Finite Group Databasebeta

← #10595422⋊C3 #13166144

Subgroups — #10595422⋊C3 #13166144

Order 1728.

348 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 58 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
1SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
4#10595422C311yesmaximal, characteristic, complemented
5#5570⋊C3—31nomaximal
6#5569⋊C3—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×Dic3⋊Dic3—41nomaximal
9Q8×C3⋊Dic3C2×C311yescharacteristic
10C3×Q8×Dic3C2×C311yescharacteristic
11C3×Q8×Dic3C2×C311yescharacteristic
12C3×C4×SL(2,𝔽3)—31no—
13C3×C4×SL(2,𝔽3)—31no—
14#63918—31no—
15C4×#627—31no—
16#63872—31no—
17#63869—31no—
18Dic3×SL(2,𝔽3)—21no—
19Dic3×SL(2,𝔽3)—21no—
20Dic3×SL(2,𝔽3)—31no—
21Dic3×SL(2,𝔽3)—31no—
22Dic3×SL(2,𝔽3)—61no—
23Dic3×SL(2,𝔽3)D311yescharacteristic
24Dic3×SL(2,𝔽3)D311yescharacteristic
25C2×C3×C3⋊Dic3—41no—
26C2×C32×Dic3—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)Q811yescharacteristic, complemented
29#5570—31no—
30#5569—31no—
31#557⋊C3—91no—
32C2×C32×Q8C22×C311yescharacteristic
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)—41no—
37C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
38C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
39C2×#627—31no—
40C4⋊(C3⋊Dic3)—31no—
41C4×C3⋊Dic3—31no—
42C3×C4⋊Dic3—31no—
43C3×C4⋊Dic3—31no—
44C3×C4⋊Dic3—121no—
45C3×Dic3⋊C4—121no—
46C3×C4×Dic3—31no—
47C3×C4×Dic3—31no—
48#3724—31no—
49Dic3⋊Dic3—32no—
50Dic3⋊Dic3—31no—
51Dic3⋊Dic3A411yescharacteristic
52C22×C33—41no—
53C3×C3⋊Dic3—81no—
54C32×Dic3—42no—
55C32×Dic3—81no—
56C3×C4×Q8—31no—
57C3×C4×Q8—31no—
58Q8×Dic3—31no—
59Q8×Dic3—31no—
60Q8×Dic3—61no—
61Q8×Dic3C3×D311yescharacteristic
62Q8×Dic3C3×D311yescharacteristic
63#890—91no—
64#890—91no—
65#890—91no—
66#888—91no—
67#870—91no—
68#869—91no—
69C4×Dic6—91no—
70C4×Dic6—91no—
71C4×SL(2,𝔽3)—31no—
72C4×SL(2,𝔽3)—31no—
73C4×SL(2,𝔽3)—61no—
74C4×SL(2,𝔽3)—61no—
75C4×SL(2,𝔽3)—91no—
76C32×Q8—31no—
77C32×Q8C3×Q811yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—121no—
80C2×C32×C4—121no—
81C2×C3⋊Dic3—31no—
82C2×C3⋊Dic3C2×A411yescharacteristic
83C2×C3×Dic3—31no—
84C2×C3×Dic3—31no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—41no—
87C2×C3×Dic3—81no—
88C2×C3×Dic3—81no—
89C2×C3×Dic3—121no—
90C2×C3×Dic3—121no—
91C2×C3×Dic3—241no—
92C2×C3×Dic3C2×A411yescharacteristic
93C2×C3×Dic3C2×A411yescharacteristic
94C3×SL(2,𝔽3)—21no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)—41no—
98C3×SL(2,𝔽3)Dic611yescharacteristic, complemented
99C3×SL(2,𝔽3)Dic611yescharacteristic, complemented
100#627—62no—
101#557—91noSylow
102C2×C33—41no—
103C2×C33—41no—
104C2×C33—41no—
105C2×C3×Q8—21no—
106C2×C3×Q8C2×C3×D311yescharacteristic
107C2×C3×Q8C2×C3×D311yescharacteristic
108C2×Dic6—91no—
109C2×Dic6—91no—
110C2×SL(2,𝔽3)—21no—
111C2×SL(2,𝔽3)—21no—
112C2×SL(2,𝔽3)—41no—
113C2×SL(2,𝔽3)D3211yescharacteristic
114C3×C4⋊C4—91no—
115C3×C4⋊C4—91no—
116C3×C4⋊C4—361no—
117C3×C42—91no—
118C3×C42—91no—
119C4⋊Dic3—31no—
120C4⋊Dic3—31no—
121C4⋊Dic3—31no—
122C4⋊Dic3—91no—
123C4⋊Dic3—92no—
124C4⋊Dic3—91no—
125C4⋊Dic3—91no—
126C4⋊Dic3—181no—
127Dic3⋊C4—31no—
128Dic3⋊C4—91no—
129Dic3⋊C4—91no—
130Dic3⋊C4—92no—
131Dic3⋊C4—91no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C4×Dic3—91no—
135C4×Dic3—91no—
136C4×Dic3—181no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—81no—
140C22×C32—81no—
141C22×C32—81no—
142C22×C32—161no—
143C22×C32C22×A411yescharacteristic
144C32×C4—31no—
145C32×C4—31no—
146C32×C4—122no—
147C32×C4—241no—
148C3⋊Dic3—21no—
149C3⋊Dic3—61no—
