Finite Group Databasebeta

← #10593653⋊C3 #13146905

Subgroups — #10593653⋊C3 #13146905

Order 1728.

356 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊#226)⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊#859)⋊C3C211yesmaximal, characteristic, complemented
4#10593653C311yesmaximal, characteristic, complemented
5C3×#5228—31nomaximal
6C2×Q8⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊#226)⋊C3—21no—
9#3757⋊C3—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C22×C3×SL(2,𝔽3)—21no—
12C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
13C2×C3⋊#226C2×C311yescharacteristic, complemented
14C2×C3×#216—31no—
15C2×C3×#216—31no—
16C3×#924—31no—
17C3⋊#859C2×C311yescharacteristic, complemented
18C3×#835—31no—
19C3×#835—31no—
20C3×#830—31no—
21C23×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic
25#5419—91no—
26C3×#451—91no—
27#5228—31no—
28C2×C32×Q8—21no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×Q8C2×D311yescharacteristic
32C22×C32×C4—31no—
33C2×C3×Dic6—31no—
34C2×C3×SL(2,𝔽3)—22no—
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
38C3⋊#226—21no—
39#3757—41no—
40C3×#217—61no—
41C3×#216—61no—
42C3×#216—62no—
43C3×#216—61no—
44C3×C4⋊Dic3—31no—
45C3×Dic3⋊C4—61no—
46C2×C3×C3⋊C8—31no—
47C2×C3×C3⋊C8—31no—
48C22×H(3,𝔽3)—41no—
49C22×H(3,𝔽3)—42no—
50C22×H(3,𝔽3)—81no—
51C22×H(3,𝔽3)—81no—
52(C3⋊Dic3)⋊C3—81no—
53(C3⋊Dic3)⋊C3—81no—
54C22×C3×Q8—31no—
55C22×C3×Q8C3×D311yescharacteristic
56C22×SL(2,𝔽3)—31no—
57C22×SL(2,𝔽3)—61no—
58C2×#226—91no—
59C2×C3×Q16—91no—
60C2×C3×Q16—91no—
61C3×C22⋊Q8—91no—
62C2×#216—31no—
63C2×#216—31no—
64#924—31no—
65#859—91no—
66C3×#103—91no—
67C3×#103—91no—
68C3×C22⋊C8—91no—
69#835—31no—
70#835—31no—
71#830—31no—
72C23×C32—41no—
73C23×C32—81no—
74C23×C32S411yescharacteristic
75C32×Q8—31no—
76C32×Q8—32no—
77C32×Q8—61no—
78C32×Q8#6711yescharacteristic
79C2×C32×C4—31no—
80C2×C32×C4—31no—
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3—41no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C3×Dic6—61no—
88C3×Dic6—61no—
89C3×SL(2,𝔽3)—21no—
90C3×SL(2,𝔽3)C3×D411yescharacteristic
91C3×C3⋊C8—61no—
92C3×C3⋊C8—61no—
93#451—91noSylow
94C2×H(3,𝔽3)—41no—
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)—41no—
97C2×H(3,𝔽3)—82no—
98C2×C3×Q8—21no—
99C2×C3×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—31no—
102C2×C3×Q8—61no—
103C2×C3×Q8—61no—
104C2×C3×Q8—62no—
105C2×C3×Q8—61no—
106C2×C3×Q8—61no—
107C2×C3×Q8—91no—
108C2×C3×Q8C2×C3×D311yescharacteristic
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—61no—
112C2×Dic6—31no—
113C2×SL(2,𝔽3)—31no—
114C2×SL(2,𝔽3)—61no—
115C2×SL(2,𝔽3)—62no—
116C2×SL(2,𝔽3)—61no—
117#226—181no—
118C3×Q16—181no—
119C3×Q16—182no—
120C3×Q16—181no—
121C2×C3×C8—91no—
122C2×C3×C8—91no—
123C3×C4⋊C4—91no—
124C3×C4⋊C4—181no—
125C3×C22⋊C4—181no—
126#217—61no—
127#217—361no—
128#216—61no—
129#216—62no—
130#216—61no—
131C4⋊Dic3—31no—
132Dic3⋊C4—61no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—31no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—41no—
138C22×C32—42no—
139C22×C32—81no—
140C22×C32—81no—
141C22×C32—82no—
142C22×C32—82no—
143C22×C32—81no—
144C22×C32—82no—
145C22×C32#22612yescomplemented
146C22×C32C2×S411yescharacteristic
