Finite Group Databasebeta

← #13167073

Subgroups — #13167073

Order 1728.

336 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 55 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#11723679C211yesmaximal, characteristic
2C3×#63870C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic
4#11722779C211yesmaximal, characteristic
5#11722779C211yesmaximal, characteristic
6#11722778C211yesmaximal, characteristic
7#11722778C211yesmaximal, characteristic
8#10593390—31nomaximal
9C2×C3⋊#622C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C3×#3750—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×Dic3⋊C4—21no—
14#88815—21no—
15#88815C2211yescharacteristic
16#88815C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×#889—61no—
21#63870—31no—
22#63870D311yescharacteristic
23#63867—61no—
24C4×#622—31no—
25#63743—31no—
26#63743—31no—
27#63742—31no—
28#63742—31no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C3⋊#622Q812yes—
37C3⋊#622Q812yes—
38#5065—91nomaximal
39C2×#622—31no—
40C2×#622—31no—
41C32×C4⋊C4—61no—
42#3750—21no—
43C4×C3⋊Dic3C2×D311yescharacteristic
44C3×C4⋊Dic3—61no—
45C3×Dic3⋊C4—21no—
46C3×Dic3⋊C4—21no—
47C3×Dic3⋊C4—41no—
48C3×Dic3⋊C4—61no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—61no—
51C3×Dic3⋊C4—61no—
52C3×Dic3⋊C4—61no—
53C3×Dic3⋊C4—121no—
54C3×C4×Dic3—61no—
55C3×C4×Dic3—61no—
56#3724—31no—
57#3724—31no—
58#3724—61no—
59#3724C2×D311yescharacteristic
60#3724C2×D311yescharacteristic
61Dic3⋊Dic3—21no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66C22×C33C2×D411yescharacteristic
67C33×C4—21no—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3—41no—
71C3×C3⋊Dic3C2×Q811yescharacteristic
72C3×C3⋊Dic3C2×Q811yescharacteristic
73C32×Dic3—41no—
74C32×Dic3—41no—
75C3×#125—91no—
76#889—61no—
77#889—181no—
78#870—91no—
79#808—91no—
80#808—91no—
81#807—91no—
82#807—91no—
83C4×C3⋊C8—91no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—41no—
87C2×C32×C4—41no—
88C2×C32×C4—61no—
89C2×C32×C4—61no—
90C2×C32×C4#6711yescharacteristic
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3#6711yescharacteristic
94C2×C3⋊Dic3C22×D311yescharacteristic
95C2×C3×Dic3—21no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—121no—
110C2×C3×Dic3—121no—
111#622—32no—
112#622—32no—
113#499—271noSylow
114C2×C33C4⋊Q811yescharacteristic
115C2×C33#13511yescharacteristic
116C2×C33#13511yescharacteristic
117C3×C4⋊C4—61no—
118C3×C4⋊C4—61no—
119C3×C4⋊C4—91no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—121no—
122C3×C4⋊C4—181no—
123C3×C4⋊C4—181no—
124C3×C42—91no—
125C4⋊Dic3—61no—
126C4⋊Dic3—91no—
127C4⋊Dic3—91no—
128C4⋊Dic3—181no—
129Dic3⋊C4—21no—
130Dic3⋊C4—61no—
131Dic3⋊C4—61no—
132Dic3⋊C4—61no—
133Dic3⋊C4—61no—
134Dic3⋊C4—61no—
135Dic3⋊C4—121no—
136Dic3⋊C4—181no—
137Dic3⋊C4—181no—
138Dic3⋊C4—181no—
139Dic3⋊C4—181no—
140C4×Dic3—61no—
141C4×Dic3—61no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—91no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32C2×#6711yescharacteristic
149C32×C4—21no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4—81no—
153C32×C4—81no—
154C32×C4—121no—
155C32×C4—121no—
156C3⋊Dic3—21no—
157C3⋊Dic3—41no—
158C3⋊Dic3—41no—
159C3⋊Dic3D3×Q811yescharacteristic
160C3⋊Dic3D3×Q811yescharacteristic
161C3×Dic3—41no—
162C3×Dic3—41no—
163C3×Dic3—42no—
164C3×Dic3—41no—
165C3×Dic3—41no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—81no—
171C3×Dic3—121no—
172C3×Dic3—121no—
173C3×Dic3—121no—
174C3×Dic3—121no—
175C3×Dic3—122no—
176C3×Dic3—121no—
177C3×Dic3—121no—
178C3×Dic3—241no—
179#125—91no—
180#125—271no—
181#107—271no—
182#107—271no—
183#106—271no—
184#106—271no—
185C4×C8—271no—
186C33#49911yescharacteristic, complemented, Sylow
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4GO+(4,𝔽2)11yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—121no—
213C2×Dic3—121no—
214C3⋊C8—92no—
215C3⋊C8—92no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32#94411yescharacteristic
229C2×C32#95011yescharacteristic
230C2×C32#95011yescharacteristic
231C2×C8—271no—
232C2×C8—271no—
233C4⋊C4—61no—
234C4⋊C4—91no—
235C4⋊C4—91no—
236C4⋊C4—181no—
237C4⋊C4—181no—
238C4⋊C4—271no—
239C4⋊C4—271no—
240C4⋊C4—541no—
241C42—91no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3C2×GO+(4,𝔽2)11yescharacteristic
247C3×C4—21no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—81no—
251C3×C4—81no—
252C3×C4—91no—
253C3×C4—91no—
254C3×C4—121no—
255C3×C4—121no—
256C3×C4—122no—
257C3×C4—121no—
258C3×C4—121no—
259C3×C4—181no—
260C3×C4—241no—
261C3×C4—361no—
262C3×C4—361no—
263Dic3—41no—
264Dic3—41no—
265Dic3—61no—
266Dic3—61no—
267Dic3—61no—
268Dic3—61no—
269Dic3—121no—
270Dic3—121no—
271Dic3—121no—
272Dic3—121no—
273Dic3—122no—
274Dic3—121no—
275Dic3—121no—
276Dic3—241no—
277C32—21no—
278C32—21no—
279C32—41no—
280C32—41no—
281C32#506511yescharacteristic, complemented
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4D32⋊D311yescharacteristic
289C8—272no—
290C8—272no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3#6422211yescharacteristic
304C2×C3#6422311yescharacteristic
305C2×C3#6423711yescharacteristic
306C22C2×D32⋊D311yescharacteristic
307C4—21no—
308C4—91no—
309C4—91no—
310C4—121no—
311C4—121no—
312C4—181no—
313C4—361no—
314C4—361no—
315C3—21no—
316C3—21no—
317C3—41no—
318C3—41no—
319C3#1059339011yescharacteristic, complemented
320C2#1172430211yescharacteristic
321C2#1172430311yescharacteristic
322C2#1172432411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#499
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
186. C33#499semidirect
281. C32#5065semidirect
319. C3#10593390semidirect

Maximal subgroups

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

subgroupindex
#117236792
C3×#638702
C4×C3⋊#6222
#117227792
#117227782
#105933903
#50659