Finite Group Databasebeta

← C2×D9×SL(2,ℤ/4) #13164141

Subgroups — C2×D9×SL(2,ℤ/4) #13164141

Order 1728.

863 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 95 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×A4×D9C211yesmaximal, characteristic
2C2×C9⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C2×C9×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4D9×SL(2,ℤ/4)C214yesmaximal, direct product factor
5C2×D3×SL(2,ℤ/4)—31nomaximal
6C2×C22⋊C4×D9—31nomaximal
7C22×C9×A4C2211yescharacteristic
8C2×A4×D9C414yescomplemented
9C2×A4×D9C2212yes—
10C2×Dic3×D9—41nomaximal
11C9⋊SL(2,ℤ/4)C2212yescomplemented
12C9×SL(2,ℤ/4)C2212yescomplemented
13C22×D3×A4—31no—
14C2×C3⋊SL(2,ℤ/4)—31no—
15C2×C3×SL(2,ℤ/4)D311yescharacteristic
16D3×SL(2,ℤ/4)—34no—
17C24×D9D311yescharacteristic
18C22×C4×D9—31no—
19C22×C4×D9—31no—
20C2×(C22⋊C9)⋊C4—61no—
21C2×C9×C22⋊C4—31no—
22C2×#3676—31no—
23C2×(C2×D9)⋊C4—31no—
24C2×(C2×D9)⋊C4—31no—
25C22⋊C4×D9—34no—
26C22⋊C4×D9—34no—
27C22×C3×D9—41no—
28C2×C9×A4C2×C412yescomplemented
29C2×C9×A4C2311yescharacteristic
30A4×D9C2×C414yescomplemented
31C2×C3⋊Dic9—41no—
32C2×C9×Dic3—41no—
33Dic3×D9—44no—
34C22×SL(2,ℤ/4)—91no—
35C2×D3×C22⋊C4—91no—
36C22×C3×A4C2×D311yescharacteristic
37C2×D3×A4—34no—
38C2×D3×A4—32no—
39C2×D3×Dic3—121no—
40C3⋊SL(2,ℤ/4)—32no—
41C3×SL(2,ℤ/4)C2×D312yes—
42C24×C9C2×D311yescharacteristic
43C23×D9—31no—
44C23×D9—31no—
45C23×D9—34no—
46C23×D9—32no—
47C23×D9Dic314yescomplemented
48C23×D9C2×D312yes—
49C22×C22⋊C9—21no—
50C22×C4×C9—31no—
51C22×C4×C9—31no—
52C22×Dic9—31no—
53C22×Dic9—31no—
54C2×C4×D9—34no—
55C2×C4×D9—34no—
56C2×C4×D9—64no—
57C2×C4×D9—61no—
58C2×C4×D9—62no—
59C2×C4×D9—61no—
60(C22⋊C9)⋊C4—62no—
61C9×C22⋊C4—32no—
62C9×C22⋊C4—32no—
63#3676—32no—
64#3676—32no—
65(C2×D9)⋊C4—34no—
66(C2×D9)⋊C4—34no—
67C22×C3×C9—41no—
68C2×C3×D9—44no—
69C2×C3×D9—42no—
70C9×A4C22×C411yescharacteristic, complemented
71C3⋊Dic9—42no—
72C9×Dic3—42no—
73C24×D3—31no—
74C23×A4—91no—
75C22×C4×D3—91no—
76C22×C4×D3—91no—
77C2×SL(2,ℤ/4)—91no—
78C2×SL(2,ℤ/4)—94no—
79C2×SL(2,ℤ/4)—181no—
80C2×SL(2,ℤ/4)D911yescharacteristic, direct product factor
81C2×C3×C22⋊C4—31no—
82C2×#217—91no—
83C2×(C2×D3)⋊C4—91no—
84C2×(C2×D3)⋊C4—91no—
85D3×C22⋊C4—94no—
86D3×C22⋊C4—94no—
87C22×C3×D3—121no—
88C2×C3×A4C4×D312yes—
89C2×C3×A4C22×D311yescharacteristic
90D3×A4—34no—
91C2×C3⋊Dic3—121no—
92C2×C3×Dic3—41no—
93D3×Dic3—124no—
94C23×C9—31no—
95C23×C9—31no—
96C23×C9—32no—
97C23×C9C2×Dic312yescomplemented
98C23×C9C22×D311yescharacteristic
99C22×D9—31no—
100C22×D9—34no—
101C22×D9—34no—
102C22×D9—32no—
103C22×D9—32no—
104C22×D9—32no—
105C22×D9—34no—
106C22×D9—38no—
107C22×D9—34no—
108C22×D9—64no—
109C22×D9—62no—
110C22×D9—61no—
111C22×D9C2×Dic314yescomplemented
112C22×D9S411yescharacteristic
113C2×C22⋊C9—22no—
114C2×C22⋊C9—21no—
115C2×C4×C9—32no—
116C2×C4×C9—32no—
117C2×C4×C9—62no—
118C2×C4×C9—61no—
119C2×C4×C9—61no—
120C2×Dic9—32no—
121C2×Dic9—32no—
122C2×Dic9—62no—
123C2×Dic9—61no—
124C2×Dic9—61no—
125C2×Dic9—241no—
126C4×D9—68no—
