Finite Group Databasebeta

← C2×#11722410 #13164577

Subgroups — C2×#11722410 #13164577

Order 1728.

654 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 87 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×C9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#11722410C214yesmaximal, direct product factor
3C2×D9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88615C211yesmaximal, characteristic, complemented
5C2×#1007⋊C3—31nomaximal
6C2×#63713C311yesmaximal, characteristic, complemented
7C2×C3×#611—41nomaximal
8C2×C9×SL(2,𝔽3)C2211yescharacteristic
9C2×C9×SL(2,𝔽3)C2212yescomplemented
10D9×SL(2,𝔽3)C2212yescomplemented
11#88615C2212yescomplemented
12C22×C3×SL(2,𝔽3)D311yescharacteristic
13#1007⋊C3—34no—
14C2×D3×SL(2,𝔽3)—31no—
15C2×C3⋊(#41⋊C3)—31no—
16C22×Q8×C9C2×C311yescharacteristic, complemented
17#63713—34no—
18#63713C2×C314yescomplemented
19C2×#3701—31no—
20C2×#3701C2×C311yescharacteristic, complemented
21C2×Q8×D9—31no—
22C2×Q8×D9C2×C311yescharacteristic, complemented
23C2×#3697—31no—
24C2×#3697—31no—
25C22×Q8⋊C9—21no—
26C23×C3×C9—41no—
27C22×C3×D9—41no—
28C3×#611—44no—
29C2×C3×Dic9—41no—
30C9×SL(2,𝔽3)C2311yescharacteristic
31C2×#1007—31no—
32C2×2−1+4⋊C3—91no—
33C2×C3×#67—121no—
34C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
35C2×C3×SL(2,𝔽3)C2×D312yes—
36D3×SL(2,𝔽3)—32no—
37C3⋊(#41⋊C3)—32no—
38C2×Q8×C9—31no—
39C2×Q8×C9—32no—
40C2×Q8×C9C22×C311yescharacteristic
41C2×Q8×C9C22×C312yescomplemented
42C22×C4×C9—31no—
43C2×#611—31no—
44C2×#611A411yescharacteristic
45#3701—34no—
46#3701—34no—
47#3701—32no—
48#3701C22×C312yescomplemented
49Q8×D9—34no—
50Q8×D9—32no—
51Q8×D9—34no—
52Q8×D9C22×C312yescomplemented
53#3697—34no—
54#3697—34no—
55#3697—34no—
56#3697—34no—
57C2×D36—31no—
58C2×D36—31no—
59C2×C4×D9—32no—
60C2×C4×D9—31no—
61C2×C4×D9—31no—
62C2×Dic18—31no—
63C2×Dic18—31no—
64C2×Q8⋊C9—21no—
65C2×Q8⋊C9—22no—
66C22×C3×C9—41no—
67C22×C3×C9—42no—
68C22×C3×C9—82no—
69C2×C3×D9—42no—
70C2×C3×D9—82no—
71C3×Dic9—42no—
72C22×C3×Q8C3×D311yescharacteristic
73#1007—34no—
74#1007—94no—
75C2×#239—31no—
76C2×#239—91no—
77C2×D3×Q8—31no—
78C2×D3×Q8—91no—
79C2×#235—91no—
80C2×#235—91no—
812−1+4⋊C3—94no—
82C2×#41⋊C3—91no—
83C22×SL(2,𝔽3)—21no—
84C22×SL(2,𝔽3)—91no—
85C22×SL(2,𝔽3)D911yescharacteristic, complemented
86C23×C32—41no—
87C22×C3×D3—121no—
88C3×#67—124no—
89C2×C3×Dic3—121no—
90C3×SL(2,𝔽3)C22×D311yescharacteristic
91C23×C9—81no—
92C23×C9C2×A411yescharacteristic
93C22×D9—31no—
94C22×D9C2×A411yescharacteristic
95Q8×C9—31no—
96Q8×C9—32no—
97Q8×C9—32no—
98Q8×C9C23×C311yescharacteristic
99C2×C4×C9—31no—
100C2×C4×C9—31no—
