, characteristic, complemented
2D12×#157C211yesmaximal, characteristic
3C3⋊D3×#215C211yesmaximal, characteristic
4#11723109C211yesmaximal, characteristic, complemented
5#11722715C211yesmaximal, characteristic, complemented
6#11722706C211yesmaximal, characteristic
7#11722704C211yesmaximal, characteristic, complemented
8#10593782—31nomaximal
9C2×D3×#157—21no—
10C3×Q8×C3⋊D3C2211yescharacteristic
11C3⋊D3×D12C2211yescharacteristic
12C4⋊(C3⋊#157)C2211yescharacteristic
13#88988C2211yescharacteristic
14C3×C4⋊#157—21no—
15C3×C4⋊#157C2211yescharacteristic
16C3×C4×#157—21no—
17C3×C4×#157C2211yescharacteristic
18(C2×D3)⋊#157—21no—
19#88862C411yescharacteristic, complemented
20C32×#215C411yescharacteristic, complemented
21#88797C411yescharacteristic, complemented
22#88792C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24Q8×#157D311yescharacteristic, complemented
25D4×#157—31no—
26SD16×C3⋊D3—31no—
27D3×#215—62no—
28#63784—31no—
29#63782—31no—
30#63769—31no—
31#63767—31no—
32C2×D3×C3⋊D3—21no—
33C2×C3⋊#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157D411yescharacteristic
36C2×C3×#157D411yescharacteristic
37D3×#157—42no—
38C33×Q8C2×C411yescharacteristic, complemented
39C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3C2311yescharacteristic
42C3×#634—21no—
43C3×#634C2×C411yescharacteristic, complemented
44C32×D12C2×C411yescharacteristic
45#6291—21no—
46C3⋊#622—21no—
47C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic, complemented
48C32×C3⋊C8C2×C411yescharacteristic, complemented
49#5026—91nomaximal
50C22×#157—61no—
51Q8×C3⋊D3C2×D311yescharacteristic
52D4×C3⋊D3—31no—
53C3×D3×Q8—62no—
54D3×D12—62no—
55C22⋊#157—61no—
56C4⋊#157—21no—
57C4⋊#157—31no—
58C4⋊#157C2×D311yescharacteristic
59C4×#157—21no—
60C4×#157C2×D311yescharacteristic
61#3787—31no—
62C32×SD16—31no—
63#3755—31no—
64#3755—62no—
65#3755—122no—
66#3754—31no—
67#3744—31no—
68C8×C3⋊D3—31no—
69C3×#215—22no—
70#3717—62no—
71#3715—62no—
72D3×C3⋊C8—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—21no—
80C3×#157—21no—
81C3×#157—41no—
82C33×C4—21no—
83C33×C4C22×C411yescharacteristic
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3#4111yescharacteristic
86C3×C4×Q8—91no—
87C2×#215—91no—
88D3×SD16—182no—
89C4×D12—91no—
90#835—91no—
91#809—91no—
92#807—91no—
93#803—91no—
94C22×C3⋊D3—61no—
95C2×D32—122no—
96C2×#157—21no—
97C2×#157—61no—
98C2×#157—122no—
99C2×#157#6711yescharacteristic
100C2×#157#6711yescharacteristic
101C32×Q8—22no—
102C32×Q8—42no—
103C32×Q8C4×D311yescharacteristic, complemented
104C32×D4—31no—
105#638—61no—
106C4⋊(C3⋊D3)—31no—
107C4⋊(C3⋊D3)—62no—
108C4⋊(C3⋊D3)—122no—
109C4×C3⋊D3—21no—
110C4×C3⋊D3C22×D311yescharacteristic
111#634—21no—
112#634C4×D311yescharacteristic, complemented
113C3×D12—22no—
114C3×C4×D3—62no—
115C3×C4×D3—122no—
116C3×Dic6—62no—
117C3×Dic6—122no—
118#626—122no—
119#622—61no—
120C32×C8—31no—
121C3⋊(C3⋊C8)—31no—
122C3⋊(C3⋊C8)—62no—
123C3⋊(C3⋊C8)—122no—
124C3×C3⋊C8—22no—
125#440—271noSylow
126C2×C33C4×D411yescharacteristic
127C3⋊(C3⋊D3)—41no—
128C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
129C3×C3⋊D3#13711yescharacteristic
130C32×D3—41no—
131C2×C3×Q8—91no—
132D3×Q8—62no—
133D3×D4—182no—
134C2×D12—91no—
135C2×C4×D3—181no—
136C3×SD16—62no—
137C3×C4⋊C4—91no—
138C3×C4⋊C4—181no—
139C3×C42—91no—
140C3×C42—181no—
141#215—91no—
142#215—91no—
143#215—91no—
144#215—182no—
145#215—362no—
146#215#15711yescharacteristic, complemented
147#214—182no—
