Finite Group Databasebeta

← #13173381

Subgroups — #13173381

Order 1728.

372 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11724517C211yesmaximal, characteristic, complemented
2#11724515C211yesmaximal, characteristic, complemented
3C3×#64125C211yesmaximal, characteristic
4#11723110C211yesmaximal, characteristic, complemented
5#11723108C211yesmaximal, characteristic, complemented
6#11723096C211yesmaximal, characteristic, complemented
7#11723092C211yesmaximal, characteristic, complemented
8#10593784—31nomaximal
9C3×#3832C2211yescharacteristic
10C3×#3830C2211yescharacteristic
11#88993—21no—
12C2×C3⋊#622—21no—
13C4×C3⋊#157C2211yescharacteristic
14#88989—21no—
15#88989C2211yescharacteristic
16#88989C2211yescharacteristic
17#88988—21no—
18#88988C2211yescharacteristic
19C33×SD16C411yescharacteristic, complemented
20C3×#3756C411yescharacteristic, complemented
21C3×#3753C411yescharacteristic, complemented
22C3×#3744C411yescharacteristic, complemented
23C3×C8×C3⋊D3C2211yescharacteristic
24#64289—31no—
25#64287—31no—
26#64125D311yescharacteristic
27C3×#916—62no—
28#63785—31no—
29#63783—31no—
30#63771—31no—
31#63767—31no—
32C2×C3⋊#157—21no—
33C33×Q8C2×C411yescharacteristic
34C33×D4C2×C411yescharacteristic
35C3×#638—21no—
36C2×C3×C3⋊Dic3—21no—
37C3×C4⋊(C3⋊D3)—21no—
38C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
39C3×C4×C3⋊D3—21no—
40C3×C4×C3⋊D3C2311yescharacteristic
41C3×#634C2×C411yescharacteristic
42C3⋊#622—21no—
43C3⋊#622—21no—
44C3⋊#622D411yescharacteristic
45C3⋊#622D411yescharacteristic
46C33×C8C2×C411yescharacteristic
47C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
48#5197—91nomaximal
49#3832C2×D311yescharacteristic
50#3830C2×D311yescharacteristic
51C3×#239—62no—
52C3×#237—62no—
53#3792—61no—
54C2×#622—61no—
55C4×#157—31no—
56#3788—31no—
57#3788—31no—
58#3788—61no—
59#3787—31no—
60#3787—61no—
61C32×SD16—22no—
62C32×SD16—42no—
63C32×SD16Dic311yescharacteristic, complemented
64#3756Dic311yescharacteristic, complemented
65#3753Dic311yescharacteristic, complemented
66#3744Dic311yescharacteristic, complemented
67C8×C3⋊D3C2×D311yescharacteristic
68C3×#216—62no—
69C3×#213—62no—
70C3×#204—62no—
71C3×D3×C8—62no—
72C22×C33—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3#4111yescharacteristic
75C3⋊#157—41no—
76C33×C4—21no—
77C33×C4C22×C411yescharacteristic
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×#135—91no—
81#948—91no—
82#948—91no—
83#916—62no—
84#837—91no—
85#837—91no—
86#819—91no—
87#815—91no—
88C2×#157—61no—
89C32×Q8—22no—
90C32×Q8—42no—
91C32×Q8C2×Dic311yescharacteristic
92C32×D4—22no—
93C32×D4—42no—
94C32×D4C2×Dic311yescharacteristic
95#638—21no—
96C2×C3⋊Dic3—21no—
97C4⋊(C3⋊D3)—21no—
98C4⋊(C3⋊D3)C2×Dic311yescharacteristic
99C4×C3⋊D3—21no—
100C4×C3⋊D3C22×D311yescharacteristic
101#634C2×Dic311yescharacteristic
102C3×#67—122no—
103C2×C3×Dic3—122no—
104C3×D12—62no—
105C3×D12—122no—
106C3×C4×D3—62no—
107C3×C4×D3—122no—
108C3×Dic6—62no—
109#622—31no—
110#622—31no—
111#622—61no—
112#622—61no—
113C32×C8—22no—
114C32×C8—42no—
115C32×C8C2×Dic311yescharacteristic
116C3⋊(C3⋊C8)C2×Dic311yescharacteristic
117C3×C3⋊C8—62no—
118#444—271noSylow
119C2×C33—41no—
120C2×C33C4×D411yescharacteristic
121C3×C3⋊D3—21no—
122C3×C3⋊D3—41no—
123C3×#41—91no—
124C3×#41—91no—
125#239—62no—
126#237—62no—
127C3×Q16—91no—
128C3×SD16—22no—
129C3×SD16—42no—
130C3×SD16—91no—
131C3×SD16#15711yescharacteristic, complemented
132C3×D8—91no—
133C2×C3×C8—91no—
134#216—62no—
135#213—62no—
136C4×Dic3—91no—
137#208—91no—
138#208—91no—
139#208—181no—
140#208—181no—
141C2×C3⋊C8—91no—
142C2×C3⋊C8—181no—
143C2×C3⋊C8—181no—
144#204—62no—
