Finite Group Databasebeta

← C23×#3118 #15670737

Subgroups — C23×#3118 #15670737

Order 2944.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#542C211yesmaximal, characteristic, complemented
2C23×#518C211yesmaximal, characteristic, complemented
3C23×#485C211yesmaximal, characteristic, complemented
4C23×#482C211yesmaximal, characteristic, complemented
5C23×#476C211yesmaximal, characteristic, complemented
6C23×#474C211yesmaximal, characteristic, complemented
7C23×#463C211yesmaximal, characteristic, complemented
8C23×#463C211yesmaximal, characteristic, complemented
9C23×#450C211yesmaximal, characteristic, complemented
10C23×#447C211yesmaximal, characteristic
11C23×#442C211yesmaximal, characteristic, complemented
12C23×#440C211yesmaximal, characteristic, complemented
13C4×SD16×C23C211yesmaximal, characteristic, complemented
14C4×D8×C23C211yesmaximal, characteristic, complemented
15C23×#433C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C2×SD16×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C2×D8×C23C2211yescharacteristic
20C2×D8×C23C2211yescharacteristic
21C23×C4⋊Q8C2211yescharacteristic, complemented
22C23×#126—21no—
23C23×#124—21no—
24C23×#124C2211yescharacteristic, complemented
25C23×#123—21no—
26C23×C22⋊Q8—21no—
27C23×C22⋊Q8C2211yescharacteristic
28C23×#121C2211yescharacteristic
29C4×Q8×C23C2211yescharacteristic, complemented
30C4×D4×C23—21no—
31C4×D4×C23—21no—
32C4×D4×C23C2211yescharacteristic
33C4×D4×C23C2211yescharacteristic
34C4×D4×C23C2211yescharacteristic
35C4×D4×C23C2211yescharacteristic
36C4×D4×C23C2211yescharacteristic, complemented
37C23×#117C2211yescharacteristic
38C23×#117C2211yescharacteristic
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C23×#107C2211yescharacteristic, complemented
42C23×#107C2211yescharacteristic, complemented
43C23×#106C2211yescharacteristic, complemented
44C23×#106C2211yescharacteristic, complemented
45C23×C4⋊C8C2211yescharacteristic, complemented
46C23×C4⋊C8C2211yescharacteristic, complemented
47C23×#103C2211yescharacteristic, complemented
48C23×#103C2211yescharacteristic, complemented
49C23×#102C2211yescharacteristic
50C23×#102C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic, complemented
54C23×#102C2211yescharacteristic, complemented
55C23×C22⋊C8C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×#97C2211yescharacteristic, complemented
58C4×C8×C23C2211yescharacteristic, complemented
59C24×C23—21no—
60C2×Q8×C23—21no—
61C2×Q8×C23C2311yescharacteristic
62C2×D4×C23—22no—
63C2×D4×C23—22no—
64C2×D4×C23—21no—
65C2×D4×C23—21no—
66C2×D4×C23C2311yescharacteristic
67C2×D4×C23C2311yescharacteristic
68C2×D4×C23C2311yescharacteristic
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23C2311yescharacteristic
75SD16×C23—22no—
76SD16×C23—22no—
77D8×C23—22no—
78D8×C23—22no—
79C2×C8×C23C2311yescharacteristic
80C2×C8×C23C2311yescharacteristic
81C2×C8×C23C2311yescharacteristic
82C2×C8×C23C2311yescharacteristic
83C4⋊C4×C23—21no—
84C4⋊C4×C23—21no—
85C4⋊C4×C23—21no—
86C4⋊C4×C23D411yescharacteristic
87C4⋊C4×C23D411yescharacteristic
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×C23—21no—
100C22⋊C4×C23—21no—
101C22⋊C4×C23—21no—
102C22⋊C4×C23D411yescharacteristic
103C22⋊C4×C23D411yescharacteristic
104C42×C23—21no—
105C42×C23—21no—
106C42×C23C2311yescharacteristic
107C42×C23C2311yescharacteristic
108C23×C23—24no—