148(C2×D3)⋊C4—181no—
149C4⋊Dic3—91no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—91no—
152#204—182no—
153D3×C8—182no—
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×#6711yescharacteristic
161C2×C3×D3—42no—
162C2×C3×D3—62no—
163D32—242no—
164D32—242no—
165#157—21no—
166#157—21no—
167#157—41no—
168#157—121no—
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4—82no—
174C32×C4C2×C4×D311yescharacteristic
175C3⋊Dic3—21no—
176C3⋊Dic3#23511yescharacteristic
177C3×Dic3—62no—
178C3×Dic3—122no—
179C2×SD16—271no—
180C4×Q8—91no—
181C4×D4—271no—
182#107—271no—
183#103—271no—
184#102—271no—
185#97—271no—
186C33#44011yescharacteristic, complemented, Sylow
187C22×D3—181no—
188C22×D3—362no—
189C3×Q8—22no—
190C3×Q8—42no—
191C3×Q8—91no—
192C3×Q8—181no—
193C3×Q8C2×#15711yescharacteristic, complemented
194C3×D4—62no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200#67—362no—
201C2×Dic3—181no—
202D12—91no—
203D12—182no—
204D12—181no—
205D12—362no—
206D12C2×#15711yescharacteristic
207C4×D3—62no—
208C4×D3—122no—
209C4×D3—362no—
210Dic6—62no—
211Dic6—122no—
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—121no—
221C2×C32C4×#6711yescharacteristic
222C3⋊D3—121no—
223C3⋊D3—242no—
224C3⋊D3—482no—
225C3⋊D3#94211yescharacteristic
226C3⋊D3#94911yescharacteristic
227C3×D3—62no—
228C3×D3—62no—
229C3×D3—82no—
230C2×Q8—91no—
231C2×D4—271no—
232C22×C4—541no—
233SD16—31no—
234SD16—271no—
235SD16—271no—
236SD16—271no—
237C2×C8—271no—
238C2×C8—271no—
239C4⋊C4—91no—
240C4⋊C4—181no—
241C4⋊C4—271no—
242C22⋊C4—541no—
243C42—91no—
244C42—181no—
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—21no—
257C3×C4—22no—
258C3×C4—42no—
259C3×C4—42no—
260C3×C4—82no—
261C3×C4—91no—
262C3×C4—181no—
263C3×C4—181no—
264C3×C4—181no—
265C3×C4—361no—
266C3×C4C22×#15711yescharacteristic
267Dic3—62no—
268Dic3—122no—
269Dic3—361no—
270C32—22no—
271C32—42no—
272C32#502611yescharacteristic, complemented
273C23—541no—
274Q8—91no—
275Q8—181no—
276Q8D3×#15711yescharacteristic, complemented
277D4—31no—
278D4—271no—
279D4—541no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4—1082no—
287C8—31no—
288C8—271no—
289C8—541no—
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—21no—
308C4—91no—
309C4—181no—
310C4—181no—
311C4—181no—
312C4—361no—
313C4—1081no—
314C4C2×D3×#15711yescharacteristic
315C3—22no—
316C3—42no—
317C3#1059378211yescharacteristic, complemented
318C2—91no—
319C2—91no—
320C2—121no—
321C2—1081no—
322C2#67×#15711yescharacteristic
subgroupG/H
centerC2#67×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×Q8C2×C4
socleC2×C33C4×D4
prime radicalC3⋊D3×D12C22
cosocleC3×C4×C3⋊D3C23
hypercenterQ8D3×#157
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×#157C2semidirect
4. #11723109C2semidirect
5. #11722715C2semidirect
7. #11722704C2semidirect
19. #88862C4semidirect
20. C32×#215C4semidirect
21. #88797C4semidirect
22. #88792C4semidirect
24. Q8×#157D3semidirect
38. C33×Q8C2×C4semidirect
43. C3×#634C2×C4semidirect
47. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
48. C32×C3⋊C8C2×C4semidirect
103. C32×Q8C4×D3semidirect
112. #634C4×D3semidirect
146. #215#157semidirect
186. C33#440semidirect
193. C3×Q8C2×#157semidirect
217. C3⋊C8C2×#157semidirect
272. C32#5026semidirect
276. Q8D3×#157semidirect
317. C3#10593782semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×#1572
D12×#1572
C3⋊D3×#2152
#117231092
#117227152
#117227062
#117227042
#105937823
#50269