Finite Group Databasebeta

← #13166562

Subgroups — #13166562

Order 1728.

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

normal subgroups only

This group has 372 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
1Q8×C3⋊#157C211yesmaximal, characteristic
2#11724245C211yesmaximal, characteristic
3C3⋊D3×#216C211yesmaximal, characteristic
4C3×#63784C211yesmaximal, characteristic
5#11722714C211yesmaximal, characteristic
6#11722709C211yesmaximal, characteristic
7#11722705C211yesmaximal, characteristic
8#10593786—31nomaximal
9C3×Q8×C3⋊D3C2211yescharacteristic
10C3⋊D3×Dic6C2211yescharacteristic
11C4⋊(C3⋊#157)—21no—
12C4⋊(C3⋊#157)C2211yescharacteristic
13C4×C3⋊#157—21no—
14C4×C3⋊#157C2211yescharacteristic
15C3×C4⋊#157C2211yescharacteristic
16C3×#3787C2211yescharacteristic
17Dic3⋊#157—21no—
18Dic3×#157—21no—
19#88863C411yescharacteristic, complemented
20C32×#216C411yescharacteristic, complemented
21#88800C411yescharacteristic, complemented
22#88798C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24Q8×#157—31no—
25Q8×#157—31no—
26Q16×C3⋊D3—31no—
27D3×#216—62no—
28#63784—31no—
29#63784D311yescharacteristic
30#63768—31no—
31#63767—31no—
32C2×C3⋊#157—21no—
33C2×C3⋊#157D411yescharacteristic
34C2×C3⋊#157D411yescharacteristic
35C2×C3×#157—21no—
36C33×Q8C2×C411yescharacteristic
37#6308C2×C411yescharacteristic
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3C2311yescharacteristic
40C3×#634—21no—
41C3×#634C2×C411yescharacteristic
42C32×Dic6C2×C411yescharacteristic
43#6293—21no—
44Dic3×C3⋊D3—21no—
45C3×#622—21no—
46C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
47C32×C3⋊C8C2×C411yescharacteristic
48#4789—91nomaximal
49Q8×C3⋊D3—31no—
50Q8×C3⋊D3C2×D311yescharacteristic
51C3×D3×Q8—62no—
52D3×Dic6—62no—
53C4⋊#157—31no—
54C4⋊#157—61no—
55C4⋊#157—61no—
56C4⋊#157C2×D311yescharacteristic
57C4×#157—31no—
58C4×#157—61no—
59C4×#157—61no—
60#3787C2×D311yescharacteristic
61C32×Q16—31no—
62#3756—31no—
63#3756—31no—
64#3756—62no—
65#3756—122no—
66#3746—31no—
67C8×C3⋊D3—31no—
68C3×#216—22no—
69#3719—62no—
70#3718—62no—
71D3×C3⋊C8—62no—
72C2×C3×C3⋊D3C2×D411yescharacteristic
73C3⋊#157—21no—
74C3⋊#157—21no—
75C3⋊#157—41no—
76C3×#157—41no—
77C33×C4—21no—
78C33×C4C22×C411yescharacteristic
79C3⋊(C3⋊Dic3)—21no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3#4111yescharacteristic
82C32×Dic3—21no—
83Q8×Dic3—91no—
84C2×#216—91no—
85D3×Q16—182no—
86#887—91no—
87C3×#103—91no—
88#816—91no—
89#815—91no—
90#808—91no—
91C2×#157—21no—
92C2×#157—31no—
93C2×#157—31no—
94C2×#157—61no—
95C32×Q8—22no—
96C32×Q8—31no—
97C32×Q8—42no—
98C32×Q8C4×D311yescharacteristic
99C4×C3⋊D3—21no—
100C4×C3⋊D3—61no—
101C4×C3⋊D3C22×D311yescharacteristic
102#634—21no—
103#634—31no—
104#634—62no—
105#634—61no—
106#634—122no—
107#634C4×D311yescharacteristic
108C3×C4×D3—62no—
109C3×C4×D3—122no—
110C3×Dic6—22no—
111C3×Dic6—62no—
112C3×Dic6—122no—
113#627—122no—
114D3×Dic3—122no—
115#622—21no—
116C32×C8—31no—
117C3⋊(C3⋊C8)—31no—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—122no—
120C3×C3⋊C8—22no—
121#441—271noSylow
122C2×C33C4×D411yescharacteristic
123C3×C3⋊D3#13711yescharacteristic
124C3×C3⋊D3#13711yescharacteristic
125C2×C3×Q8—91no—
126D3×Q8—62no—
127D3×Q8—182no—
128C2×Dic6—91no—
129C3×Q16—62no—
130C2×C3×C8—91no—
131C3×C4⋊C4—91no—
132#216—91no—
133#216—91no—
134#216—91no—
135#216—182no—
136#216—182no—
