Finite Group Databasebeta

← C22×#1842⋊C3 #72211184

Subgroups — C22×#1842⋊C3 #72211184

Order 1536.

1740 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1740 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×C42⋊C3C211yesmaximal, characteristic
2C2×#1842⋊C3C216yesmaximal, direct product factor
3C22×#1842C311yesmaximal, characteristic, complemented, Sylow
4C22×C8×A4—41nomaximal
5C23×C42⋊C3C411yescharacteristic
6C2×C4×C42⋊C3C413yes—
7C2×C4×C42⋊C3C2213yes—
8#1842⋊C3C2214yesdirect product factor
9C23×C4⋊C8—31no—
10C23×C4⋊C8—31no—
11C22×C43C2×C311yescharacteristic
12C2×#1842—36no—
13C2×#1842C2×C316yescomplemented
14C22×C4×A4—41no—
15C2×C8×A4—46no—
16C22×C42⋊C3C814yescomplemented
17C22×C42⋊C3C2×C413yes—
18C4×C42⋊C3C2×C413yes—
19C4×C42⋊C3C2311yescharacteristic
20C24×C8—31no—
21C24×C8A411yescharacteristic
22C23×C42—31no—
23C23×C42—31no—
24C23×C42C3×C411yescharacteristic
25C22×C4⋊C8—36no—
26C22×C4⋊C8—36no—
27C22×C4⋊C8—36no—
28C22×C4⋊C8—36no—
29C22×C4⋊C8—62no—
30C22×C4⋊C8—66no—
31C22×C4⋊C8—62no—
32C22×C4⋊C8—66no—
33C2×C43—33no—
34C2×C43—33no—
35C2×C43C3×C413yes—
36C2×C43C22×C313yes—
37#1842—312no—
38#1842—38no—
39#1842C22×C314yescomplemented
40C23×A4—41no—
41C2×C4×A4—43no—
42C2×C4×A4—43no—
43C22×C3×C8—161no—
44C8×A4—44no—
45C2×C42⋊C3C2×C816yescomplemented
46C2×C42⋊C3C22×C411yescharacteristic
47C24×C4—31no—
48C24×C4—31no—
49C24×C4C2×A411yescharacteristic
50C23×C8—36no—
51C23×C8—61no—
52C23×C8—66no—
53C23×C8—63no—
54C23×C8—61no—
55C23×C8—62no—
56C23×C8—63no—
57C23×C8C2×A416yes—
58C22×C42—34no—
59C22×C42—33no—
60C22×C42—32no—
61C22×C42—36no—
62C22×C42—36no—
63C22×C42—32no—
64C22×C42—33no—
65C22×C42—33no—
66C22×C42—33no—
67C22×C42—33no—
68C22×C42—33no—
69C22×C42—33no—
70C22×C42—36no—
71C22×C42—68no—
72C22×C42C3×C814yescomplemented
73C22×C42C2×C3×C413yes—
74C2×C4⋊C8—34no—
75C2×C4⋊C8—312no—
76C2×C4⋊C8—34no—
77C2×C4⋊C8—312no—
78C2×C4⋊C8—612no—
79C2×C4⋊C8—612no—
80C2×C4⋊C8—612no—
81C2×C4⋊C8—612no—
82C2×C4⋊C8—612no—
83C2×C4⋊C8—612no—
84C2×C4⋊C8—612no—
85C2×C4⋊C8—612no—
86C43—33no—
87C43—36no—
88C43—33no—
89C43—32no—
90C43—36no—
91C43C2×C3×C413yes—
92C43C23×C311yescharacteristic
93C22×A4—44no—
94C22×A4—43no—
95C22×C3×C4—161no—
96C4×A4—43no—
97C4×A4—41no—
98C2×C3×C8—166no—
99C42⋊C3C22×C811yescharacteristic, complemented
100C25C4×A411yescharacteristic
101C23×C4—36no—
102C23×C4—32no—
103C23×C4—38no—
104C23×C4—34no—
105C23×C4—34no—
106C23×C4—33no—
107C23×C4—33no—
108C23×C4—31no—
109C23×C4—33no—
110C23×C4—31no—
111C23×C4—33no—
112C23×C4—36no—
113C23×C4—32no—
114C23×C4—33no—
115C23×C4—33no—
116C23×C4—68no—
117C23×C4C4×A413yes—
118C23×C4C22×A413yes—
119C22×C8—34no—
120C22×C8—41no—
121C22×C8—66no—
122C22×C8—612no—
123C22×C8—66no—
124C22×C8—66no—
125C22×C8—612no—
126C22×C8—66no—
127C22×C8—123no—
128C22×C8—122no—
129C22×C8—126no—
130C22×C8—123no—
131C22×C8—126no—
132C22×C8—121no—
133C22×C8C22×A414yes—
134C2×C42—33no—
135C2×C42—312no—
136C2×C42—36no—
137C2×C42—32no—
138C2×C42—36no—
139C2×C42—36no—
140C2×C42—36no—
141C2×C42—36no—
142C2×C42—36no—
143C2×C42—36no—
144C2×C42—312no—
145C2×C42—36no—
146C2×C42—312no—
147C2×C42—36no—
148C2×C42—36no—
149C2×C42—36no—
150C2×C42—36no—
151C2×C42—36no—
152C2×C42—312no—
153C2×C42—33no—
154C2×C42—36no—
155C2×C42—33no—
156C2×C42—31no—
157C2×C42—33no—
158C2×C42—33no—
159C2×C42—31no—
160C2×C42—36no—
161C2×C42—36no—
162C2×C42—612no—
163C2×C42—612no—
164C2×C42—624no—
165C2×C42C2×C3×C816yescomplemented
166C2×C42C22×C3×C411yescharacteristic
167C4⋊C8—68no—
