Finite Group Databasebeta

← C2×#14518201 #72252763

Subgroups — C2×#14518201 #72252763

Order 3888.

714 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 80 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×#11757845C211yesmaximal, characteristic, complemented
2#14518201C212yesmaximal, direct product factor
3C23×C3×AGL(1,ℤ/9)C311yesmaximal, characteristic
4C2×D3×#1094—31nomaximal
5C2×D9⋊(C22⋊C9)C312yesmaximal, complemented
6C2×D9×C22⋊C9C311yesmaximal, characteristic, complemented
7C2×#92265—41nomaximal
8#11757845C2211yescharacteristic, complemented
9C23×C3×3−1+2C2×C311yescharacteristic
10C22×C3×AGL(1,ℤ/9)—31no—
11C22×C3×AGL(1,ℤ/9)—31no—
12C22×C3×AGL(1,ℤ/9)—32no—
13C22×C3×AGL(1,ℤ/9)C2×C312yes—
14C2×C3×#1094—21no—
15C2×C3×#1094D311yescharacteristic
16D3×#1094—32no—
17C2×C9⋊(C22⋊C9)C2×C312yescomplemented
18C2×(C22⋊C9)⋊C9—22no—
19C2×(C22⋊C9)⋊C9—21no—
20C2×C9×C22⋊C9C2×C311yescharacteristic, complemented
21D9×C22⋊C9C2×C312yescomplemented
22D9⋊(C22⋊C9)C2×C314yescomplemented
23C2×#6900—41no—
24#92265—42no—
25C23×C32×D3—31no—
26C23×AGL(1,ℤ/9)—31no—
27C23×C3×D9C3211yescharacteristic
28C22×#1094—91no—
29C2×D3×C22⋊C9—31no—
30C2×D3×C22⋊C9—32no—
31C22×C3×3−1+2—31no—
32C22×C3×3−1+2—31no—
33C22×C3×3−1+2C22×C311yescharacteristic
34C2×D3×3−1+2—121no—
35C2×C3×AGL(1,ℤ/9)—32no—
36C2×C3×AGL(1,ℤ/9)—31no—
37C2×C3×AGL(1,ℤ/9)—32no—
38C2×C3×AGL(1,ℤ/9)—32no—
39C2×C3×AGL(1,ℤ/9)—32no—
40C2×C3×AGL(1,ℤ/9)A411yescharacteristic
41C3×#1094—21no—
42C3×#1094C2×D311yescharacteristic
43C2×D9⋊C9—42no—
44C2×C9×D9—41no—
45C9⋊(C22⋊C9)C22×C312yescomplemented
46(C22⋊C9)⋊C9—22no—
47(C22⋊C9)⋊C9—21no—
48C9×C22⋊C9C22×C311yescharacteristic, complemented
49#6900—41noSylow
50C23×C33C3×D311yescharacteristic
51C22×C32×D3—32no—
52C22×C32×D3—91no—
53C22×C32×D3—92no—
54C22×C32×D3—91no—
55C23×3−1+2—31no—
56C23×3−1+2—61no—
57C23×C3×C9—21no—
58C23×C3×C9C2×C3211yescharacteristic
59C22×AGL(1,ℤ/9)—33no—
60C22×AGL(1,ℤ/9)—36no—
61C22×AGL(1,ℤ/9)—33no—
62C22×AGL(1,ℤ/9)—32no—
63C22×C3×D9—31no—
64C22×C3×D9—31no—
65C22×C3×D9—32no—
66C22×C3×D9C2×C3212yes—
67C2×#1094—31no—
68C2×#1094—61no—
69C2×#1094—63no—
70C2×#1094—92no—
71C2×C3×C22⋊C9—21no—
72C2×C3×C22⋊C9—22no—
73C2×C3×C22⋊C9C3×D311yescharacteristic
74C2×C3×C22⋊C9C3×D312yes—
75D3×C22⋊C9—34no—
76D3×C22⋊C9—32no—
77C2×C3×3−1+2—31no—
78C2×C3×3−1+2—31no—
79C2×C3×3−1+2—41no—
80C2×C3×3−1+2—81no—
81C2×C3×3−1+2C2×A411yescharacteristic
82D3×3−1+2—122no—
83C3×AGL(1,ℤ/9)—32no—
84C3×AGL(1,ℤ/9)C2×A412yes—