147C32×C4—31no—
148C32×C4—31no—
149C32×C4—61no—
150C3⋊Dic3—81no—
151C3⋊Dic3—81no—
152C3×Dic3—121no—
153C3×Dic3—121no—
154C22×Q8(C3⋊D3)⋊C311yescharacteristic
155C2×Q16—91no—
156C2×Q16—91no—
157C22⋊Q8—91no—
158#103—91no—
159#103—91no—
160C22⋊C8—91no—
161H(3,𝔽3)—41noSylow
162C23×C3—31no—
163C23×C3—121no—
164C23×C3—241no—
165C23×C3C3×S411yescharacteristic
166C3×Q8—31no—
167C3×Q8—32no—
168C3×Q8—31no—
169C3×Q8—32no—
170C3×Q8—31no—
171C3×Q8—61no—
172C3×Q8—62no—
173C3×Q8—61no—
174C3×Q8—61no—
175C3×Q8—62no—
176C3×Q8—181no—
177C3×Q8—181no—
178C3×Q8C3×#6711yescharacteristic
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—361no—
196C2×Dic3—361no—
197Dic6—61no—
198Dic6—61no—
199SL(2,𝔽3)—31no—
200SL(2,𝔽3)—61no—
201C3×C8—181no—
202C3×C8—181no—
203C3⋊C8—61no—
204C3⋊C8—61no—
205C2×C32—22no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—82no—
210C2×C32—84no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32—81no—
214C2×C32C2×#22611yescharacteristic
215C2×C32#98311yescharacteristic
216C2×C32GL(2,ℤ/4)11yescharacteristic
217C2×Q8—21no—
218C2×Q8—31no—
219C2×Q8—61no—
220C2×Q8—91no—
221C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
222C22×C4—31no—
223Q16—181no—
224Q16—182no—
225Q16—181no—
226C2×C8—91no—
227C2×C8—91no—
228C4⋊C4—91no—
229C4⋊C4—181no—
230C22⋊C4—181no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—32no—
234C22×C3—31no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—122no—
238C22×C3—121no—
239C22×C3—241no—
240C22×C3—241no—
241C22×C3—242no—
242C22×C3—242no—
243C22×C3—242no—
244C22×C3—241no—
245C22×C3C3×#22612yes—
246C22×C3C2×C3×S411yescharacteristic
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—62no—
255C3×C4—61no—
256C3×C4—361no—
257C3×C4—361no—
258Dic3—121no—
259Dic3—121no—
260Dic3—721no—
261Dic3—721no—
262C32—41no—
263C32—81no—
264C32#541911yescharacteristic, complemented
265C23(C3⋊S4)⋊C311yescharacteristic
266Q8—31no—
267Q8—32no—
268Q8—61no—
269Q8—181no—
270Q8—181no—
271Q8#638⋊C311yescharacteristic
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—181no—
277C2×C4—181no—
278C8—181no—
279C8—181no—
280C2×C3—22no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—62no—
285C2×C3—121no—
286C2×C3—121no—
287C2×C3—121no—
288C2×C3—242no—
289C2×C3—244no—
290C2×C3—241no—
291C2×C3—241no—
292C2×C3—241no—
293C2×C3C2×C3×#22611yescharacteristic
294C2×C3C3×#98311yescharacteristic
295C2×C3C3×GL(2,ℤ/4)11yescharacteristic
296C22—21no—
297C22—21no—
298C22(C3⋊#226)⋊C312yescomplemented
299C22C2×(C3⋊S4)⋊C311yescharacteristic
300C4—31no—
301C4—31no—
302C4—61no—
303C4—361no—
304C4—361no—
305C3—31no—
306C3—121no—
307C3—241no—
308C3C3×#541911yescharacteristic
309C2—22no—
310C2C2×(C3⋊#226)⋊C311yescharacteristic
311C2#64264⋊C311yescharacteristic
312C2#64269⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC22×C32×Q8D3
prime radicalC22×Q8⋊H(3,𝔽3)C2
hypercenterC23(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊#226)⋊C3C2semidirect
3. (C3⋊#859)⋊C3C2semidirect
4. #10593653C3semidirect
13. C2×C3⋊#226C2×C3semidirect
17. C3⋊#859C2×C3semidirect
145. C22×C32#226semidirect
264. C32#5419semidirect
298. C22(C3⋊#226)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×(C3⋊#226)⋊C32
(C3⋊#859)⋊C32
#105936533
C3×#52283
#3757⋊C34