Finite Group Databasebeta

← #13165740

Subgroups — #13165740

Order 1728.

336 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 54 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#11723018C211yesmaximal, characteristic
2#11723015C211yesmaximal, characteristic
3#11722796C211yesmaximal, characteristic
4#11722779C211yesmaximal, characteristic
5#11722779C211yesmaximal, characteristic
6C3×#63742C211yesmaximal, characteristic
7Dic3×#622C211yesmaximal, characteristic
8#10593390—31nomaximal
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88815C2211yescharacteristic
12#88815C2211yescharacteristic
13#88807—21no—
14Dic3⋊(C3⋊Dic3)—21no—
15Dic3×C3⋊Dic3C2211yescharacteristic
16C3×#3724—21no—
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20#63870—31no—
21#63870—31no—
22#63846—61no—
23#63845—61no—
24C4×#622—31no—
25#63743—31no—
26#63743—31no—
27#63742—31no—
28#63742D311yescharacteristic
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36C3⋊#622Q812yes—
37C3×#622Q812yes—
38#4858—91nomaximal
39C2×#622—31no—
40C2×#622C2×D311yescharacteristic
41C4⋊(C3⋊Dic3)—61no—
42#3750—61no—
43C4×C3⋊Dic3—31no—
44C3×C4⋊Dic3—61no—
45C3×Dic3⋊C4—61no—
46C3×Dic3⋊C4—61no—
47C3×Dic3⋊C4—61no—
48C3×Dic3⋊C4—61no—
49C3×Dic3⋊C4—61no—
50#3724—21no—
51#3724—31no—
52#3724—31no—
53#3724C2×D311yescharacteristic
54#3724C2×D311yescharacteristic
55Dic3⋊Dic3—21no—
56Dic3⋊Dic3—61no—
57Dic3⋊Dic3—61no—
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—61no—
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—121no—
63Dic3⋊Dic3—121no—
64Dic32—61no—
65Dic32—61no—
66C22×C33C2×D411yescharacteristic
67C3⋊(C3⋊Dic3)—21no—
68C3×C3⋊Dic3—41no—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3C2×Q811yescharacteristic
71C3×C3⋊Dic3C2×Q811yescharacteristic
72C32×Dic3—21no—
73C32×Dic3—41no—
74C32×Dic3—41no—
75#890—91no—
76#890—91no—
77#889—181no—
78#889—181no—
79C3×#106—91no—
80#817—91no—
81C8×Dic3—91no—
82#807—91no—
83#807—91no—
84C2×C32×C4—31no—
85C2×C32×C4—61no—
86C2×C32×C4—61no—
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—61no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—121no—
93C2×C3⋊Dic3—121no—
94C2×C3⋊Dic3C22×D311yescharacteristic
95C2×C3×Dic3—21no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—41no—
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#622Dic612yes—
113#499—271noSylow
114C2×C33C4⋊Q811yescharacteristic
115C2×C33#13511yescharacteristic
116C2×C33#13511yescharacteristic
117C2×C3×C8—91no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4—91no—
120C3×C4⋊C4—181no—
121C3×C4⋊C4—181no—
122C4⋊Dic3—61no—
123C4⋊Dic3—91no—
124C4⋊Dic3—91no—
125C4⋊Dic3—181no—
126C4⋊Dic3—181no—
127C4⋊Dic3—181no—
128C4⋊Dic3—361no—
129Dic3⋊C4—61no—
130Dic3⋊C4—61no—
131Dic3⋊C4—61no—
132Dic3⋊C4—61no—
133Dic3⋊C4—61no—
134Dic3⋊C4—181no—
135Dic3⋊C4—181no—
136Dic3⋊C4—181no—
137Dic3⋊C4—181no—
138Dic3⋊C4—181no—
139Dic3⋊C4—361no—
140C4×Dic3—91no—
141C4×Dic3—181no—
142C4×Dic3—181no—
143C2×C3⋊C8—91no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32D3×D411yescharacteristic
149C32×C4—61no—
150C32×C4—121no—
151C32×C4—121no—
152C3⋊Dic3—41no—
153C3⋊Dic3—41no—
154C3⋊Dic3—61no—
155C3⋊Dic3—121no—
156C3⋊Dic3—121no—
157C3⋊Dic3—241no—
158C3⋊Dic3—241no—
159C3⋊Dic3C2×Dic611yescharacteristic
160C3⋊Dic3D3×Q811yescharacteristic
161C3×Dic3—41no—
162C3×Dic3—41no—
163C3×Dic3—41no—
164C3×Dic3—41no—
165C3×Dic3—42no—
166C3×Dic3—41no—
167C3×Dic3—41no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—81no—
173C3×Dic3—121no—
174C3×Dic3—121no—
175C3×Dic3—122no—
176C3×Dic3—121no—
177C3×Dic3—121no—
178C3×Dic3—241no—
179#125—271no—
180#125—271no—
181#107—271no—
182#107—271no—
183#106—91no—
184#106—271no—
185C4×C8—271no—
186C33#49911yescharacteristic, complemented, Sylow
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—91no—
194C2×C3×C4—121no—
195C2×C3×C4—121no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—91no—
207C2×Dic3—121no—
208C2×Dic3—121no—
209C2×Dic3—181no—
210C2×Dic3—181no—
211C2×Dic3—361no—
212C2×Dic3—361no—
213C2×Dic3GO+(4,𝔽2)11yescharacteristic
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#88811yescharacteristic
229C2×C32#91611yescharacteristic
230C2×C32#91611yescharacteristic
231C2×C8—91no—
232C2×C8—271no—
233C4⋊C4—91no—
234C4⋊C4—91no—
235C4⋊C4—181no—
236C4⋊C4—181no—
237C4⋊C4—271no—
238C4⋊C4—271no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C42—271no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3C2×GO+(4,𝔽2)11yescharacteristic
247C3×C4—91no—
248C3×C4—91no—
249C3×C4—121no—
250C3×C4—121no—
251C3×C4—121no—
252C3×C4—121no—
253C3×C4—122no—
254C3×C4—121no—
255C3×C4—121no—
256C3×C4—241no—
257C3×C4—361no—
258C3×C4—361no—
259Dic3—21no—
260Dic3—41no—
261Dic3—41no—
262Dic3—61no—
263Dic3—61no—
264Dic3—61no—
265Dic3—61no—
266Dic3—121no—
267Dic3—121no—
268Dic3—122no—
269Dic3—121no—
270Dic3—121no—
271Dic3—181no—
272Dic3—241no—
273Dic3—361no—
274Dic3—361no—
275Dic3—721no—
276Dic3—721no—
277C32—21no—
278C32—21no—
279C32—41no—
280C32—41no—
281C32#485811yescharacteristic, complemented
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—271no—
289C8—92no—
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
306C22D3×GO+(4,𝔽2)11yescharacteristic
307C4—61no—
308C4—91no—
309C4—91no—
310C4—121no—
311C4—121no—
312C4—361no—
313C4—361no—
314C4—541no—
315C3—21no—
316C3—21no—
317C3—41no—
318C3—41no—
319C3#1059339011yescharacteristic, complemented
320C2#1172412511yescharacteristic
321C2#1172413311yescharacteristic
322C2#1172413611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#499
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
186. C33#499semidirect
281. C32#4858semidirect
319. C3#10593390semidirect

Maximal subgroups

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

subgroupindex
#117230182
#117230152
#117227962
#117227792
C3×#637422
Dic3×#6222
#105933903
#48589