101C2×C4×C9—32no—
102C2×C4×C9—32no—
103#611—34no—
104#611C2×A414yes—
105C2×Dic9—31no—
106C2×Dic9C2×A411yescharacteristic
107D36—32no—
108D36—32no—
109D36—34no—
110C4×D9—38no—
111C4×D9—32no—
112C4×D9—32no—
113C4×D9—32no—
114C4×D9—32no—
115Dic18—32no—
116Dic18—34no—
117Dic18—32no—
118Q8⋊C9—21no—
119C2×2−1+4—91noSylow
120C2×C3×C9—42no—
121C2×C3×C9—41no—
122C2×C3×C9—82no—
123C3×D9—82no—
124C2×C3×Q8—31no—
125C2×C3×Q8—32no—
126C2×C3×Q8C2×C3×D311yescharacteristic
127C2×C3×Q8C2×C3×D312yes—
128C2×C3×D4—361no—
129C22×C3×C4—31no—
130C2×#67—31no—
131C2×#67—91no—
132#239—32no—
133#239—94no—
134#239—94no—
135#239—92no—
136D3×Q8—32no—
137D3×Q8—94no—
138D3×Q8—92no—
139D3×Q8—94no—
140#235—94no—
141#235—94no—
142#235—94no—
143#235—94no—
144C2×D12—91no—
145C2×D12—91no—
146C2×C4×D3—91no—
147C2×C4×D3—91no—
148C2×C4×D3—92no—
149C2×Dic6—91no—
150C2×Dic6—91no—
151#41⋊C3—92no—
152C2×SL(2,𝔽3)—21no—
153C2×SL(2,𝔽3)—22no—
154C2×SL(2,𝔽3)—92no—
155C2×SL(2,𝔽3)C2×D911yescharacteristic
156C2×SL(2,𝔽3)C2×D912yescomplemented
157C22×C32—41no—
158C22×C32—42no—
159C22×C32—82no—
160C2×C3×D3—122no—
161C2×C3×D3—242no—
162C3×Dic3—122no—
163C22×C9—22no—
164C22×C9—84no—
165C22×C9—82no—
166C22×C9—81no—
167C22×C9C22×A412yes—
168C22×C9C22×A411yescharacteristic
169C2×D9—22no—
170C2×D9—32no—
171C2×D9—62no—
172C2×D9C22×A412yes—
173C4×C9—31no—
174C4×C9—32no—
175C4×C9—31no—
176Dic9—32no—
177Dic9C22×A412yes—
1782−1+4—94no—
1792−1+4—274no—
180C2×#41—91no—
181C2×#41—271no—
182C2×#41—271no—
183C2×#41—271no—
184C22×Q8—91no—
185C22×Q8—271no—
186C22×Q8C3×D911yescharacteristic, complemented
187C3×C9—41noSylow
188C23×C3—41no—
189C23×C3—81no—
190C23×C3—361no—
191C23×C3D3×A411yescharacteristic
192C22×D3—31no—
193C22×D3—91no—
194C3×Q8—31no—
195C3×Q8—32no—
196C3×Q8—32no—
197C3×Q8C22×C3×D311yescharacteristic
198C3×D4—364no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—361no—
204#67—34no—
205#67—94no—
206C2×Dic3—31no—
207C2×Dic3—91no—
208D12—94no—
209D12—92no—
210D12—92no—
211C4×D3—92no—
212C4×D3—92no—
213C4×D3—92no—
214C4×D3—92no—
215C4×D3—98no—
216Dic6—94no—
217Dic6—92no—
218Dic6—92no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)C22×D911yescharacteristic
221C2×C32—42no—
222C2×C32—41no—
223C2×C32—82no—
224C3×D3—242no—
225C2×C9—22no—
226C2×C9—84no—
227C2×C9—82no—
228C2×C9—81no—
229C2×C92−1+4⋊C312yescomplemented
230C2×C9C23×A411yescharacteristic
231D9—22no—
232D9—62no—
233#41—92no—
234#41—274no—
235#41—274no—
236#41—274no—
237#41—274no—
238#41—274no—
239#41—274no—
240#41—272no—
241C2×Q8—31no—
242C2×Q8—32no—
243C2×Q8—92no—
244C2×Q8—271no—
245C2×Q8—271no—
246C2×Q8—274no—
247C2×Q8—274no—