150C3×Dic3—21no—
151C3×Dic3—42no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—81no—
155C3×Dic3—82no—
156C3×Dic3—82no—
157C3×Dic3—241no—
158C3×Dic3—241no—
159C3×Dic3—242no—
160C3×Dic3#41⋊C312yes—
161C4⋊Q8—271no—
162#125—271no—
163#125—271no—
164C4×Q8—31no—
165C4×Q8—31no—
166C4×Q8—91no—
167C4×Q8—271no—
168C33—41noSylow
169C3×Q8—21no—
170C3×Q8—31no—
171C3×Q8—31no—
172C3×Q8—61no—
173C3×Q8C3×Dic611yescharacteristic
174C3×Q8C3×Dic611yescharacteristic
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—61no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×C3×C4—121no—
183C2×C3×C4—121no—
184C2×C3×C4—241no—
185C2×C3×C4—241no—
186C2×C3×C4—361no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—61no—
192C2×Dic3—91no—
193C2×Dic3—91no—
194C2×Dic3—181no—
195C2×Dic3D3×A411yescharacteristic
196C2×Dic3D3×A411yescharacteristic
197Dic6—182no—
198Dic6—182no—
199SL(2,𝔽3)—21no—
200SL(2,𝔽3)—21no—
201SL(2,𝔽3)—41no—
202SL(2,𝔽3)#62711yescharacteristic, complemented
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—81no—
210C2×C32—81no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32—81no—
214C2×C32—81no—
215C2×C32—81no—
216C2×C32—81no—
217C2×C32—81no—
218C2×C32—161no—
219C2×C32—161no—
220C2×C32—161no—
221C2×C32Q8×A411yescharacteristic
222C2×C32C2×#41⋊C311yescharacteristic
223C2×C322−1+4⋊C311yescharacteristic
224C2×Q8—271no—
225C2×Q8C3×D3211yescharacteristic
226C4⋊C4—91no—
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4—271no—
230C4⋊C4—271no—
231C4⋊C4—271no—
232C4⋊C4—272no—
233C42—91no—
234C42—91no—
235C42—271no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—81no—
239C22×C3—81no—
240C22×C3—161no—
241C22×C3C2×D3×A411yescharacteristic
242C22×C3C2×D3×A411yescharacteristic
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—32no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—122no—
252C3×C4—181no—
253C3×C4—181no—
254C3×C4—242no—
255C3×C4—241no—
256C3×C4—242no—
257C3×C4—721no—
258Dic3—21no—
259Dic3—62no—
260Dic3—61no—
261Dic3—61no—
262Dic3—61no—
263Dic3—61no—
264Dic3—181no—
265Dic3—181no—
266Dic3—361no—
267Dic3C3⋊(#41⋊C3)12yes—
268C32—41no—
269C32—41no—
270C32—81no—
271C32—81no—
272C32—81no—
273C32—161no—
274C32#557⋊C311yescharacteristic, complemented
275Q8—31no—
276Q8—542no—
277Q8C3×#62711yescharacteristic
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—271no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3—161no—
298C2×C3—161no—
299C2×C3—161no—
300C2×C3A4×Dic611yescharacteristic
301C2×C3A4×Dic611yescharacteristic
302C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
303C2×C3#1007⋊C311yescharacteristic
304C2×C3D3×#41⋊C311yescharacteristic
305C2×C3#1010⋊C311yescharacteristic
306C22D32×A411yescharacteristic
307C4—32no—
308C4—31no—
309C4—31no—
310C4—61no—
311C4—181no—
312C4—181no—
313C4—181no—
314C4—541no—
315C3—21no—
316C3—41no—
317C3—81no—
318C3—81no—
319C3—161no—
320C3#5569⋊C311yescharacteristic, complemented
321C3#5570⋊C311yescharacteristic, complemented
322C2D3×C3⋊(#41⋊C3)11yescharacteristic
323C2#64315⋊C311yescharacteristic
324C2A4×#62711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C32×SL(2,𝔽3)C22
hypocenterC32×Q8C3×Q8
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #10595422C3semidirect
28. C32×SL(2,𝔽3)Q8semidirect
98. C3×SL(2,𝔽3)Dic6semidirect
99. C3×SL(2,𝔽3)Dic6semidirect
202. SL(2,𝔽3)#627semidirect
274. C32#557⋊C3semidirect
320. C3#5569⋊C3semidirect
321. C3#5570⋊C3semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
C3×Dic3×SL(2,𝔽3)2
#105954223
#5570⋊C33
#5569⋊C33
C3×Dic3⋊Dic34