85C2×C9⋊C9—42no—
86C2×C9⋊C9—81no—
87C2×C9⋊C9—82no—
88C2×C92—41no—
89D9⋊C9—44no—
90C9×D9—42no—
91C24×C32—91no—
92C23×C3×D3—31no—
93C23×C3×D3—91no—
94C23×D93−1+211yescharacteristic, complemented
95C22×C22⋊C9—92no—
96C22×C22⋊C9—91no—
97C22×C33—31no—
98C22×C33—31no—
99C22×C33C2×C3×D311yescharacteristic
100C2×C32×D3—31no—
101C2×C32×D3—92no—
102C2×C32×D3—92no—
103C2×C32×D3—92no—
104C2×C32×D3—91no—
105C2×C32×D3—92no—
106C22×3−1+2—33no—
107C22×3−1+2—33no—
108C22×3−1+2—31no—
109C22×3−1+2—61no—
110C22×3−1+2—63no—
111C22×3−1+2—63no—
112C22×3−1+2—361no—
113C22×C3×C9—21no—
114C22×C3×C9—31no—
115C22×C3×C9—31no—
116C22×C3×C9—61no—
117C22×C3×C9—61no—
118C22×C3×C9C22×C3211yescharacteristic
119C2×AGL(1,ℤ/9)—36no—
120C2×AGL(1,ℤ/9)—36no—
121C2×AGL(1,ℤ/9)—33no—
122C2×AGL(1,ℤ/9)—36no—
123C2×AGL(1,ℤ/9)—36no—
124C2×AGL(1,ℤ/9)—31no—
125C2×D3×C9—121no—
126C2×D3×C9—122no—
127C2×C3×D9—32no—
128C2×C3×D9—31no—
129C2×C3×D9—32no—
130C2×C3×D9—32no—
131C2×C3×D9—32no—
132C2×C3×D9C3×A411yescharacteristic
133#1094—31no—
134#1094—61no—
135#1094—63no—
136C3×C22⋊C9—21no—
137C3×C22⋊C9—22no—
138C3×C22⋊C9C2×C3×D311yescharacteristic
139C3×C22⋊C9C2×C3×D312yes—
140C3×3−1+2—41no—
141C3×3−1+2—81no—
142C3×3−1+2C22×A411yescharacteristic
143C9⋊C9—42no—
144C9⋊C9—81no—
145C9⋊C9—82no—
146C92—41no—
147C23×C32—31no—
148C23×C32—31no—
149C23×C32—61no—
150C23×C32—92no—
151C23×C32—271no—
152C23×C32—271no—
153C23×C32—272no—
154C23×C32C32×D311yescharacteristic
155C22×C3×D3—32no—
156C22×C3×D3—92no—
157C22×C3×D3—91no—
158C22×C3×D3—93no—
159C22×C3×D3—96no—
160C22×C3×D3—93no—
161C22×C3×D3—92no—
162C22×C3×D3—91no—
163C23×C9—21no—
164C23×C9—61no—
165C23×C9C2×3−1+211yescharacteristic, complemented
166C22×D9—31no—
167C22×D9—31no—
168C22×D9—32no—
169C22×D9C2×3−1+212yescomplemented
170C2×C22⋊C9—23no—
171C2×C22⋊C9—21no—
172C2×C22⋊C9—32no—
173C2×C22⋊C9—62no—
174C2×C22⋊C9—94no—
175C2×C22⋊C9—92no—
176C2×C22⋊C9AGL(1,ℤ/9)11yescharacteristic, complemented
177C2×C33—31no—
178C2×C33—31no—
179C2×C33D3×A411yescharacteristic
180C32×D3—32no—
181C32×D3—92no—
182C2×3−1+2—33no—
183C2×3−1+2—33no—
184C2×3−1+2—31no—
185C2×3−1+2—61no—
186C2×3−1+2—63no—
187C2×3−1+2—63no—
188C2×3−1+2—121no—
189C2×3−1+2—241no—
190C2×3−1+2—243no—
191C2×3−1+2—362no—
192C2×C3×C9—21no—
193C2×C3×C9—31no—
194C2×C3×C9—31no—
195C2×C3×C9—41no—
196C2×C3×C9—42no—
197C2×C3×C9—61no—
198C2×C3×C9—61no—
199C2×C3×C9—81no—
200C2×C3×C9—82no—
201C2×C3×C9C2×C3×A411yescharacteristic
202AGL(1,ℤ/9)—36no—