248C2×Q8—272no—
249C2×Q8C2×C3×D911yescharacteristic
250C2×Q8C2×C3×D912yescomplemented
251C2×D4—91no—
252C2×D4—271no—
253C2×D4—271no—
254C2×D4—271no—
255C22×C4—31no—
256C22×C4—271no—
257C22×C4—271no—
258C22×C4—272no—
259C22×C3—22no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—82no—
263C22×C3—84no—
264C22×C3—82no—
265C22×C3—81no—
266C22×C3—362no—
267C22×C3—722no—
268C22×C3C2×D3×A412yes—
269C22×C3C2×D3×A411yescharacteristic
270C2×D3—32no—
271C2×D3—62no—
272C2×D3—92no—
273C2×D3—182no—
274C3×C4—31no—
275C3×C4—32no—
276C3×C4—31no—
277C3×C4—362no—
278Dic3—32no—
279Dic3—92no—
280C32—41no—
281C9—81no—
282C9C2×2−1+4⋊C311yescharacteristic, complemented
283C23—91no—
284C23—271no—
285C23A4×D911yescharacteristic
286Q8—31no—
287Q8—32no—
288Q8—32no—
289Q8—274no—
290Q8—272no—
291Q8—272no—
292Q8C22×C3×D911yescharacteristic
293D4—94no—
294D4—274no—
295D4—272no—
296D4—272no—
297D4—274no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—91no—
303C2×C4—272no—
304C2×C4—272no—
305C2×C4—272no—
306C2×C4—272no—
307C2×C4—278no—
308C2×C4—271no—
309C2×C3—22no—
310C2×C3—41no—
311C2×C3—42no—
312C2×C3—82no—
313C2×C3—84no—
314C2×C3—81no—
315C2×C3—82no—
316C2×C3—722no—
317C2×C3#1007⋊C312yes—
318C2×C3C22×D3×A411yescharacteristic
319D3—62no—
320D3—182no—
321C22—22no—
322C22—92no—
323C22—182no—
324C22—272no—
325C22—542no—
326C22C2×A4×D912yes—
327C22C2×A4×D911yescharacteristic
328C4—31no—
329C4—32no—
330C4—31no—
331C4—92no—
332C4—272no—
333C3—41no—
334C3—81no—
335C3C2×#1007⋊C311yescharacteristic
336C2—22no—
337C2—182no—
338C2—542no—
339C2#1172241012yesdirect product factor
340C2C22×A4×D911yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, cosocle, hypocenterQ8×C9C23×C3
FrattiniC2×C3C22×D3×A4
FittingC22×Q8×C9C2×C3
socleC22×C3C2×D3×A4
hypercenterC23A4×D9
LCSG ▷ Q8×C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C9×SL(2,𝔽3)C2semidirect
2. #11722410C2direct (N × H)
3. C2×D9×SL(2,𝔽3)C2semidirect
4. C2×#88615C2semidirect
6. C2×#63713C3semidirect
9. C2×C9×SL(2,𝔽3)C22semidirect
10. D9×SL(2,𝔽3)C22semidirect
11. #88615C22semidirect
16. C22×Q8×C9C2×C3semidirect
18. #63713C2×C3semidirect
20. C2×#3701C2×C3semidirect
22. C2×Q8×D9C2×C3semidirect
41. C2×Q8×C9C22×C3semidirect
48. #3701C22×C3semidirect
52. Q8×D9C22×C3semidirect
85. C22×SL(2,𝔽3)D9semidirect
156. C2×SL(2,𝔽3)C2×D9semidirect
186. C22×Q8C3×D9semidirect
229. C2×C92−1+4⋊C3semidirect
250. C2×Q8C2×C3×D9semidirect
282. C9C2×2−1+4⋊C3semidirect
339. C2#11722410direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C9×SL(2,𝔽3)2
#117224102
C2×D9×SL(2,𝔽3)2
C2×#886152
C2×#1007⋊C33
C2×#637133
C2×C3×#6114