Finite Group Databasebeta

← C2×#11722438 #13164631

Subgroups — C2×#11722438 #13164631

Order 1728.

1035 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1035 conjugacy classes of subgroups — too many to draw. Only its 82 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
2#11722438C214yesmaximal, direct product factor
3C2×#88623C212yesmaximal, complemented
4C2×#64367C311yesmaximal, characteristic, complemented
5C2×#1010⋊C3—31nomaximal
6C2×C4⋊((C3⋊D3)⋊C3)—41nomaximal
7#41⋊H(3,𝔽3)C2212yescomplemented
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9#88623C2214yescomplemented
10C2×C32×#41C2×C311yescharacteristic, complemented
11#64367—34no—
12#64367C2×C314yescomplemented
13C2×Q8×C3⋊D3C2×C312yescomplemented
14C2×#3830—32no—
15C2×#3828—31no—
16C22×#634—31no—
17C2×C3×#41⋊C3—21no—
18C2×C3×#41⋊C3D311yescharacteristic, complemented
19#1010⋊C3—34no—
20C2×D3×SL(2,𝔽3)—32no—
21C22×(C3⋊D3)⋊C3—42no—
22C2×C4×H(3,𝔽3)—41no—
23C4⋊((C3⋊D3)⋊C3)—44no—
24Q8⋊H(3,𝔽3)C2311yescharacteristic
25C2×#1010—31no—
26C2×#1010—91no—
27C2×2−1+4⋊C3—91no—
28C32×#41—32no—
29C32×#41C22×C312yescomplemented
30C2×C32×Q8C22×C311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×#638—32no—
34C22×C3⋊Dic3—32no—
35Q8×C3⋊D3—34no—
36Q8×C3⋊D3C22×C314yescomplemented
37#3830—38no—
38#3830—38no—
39#3828—34no—
40#3828—34no—
41C2×C4⋊(C3⋊D3)A411yescharacteristic
42C2×C4×C3⋊D3—32no—
43C2×#634—34no—
44C2×#634—31no—
45C2×#634—32no—
46C2×#634—34no—
47C2×C3×D12—121no—
48C3×#41⋊C3—22no—
49C3×#41⋊C3C2×D312yescomplemented
50C2×C3×SL(2,𝔽3)—21no—
51C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
52D3×SL(2,𝔽3)—34no—
53C22×H(3,𝔽3)—41no—
54C2×(C3⋊D3)⋊C3—44no—
55C2×(C3⋊D3)⋊C3—84no—
56C4×H(3,𝔽3)—42no—
57C2×C3×#41—31no—
58C2×C3×#41C3×D311yescharacteristic
59#1010—34no—
60#1010—94no—
61#1010—912no—
62#1010—94no—
63C2×D3×Q8—32no—
64C2×D3×Q8—92no—
65C2×#237—96no—
66C2×#237—92no—
67C2×#235—91no—
68C2×#235—93no—
69C22×Dic6—93no—
70C22×Dic6—91no—
712−1+4⋊C3—94no—
72C2×#41⋊C3—31no—
73C2×#41⋊C3—61no—
74C22×SL(2,𝔽3)—92no—
75C23×C32—31no—
76C22×C3⋊D3C2×A412yes—
77C22×C3×D3—122no—
78C32×Q8—31no—
79C32×Q8C23×C311yescharacteristic
80C32×D4—32no—
81C32×D4—32no—
82C2×C32×C4—31no—
83C2×C32×C4—32no—
84C2×C32×C4—32no—
85C2×C32×C4—41no—
86C2×C32×C4—81no—
87C2×C32×C4C2×A411yescharacteristic
88#638—38no—
89C2×C3⋊Dic3—38no—
90C2×C3⋊Dic3—32no—
91C4⋊(C3⋊D3)C2×A414yes—
92C4×C3⋊D3—34no—
93C4×C3⋊D3—34no—
94#634—34no—
95#634—34no—
96#634—34no—
97C3×D12—124no—
