Finite Group Databasebeta

← C23×#3516 #15671135

Subgroups — C23×#3516 #15671135

Order 2944.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#557C211yesmaximal, characteristic, complemented
2C23×#556C213yesmaximal, complemented
3C23×#554C213yesmaximal, complemented
4C23×#553C213yesmaximal, complemented
5C23×#552C213yesmaximal, complemented
6D4×Q8×C23C211yesmaximal, characteristic, complemented
7C23×#548C211yesmaximal, characteristic, complemented
8C23×#548C211yesmaximal, characteristic, complemented
9C23×#542C213yesmaximal, complemented
10C23×#541C213yesmaximal, complemented
11C23×#529C213yesmaximal, complemented
12C23×#529C213yesmaximal, complemented
13C23×#520C211yesmaximal, characteristic, complemented
14C23×#519C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C23C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic
18C22×Q8×C23C2211yescharacteristic
19C23×C4⋊Q8C2213yes—
20C23×C4⋊Q8C2213yes—
21C23×#126C2213yes—
22C23×#126C2216yes—
23C23×#126C2213yes—
24C23×#126C2216yes—
25C23×#125C2216yes—
26C23×#125C2213yes—
27C23×#125C2213yes—
28C23×#125C2213yes—
29C23×#124C2213yes—
30C23×#124C2213yes—
31C23×#124C2213yes—
32C23×#123C2216yes—
33C23×#123C2211yescharacteristic
34C23×#123C2213yes—
35C23×C22⋊Q8C2213yes—
36C23×C22⋊Q8C2213yes—
37C23×C22⋊Q8C2213yes—
38C23×C22⋊Q8C2213yes—
39C23×C22⋊Q8C2213yes—
40C23×C22⋊Q8C2216yes—
41C23×#121C2213yes—
42C23×#121C2213yes—
43C23×#121C2213yes—
44C4×Q8×C23C2211yescharacteristic
45C4×Q8×C23C2211yescharacteristic
46C4×Q8×C23C2211yescharacteristic
47C4×Q8×C23C2213yes—
48C4×Q8×C23C2213yes—
49C4×Q8×C23C2213yes—
50C4×Q8×C23C2211yescharacteristic
51C4×Q8×C23C2213yes—
52C4×Q8×C23C2213yes—
53C4×D4×C23C2213yes—
54C4×D4×C23C2213yes—
55C4×D4×C23C2213yes—
56C4×D4×C23C2216yes—
57C4×D4×C23C2213yes—
58C4×D4×C23C2213yes—
59C23×#117C2213yes—
60C23×#117C2216yes—
61C23×#117C2213yes—
62C23×#117C2213yes—
63C2×C4⋊C4×C23C2213yes—
64C2×C4⋊C4×C23C2213yes—
65C2×C42×C23C2213yes—
66#41×C23—24no—
67C2×Q8×C23—24no—
68C2×Q8×C23—24no—
69C2×Q8×C23C2313yes—
70C2×Q8×C23C2313yes—
71C2×Q8×C23C2313yes—
72C2×Q8×C23C2311yescharacteristic
73C2×D4×C23C2313yes—
74C2×D4×C23C2311yescharacteristic
75C2×D4×C23C2313yes—
76C2×D4×C23C2313yes—
77C22×C4×C23C2311yescharacteristic
78C22×C4×C23C2313yes—
79C22×C4×C23C2313yes—
80C22×C4×C23C2313yes—
81C22×C4×C23C2313yes—
82C22×C4×C23C2316yes—
83C4⋊C4×C23C2313yes—
84C4⋊C4×C23C2313yes—
85C4⋊C4×C23C2313yes—
86C4⋊C4×C23C2313yes—
87C4⋊C4×C23C2316yes—
88C4⋊C4×C23C2311yescharacteristic
89C4⋊C4×C23C2313yes—
90C4⋊C4×C23C2313yes—
91C4⋊C4×C23C2313yes—
92C4⋊C4×C23C2313yes—
93C4⋊C4×C23C2316yes—
94C4⋊C4×C23C2313yes—
95C4⋊C4×C23C2313yes—
96C4⋊C4×C23C2316yes—
97C4⋊C4×C23C2313yes—
98C4⋊C4×C23C2311yescharacteristic
99C4⋊C4×C23C2313yes—
100C22⋊C4×C23C2313yes—
101C22⋊C4×C23C2311yescharacteristic
102C22⋊C4×C23C2313yes—
103C22⋊C4×C23C2313yes—
104C22⋊C4×C23C2313yes—
105C22⋊C4×C23C2316yes—
106C22⋊C4×C23C2311yescharacteristic
107C22⋊C4×C23C2313yes—
108C22⋊C4×C23C2311yescharacteristic
109C22⋊C4×C23C2316yes—
110C22⋊C4×C23C2313yes—
111C22⋊C4×C23C2313yes—
112C42×C23C2313yes—
113C42×C23C2313yes—
114C42×C23C2313yes—
115C42×C23C2313yes—
116C42×C23C2316yes—
117C42×C23C2313yes—
118C42×C23C2313yes—
119C23×C23C2411yescharacteristic
120C23×C23C2413yes—