137#216—362no—
138#216#15711yescharacteristic, complemented
139C4⋊Dic3—91no—
140C4⋊Dic3—181no—
141Dic3⋊C4—181no—
142C4×Dic3—91no—
143C4×Dic3—181no—
144C4×Dic3—181no—
145C2×C3⋊C8—91no—
146Dic12—182no—
147D3×C8—182no—
148C2×C3⋊D3D3×D411yescharacteristic
149C2×C3×D3—62no—
150#157—41no—
151#157—61no—
152#157—61no—
153#157—121no—
154C32×C4—22no—
155C32×C4—21no—
156C32×C4—42no—
157C32×C4—42no—
158C32×C4—61no—
159C32×C4—82no—
160C32×C4C2×C4×D311yescharacteristic
161C3⋊Dic3—21no—
162C3⋊Dic3—61no—
163C3⋊Dic3—122no—
164C3⋊Dic3—242no—
165C3⋊Dic3#23711yescharacteristic
166C3×Dic3—42no—
167C3×Dic3—62no—
168C3×Dic3—122no—
169C2×Q16—271no—
170C4×Q8—271no—
171C4×Q8—271no—
172#106—271no—
173#103—91no—
174#103—271no—
175#97—271no—
176C33#44111yescharacteristic, complemented, Sylow
177C3×Q8—22no—
178C3×Q8—42no—
179C3×Q8—62no—
180C3×Q8—91no—
181C3×Q8—181no—
182C3×Q8C2×#15711yescharacteristic
183C2×C3×C4—91no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×Dic3—91no—
187C2×Dic3—91no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C4×D3—62no—
191C4×D3—122no—
192C4×D3—362no—
193Dic6—62no—
194Dic6—91no—
195Dic6—122no—
196Dic6—182no—
197Dic6—181no—
198Dic6—362no—
199Dic6—362no—
200Dic6C2×#15711yescharacteristic
201C3×C8—62no—
202C3×C8—181no—
203C3⋊C8—91no—
204C3⋊C8—182no—
205C3⋊C8—362no—
206C3⋊C8C2×#15711yescharacteristic
207C2×C32—22no—
208C2×C32—42no—
209C2×C32#67⋊C411yescharacteristic
210C3⋊D3#91511yescharacteristic
211C3⋊D3#91811yescharacteristic
212C3×D3—62no—
213C3×D3—62no—
214C2×Q8—91no—
215C2×Q8—271no—
216Q16—31no—
217Q16—271no—
218Q16—271no—
219Q16—271no—
220C2×C8—91no—
221C2×C8—271no—
222C4⋊C4—91no—
223C4⋊C4—271no—
224C4⋊C4—541no—
225C4⋊C4—541no—
226C42—271no—
227C42—541no—
228C42—541no—
229C22×C3—91no—
230C2×D3—62no—
231C3×C4—21no—
232C3×C4—22no—
233C3×C4—42no—
234C3×C4—42no—
235C3×C4—82no—
236C3×C4—91no—
237C3×C4—122no—
238C3×C4—181no—
239C3×C4—361no—
240C3×C4C22×#15711yescharacteristic
241Dic3—21no—
242Dic3—62no—
243Dic3—122no—
244Dic3—181no—
245Dic3—181no—
246Dic3—181no—
247Dic3—362no—
248Dic3—361no—
249Dic3—722no—
250C32—22no—
251C32—42no—
252C32#478911yescharacteristic, complemented
253Q8—31no—
254Q8—91no—
255Q8—181no—
256Q8—271no—
257Q8—541no—
258Q8D3×#15711yescharacteristic
259C2×C4—91no—
260C2×C4—181no—
261C2×C4—181no—
262C2×C4—271no—
263C2×C4—271no—
264C2×C4—541no—
265C2×C4—541no—
266C8—31no—
267C8—181no—
268C8—271no—
269C2×C3—22no—
270C2×C3—42no—
271C2×C3—91no—
272C2×C3—91no—
273C2×C3D4×#15711yescharacteristic
274D3—62no—
275D3—62no—
276C22—91no—
277C4—21no—
278C4—61no—
279C4—91no—
280C4—181no—
281C4—361no—
282C4—541no—
283C4—541no—
284C4—541no—
285C4—1081no—
286C4C2×D3×#15711yescharacteristic
287C3—22no—
288C3—42no—
289C3#1059378611yescharacteristic, complemented
290C2—91no—
291C2—91no—
292C2#67⋊#15711yescharacteristic
subgroupG/H
centerC2#67⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×Q8C2×C4
socleC2×C33C4×D4
prime radicalC2×C3×C3⋊D3C2×D4
cosocleC3×C4×C3⋊D3C23
hypercenterQ8D3×#157
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
19. #88863C4semidirect
20. C32×#216C4semidirect
21. #88800C4semidirect
22. #88798C4semidirect
138. #216#157semidirect
176. C33#441semidirect
252. C32#4789semidirect
289. C3#10593786semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊#1572
#117242452
C3⋊D3×#2162
C3×#637842
#117227142
#117227092
#117227052
#105937863
#47899