Finite Group Databasebeta

← 3−1+2×#834 #72220083

Subgroups — 3−1+2×#834 #72220083

Order 2592.

713 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 147 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×C3×D4×3−1+2C211yesmaximal, characteristic
23−1+2×#217C212yesmaximal, complemented
3C32×#834C311yesmaximal, characteristic
43−1+2×#99—31nomaximal
5C9×#834C313yesmaximal, complemented
6C23×C3×3−1+2C411yescharacteristic, complemented
7C23×C3×3−1+2C2211yescharacteristic
8C3×D4×3−1+2—22no—
9C2×C3×C4×3−1+2C411yescharacteristic, complemented
10C2×Dic3×3−1+2—22no—
11C2×C33×D4C2×C311yescharacteristic
12C32×#217C2×C312yes—
13C2×D4×3−1+2—21no—
14C2×D4×3−1+2—23no—
15C2×D4×3−1+2D311yescharacteristic
16C2×C3×D4×C9C2×C313yes—
17C22⋊C4×3−1+2—32no—
18C9×#217C2×C316yescomplemented
19C22×C3×3−1+2—21no—
20C22×C3×3−1+2—41no—
21C22×C3×3−1+2—41no—
22C22×C3×3−1+2C2×C411yescharacteristic
23C22×C3×3−1+2D412yes—
24C3×C4×3−1+2—21no—
25Dic3×3−1+2—42no—
26C32×#99—31no—
27C3×#834—31no—
28C3×#834C3211yescharacteristic
29C9×#99—33no—
30C23×C33C3×C411yescharacteristic
31C23×C33C22×C311yescharacteristic
32C33×D4—22no—
33C2×C33×C4C3×C411yescharacteristic
34C2×C32×Dic3—22no—
35C23×3−1+2—21no—
36C23×3−1+2—21no—
37C23×3−1+2—23no—
38C23×3−1+2—23no—
39C23×3−1+2Dic311yescharacteristic, complemented
40C23×3−1+2C2×D311yescharacteristic
41C23×C3×C9C3×C413yescomplemented
42C23×C3×C9C22×C313yes—
43D4×3−1+2—24no—
44D4×3−1+2—212no—
45D4×3−1+2—22no—
46C3×D4×C9—26no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2—23no—
49C2×C4×3−1+2—62no—
50C2×C4×3−1+2Dic311yescharacteristic, complemented
51C2×C3×C4×C9C3×C413yescomplemented
52C2×C9×Dic3—26no—
53C2×C3×3−1+2—21no—
54C2×C3×3−1+2—22no—
55C2×C3×3−1+2—41no—
56C2×C3×3−1+2C22⋊C411yescharacteristic
57C2×C32×D4—21no—
58C2×C32×D4—31no—
59C2×C32×D4—61no—
60C2×C32×D4C3×D311yescharacteristic
61C2×C32×D4C2×C3211yescharacteristic
62C32×C22⋊C4—32no—
63C3×#217—32no—
64C3×#217C2×C3212yes—
65C2×D4×C9—23no—
66C2×D4×C9C3×D313yes—
67C9×C22⋊C4—36no—
68C22×C33—21no—
69C22×C33—41no—
70C22×C33—41no—
71C22×C33C2×C3×C411yescharacteristic
72C22×C33C3×D412yes—
73C33×C4—21no—
74C32×Dic3—42no—
75C22×3−1+2—21no—
76C22×3−1+2—23no—
77C22×3−1+2—22no—
78C22×3−1+2—26no—
79C22×3−1+2—22no—
80C22×3−1+2—26no—
81C22×3−1+2—21no—
82C22×3−1+2—41no—
83C22×3−1+2—46no—
84C22×3−1+2—42no—
85C22×3−1+2—46no—
86C22×3−1+2—42no—
87C22×3−1+2—41no—
88C22×3−1+2C2×Dic311yescharacteristic
89C22×3−1+2#6712yes—
90C22×C3×C9—23no—
91C22×C3×C9—43no—
92C22×C3×C9—43no—
93C22×C3×C9C2×C3×C413yes—
94C22×C3×C9C3×D416yes—
95C4×3−1+2—22no—
96C4×3−1+2—26no—
97C4×3−1+2—21no—
98C4×3−1+2—122no—
99C3×C4×C9—23no—
100C9×Dic3—46no—
101C3×#99—31no—
102C3×#99—91no—
103#8343−1+211yescharacteristic, direct product factor
104C3×3−1+2#9911yescharacteristic, complemented, Sylow
105C23×C32—21no—
106C23×C32—21no—
107C23×C32—31no—
108C23×C32—31no—
109C23×C32—61no—
110C23×C32—61no—