121C23×C23C2411yescharacteristic
122Q8×C23—26no—
123Q8×C23—26no—
124Q8×C23—43no—
125Q8×C23#4114yes—
126D4×C23—26no—
127D4×C23—22no—
128D4×C23—26no—
129D4×C23—43no—
130C2×C4×C23—22no—
131C2×C4×C23—26no—
132C2×C4×C23—26no—
133C2×C4×C23—26no—
134C2×C4×C23—46no—
135C2×C4×C23—43no—
136C2×C4×C23C2411yescharacteristic
137C2×C4×C23C2411yescharacteristic
138C2×C4×C23C2413yes—
139C2×C4×C23C2411yescharacteristic
140C2×C4×C23C2413yes—
141C2×C4×C23C2411yescharacteristic
142C2×C4×C23C2413yes—
143C2×C4×C23C2413yes—
144C2×C4×C23C2413yes—
145C2×C4×C23C2413yes—
146C2×C4×C23C2411yescharacteristic
147C2×C4×C23C2413yes—
148#3516C2311yesmaximal, characteristic, direct product factor, Sylow
149C22×C23—21no—
150C22×C23—21no—
151C22×C23—21no—
152C22×C23—23no—
153C22×C23—21no—
154C22×C23—23no—
155C22×C23—23no—
156C22×C23—21no—
157C22×C232−1+412yes—
158C22×C23C2511yescharacteristic
159C4×C23—21no—
160C4×C23—23no—
161C4×C23—23no—
162C4×C23—21no—
163C4×C23—21no—
164C4×C23—23no—
165C4×C23—21no—
166C4×C23—23no—
167C4×C23—23no—
168C4×C23—23no—
169C4×C23—21no—
170C4×C23C2×#4116yes—
171#557C2×C2311yescharacteristic, complemented
172#556C2×C2313yescomplemented
173#554C2×C2313yescomplemented
174#553C2×C2313yescomplemented
175#552C2×C2313yescomplemented
176D4×Q8C2×C2311yescharacteristic, complemented
177#548C2×C2311yescharacteristic, complemented
178#548C2×C2311yescharacteristic, complemented
179#542C2×C2313yescomplemented
180#541C2×C2313yescomplemented
181#529C2×C2313yescomplemented
182#529C2×C2313yescomplemented
183#520C2×C2311yescharacteristic, complemented
184#519C2×C2311yescharacteristic, complemented
185C2×C4×Q8C2×C2311yescharacteristic, complemented
186C2×C23—22no—
187C2×C23—41no—
188C2×C23—43no—
189C2×C23C22×#4111yescharacteristic
190C2×C23C2×2−1+411yescharacteristic
191C2×C23#58511yescharacteristic
192C2×#41C22×C2311yescharacteristic
193C22×Q8C22×C2311yescharacteristic
194C22×Q8C22×C2311yescharacteristic
195C4⋊Q8C22×C2313yes—
196C4⋊Q8C22×C2313yes—
197#126C22×C2316yes—
198#126C22×C2313yes—
199#126C22×C2313yes—
200#126C22×C2316yes—
201#125C22×C2313yes—
202#125C22×C2313yes—
203#125C22×C2316yes—
204#125C22×C2313yes—
205#124C22×C2313yes—
206#124C22×C2313yes—
207#124C22×C2313yes—
208#123C22×C2311yescharacteristic
209#123C22×C2313yes—
210#123C22×C2316yes—
211C22⋊Q8C22×C2313yes—
212C22⋊Q8C22×C2313yes—
213C22⋊Q8C22×C2313yes—
214C22⋊Q8C22×C2313yes—
215C22⋊Q8C22×C2313yes—
216C22⋊Q8C22×C2316yes—
217#121C22×C2313yes—
218#121C22×C2313yes—
219#121C22×C2313yes—
220C4×Q8C22×C2311yescharacteristic
221C4×Q8C22×C2313yes—
222C4×Q8C22×C2311yescharacteristic
223C4×Q8C22×C2313yes—
224C4×Q8C22×C2313yes—
225C4×Q8C22×C2311yescharacteristic
226C4×Q8C22×C2313yes—
227C4×Q8C22×C2311yescharacteristic
228C4×Q8C22×C2313yes—
229C4×D4C22×C2313yes—
230C4×D4C22×C2313yes—
231C4×D4C22×C2313yes—
232C4×D4C22×C2313yes—
233C4×D4C22×C2313yes—
234C4×D4C22×C2316yes—
235#117C22×C2313yes—
236#117C22×C2313yes—
237#117C22×C2313yes—
238#117C22×C2316yes—
239C2×C4⋊C4C22×C2313yes—
240C2×C4⋊C4C22×C2313yes—
241C2×C42C22×C2313yes—
242C23#351611yescharacteristic, direct product factor, Sylow
243#41—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8C23×C2313yes—
247C2×Q8C23×C2313yes—
248C2×Q8C23×C2313yes—
249C2×Q8C23×C2311yescharacteristic