145D3×C8—62no—
146C22×C32—21no—
147C22×C32—42no—
148C22×C32—82no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3#23711yescharacteristic
151C2×C3×D3—62no—
152C2×C3×D3—122no—
153#157—121no—
154C32×C4—22no—
155C32×C4—21no—
156C32×C4—42no—
157C32×C4—42no—
158C32×C4—82no—
159C32×C4C22×Dic311yescharacteristic
160C3⋊Dic3—21no—
161C3⋊Dic3D3×D411yescharacteristic
162C3×Dic3—62no—
163C3×Dic3—122no—
164#135—91no—
165#131—271no—
166#131—271no—
167#108—271no—
168#104—271no—
169#104—271no—
170#97—271no—
171C33#44411yescharacteristic, complemented, Sylow
172C3×Q8—22no—
173C3×Q8—42no—
174C3×Q8—91no—
175C3×Q8C2×#15711yescharacteristic
176C3×D4—22no—
177C3×D4—42no—
178C3×D4—91no—
179C3×D4—181no—
180C3×D4—181no—
181C3×D4C2×#15711yescharacteristic
182C2×C3×C4—91no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185#67—122no—
186C2×Dic3—122no—
187C2×Dic3—181no—
188D12—62no—
189D12—122no—
190C4×D3—62no—
191C4×D3—122no—
192Dic6—62no—
193C3×C8—22no—
194C3×C8—42no—
195C3×C8—91no—
196C3×C8C2×#15711yescharacteristic
197C3⋊C8—62no—
198C3⋊C8—91no—
199C3⋊C8—91no—
200C3⋊C8—181no—
201C3⋊C8—181no—
202C2×C32—22no—
203C2×C32—42no—
204C2×C32—41no—
205C2×C32—82no—
206C2×C32—162no—
207C2×C32D4×Dic311yescharacteristic
208C3⋊D3—21no—
209C3⋊D3—41no—
210C3×D3—122no—
211C3×D3—242no—
212#41—91no—
213#41—91no—
214Q16—91no—
215SD16—91no—
216SD16C3⋊#15711yescharacteristic, complemented
217D8—91no—
218M16—271no—
219M16—271no—
220M16—541no—
221M16—541no—
222C2×C8—91no—
223C2×C8—271no—
224C2×C8—541no—
225C2×C8—541no—
226C42—271no—
227C22×C3—21no—
228C22×C3—42no—
229C22×C3—82no—
230C22×C3—91no—
231C22×C3—181no—
232C2×D3—62no—
233C2×D3—122no—
234C3×C4—21no—
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—42no—
238C3×C4—82no—
239C3×C4—91no—
240C3×C4—181no—
241C3×C4C22×#15711yescharacteristic
242Dic3—62no—
243Dic3—122no—
244Dic3—361no—
245C32—22no—
246C32—42no—
247C32#519711yescharacteristic, complemented
248Q8—91no—
249Q8C2×C3⋊#15711yescharacteristic
250D4—91no—
251D4—181no—
252D4—181no—
253D4C2×C3⋊#15711yescharacteristic
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—541no—
258C8—91no—
259C8—271no—
260C8—271no—
261C8—541no—
262C8—541no—
263C8C2×C3⋊#15711yescharacteristic
264C2×C3—22no—
265C2×C3—41no—
266C2×C3—42no—
267C2×C3—82no—
268C2×C3—162no—
269C2×C3—181no—
270C2×C3—361no—
271C2×C3D4×#15711yescharacteristic
272D3—122no—
273D3—242no—
274C22—21no—
275C22—91no—
276C22—181no—
277C4—21no—
278C4—91no—
279C4—181no—
280C4—1081no—
281C4C22×C3⋊#15711yescharacteristic
282C3—22no—
283C3—42no—
284C3#1059378411yescharacteristic, complemented
285C2—41no—
286C2—181no—
287C2—361no—
288C2D4×C3⋊#15711yescharacteristic
subgroupG/H
centerC2D4×C3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×SD16C4
socleC2×C33C4×D4
prime radicalC3×#64125C2
cosocleC3×C4×C3⋊D3C23
hypercenterSD16C3⋊#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ SD16

Complements

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

NH ≅ G/Ntype
1. #11724517C2semidirect
2. #11724515C2semidirect
4. #11723110C2semidirect
5. #11723108C2semidirect
6. #11723096C2semidirect
7. #11723092C2semidirect
19. C33×SD16C4semidirect
20. C3×#3756C4semidirect
21. C3×#3753C4semidirect
22. C3×#3744C4semidirect
63. C32×SD16Dic3semidirect
64. #3756Dic3semidirect
65. #3753Dic3semidirect
66. #3744Dic3semidirect
131. C3×SD16#157semidirect
171. C33#444semidirect
216. SD16C3⋊#157semidirect
247. C32#5197semidirect
284. C3#10593784semidirect

Maximal subgroups

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

subgroupindex
#117245172
#117245152
C3×#641252
#117231102
#117231082
#117230962
#117230922
#105937843
#51979