168C4⋊C8—624no—
169C4⋊C8—68no—
170C4⋊C8—624no—
171C23×C3—161no—
172C2×A4—41no—
173C2×A4—46no—
174C2×C3×C4—163no—
175C2×C3×C4—163no—
176C3×C8—164no—
177C24—33no—
178C24—34no—
179C24—31no—
180C24C8×A414yes—
181C24C2×C4×A413yes—
182C22×C4—36no—
183C22×C4—36no—
184C22×C4—36no—
185C22×C4—36no—
186C22×C4—312no—
187C22×C4—312no—
188C22×C4—32no—
189C22×C4—324no—
190C22×C4—324no—
191C22×C4—38no—
192C22×C4—38no—
193C22×C4—36no—
194C22×C4—36no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—33no—
198C22×C4—36no—
199C22×C4—312no—
200C22×C4—36no—
201C22×C4—33no—
202C22×C4—33no—
203C22×C4—33no—
204C22×C4—33no—
205C22×C4—33no—
206C22×C4—33no—
207C22×C4—36no—
208C22×C4—33no—
209C22×C4—33no—
210C22×C4—36no—
211C22×C4—36no—
212C22×C4—36no—
213C22×C4—36no—
214C22×C4—33no—
215C22×C4—31no—
216C22×C4—612no—
217C22×C4—612no—
218C22×C4—624no—
219C22×C4—68no—
220C22×C4#127⋊C311yescharacteristic
221C22×C4C2×C4×A413yes—
222C22×C4C23×A411yescharacteristic
223C2×C8—46no—
224C2×C8—64no—
225C2×C8—64no—
226C2×C8—68no—
227C2×C8—126no—
228C2×C8—126no—
229C2×C8—1212no—
230C2×C8—126no—
231C42—36no—
232C42—31no—
233C42—32no—
234C42—36no—
235C42—36no—
236C42—312no—
237C42—36no—
238C42—36no—
239C42—312no—
240C42—36no—
241C42—32no—
242C42—36no—
243C42—36no—
244C42—38no—
245C42—66no—
246C42—62no—
247C42—612no—
248C42—612no—
249C42C22×C3×C811yescharacteristic, complemented
250C22×C3—164no—
251C22×C3—163no—
252A4—41no—
253C3×C4—161no—
254C3×C4—163no—
255C23—33no—
256C23—36no—
257C23—312no—
258C23—32no—
259C23—312no—
260C23—34no—
261C23—34no—
262C23—33no—
263C23—33no—
264C23#384⋊C311yescharacteristic
265C23C2×C8×A416yes—
266C23C22×C4×A411yescharacteristic
267C2×C4—32no—
268C2×C4—36no—
269C2×C4—32no—
270C2×C4—36no—
271C2×C4—312no—
272C2×C4—324no—
273C2×C4—312no—
274C2×C4—312no—
275C2×C4—324no—
276C2×C4—312no—
277C2×C4—36no—
278C2×C4—36no—
279C2×C4—31no—
280C2×C4—36no—
281C2×C4—31no—
282C2×C4—33no—
283C2×C4—33no—
284C2×C4—31no—
285C2×C4—33no—
286C2×C4—33no—
287C2×C4—36no—
288C2×C4—33no—
289C2×C4—33no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—36no—
293C2×C4—66no—
294C2×C4—62no—
295C2×C4—612no—
296C2×C4—612no—
297C2×C4—612no—
298C2×C4—612no—
299C2×C4#384⋊C313yes—
300C2×C4C2×#127⋊C313yes—
301C8—44no—
302C8—124no—
303C2×C3—166no—
304C2×C3—161no—
305C22—31no—
306C22—36no—
307C22—36no—
308C22—38no—
309C22—36no—
310C22—36no—
311C22—312no—
312C22—31no—
313C22—33no—
314C22#1842⋊C314yesdirect product factor
315C22C2×#384⋊C313yes—
316C22C22×C8×A411yescharacteristic
317C4—32no—
318C4—32no—
319C4—312no—
320C4—31no—
321C4—33no—
322C4—62no—
323C4—66no—
324C4C2×#384⋊C313yes—
325C4C22×#127⋊C311yescharacteristic
326C3—161noSylow
327C2—31no—
328C2—31no—
329C2—36no—
330C2C2×#1842⋊C316yesdirect product factor
331C2C22×#384⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22×C4#127⋊C3
commutator, hypocenterC42C22×C3×C8
FrattiniC22×C4C23×A4
FittingC22×#1842C3
socleC25C4×A4
prime radicalC23×C42⋊C3C4
cosocleC43C23×C3
LCSG ▷ C42
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#1842⋊C3C2direct (N × H)
3. C22×#1842C3semidirect
8. #1842⋊C3C22direct (N × H)
13. C2×#1842C2×C3semidirect
16. C22×C42⋊C3C8semidirect
39. #1842C22×C3semidirect
45. C2×C42⋊C3C2×C8semidirect
72. C22×C42C3×C8semidirect
99. C42⋊C3C22×C8semidirect
165. C2×C42C2×C3×C8semidirect
249. C42C22×C3×C8semidirect
314. C22#1842⋊C3direct (N × H)
330. C2C2×#1842⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C42⋊C32
C2×#1842⋊C32
C22×#18423
C22×C8×A44