Finite Group Databasebeta

← #13166538

Subgroups — #13166538

Order 1728.

416 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 60 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
1D4×C3⋊#157C211yesmaximal, characteristic
2#11724245C211yesmaximal, characteristic, complemented
3C3⋊D3×#214C211yesmaximal, characteristic
4C3×#63782C211yesmaximal, characteristic
5#11722715C211yesmaximal, characteristic, complemented
6#11722709C211yesmaximal, characteristic, complemented
7#11722705C211yesmaximal, characteristic, complemented
8#10593782—31nomaximal
9C22×C3⋊#157—21no—
10C3×D4×C3⋊D3C2211yescharacteristic
11C3⋊D3×Dic6C2211yescharacteristic
12#88994—21no—
13C4⋊(C3⋊#157)C2211yescharacteristic
14C4×C3⋊#157C2211yescharacteristic
15C3×C4⋊#157C2211yescharacteristic
16C3×#3787C2211yescharacteristic
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×#157—31no—
26SD16×C3⋊D3—31no—
27D3×#214—62no—
28#63784—31no—
29#63782D311yescharacteristic
30#63769—31no—
31#63767—31no—
32C22×C3×C3⋊D3—21no—
33C2×C3⋊#157—21no—
34C2×C3⋊#157—42no—
35C2×C3⋊#157D411yescharacteristic
36C2×C3⋊#157D411yescharacteristic
37C2×C3×#157—21no—
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
49#4757—91nomaximal
50C22×#157—61no—
51Q8×C3⋊D3—31no—
52D4×C3⋊D3C2×D311yescharacteristic
53C3×D3×D4—62no—
54D3×Dic6—62no—
55C22⋊#157—61no—
56C4⋊#157—31no—
57C4⋊#157—61no—
58C4⋊#157C2×D311yescharacteristic
59C4×#157—31no—
60C4×#157—61no—
61#3787C2×D311yescharacteristic
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—21no—
79C3⋊#157—21no—
80C3⋊#157—41no—
81C3×#157—41no—
82C33×C4C22×C411yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3#4111yescharacteristic
85C32×Dic3—21no—
86D4×Dic3—91no—
87C2×#214—91no—
88D3×SD16—182no—
89#887—91no—
90C3×#102—91no—
91#816—91no—
92#815—91no—
93#807—91no—
94C22×C3⋊D3—21no—
95C22×C3×D3—122no—
96C2×#157—21no—
97C2×#157—31no—
98C2×#157—31no—
99C2×#157—61no—
100C2×#157—122no—
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—21no—
120C32×C8—31no—
121C3⋊(C3⋊C8)—31no—
122C3⋊(C3⋊C8)—62no—
123C3⋊(C3⋊C8)—122no—
124C3×C3⋊C8—22no—
125#440—271noSylow
126C2×C33—41no—
127C2×C33C4×D411yescharacteristic
128C3×C3⋊D3—41no—
129C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
130C3×C3⋊D3#13711yescharacteristic
131C2×C3×D4—91no—
132C22×Dic3—181no—
133D3×Q8—182no—
134D3×D4—62no—
135C2×Dic6—91no—
136C3×SD16—62no—
137C2×C3×C8—91no—
138C3×C4⋊C4—91no—
139#217—181no—
140#215—182no—
141#214—91no—
142#214—91no—
143#214—91no—
144#214—182no—
145#214—362no—
146#214#15711yescharacteristic, complemented
147C4⋊Dic3—91no—
148Dic3⋊C4—181no—
149C4×Dic3—91no—
150C4×Dic3—181no—
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⋊D3D3×D411yescharacteristic
161C2×C3×D3—62no—
162C2×C3×D3—122no—
163C2×C3×D3—242no—
164C2×C3×D3—242no—
165#157—41no—
166#157—61no—
167#157—61no—
168#157—121no—
169C32×C4—22no—
170C32×C4—42no—
171C32×C4—61no—
172C32×C4C2×C4×D311yescharacteristic
173C3⋊Dic3—61no—
174C3⋊Dic3—122no—
175C3⋊Dic3—242no—
176C3⋊Dic3#23711yescharacteristic
177C3×Dic3—42no—
178C3×Dic3—62no—
179C2×SD16—271no—
180C4×Q8—271no—
181C4×D4—271no—
182#107—271no—
183#103—271no—
184#102—91no—
185#97—271no—
186C33#44011yescharacteristic, 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—181no—
197#67—122no—
198C2×Dic3—91no—
199C2×Dic3—91no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—362no—
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—181no—
214C3⋊C8—91no—
215C3⋊C8—182no—
216C3⋊C8—362no—
217C3⋊C8C2×#15711yescharacteristic, complemented
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—41no—
221C2×C32—82no—
222C2×C32—162no—
223C2×C32#67⋊C411yescharacteristic
224C3⋊D3—41no—
225C3⋊D3#91111yescharacteristic
226C3⋊D3#91511yescharacteristic
227C3×D3—62no—
228C3×D3—62no—
229C3×D3—242no—
230C2×Q8—271no—
231C2×D4—91no—
232C22×C4—541no—
233SD16—31no—
234SD16—271no—
235SD16—271no—
236SD16—271no—
237C2×C8—91no—
238C2×C8—271no—
239C4⋊C4—91no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C22⋊C4—541no—
243C42—271no—
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—91no—
259C3×C4—122no—
260C3×C4—361no—
261C3×C4C22×#15711yescharacteristic
262Dic3—21no—
263Dic3—62no—
264Dic3—181no—
265Dic3—181no—
266Dic3—181no—
267Dic3—362no—
268Dic3—361no—
269Dic3—722no—
270C32—22no—
271C32—42no—
272C32#475711yescharacteristic, complemented
273C23—181no—
274Q8—31no—
275Q8—271no—
276Q8—541no—
277D4—91no—
278D4—181no—
279D4D3×#15711yescharacteristic
280C2×C4—91no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4—541no—
285C2×C4—541no—
286C2×C4—1082no—
287C8—31no—
288C8—181no—
289C8—271no—
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—91no—
309C4—361no—
310C4—541no—
311C4—541no—
312C4—541no—
313C4—1081no—
314C4C2×D3×#15711yescharacteristic
315C3—22no—
316C3—42no—
317C3#1059378211yescharacteristic, 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
hypocenterC33#440
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. #11724245C2semidirect
5. #11722715C2semidirect
6. #11722709C2semidirect
7. #11722705C2semidirect
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
146. #214#157semidirect
186. C33#440semidirect
211. Dic6C2×#157semidirect
217. C3⋊C8C2×#157semidirect
272. C32#4757semidirect
317. C3#10593782semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊#1572
#117242452
C3⋊D3×#2142
C3×#637822
#117227152
#117227092
#117227052
#105937823
#47579