250C2×D4C23×C2313yes—
251C2×D4C23×C2311yescharacteristic
252C2×D4C23×C2313yes—
253C2×D4C23×C2313yes—
254C22×C4C23×C2311yescharacteristic
255C22×C4C23×C2313yes—
256C22×C4C23×C2313yes—
257C22×C4C23×C2313yes—
258C22×C4C23×C2313yes—
259C22×C4C23×C2316yes—
260C4⋊C4C23×C2313yes—
261C4⋊C4C23×C2313yes—
262C4⋊C4C23×C2313yes—
263C4⋊C4C23×C2313yes—
264C4⋊C4C23×C2316yes—
265C4⋊C4C23×C2311yescharacteristic
266C4⋊C4C23×C2313yes—
267C4⋊C4C23×C2313yes—
268C4⋊C4C23×C2313yes—
269C4⋊C4C23×C2313yes—
270C4⋊C4C23×C2316yes—
271C4⋊C4C23×C2313yes—
272C4⋊C4C23×C2313yes—
273C4⋊C4C23×C2316yes—
274C4⋊C4C23×C2313yes—
275C4⋊C4C23×C2311yescharacteristic
276C4⋊C4C23×C2313yes—
277C22⋊C4C23×C2313yes—
278C22⋊C4C23×C2311yescharacteristic
279C22⋊C4C23×C2313yes—
280C22⋊C4C23×C2313yes—
281C22⋊C4C23×C2313yes—
282C22⋊C4C23×C2316yes—
283C22⋊C4C23×C2311yescharacteristic
284C22⋊C4C23×C2313yes—
285C22⋊C4C23×C2311yescharacteristic
286C22⋊C4C23×C2316yes—
287C22⋊C4C23×C2313yes—
288C22⋊C4C23×C2313yes—
289C42C23×C2313yes—
290C42C23×C2313yes—
291C42C23×C2313yes—
292C42C23×C2313yes—
293C42C23×C2316yes—
294C42C23×C2313yes—
295C42C23×C2313yes—
296C23C24×C2311yescharacteristic
297C23C24×C2313yes—
298C23C24×C2311yescharacteristic
299Q8—26no—
300Q8—26no—
301Q8—43no—
302Q8#41×C2314yes—
303D4—26no—
304D4—22no—
305D4—26no—
306D4—43no—
307C2×C4—22no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—46no—
312C2×C4—43no—
313C2×C4C24×C2311yescharacteristic
314C2×C4C24×C2311yescharacteristic
315C2×C4C24×C2313yes—
316C2×C4C24×C2311yescharacteristic
317C2×C4C24×C2313yes—
318C2×C4C24×C2311yescharacteristic
319C2×C4C24×C2313yes—
320C2×C4C24×C2313yes—
321C2×C4C24×C2313yes—
322C2×C4C24×C2313yes—
323C2×C4C24×C2311yescharacteristic
324C2×C4C24×C2313yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22C23×2−1+412yes—
334C22C25×C2311yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—21no—
342C4—23no—
343C4—23no—
344C4—23no—
345C4—21no—
346C4C2×#41×C2316yes—
347C2—22no—
348C2—41no—
349C2—43no—
350C2C22×#41×C2311yescharacteristic
351C2C2×C23×2−1+411yescharacteristic
352C2C23×#58511yescharacteristic
subgroupG/H
center, socleC22×C23C25
commutator, Frattini, cosocleC22C25×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#557C2semidirect
2. C23×#556C2semidirect
3. C23×#554C2semidirect
4. C23×#553C2semidirect
5. C23×#552C2semidirect
6. D4×Q8×C23C2semidirect
7. C23×#548C2semidirect
8. C23×#548C2semidirect
9. C23×#542C2semidirect
10. C23×#541C2semidirect
11. C23×#529C2semidirect
12. C23×#529C2semidirect
13. C23×#520C2semidirect
14. C23×#519C2semidirect
15. C2×C4×Q8×C23C2semidirect
148. #3516C23direct (N × H)
171. #557C2×C23semidirect
172. #556C2×C23semidirect
173. #554C2×C23semidirect
174. #553C2×C23semidirect
175. #552C2×C23semidirect
176. D4×Q8C2×C23semidirect
177. #548C2×C23semidirect
178. #548C2×C23semidirect
179. #542C2×C23semidirect
180. #541C2×C23semidirect
181. #529C2×C23semidirect
182. #529C2×C23semidirect
183. #520C2×C23semidirect
184. #519C2×C23semidirect
185. C2×C4×Q8C2×C23semidirect
242. C23#3516direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5572
C23×#5562
C23×#5542
C23×#5532
C23×#5522
D4×Q8×C232
C23×#5482
C23×#5422
C23×#5412
C23×#5292
C23×#5202
C23×#5192
C2×C4×Q8×C232
#351623