109C23×C23—21no—
110C23×C23—21no—
111C23×C23—21no—
112C23×C23—81no—
113C23×C23—81no—
114C23×C23C2×D411yescharacteristic
115Q8×C23—21no—
116Q8×C23—21no—
117Q8×C23—81no—
118D4×C23—24no—
119D4×C23—21no—
120D4×C23—21no—
121D4×C23—21no—
122D4×C23—21no—
123D4×C23—81no—
124D4×C23#4112yes—
125D4×C23#4112yes—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—81no—
134C2×C4×C23—81no—
135C2×C4×C23—81no—
136C2×C4×C23—81no—
137C2×C4×C23—81no—
138C2×C4×C23C2×D411yescharacteristic
139C2×C4×C23C2×D411yescharacteristic
140C2×C4×C23C2×D411yescharacteristic
141C2×C4×C23C2×D411yescharacteristic
142C2×C4×C23C2×D411yescharacteristic
143C2×C4×C23C2411yescharacteristic
144C8×C23—21no—
145C8×C23—21no—
146C8×C23—21no—
147C8×C23—21no—
148#3118C2311yesmaximal, characteristic, direct product factor, Sylow
149C22×C23—21no—
150C22×C23—22no—
151C22×C23—22no—
152C22×C23—21no—
153C22×C23—21no—
154C22×C23—41no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23—41no—
161C22×C23—84no—
162C22×C23C22×D411yescharacteristic
163C4×C23—21no—
164C4×C23—21no—
165C4×C23—21no—
166C4×C23—21no—
167C4×C23—21no—
168C4×C23—41no—
169C4×C23—41no—
170C4×C23—41no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23C2×#4111yescharacteristic
174C4×C23C2×#4111yescharacteristic
175#542C2×C2311yescharacteristic, complemented
176#518C2×C2311yescharacteristic, complemented
177#485C2×C2311yescharacteristic, complemented
178#482C2×C2311yescharacteristic, complemented
179#476C2×C2311yescharacteristic, complemented
180#474C2×C2311yescharacteristic, complemented
181#463C2×C2311yescharacteristic, complemented
182#463C2×C2311yescharacteristic, complemented
183#450C2×C2311yescharacteristic, complemented
184#447C2×C2311yescharacteristic
185#442C2×C2311yescharacteristic, complemented
186#440C2×C2311yescharacteristic, complemented
187C4×SD16C2×C2311yescharacteristic, complemented
188C4×D8C2×C2311yescharacteristic, complemented
189#433C2×C2311yescharacteristic, complemented
190C2×C23—41no—
191C2×C23—42no—
192C2×C23—42no—
193C2×C23—81no—
194C2×C23#52511yescharacteristic
195C2×C23#57611yescharacteristic
196C2×C23#57711yescharacteristic
197C22×D4C22×C2311yescharacteristic
198C2×SD16C22×C2311yescharacteristic
199C2×SD16C22×C2311yescharacteristic
200C2×D8C22×C2311yescharacteristic
201C2×D8C22×C2311yescharacteristic
202C4⋊Q8C22×C2311yescharacteristic, complemented
203#126—21no—
204#124—21no—
205#124C22×C2311yescharacteristic, complemented
206#123—21no—
207C22⋊Q8—21no—
208C22⋊Q8C22×C2311yescharacteristic
209#121C22×C2311yescharacteristic
210C4×Q8C22×C2311yescharacteristic, complemented
211C4×D4—21no—
212C4×D4—21no—
213C4×D4C22×C2311yescharacteristic
214C4×D4C22×C2311yescharacteristic
215C4×D4C22×C2311yescharacteristic
216C4×D4C22×C2311yescharacteristic
217C4×D4C22×C2311yescharacteristic, complemented
218#117C22×C2311yescharacteristic
219#117C22×C2311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222#107C22×C2311yescharacteristic, complemented
223#107C22×C2311yescharacteristic, complemented
224#106C22×C2311yescharacteristic, complemented
225#106C22×C2311yescharacteristic, complemented
226C4⋊C8C22×C2311yescharacteristic, complemented
227C4⋊C8C22×C2311yescharacteristic, complemented
228#103C22×C2311yescharacteristic, complemented
229#103C22×C2311yescharacteristic, complemented
230#102C22×C2311yescharacteristic