98C3×SL(2,𝔽3)—21no—
99C3×SL(2,𝔽3)C22×D311yescharacteristic
100C2×2−1+4—91noSylow
101C2×H(3,𝔽3)—42no—
102C2×H(3,𝔽3)—41no—
103(C3⋊D3)⋊C3—84no—
104C3×#41—36no—
105C3×#41—32no—
106C3×#41—32no—
107C3×#41C2×C3×D312yes—
108C2×C3×Q8—31no—
109C2×C3×Q8C2×C3×D311yescharacteristic
110C2×C3×D4—31no—
111C2×C3×D4—33no—
112C2×C3×D4—361no—
113C22×C3×C4—33no—
114C22×C3×C4—31no—
115C2×#67—92no—
116C2×#67—96no—
117C22×Dic3—96no—
118C22×Dic3—92no—
119D3×Q8—34no—
120D3×Q8—94no—
121D3×Q8—912no—
122D3×Q8—94no—
123#237—98no—
124#237—924no—
125#237—924no—
126#237—98no—
127#235—94no—
128#235—94no—
129#235—912no—
130#235—912no—
131C2×D12—31no—
132C2×D12—91no—
133C2×C4×D3—92no—
134C2×C4×D3—96no—
135C2×Dic6—93no—
136C2×Dic6—96no—
137C2×Dic6—912no—
138C2×Dic6—912no—
139C2×Dic6—94no—
140C2×Dic6—91no—
141C2×Dic6—92no—
142C2×Dic6—94no—
143#41⋊C3—32no—
144#41⋊C3—62no—
145C2×SL(2,𝔽3)—31no—
146C2×SL(2,𝔽3)—61no—
147C2×SL(2,𝔽3)—94no—
148C22×C32—32no—
149C22×C32—41no—
150C22×C32—62no—
151C22×C32—81no—
152C22×C32C22×A411yescharacteristic
153C2×C3⋊D3—24no—
154C2×C3⋊D3C22×A414yes—
155C2×C3×D3—124no—
156C2×C3×D3—244no—
157C32×C4—31no—
158C32×C4—31no—
159C32×C4—42no—
160C32×C4—82no—
161C32×C4C22×A412yes—
162C3⋊Dic3—34no—
1632−1+4—94no—
1642−1+4—274no—
165C2×#41—271no—
166C2×#41—272no—
167C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
168C22×Q8—92no—
169C22×Q8—271no—
170H(3,𝔽3)—41noSylow
171C23×C3—33no—
172C23×C3—31no—
173C23×C3—362no—
174C22×D3—32no—
175C22×D3—92no—
176C3×Q8—31no—
177C3×Q8—33no—
178C3×Q8—31no—
179C3×Q8C22×C3×D311yescharacteristic
180C3×D4—32no—
181C3×D4—36no—
182C3×D4—36no—
183C3×D4—32no—
184C3×D4—364no—
185C2×C3×C4—33no—
186C2×C3×C4—36no—
187C2×C3×C4—36no—
188C2×C3×C4—31no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—31no—
192C2×C3×C4—121no—
193C2×C3×C4—241no—
194C2×C3×C4D3×A411yescharacteristic
195#67—98no—
196#67—924no—
197C2×Dic3—96no—
198C2×Dic3—924no—
199C2×Dic3—98no—
200C2×Dic3—92no—
201D12—34no—
202D12—94no—
203C4×D3—94no—
204C4×D3—94no—
205C4×D3—912no—
206C4×D3—912no—
207Dic6—912no—
208Dic6—912no—
209Dic6—912no—
210Dic6—94no—
211Dic6—94no—
212Dic6—94no—
213SL(2,𝔽3)—31no—
214SL(2,𝔽3)—61no—
215C2×C32—42no—
216C2×C32—41no—
217C2×C32—62no—
218C2×C32—82no—
219C2×C32—81no—
220C2×C322−1+4⋊C312yescomplemented
221C2×C32C23×A411yescharacteristic
222C3⋊D3—24no—
223C3×D3—244no—
224#41—32no—
225#41—274no—
226#41—274no—
227#41—278no—
228#41—278no—
229#41C2×(C3⋊D3)⋊C312yescomplemented
230C2×Q8—94no—
