Finite Group Databasebeta

← #10593661⋊C3 #13146925

Subgroups — #10593661⋊C3 #13146925

Order 1728.

356 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×(C3⋊GL(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊#859)⋊C3C211yesmaximal, characteristic, complemented
4#10593661C311yesmaximal, characteristic, complemented
5C3×#5238—31nomaximal
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊GL(2,𝔽3))⋊C3—21no—
9#3752⋊C3—41nomaximal
10C2×C32×#41D311yescharacteristic, complemented
11C2×C3×#41⋊C3—21no—
12C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
13C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
14C2×C3×#215—31no—
15C2×C3×#213—31no—
16C3×#930—31no—
17C3⋊#859C2×C311yescharacteristic, complemented
18C3×#835—31no—
19C3×#832—31no—
20C3×#830—31no—
21C22×(C3⋊D3)⋊C3—41no—
22C2×C4×H(3,𝔽3)—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
25#5427—91no—
26C3×#449—91no—
27#5238—31no—
28C32×#41—21no—
29C32×#41—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×C3×#67—61no—
34C2×C3×D12—31no—
35C3×#41⋊C3—22no—
36C3×#41⋊C3—21no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
39C3⋊GL(2,𝔽3)—21no—
40#3752—41no—
41C3×#215—61no—
42C3×#215—61no—
43C3×#213—62no—
44C3×(C2×D3)⋊C4—61no—
45C3×C4⋊Dic3—31no—
46C2×C3×C3⋊C8—31no—
47C2×C3×C3⋊C8—31no—
48C22×H(3,𝔽3)—41no—
49C2×(C3⋊D3)⋊C3—42no—
50C2×(C3⋊D3)⋊C3—81no—
51C2×(C3⋊D3)⋊C3—81no—
52C4×H(3,𝔽3)—81no—
53(C3⋊Dic3)⋊C3—81no—
54C2×C3×#41—31no—
55C2×C3×#41C3×D311yescharacteristic
56C2×#41⋊C3—31no—
57C2×#41⋊C3—61no—
58C2×GL(2,𝔽3)—91no—
59C2×C3×SD16—91no—
60C2×C3×D8—91no—
61C3×#121—91no—
62C2×#215—31no—
63C2×#213—31no—
64#930—31no—
65#859—91no—
66C3×#103—91no—
67C3×#102—91no—
68C3×C22⋊C8—91no—
69#835—31no—
70#832—31no—
71#830—31no—
72C23×C32—31no—
73C22×C3⋊D3—41no—
74C22×C3×D3—61no—
75C32×Q8—31no—
76C32×Q8#6711yescharacteristic
77C32×D4—32no—
78C32×D4—61no—
79C2×C32×C4—31no—
80C2×C32×C4—41no—
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83C2×C32×C4—81no—
84C2×C32×C4S411yescharacteristic
85C2×C3⋊Dic3—41no—
86C3×#67—122no—
87C2×C3×Dic3—61no—
88C3×D12—61no—
89C3×D12—61no—
90C3×SL(2,𝔽3)—21no—
91C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
92C3×C3⋊C8—61no—
93C3×C3⋊C8—61no—
94#449—91noSylow
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)—41no—
97C2×H(3,𝔽3)—41no—
98(C3⋊D3)⋊C3—82no—
99C3×#41—21no—
100C3×#41—61no—
101C3×#41—62no—
102C3×#41—61no—
103C3×#41—61no—
104C2×C3×Q8—31no—
105C2×C3×Q8C2×C3×D311yescharacteristic
106C2×C3×D4—31no—
107C2×C3×D4—31no—
108C2×C3×D4—61no—
109C2×C3×D4—91no—
110C2×C3×D4—181no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—61no—
114C2×#67—61no—
115C2×D12—31no—
116#41⋊C3—61no—
117#41⋊C3—62no—
118C2×SL(2,𝔽3)—31no—
119C2×SL(2,𝔽3)—61no—
120GL(2,𝔽3)—181no—
121C3×SD16—181no—
122C3×SD16—181no—
123C3×D8—182no—
124C2×C3×C8—91no—
125C2×C3×C8—91no—
126C3×C4⋊C4—91no—
127C3×C22⋊C4—181no—
128#215—61no—
129#215—61no—
130#213—62no—
131(C2×D3)⋊C4—61no—
132(C2×D3)⋊C4—361no—
133C4⋊Dic3—31no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—31no—
136C22×C32—41no—
137C22×C32—61no—
138C22×C32—61no—
139C22×C32—61no—
140C22×C32—81no—
141C22×C32C2×S411yescharacteristic
142C2×C3⋊D3—42no—
143C2×C3⋊D3—81no—
144C2×C3⋊D3—81no—
145C2×C3×D3—121no—
146C2×C3×D3—121no—
147C2×C3×D3—121no—
148C32×C4—21no—
149C32×C4—31no—
150C32×C4—31no—
151C32×C4—81no—
152C32×C4—82no—
153C3⋊Dic3—81no—
154C3×Dic3—121no—
