Finite Group Databasebeta

← C2×#11722244 #13164150

Subgroups — C2×#11722244 #13164150

Order 1728.

527 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 71 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×A4×Dic9C211yesmaximal, characteristic
2C2×C9⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C9×SL(2,ℤ/4)C211yesmaximal, characteristic
4#11722244C214yesmaximal, direct product factor
5C2×#64204—31nomaximal
6C2×#63440—31nomaximal
7C22×C9×A4C2211yescharacteristic
8C2×#6189—41nomaximal
9A4×Dic9C2212yes—
10C9⋊SL(2,ℤ/4)C2212yes—
11C9×SL(2,ℤ/4)C2212yes—
12C2×Dic3×A4—31no—
13C2×C3⋊SL(2,ℤ/4)—31no—
14C2×C3×SL(2,ℤ/4)D311yescharacteristic
15#64204—34no—
16C23×Dic9D311yescharacteristic
17C22×Dic18—31no—
18C2×(C22⋊C9)⋊C4—61no—
19C2×C9×C22⋊C4—31no—
20C2×#3676—31no—
21C2×C4⋊Dic9—31no—
22C2×Dic9⋊C4—31no—
23C2×Dic9⋊C4—31no—
24#63440—34no—
25#63440—34no—
26C2×C9×A4Q812yescomplemented
27C2×C9×A4C2311yescharacteristic
28C2×C3⋊Dic9—41no—
29C2×C9×Dic3—41no—
30C2×C3×Dic9—41no—
31#6189—44no—
32C2×A4⋊Q8—91no—
33C2×#878—91no—
34C22×C3×A4C2×D311yescharacteristic
35C2×#627—121no—
36Dic3×A4—32no—
37C3⋊SL(2,ℤ/4)—32no—
38C3×SL(2,ℤ/4)C2×D312yes—
39C24×C9C2×D311yescharacteristic
40C22×C22⋊C9—21no—
41C22×C4×C9—31no—
42C22×C4×C9—31no—
43C22×Dic9—31no—
44C22×Dic9—32no—
45C22×Dic9—31no—
46C22×Dic9—31no—
47C22×Dic9—31no—
48C22×Dic9C2×D312yes—
49C2×Dic18—34no—
50C2×Dic18—61no—
51C2×Dic18—62no—
52C2×Dic18—61no—
53(C22⋊C9)⋊C4—62no—
54C9×C22⋊C4—32no—
55C9×C22⋊C4—32no—
56#3676—32no—
57#3676—32no—
58C4⋊Dic9—34no—
59Dic9⋊C4—34no—
60Dic9⋊C4—34no—
61C22×C3×C9—41no—
62C9×A4C2×Q811yescharacteristic, complemented
63C3⋊Dic9—42no—
64C9×Dic3—42no—
65C3×Dic9—42no—
66C23×Dic3—31no—
67C22×Dic6—91no—
68C2×C4×A4—91no—
69C2×SL(2,ℤ/4)—91no—
70C2×SL(2,ℤ/4)—181no—
71C2×SL(2,ℤ/4)D911yescharacteristic
72A4⋊Q8—94no—
73C2×C3×C22⋊C4—31no—
74C2×#217—91no—
75C2×C4⋊Dic3—91no—
76C2×Dic3⋊C4—91no—
77C2×Dic3⋊C4—91no—
78#878—94no—
79#878—94no—
80C2×C3×A4Dic612yes—
81C2×C3×A4C22×D311yescharacteristic
82C2×C3⋊Dic3—121no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—121no—
85#627—124no—
86C23×C9—31no—
87C23×C9—31no—
88C23×C9—32no—
89C23×C9Dic612yescomplemented
90C23×C9C22×D311yescharacteristic
91C2×C22⋊C9—22no—
92C2×C22⋊C9—21no—
93C2×C4×C9—32no—
94C2×C4×C9—32no—
95C2×C4×C9—61no—
96C2×C4×C9—62no—
97C2×C4×C9—61no—
98C2×Dic9—31no—
99C2×Dic9—32no—
100C2×Dic9—32no—
101C2×Dic9—32no—
102C2×Dic9—32no—
103C2×Dic9—32no—
104C2×Dic9—61no—
105C2×Dic9—61no—
106C2×Dic9—62no—
107C2×Dic9—61no—
108C2×Dic9—241no—
109C2×Dic9S411yescharacteristic
110Dic18—64no—
111Dic18—64no—
112C2×C22⋊Q8—271noSylow
113C2×C3×C9—42no—
114C2×C3×C9—41no—
115C24×C3D3211yescharacteristic
116C22×A4—21no—
117C22×A4C2×D911yescharacteristic
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×Dic3—32no—
121C22×Dic3—91no—
122C22×Dic3—92no—
123C22×Dic3—91no—
124C22×Dic3—91no—
125C22×Dic3—91no—
126C2×Dic6—94no—
127C2×Dic6—181no—
128C2×Dic6—182no—
129C2×Dic6—181no—
130C2×Dic6—361no—
131C4×A4—92no—
132SL(2,ℤ/4)—92no—
133SL(2,ℤ/4)—182no—
134SL(2,ℤ/4)C2×D912yes—
135C3×C22⋊C4—32no—
136C3×C22⋊C4—32no—
137#217—92no—
138#217—92no—
139C4⋊Dic3—94no—
140Dic3⋊C4—94no—
141Dic3⋊C4—94no—
142C22×C32—41no—
143C3×A4C2×Dic611yescharacteristic
144C3⋊Dic3—122no—
145C3×Dic3—42no—
146C3×Dic3—122no—
147C22×C9—32no—
148C22×C9—31no—
149C22×C9—32no—
150C22×C9—32no—
151C22×C9—31no—
152C22×C9—31no—
153C22×C9—61no—
154C22×C9—81no—
155C22×C9C2×Dic611yescharacteristic, complemented
156C22×C9C2×S411yescharacteristic
157C22⋊C9—21no—
158C4×C9—62no—
159C4×C9—62no—
160Dic9—32no—