127C4×D9—64no—
128C4×D9—64no—
129C22×C22⋊C4—271noSylow
130C2×C3×C9—42no—
131C2×C3×C9—41no—
132C3×D9—44no—
133C24×C3D3211yescharacteristic
134C23×D3—34no—
135C23×D3—32no—
136C23×D3—91no—
137C23×D3—92no—
138C23×D3—91no—
139C23×D3—94no—
140C22×A4—21no—
141C22×A4—94no—
142C22×A4—92no—
143C22×A4C2×D911yescharacteristic
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×Dic3—91no—
147C22×Dic3—91no—
148C22×Dic3—361no—
149C2×C4×D3—94no—
150C2×C4×D3—94no—
151C2×C4×D3—181no—
152C2×C4×D3—182no—
153C2×C4×D3—184no—
154C2×C4×D3—181no—
155SL(2,ℤ/4)—92no—
156SL(2,ℤ/4)—182no—
157SL(2,ℤ/4)C2×D912yesdirect product factor
158C3×C22⋊C4—32no—
159C3×C22⋊C4—32no—
160#217—92no—
161#217—92no—
162(C2×D3)⋊C4—94no—
163(C2×D3)⋊C4—94no—
164C22×C32—41no—
165C2×C3×D3—122no—
166C2×C3×D3—124no—
167C3×A4C2×C4×D311yescharacteristic
168C3⋊Dic3—122no—
169C3×Dic3—42no—
170C22×C9—32no—
171C22×C9—31no—
172C22×C9—32no—
173C22×C9—32no—
174C22×C9—31no—
175C22×C9—31no—
176C22×C9—61no—
177C22×C9—81no—
178C22×C9C22×Dic311yescharacteristic, complemented
179C22×C9C2×S411yescharacteristic
180C2×D9—34no—
181C2×D9—32no—
182C2×D9—34no—
183C2×D9—34no—
184C2×D9—38no—
185C2×D9—34no—
186C2×D9—62no—
187C2×D9—64no—
188C2×D9SL(2,ℤ/4)14yesdirect product factor
189C2×D9C2×S412yes—
190C22⋊C9—21no—
191C4×C9—62no—
192C4×C9—62no—
193Dic9—62no—
194Dic9—62no—
195Dic9—242no—
196C25—91no—
197C23×C4—271no—
198C23×C4—271no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—274no—
201C2×C22⋊C4—274no—
202C2×C22⋊C4—271no—
203C2×C22⋊C4—271no—
204C2×C22⋊C4—271no—
205C3×C9—41noSylow
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—32no—
209C23×C3—361no—
210C23×C3D3×Dic312yes—
211C23×C3C2×D3211yescharacteristic
212C22×D3—31no—
213C22×D3—34no—
214C22×D3—92no—
215C22×D3—92no—
216C22×D3—94no—
217C22×D3—94no—
218C22×D3—91no—
219C22×D3—92no—
220C22×D3—94no—
221C22×D3—98no—
222C22×D3—94no—
223C22×D3—184no—
224C22×D3—181no—
225C22×D3—182no—
226C2×A4—22no—
227C2×A4—21no—
228C2×A4—94no—
229C2×A4C4×D912yescomplemented
230C2×A4C22×D911yescharacteristic
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—62no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×Dic3—41no—
237C2×Dic3—92no—
238C2×Dic3—92no—
239C2×Dic3—182no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C2×Dic3—364no—
243C2×Dic3—361no—
244C2×Dic3—721no—
245C4×D3—184no—
246C4×D3—188no—
247C4×D3—184no—
248C2×C32—41no—
249C2×C32—42no—
250C3×D3—124no—
251C2×C9—31no—
252C2×C9—31no—
253C2×C9—32no—
254C2×C9—81no—
255C2×C9—82no—
256C2×C9C2×SL(2,ℤ/4)12yescomplemented
257C2×C9C22×S411yescharacteristic
258D9—34no—
259D9C2×SL(2,ℤ/4)14yesdirect product factor
260C24—94no—
261C24—92no—
262C24—271no—
263C24—271no—
264C24—272no—
265C24—274no—
266C24D3×D911yescharacteristic
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—274no—
270C22×C4—274no—
271C22×C4—271no—
272C22×C4—271no—
273C22×C4—541no—
274C22×C4—542no—
275C22×C4—544no—
276C22×C4—541no—
277C22⋊C4—32no—
278C22⋊C4—32no—
279C22⋊C4—274no—
280C22⋊C4—274no—
281C22⋊C4—272no—