155C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
156C2×SD16—91no—
157C2×D8—91no—
158#121—91no—
159#103—91no—
160#102—91no—
161C22⋊C8—91no—
162H(3,𝔽3)—41noSylow
163C23×C3—31no—
164C23×C3—31no—
165C23×C3—61no—
166C23×C3—181no—
167C22×D3—61no—
168C22×D3—361no—
169C3×Q8—31no—
170C3×Q8—31no—
171C3×Q8—31no—
172C3×Q8—61no—
173C3×Q8C3×#6711yescharacteristic
174C3×D4—32no—
175C3×D4—32no—
176C3×D4—62no—
177C3×D4—61no—
178C3×D4—61no—
179C3×D4—62no—
180C3×D4—181no—
181C3×D4—181no—
182C3×D4—362no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—121no—
194C2×C3×C4—181no—
195C2×C3×C4—241no—
196C2×C3×C4C3×S411yescharacteristic
197#67—122no—
198C2×Dic3—61no—
199C2×Dic3—361no—
200D12—61no—
201D12—61no—
202SL(2,𝔽3)—31no—
203SL(2,𝔽3)—61no—
204C3×C8—181no—
205C3×C8—181no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32—81no—
214C2×C32—121no—
215C2×C32#98511yescharacteristic
216C2×C32#98611yescharacteristic
217C2×C32GL(2,ℤ/4)11yescharacteristic
218C3⋊D3—82no—
219C3×D3—241no—
220#41—21no—
221#41—61no—
222C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
223C2×D4—31no—
224C2×D4—91no—
225C2×D4—181no—
226C22×C4—31no—
227SD16—181no—
228SD16—181no—
229D8—182no—
230C2×C8—91no—
231C2×C8—91no—
232C4⋊C4—91no—
233C22⋊C4—181no—
234C22×C3—31no—
235C22×C3—62no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—121no—
243C22×C3—121no—
244C22×C3—121no—
245C22×C3—241no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3—361no—
249C22×C3C2×C3×S411yescharacteristic
250C2×D3—121no—
251C2×D3—121no—
252C2×D3—121no—
253C2×D3—362no—
254C2×D3—721no—
255C2×D3—721no—
256C3×C4—21no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—241no—
265C3×C4—242no—
266C3×C4—361no—
267Dic3—121no—
268Dic3—721no—
269C32—41no—
270C32—81no—
271C32#542711yescharacteristic, complemented
272C23—31no—
273C23—181no—
274Q8—31no—
275Q8#638⋊C311yescharacteristic
276D4—32no—
277D4—61no—
278D4—181no—
279D4—181no—
280D4—362no—
281C2×C4—31no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—181no—
285C2×C4(C3⋊S4)⋊C311yescharacteristic
286C8—181no—
287C8—181no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—121no—
292C2×C3—122no—
293C2×C3—121no—
294C2×C3—121no—
295C2×C3—121no—
296C2×C3—121no—
297C2×C3—241no—
298C2×C3—241no—
299C2×C3—241no—
300C2×C3—721no—
301C2×C3C3×#98511yescharacteristic
302C2×C3C3×#98611yescharacteristic
303C2×C3C3×GL(2,ℤ/4)11yescharacteristic
304D3—241no—
305D3—722no—
306C22—61no—
307C22—61no—
308C22—61no—
309C22—361no—
310C22—361no—
311C22—361no—
312C22C2×(C3⋊S4)⋊C311yescharacteristic
313C4—21no—
314C4—31no—
315C4—31no—
316C4—361no—
317C3—31no—
318C3—121no—
319C3—241no—
320C3C3×#542711yescharacteristic
321C2—121no—
322C2—721no—
323C2#64266⋊C311yescharacteristic
324C2#64267⋊C311yescharacteristic
325C2#64269⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC2×C32×#41D3
hypercenterC2×C4(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. C2×(C3⋊GL(2,𝔽3))⋊C3C2semidirect
3. (C3⋊#859)⋊C3C2semidirect
4. #10593661C3semidirect
10. C2×C32×#41D3semidirect
12. C2×C3×#41⋊C3C2×C3semidirect
13. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
17. C3⋊#859C2×C3semidirect
24. Q8⋊H(3,𝔽3)D4semidirect
91. C3×SL(2,𝔽3)C3×D4semidirect
155. C2×#41(C3⋊D3)⋊C3semidirect
271. C32#5427semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
C2×(C3⋊GL(2,𝔽3))⋊C32
(C3⋊#859)⋊C32
#105936613
C3×#52383
#3752⋊C34