161Dic9—62no—
162Dic9—62no—
163Dic9—242no—
164Dic9C2×S412yes—
165C22×Q8—271no—
166C23×C4—91no—
167C22⋊Q8—274no—
168C22⋊Q8—274no—
169C2×C4⋊C4—271no—
170C2×C4⋊C4—271no—
171C2×C4⋊C4—271no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—271no—
174C3×C9—41noSylow
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—32no—
178C23×C3#62712yes—
179C23×C3C2×D3211yescharacteristic
180C2×A4—22no—
181C2×A4—21no—
182C2×A4Dic1812yescomplemented
183C2×A4C22×D911yescharacteristic
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—61no—
187C2×C3×C4—62no—
188C2×C3×C4—61no—
189C2×C3×C4—361no—
190C2×Dic3—31no—
191C2×Dic3—41no—
192C2×Dic3—92no—
193C2×Dic3—92no—
194C2×Dic3—91no—
195C2×Dic3—92no—
196C2×Dic3—92no—
197C2×Dic3—92no—
198C2×Dic3—181no—
199C2×Dic3—181no—
200C2×Dic3—182no—
201C2×Dic3—181no—
202C2×Dic3—361no—
203C2×Dic3—721no—
204Dic6—184no—
205Dic6—184no—
206Dic6—364no—
207C2×C32—41no—
208C2×C32—42no—
209C2×C9—31no—
210C2×C9—31no—
211C2×C9—32no—
212C2×C9—81no—
213C2×C9—82no—
214C2×C9A4⋊Q812yescomplemented
215C2×C9C22×S411yescharacteristic
216C24D3×D911yescharacteristic
217C2×Q8—274no—
218C2×Q8—541no—
219C2×Q8—542no—
220C2×Q8—541no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—92no—
224C22×C4—271no—
225C22×C4—271no—
226C22×C4—272no—
227C22×C4—271no—
228C22×C4—271no—
229C4⋊C4—274no—
230C4⋊C4—274no—
231C4⋊C4—274no—
232C22⋊C4—32no—
233C22⋊C4—32no—
234C22⋊C4—272no—
235C22⋊C4—272no—
236C22×C3—32no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—32no—
241C22×C3—32no—
242C22×C3—41no—
243C22×C3—61no—
244C22×C3—81no—
245C22×C3C2×#62711yescharacteristic
246C22×C3D3×S411yescharacteristic
247A4—21no—
248A4C2×Dic1811yescharacteristic, complemented
249C3×C4—62no—
250C3×C4—62no—
251C3×C4—362no—
252Dic3—32no—
253Dic3—42no—
254Dic3—92no—
255Dic3—182no—
256Dic3—182no—
257Dic3—362no—
258Dic3—722no—
259C32—41no—
260C9—81no—
261C9C2×A4⋊Q811yescharacteristic, complemented
262C23—31no—
263C23—31no—
264C23—32no—
265C23#618912yescomplemented
266C23C2×D3×D911yescharacteristic
267Q8—544no—
268Q8—544no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—61no—
274C2×C4—91no—
275C2×C4—272no—
276C2×C4—272no—
277C2×C4—271no—
278C2×C4—272no—
279C2×C4—272no—
280C2×C4—272no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C4—542no—
284C2×C4—541no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—32no—
288C2×C3—42no—
289C2×C3—41no—
290C2×C3—82no—
291C2×C3—81no—
292C2×C3#6420412yes—
293C2×C3C2×D3×S411yescharacteristic
294C22—32no—
295C22—31no—
296C22—32no—
297C22—32no—
298C22—31no—
299C22—31no—
300C22—61no—
301C22C2×#618911yescharacteristic, complemented
302C22D9×S411yescharacteristic
303C4—62no—
304C4—62no—
305C4—92no—
306C4—272no—
307C4—542no—
308C4—542no—
309C3—41no—
310C3—81no—
311C3C2×#6420411yescharacteristic
312C2—31no—
313C2—31no—
314C2—32no—
315C2#1172224412yesdirect product factor
316C2C2×D9×S411yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, cosocleC2×C9×A4C23
FrattiniC2×C3C2×D3×S4
FittingC24×C9C2×D3
socleC24×C3D32
prime radicalC22×C3×A4C2×D3
hypocenterC9×A4C2×Q8
LCSG ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #11722244C2direct (N × H)
26. C2×C9×A4Q8semidirect
62. C9×A4C2×Q8semidirect
89. C23×C9Dic6semidirect
155. C22×C9C2×Dic6semidirect
182. C2×A4Dic18semidirect
214. C2×C9A4⋊Q8semidirect
248. A4C2×Dic18semidirect
261. C9C2×A4⋊Q8semidirect
265. C23#6189semidirect
301. C22C2×#6189semidirect
315. C2#11722244direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×Dic92
C2×C9⋊SL(2,ℤ/4)2
C2×C9×SL(2,ℤ/4)2
#117222442
C2×#642043
C2×#634403
C2×#61894