203AGL(1,ℤ/9)—32no—
204D3×C9—122no—
205D3×C9—124no—
206C3×D9—32no—
207C3×D9C2×C3×A412yes—
208C24×C3—91no—
209C24×C3—271no—
210C23×D3—31no—
211C22×C32—31no—
212C22×C32—31no—
213C22×C32—33no—
214C22×C32—33no—
215C22×C32—31no—
216C22×C32—31no—
217C22×C32—61no—
218C22×C32—91no—
219C22×C32—91no—
220C22×C32—91no—
221C22×C32—181no—
222C22×C32—181no—
223C22×C32—271no—
224C22×C32—272no—
225C22×C32—272no—
226C22×C32—272no—
227C22×C32—272no—
228C22×C32C2×C32×D311yescharacteristic
229C2×C3×D3—31no—
230C2×C3×D3—91no—
231C2×C3×D3—92no—
232C2×C3×D3—92no—
233C2×C3×D3—92no—
234C2×C3×D3—96no—
235C2×C3×D3—96no—
236C2×C3×D3—96no—
237C2×C3×D3—93no—
238C2×C3×D3—96no—
239C2×C3×D3—91no—
240C2×C3×D3—92no—
241C22×C9—21no—
242C22×C9—31no—
243C22×C9—31no—
244C22×C9—61no—
245C22×C9—61no—
246C22×C9—63no—
247C22×C9—63no—
248C22×C9—61no—
249C22×C9—361no—
250C22×C9—362no—
251C22×C9C22×3−1+211yescharacteristic, complemented
252C2×D9—32no—
253C2×D9—31no—
254C2×D9—32no—
255C2×D9—32no—
256C2×D9—32no—
257C2×D9#109411yescharacteristic, complemented
258C22⋊C9—23no—
259C22⋊C9—21no—
260C22⋊C9—32no—
261C22⋊C9—62no—
262C22⋊C9C2×AGL(1,ℤ/9)11yescharacteristic, complemented
263C33C2×D3×A411yescharacteristic
2643−1+2—31no—
2653−1+2—61no—
2663−1+2—121no—
2673−1+2—241no—
2683−1+2—243no—
269C3×C9—21no—
270C3×C9—41no—
271C3×C9—42no—
272C3×C9—81no—
273C3×C9—82no—
274C3×C9C22×C3×A411yescharacteristic
275C23×C3—21no—
276C23×C3—92no—
277C23×C3—91no—
278C23×C3—272no—
279C23×C3—271no—
280C23×C3—272no—
281C23×C3—271no—
282C23×C3—273no—
283C23×C3—276no—
284C23×C3—273no—
285C23×C3C3×AGL(1,ℤ/9)11yescharacteristic
286C23×C3D3×3−1+211yescharacteristic
287C22×D3—32no—
288C22×D3—91no—
289C22×D3—92no—
290C22×D3—91no—
291C2×C32—31no—
292C2×C32—31no—
293C2×C32—33no—
294C2×C32—33no—
295C2×C32—31no—
296C2×C32—31no—
297C2×C32—61no—
298C2×C32—91no—
299C2×C32—91no—
300C2×C32—92no—
301C2×C32—181no—
302C2×C32—181no—
303C2×C32—272no—
304C2×C32C3×D3×A411yescharacteristic
305C3×D3—32no—
306C3×D3—92no—
307C3×D3—96no—
308C3×D3—92no—
309C2×C9—21no—
310C2×C9—31no—
311C2×C9—31no—
312C2×C9—41no—
313C2×C9—61no—
314C2×C9—61no—
315C2×C9—63no—
316C2×C9—63no—
317C2×C9—61no—
318C2×C9—83no—
319C2×C9—81no—
320C2×C9—122no—
321C2×C9—242no—
322C2×C9—362no—
323C2×C9—364no—
324C2×C9C2×#109411yescharacteristic, complemented
325D9—32no—
326D9C2×#109412yescomplemented
327C24—91noSylow
328C22×C3—21no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—93no—
336C22×C3—93no—
337C22×C3—91no—