282C22⋊C4—272no—
283C22×C3—32no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—32no—
288C22×C3—32no—
289C22×C3—41no—
290C22×C3—61no—
291C22×C3—81no—
292C22×C3—362no—
293C22×C3—364no—
294C22×C3C2×D3×Dic311yescharacteristic
295C22×C3D3×S411yescharacteristic
296C2×D3—32no—
297C2×D3—34no—
298C2×D3—94no—
299C2×D3—94no—
300C2×D3—98no—
301C2×D3—92no—
302C2×D3—94no—
303C2×D3—94no—
304C2×D3—182no—
305C2×D3—184no—
306A4—21no—
307A4C2×C4×D911yescharacteristic, complemented
308C3×C4—62no—
309C3×C4—62no—
310Dic3—42no—
311Dic3—182no—
312Dic3—182no—
313Dic3—362no—
314Dic3—722no—
315C32—41no—
316C9—81no—
317C9C22×SL(2,ℤ/4)11yescharacteristic, complemented
318C23—31no—
319C23—31no—
320C23—32no—
321C23—91no—
322C23—94no—
323C23—272no—
324C23—274no—
325C23—272no—
326C23—274no—
327C23—272no—
328C23—278no—
329C23—274no—
330C23—274no—
331C23—271no—
332C23—542no—
333C23—541no—
334C23—544no—
335C23Dic3×D912yescomplemented
336C23C2×D3×D911yescharacteristic
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—272no—
343C2×C4—272no—
344C2×C4—544no—
345C2×C4—548no—
346C2×C4—544no—
347C2×C4—542no—
348C2×C4—541no—
349C2×C4—541no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—32no—
353C2×C3—42no—
354C2×C3—41no—
355C2×C3—82no—
356C2×C3—81no—
357C2×C3—364no—
358C2×C3D3×SL(2,ℤ/4)12yes—
359C2×C3C2×D3×S411yescharacteristic
360D3—34no—
361D3—94no—
362C22—32no—
363C22—31no—
364C22—32no—
365C22—32no—
366C22—31no—
367C22—31no—
368C22—61no—
369C22—94no—
370C22—92no—
371C22—274no—
372C22—274no—
373C22—278no—
374C22—274no—
375C22—274no—
376C22—272no—
377C22—542no—
378C22—544no—
379C22C2×Dic3×D911yescharacteristic, complemented
380C22D9×S411yescharacteristic
381C4—62no—
382C4—62no—
383C4—542no—
384C4—542no—
385C3—41no—
386C3—81no—
387C3C2×D3×SL(2,ℤ/4)11yescharacteristic
388C2—31no—
389C2—31no—
390C2—32no—
391C2—94no—
392C2—274no—
393C2D9×SL(2,ℤ/4)12yesdirect product factor
394C2C2×D9×S411yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, hypocenterC9×A4C22×C4
FrattiniC2×C3C2×D3×S4
FittingC24×C9C2×D3
socleC24×C3D32
prime radicalC22×A4×D9C2
cosocleC2×C9×A4C23
LCSG ▷ C9×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C9⋊SL(2,ℤ/4)C2semidirect
3. C2×C9×SL(2,ℤ/4)C2semidirect
4. D9×SL(2,ℤ/4)C2direct (N × H)
8. C2×A4×D9C4semidirect
11. C9⋊SL(2,ℤ/4)C22semidirect
12. C9×SL(2,ℤ/4)C22semidirect
28. C2×C9×A4C2×C4semidirect
30. A4×D9C2×C4semidirect
47. C23×D9Dic3semidirect
70. C9×A4C22×C4semidirect
80. C2×SL(2,ℤ/4)D9direct (N × H)
97. C23×C9C2×Dic3semidirect
111. C22×D9C2×Dic3semidirect
157. SL(2,ℤ/4)C2×D9direct (N × H)
178. C22×C9C22×Dic3semidirect
188. C2×D9SL(2,ℤ/4)direct (N × H)
229. C2×A4C4×D9semidirect
256. C2×C9C2×SL(2,ℤ/4)semidirect
259. D9C2×SL(2,ℤ/4)direct (N × H)
307. A4C2×C4×D9semidirect
317. C9C22×SL(2,ℤ/4)semidirect
335. C23Dic3×D9semidirect
379. C22C2×Dic3×D9semidirect
393. C2D9×SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4×D92
C2×C9⋊SL(2,ℤ/4)2
C2×C9×SL(2,ℤ/4)2
D9×SL(2,ℤ/4)2
C2×D3×SL(2,ℤ/4)3
C2×C22⋊C4×D93
C2×Dic3×D94