Finite Group Databasebeta

← C23×#3351 #15670970

Subgroups — C23×#3351 #15670970

Order 2944.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#553C211yesmaximal, characteristic
2C23×#552C211yesmaximal, characteristic, complemented
3C23×#499C211yesmaximal, characteristic, complemented
4C23×#485C211yesmaximal, characteristic, complemented
5C23×#483C211yesmaximal, characteristic, complemented
6C23×#482C211yesmaximal, characteristic, complemented
7C23×#481C211yesmaximal, characteristic, complemented
8C23×#479C211yesmaximal, characteristic, complemented
9C23×#478C211yesmaximal, characteristic, complemented
10C23×#468C211yesmaximal, characteristic, complemented
11C23×#468C211yesmaximal, characteristic, complemented
12C4×SD16×C23C211yesmaximal, characteristic, complemented
13C23×#436C211yesmaximal, characteristic, complemented
14C23×#428C211yesmaximal, characteristic, complemented
15C23×#428C211yesmaximal, characteristic, complemented
16C2×SD16×C23C2211yescharacteristic
17C2×M16×C23C2211yescharacteristic
18C2×M16×C23C2211yescharacteristic
19C23×#126—21no—
20C23×#126—21no—
21C23×#125—21no—
22C23×#125C2211yescharacteristic
23C23×#125C2211yescharacteristic
24C23×#123—21no—
25C23×#123—21no—
26C23×C22⋊Q8C2211yescharacteristic
27C23×C22⋊Q8C2211yescharacteristic
28C23×#121—21no—
29C23×#121C2211yescharacteristic
30C23×#121C2211yescharacteristic
31C4×Q8×C23C2211yescharacteristic
32C4×D4×C23—21no—
33C4×D4×C23C2211yescharacteristic
34C4×D4×C23C2211yescharacteristic
35C23×#117—21no—
36C23×#117C2211yescharacteristic
37C23×#117C2211yescharacteristic
38C2×C4⋊C4×C23C2211yescharacteristic
39C2×C4⋊C4×C23C2211yescharacteristic
40C23×#107C2211yescharacteristic
41C23×#107C2211yescharacteristic
42C23×#107C2211yescharacteristic
43C23×#107C2211yescharacteristic
44C23×#106C2211yescharacteristic
45C23×#106C2211yescharacteristic
46C23×#106C2211yescharacteristic
47C23×#106C2211yescharacteristic
48C23×#106C2211yescharacteristic
49C23×C4⋊C8C2211yescharacteristic
50C23×#103C2211yescharacteristic
51C23×#103C2211yescharacteristic
52C23×#103C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×#102C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×C22⋊C8C2211yescharacteristic
58C4×C8×C23C2211yescharacteristic
59C2×Q8×C23C2311yescharacteristic
60C2×D4×C23—21no—
61C2×D4×C23—21no—
62C2×D4×C23D411yescharacteristic
63C2×D4×C23C2311yescharacteristic
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69SD16×C23—22no—
70M16×C23—22no—
71M16×C23—22no—
72C2×C8×C23C2311yescharacteristic
73C2×C8×C23C2311yescharacteristic
74C2×C8×C23C2311yescharacteristic
75C2×C8×C23C2311yescharacteristic
76C4⋊C4×C23—21no—
77C4⋊C4×C23—21no—
78C4⋊C4×C23—21no—
79C4⋊C4×C23—21no—
80C4⋊C4×C23—21no—
81C4⋊C4×C23—21no—
82C4⋊C4×C23—21no—
83C4⋊C4×C23—21no—
84C4⋊C4×C23—21no—
85C4⋊C4×C23D411yescharacteristic
86C4⋊C4×C23C2311yescharacteristic
87C4⋊C4×C23C2311yescharacteristic
88C4⋊C4×C23C2311yescharacteristic
89C4⋊C4×C23C2311yescharacteristic
90C4⋊C4×C23C2311yescharacteristic
91C4⋊C4×C23C2311yescharacteristic
92C22⋊C4×C23—21no—
93C22⋊C4×C23—21no—
94C22⋊C4×C23—21no—
95C22⋊C4×C23—21no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C22⋊C4×C23—21no—
99C22⋊C4×C23D411yescharacteristic
100C22⋊C4×C23D411yescharacteristic
101C42×C23—21no—
102C42×C23—21no—
103C42×C23—21no—
104C42×C23C2311yescharacteristic
105C23×C23—21no—
106C23×C23C2×D411yescharacteristic
107C23×C23C2×D411yescharacteristic
108Q8×C23—21no—
109Q8×C23—21no—
110D4×C23—21no—
111D4×C23—21no—
112D4×C23—41no—
113D4×C23—42no—
114D4×C23—81no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—42no—
127C2×C4×C23—81no—
128C2×C4×C23—81no—
129C2×C4×C23C2×D411yescharacteristic
130C2×C4×C23C2×D411yescharacteristic
131C2×C4×C23C2×D411yescharacteristic
132C2×C4×C23C2×D411yescharacteristic
133C2×C4×C23C2411yescharacteristic
134C8×C23—21no—
135C8×C23—21no—
136C8×C23—21no—
137C8×C23#4112yes—
138#3351C2311yesmaximal, characteristic, direct product factor, Sylow
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23—41no—
148C22×C23C22×D411yescharacteristic
149C4×C23—21no—
150C4×C23—21no—
151C4×C23—21no—
152C4×C23—21no—
153C4×C23—41no—
154C4×C23—41no—
155C4×C23—41no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23C2×#4111yescharacteristic
161C4×C232−1+411yescharacteristic
162#553C2×C2311yescharacteristic
163#552C2×C2311yescharacteristic, complemented
164#499C2×C2311yescharacteristic, complemented
165#485C2×C2311yescharacteristic, complemented
166#483C2×C2311yescharacteristic, complemented
167#482C2×C2311yescharacteristic, complemented
168#481C2×C2311yescharacteristic, complemented