111C23×C32C3×Dic311yescharacteristic
112C23×C32C32×C411yescharacteristic
113C23×C32C2×C3×D311yescharacteristic
114C23×C32C22×C3211yescharacteristic
115C32×D4—22no—
116C32×D4—24no—
117C32×D4—22no—
118C32×D4—62no—
119C32×D4—64no—
120C2×C32×C4—21no—
121C2×C32×C4—31no—
122C2×C32×C4—61no—
123C2×C32×C4—62no—
124C2×C32×C4C3×Dic311yescharacteristic
125C2×C32×C4C32×C411yescharacteristic
126C2×C3×Dic3—22no—
127C2×C3×Dic3—62no—
128C23×C9—23no—
129C23×C9—23no—
130C23×C9C3×Dic313yescomplemented
131C23×C9C2×C3×D313yes—
132D4×C9—212no—
133D4×C9—26no—
134C2×C4×C9—23no—
135C2×C4×C9—66no—
136C2×C4×C9C3×Dic313yescomplemented
137C2×C33—21no—
138C2×C33—22no—
139C2×C33—41no—
140C2×C33C3×C22⋊C411yescharacteristic
141C2×3−1+2—22no—
142C2×3−1+2—26no—
143C2×3−1+2—22no—
144C2×3−1+2—21no—
145C2×3−1+2—21no—
146C2×3−1+2—23no—
147C2×3−1+2—46no—
148C2×3−1+2—42no—
149C2×3−1+2—42no—
150C2×3−1+2—46no—
151C2×3−1+2—41no—
152C2×3−1+2#21711yescharacteristic
153C2×C3×C9—23no—
154C2×C3×C9—26no—
155C2×C3×C9—43no—
156C2×C3×C9C3×C22⋊C413yes—
157C2×C3×D4—21no—
158C2×C3×D4—31no—
159C2×C3×D4—61no—
160C2×C3×D4C2×3−1+211yescharacteristic
161C2×C3×D4C32×D311yescharacteristic
162C3×C22⋊C4—32no—
163C3×C22⋊C4—92no—
164#217C2×3−1+212yescomplemented
165C22×C32—21no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—31no—
171C22×C32—32no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32—42no—
177C22×C32—42no—
178C22×C32—61no—
179C22×C32—62no—
180C22×C32—61no—
181C22×C32—62no—
182C22×C32—121no—
183C22×C32—121no—
184C22×C32—122no—
185C22×C32—122no—
186C22×C32C2×C3×Dic311yescharacteristic
187C22×C32C3×#6712yes—
188C22×C32C2×C32×C411yescharacteristic
189C22×C32C32×D412yes—
190C32×C4—21no—
191C32×C4—22no—
192C32×C4—21no—
193C32×C4—61no—
194C32×C4—62no—
195C32×C4—122no—
196C3×Dic3—42no—
197C3×Dic3—122no—
198C22×C9—26no—
199C22×C9—23no—
200C22×C9—23no—
201C22×C9—26no—
202C22×C9—43no—
203C22×C9—43no—
204C22×C9—46no—
205C22×C9—46no—
206C22×C9C2×C3×Dic313yes—
207C22×C9C3×#6716yes—
208C4×C9—26no—
209C4×C9—23no—
210C4×C9—126no—
211#99—31noSylow
212C33C3×#9911yescharacteristic
2133−1+2—21no—
2143−1+2—23no—
2153−1+2#83411yescharacteristic, direct product factor
216C3×C9C3×#9913yescomplemented
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—61no—
222C23×C3—61no—
223C23×C3C4×3−1+211yescharacteristic, complemented
224C23×C3C22×3−1+211yescharacteristic
225C23×C3C32×Dic311yescharacteristic
226C23×C3C2×C32×D311yescharacteristic
227C3×D4—24no—
228C3×D4—22no—
229C3×D4—22no—
230C3×D4—62no—
231C3×D4—64no—
232C2×C3×C4—21no—
233C2×C3×C4—31no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—182no—
237C2×C3×C4C4×3−1+211yescharacteristic, complemented
238C2×C3×C4C32×Dic311yescharacteristic
239C2×Dic3—22no—
240C2×C32—21no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—21no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—31no—
247C2×C32—42no—
248C2×C32—41no—
249C2×C32—42no—