338C22×C3—91no—
339C22×C3—271no—
340C22×C3—272no—
341C22×C3—272no—
342C22×C3—272no—
343C22×C3—272no—
344C22×C3—276no—
345C22×C3—276no—
346C22×C3—276no—
347C22×C3—276no—
348C22×C3—273no—
349C22×C3—271no—
350C22×C3C2×C3×AGL(1,ℤ/9)11yescharacteristic
351C22×C3C2×D3×3−1+211yescharacteristic
352C2×D3—31no—
353C2×D3—92no—
354C2×D3—92no—
355C2×D3—92no—
356C2×D3—91no—
357C2×D3—92no—
358C32—31no—
359C32—31no—
360C32—61no—
361C32C2×C3×D3×A411yescharacteristic
362C9—21no—
363C9—41no—
364C9—61no—
365C9—83no—
366C9—81no—
367C9—122no—
368C9—242no—
369C9C22×#109411yescharacteristic, complemented
370C23—92no—
371C23—271no—
372C23—272no—
373C23—271no—
374C23#9226511yescharacteristic, complemented
375C2×C3—21no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—93no—
383C2×C3—93no—
384C2×C3—92no—
385C2×C3—91no—
386C2×C3—272no—
387C2×C3—276no—
388C2×C3—272no—
389C2×C3A4×AGL(1,ℤ/9)11yescharacteristic
390C2×C3D3×#109411yescharacteristic
391D3—32no—
392D3—92no—
393C22—31no—
394C22—31no—
395C22—91no—
396C22—272no—
397C22—272no—
398C22—272no—
399C22—272no—
400C22—271no—
401C22C2×#9226511yescharacteristic, complemented
402C3—21no—
403C3—91no—
404C3C2×A4×AGL(1,ℤ/9)11yescharacteristic
405C3C2×D3×#109411yescharacteristic
406C2—31no—
407C2—31no—
408C2—92no—
409C2—272no—
410C2#1451820111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3A4×AGL(1,ℤ/9)
commutator, cosocleC22×C3×C9C22×C32
FrattiniC32C2×C3×D3×A4
FittingC23×C3×3−1+2C2×C3
socleC23×C32C32×D3
prime radicalC23×C3×AGL(1,ℤ/9)C3
hypocenterC22×C9C22×3−1+2
LCSG ▷ C22×C3×C9 ▷ C22×C9
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×#11757845C2semidirect
2. #14518201C2direct (N × H)
5. C2×D9⋊(C22⋊C9)C3semidirect
6. C2×D9×C22⋊C9C3semidirect
8. #11757845C22semidirect
17. C2×C9⋊(C22⋊C9)C2×C3semidirect
20. C2×C9×C22⋊C9C2×C3semidirect
21. D9×C22⋊C9C2×C3semidirect
22. D9⋊(C22⋊C9)C2×C3semidirect
45. C9⋊(C22⋊C9)C22×C3semidirect
48. C9×C22⋊C9C22×C3semidirect
94. C23×D93−1+2semidirect
165. C23×C9C2×3−1+2semidirect
169. C22×D9C2×3−1+2semidirect
176. C2×C22⋊C9AGL(1,ℤ/9)semidirect
251. C22×C9C22×3−1+2semidirect
257. C2×D9#1094semidirect
262. C22⋊C9C2×AGL(1,ℤ/9)semidirect
324. C2×C9C2×#1094semidirect
326. D9C2×#1094semidirect
369. C9C22×#1094semidirect
374. C23#92265semidirect
401. C22C2×#92265semidirect
410. C2#14518201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117578452
#145182012
C23×C3×AGL(1,ℤ/9)3
C2×D3×#10943
C2×D9⋊(C22⋊C9)3
C2×D9×C22⋊C93
C2×#922654