169#479C2×C2311yescharacteristic, complemented
170#478C2×C2311yescharacteristic, complemented
171#468C2×C2311yescharacteristic, complemented
172#468C2×C2311yescharacteristic, complemented
173C4×SD16C2×C2311yescharacteristic, complemented
174#436C2×C2311yescharacteristic, complemented
175#428C2×C2311yescharacteristic, complemented
176#428C2×C2311yescharacteristic, complemented
177C2×C23—41no—
178C2×C23—41no—
179C2×C23—81no—
180C2×C23#54711yescharacteristic
181C2×C23#57511yescharacteristic
182C2×C23#57711yescharacteristic
183C2×SD16C22×C2311yescharacteristic
184C2×M16C22×C2311yescharacteristic
185C2×M16C22×C2311yescharacteristic
186#126—21no—
187#126—21no—
188#125—21no—
189#125C22×C2311yescharacteristic
190#125C22×C2311yescharacteristic
191#123—21no—
192#123—21no—
193C22⋊Q8C22×C2311yescharacteristic
194C22⋊Q8C22×C2311yescharacteristic
195#121—21no—
196#121C22×C2311yescharacteristic
197#121C22×C2311yescharacteristic
198C4×Q8C22×C2311yescharacteristic
199C4×D4—21no—
200C4×D4C22×C2311yescharacteristic
201C4×D4C22×C2311yescharacteristic
202#117—21no—
203#117C22×C2311yescharacteristic
204#117C22×C2311yescharacteristic
205C2×C4⋊C4C22×C2311yescharacteristic
206C2×C4⋊C4C22×C2311yescharacteristic
207#107C22×C2311yescharacteristic
208#107C22×C2311yescharacteristic
209#107C22×C2311yescharacteristic
210#107C22×C2311yescharacteristic
211#106C22×C2311yescharacteristic
212#106C22×C2311yescharacteristic
213#106C22×C2311yescharacteristic
214#106C22×C2311yescharacteristic
215#106C22×C2311yescharacteristic
216C4⋊C8C22×C2311yescharacteristic
217#103C22×C2311yescharacteristic
218#103C22×C2311yescharacteristic
219#103C22×C2311yescharacteristic
220#102C22×C2311yescharacteristic
221#102C22×C2311yescharacteristic
222#102C22×C2311yescharacteristic
223C22⋊C8C22×C2311yescharacteristic
224C22⋊C8C22×C2311yescharacteristic
225C4×C8C22×C2311yescharacteristic
226C23#335111yescharacteristic, direct product factor, Sylow
227C2×Q8C23×C2311yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4D4×C2311yescharacteristic
231C2×D4C23×C2311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C23×C2311yescharacteristic
236C22×C4C23×C2311yescharacteristic
237SD16—22no—
238M16—22no—
239M16—22no—
240C2×C8C23×C2311yescharacteristic
241C2×C8C23×C2311yescharacteristic
242C2×C8C23×C2311yescharacteristic
243C2×C8C23×C2311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4D4×C2311yescharacteristic
254C4⋊C4C23×C2311yescharacteristic
255C4⋊C4C23×C2311yescharacteristic
256C4⋊C4C23×C2311yescharacteristic
257C4⋊C4C23×C2311yescharacteristic
258C4⋊C4C23×C2311yescharacteristic
259C4⋊C4C23×C2311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4D4×C2311yescharacteristic
268C22⋊C4D4×C2311yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42C23×C2311yescharacteristic
273C23—21no—
274C23C2×D4×C2311yescharacteristic
275C23C2×D4×C2311yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—21no—
279D4—21no—
280D4—41no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4C2×D4×C2311yescharacteristic
298C2×C4C2×D4×C2311yescharacteristic
299C2×C4C2×D4×C2311yescharacteristic
300C2×C4C2×D4×C2311yescharacteristic
301C2×C4C24×C2311yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8#41×C2312yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C22×D4×C2311yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4C2×#41×C2311yescharacteristic
328C4C23×2−1+411yescharacteristic
329C2—41no—
330C2—41no—
331C2—81no—
332C2C23×#54711yescharacteristic
333C2C23×#57511yescharacteristic
334C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#553C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#552C2semidirect
3. C23×#499C2semidirect
4. C23×#485C2semidirect
5. C23×#483C2semidirect
6. C23×#482C2semidirect
7. C23×#481C2semidirect
8. C23×#479C2semidirect
9. C23×#478C2semidirect
10. C23×#468C2semidirect
11. C23×#468C2semidirect
12. C4×SD16×C23C2semidirect
13. C23×#436C2semidirect
14. C23×#428C2semidirect
15. C23×#428C2semidirect
138. #3351C23direct (N × H)
163. #552C2×C23semidirect
164. #499C2×C23semidirect
165. #485C2×C23semidirect
166. #483C2×C23semidirect
167. #482C2×C23semidirect
168. #481C2×C23semidirect
169. #479C2×C23semidirect
170. #478C2×C23semidirect
171. #468C2×C23semidirect
172. #468C2×C23semidirect
173. C4×SD16C2×C23semidirect
174. #436C2×C23semidirect
175. #428C2×C23semidirect
176. #428C2×C23semidirect
226. C23#3351direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5532
C23×#5522
C23×#4992
C23×#4852
C23×#4832
C23×#4822
C23×#4812
C23×#4792
C23×#4782
C23×#4682
C4×SD16×C232
C23×#4362
C23×#4282
#335123