250C2×C32—41no—
251C2×C32—61no—
252C2×C32—61no—
253C2×C32—62no—
254C2×C32—62no—
255C2×C32—122no—
256C2×C32—121no—
257C2×C32—122no—
258C2×C32C3×#21711yescharacteristic
259C2×C32C32×C22⋊C411yescharacteristic
260C2×C9—26no—
261C2×C9—23no—
262C2×C9—26no—
263C2×C9—23no—
264C2×C9—46no—
265C2×C9—46no—
266C2×C9—43no—
267C2×C9C3×#21713yes—
268C2×D4D3×3−1+211yescharacteristic
269C22⋊C4—32no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—31no—
276C22×C3—32no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—61no—
284C22×C3—62no—
285C22×C3—61no—
286C22×C3—62no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—122no—
290C22×C3—122no—
291C22×C3C2×C4×3−1+211yescharacteristic
292C22×C3D4×3−1+212yes—
293C22×C3C2×C32×Dic311yescharacteristic
294C22×C3C32×#6712yes—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4—122no—
301C3×C4—362no—
302Dic3—42no—
303C32—21no—
304C32—31no—
305C32—61no—
306C32C3×#83411yescharacteristic
307C32C32×#9911yescharacteristic
308C9—23no—
309C9C3×#83413yescomplemented
310C23Dic3×3−1+211yescharacteristic, complemented
311C23C2×D3×3−1+211yescharacteristic
312D4—22no—
313C2×C4—62no—
314C2×C4Dic3×3−1+211yescharacteristic, complemented
315C2×C3—22no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—31no—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—62no—
327C2×C3—61no—
328C2×C3—62no—
329C2×C3—61no—
330C2×C3—122no—
331C2×C3—121no—
332C2×C3—122no—
333C2×C3C22⋊C4×3−1+211yescharacteristic
334C2×C3C32×#21711yescharacteristic
335C22—21no—
336C22—41no—
337C22—41no—
338C22C2×Dic3×3−1+211yescharacteristic
339C22#67×3−1+212yes—
340C4—21no—
341C4—122no—
342C3—21no—
343C3—31no—
344C3—61no—
345C33−1+2×#9911yescharacteristic, complemented
346C3C32×#83411yescharacteristic
347C2—22no—
348C2—21no—
349C2—41no—
350C23−1+2×#21711yescharacteristic
subgroupG/H
centerC2×C3C32×#217
commutatorC22×C32C2×C32×C4
FrattiniC23×C3C2×C32×D3
FittingC2×C3×D4×3−1+2C2
socleC2×C32C32×C22⋊C4
prime radicalC2×C33×D4C2×C3
cosocleC23×C32C22×C32
hypercenterC2×D4×3−1+2D3
hypocenterC33−1+2×#99
LCSG ▷ C22×C32 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×3−1+2 ◁ C2×D4×3−1+2

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#217C2semidirect
5. C9×#834C3semidirect
6. C23×C3×3−1+2C4semidirect
9. C2×C3×C4×3−1+2C4semidirect
18. C9×#217C2×C3semidirect
39. C23×3−1+2Dic3semidirect
41. C23×C3×C9C3×C4semidirect
50. C2×C4×3−1+2Dic3semidirect
51. C2×C3×C4×C9C3×C4semidirect
103. #8343−1+2direct (N × H)
104. C3×3−1+2#99semidirect
130. C23×C9C3×Dic3semidirect
136. C2×C4×C9C3×Dic3semidirect
164. #217C2×3−1+2semidirect
215. 3−1+2#834direct (N × H)
216. C3×C9C3×#99semidirect
223. C23×C3C4×3−1+2semidirect
237. C2×C3×C4C4×3−1+2semidirect
309. C9C3×#834semidirect
310. C23Dic3×3−1+2semidirect
314. C2×C4Dic3×3−1+2semidirect
345. C33−1+2×#99semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×3−1+22
3−1+2×#2172
C32×#8343
3−1+2×#993
C9×#8343