231C2×Q8—274no—
232C2×Q8—274no—
233C2×Q8—272no—
234C2×Q8—271no—
235C2×Q8—274no—
236C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
237C2×D4—31no—
238C2×D4—91no—
239C2×D4—272no—
240C22×C4—31no—
241C22×C4—272no—
242C22×C4—272no—
243C22×C3—36no—
244C22×C3—32no—
245C22×C3—31no—
246C22×C3—66no—
247C22×C3—62no—
248C22×C3—121no—
249C22×C3—241no—
250C22×C3—364no—
251C22×C3—724no—
252C22×C3C2×D3×A411yescharacteristic
253C2×D3—34no—
254C2×D3—64no—
255C2×D3—94no—
256C2×D3—184no—
257C3×C4—33no—
258C3×C4—33no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—32no—
262C3×C4—122no—
263C3×C4—242no—
264C3×C4C2×D3×A412yes—
265Dic3—912no—
266Dic3—94no—
267C32—41no—
268C32—81no—
269C32C2×2−1+4⋊C311yescharacteristic, complemented
270C23—31no—
271C23—92no—
272Q8—31no—
273Q8—274no—
274Q8—274no—
275Q8—274no—
276Q8C22×(C3⋊D3)⋊C311yescharacteristic
277D4—32no—
278D4—32no—
279D4—94no—
280D4—278no—
281C2×C4—31no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—274no—
285C2×C4—274no—
286C2×C4—278no—
287C2×C4—272no—
288C2×C4(C3⋊D3)⋊A411yescharacteristic
289C2×C3—31no—
290C2×C3—32no—
291C2×C3—66no—
292C2×C3—62no—
293C2×C3—121no—
294C2×C3—122no—
295C2×C3—241no—
296C2×C3—242no—
297C2×C3—724no—
298C2×C3#1010⋊C312yes—
299C2×C3C22×D3×A411yescharacteristic
300D3—64no—
301D3—184no—
302C22—32no—
303C22—62no—
304C22—94no—
305C22—184no—
306C22C2×(C3⋊D3)⋊A411yescharacteristic
307C4—31no—
308C4—31no—
309C4—274no—
310C4C2×(C3⋊D3)⋊A412yes—
311C3—31no—
312C3—121no—
313C3—241no—
314C3C2×#1010⋊C311yescharacteristic
315C2—62no—
316C2—184no—
317C2#1172243812yesdirect product factor
318C2C22×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2×C3C22×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C3C2×D3×A4
hypercenterC2×C4(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
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. #11722438C2direct (N × H)
3. C2×#88623C2semidirect
4. C2×#64367C3semidirect
7. #41⋊H(3,𝔽3)C22semidirect
9. #88623C22semidirect
10. C2×C32×#41C2×C3semidirect
12. #64367C2×C3semidirect
13. C2×Q8×C3⋊D3C2×C3semidirect
18. C2×C3×#41⋊C3D3semidirect
29. C32×#41C22×C3semidirect
36. Q8×C3⋊D3C22×C3semidirect
49. C3×#41⋊C3C2×D3semidirect
167. C2×#41(C3⋊D3)⋊C3semidirect
220. C2×C322−1+4⋊C3semidirect
229. #41C2×(C3⋊D3)⋊C3semidirect
269. C32C2×2−1+4⋊C3semidirect
317. C2#11722438direct (N × H)

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
#117224382
C2×#886232
C2×#643673
C2×#1010⋊C33
C2×C4⋊((C3⋊D3)⋊C3)4