231#102C22×C2311yescharacteristic
232#102C22×C2311yescharacteristic
233#102C22×C2311yescharacteristic
234#102C22×C2311yescharacteristic, complemented
235#102C22×C2311yescharacteristic, complemented
236C22⋊C8C22×C2311yescharacteristic
237C22⋊C8C22×C2311yescharacteristic
238#97C22×C2311yescharacteristic, complemented
239C4×C8C22×C2311yescharacteristic, complemented
240C23#311811yescharacteristic, direct product factor, Sylow
241C24—21no—
242C2×Q8—21no—
243C2×Q8C23×C2311yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C23×C2311yescharacteristic
249C2×D4C23×C2311yescharacteristic
250C2×D4C23×C2311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4C23×C2311yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—22no—
260D8—22no—
261C2×C8C23×C2311yescharacteristic
262C2×C8C23×C2311yescharacteristic
263C2×C8C23×C2311yescharacteristic
264C2×C8C23×C2311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4D4×C2311yescharacteristic
269C4⋊C4D4×C2311yescharacteristic
270C4⋊C4C23×C2311yescharacteristic
271C4⋊C4C23×C2311yescharacteristic
272C4⋊C4C23×C2311yescharacteristic
273C4⋊C4C23×C2311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4D4×C2311yescharacteristic
285C22⋊C4D4×C2311yescharacteristic
286C42—21no—
287C42—21no—
288C42C23×C2311yescharacteristic
289C42C23×C2311yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—81no—
295C23—81no—
296C23C2×D4×C2311yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—81no—
306D4#41×C2312yes—
307D4#41×C2312yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4C2×D4×C2311yescharacteristic
321C2×C4C2×D4×C2311yescharacteristic
322C2×C4C2×D4×C2311yescharacteristic
323C2×C4C2×D4×C2311yescharacteristic
324C2×C4C2×D4×C2311yescharacteristic
325C2×C4C24×C2311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—84no—
343C22C22×D4×C2311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4C2×#41×C2311yescharacteristic
355C4C2×#41×C2311yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—81no—
360C2C23×#52511yescharacteristic
361C2C23×#57611yescharacteristic
362C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#447C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#542C2semidirect
2. C23×#518C2semidirect
3. C23×#485C2semidirect
4. C23×#482C2semidirect
5. C23×#476C2semidirect
6. C23×#474C2semidirect
7. C23×#463C2semidirect
8. C23×#463C2semidirect
9. C23×#450C2semidirect
11. C23×#442C2semidirect
12. C23×#440C2semidirect
13. C4×SD16×C23C2semidirect
14. C4×D8×C23C2semidirect
15. C23×#433C2semidirect
21. C23×C4⋊Q8C22semidirect
24. C23×#124C22semidirect
29. C4×Q8×C23C22semidirect
36. C4×D4×C23C22semidirect
41. C23×#107C22semidirect
42. C23×#107C22semidirect
43. C23×#106C22semidirect
44. C23×#106C22semidirect
45. C23×C4⋊C8C22semidirect
46. C23×C4⋊C8C22semidirect
47. C23×#103C22semidirect
48. C23×#103C22semidirect
53. C23×#102C22semidirect
54. C23×#102C22semidirect
57. C23×#97C22semidirect
58. C4×C8×C23C22semidirect
148. #3118C23direct (N × H)
175. #542C2×C23semidirect
176. #518C2×C23semidirect
177. #485C2×C23semidirect
178. #482C2×C23semidirect
179. #476C2×C23semidirect
180. #474C2×C23semidirect
181. #463C2×C23semidirect
182. #463C2×C23semidirect
183. #450C2×C23semidirect
185. #442C2×C23semidirect
186. #440C2×C23semidirect
187. C4×SD16C2×C23semidirect
188. C4×D8C2×C23semidirect
189. #433C2×C23semidirect
202. C4⋊Q8C22×C23semidirect
205. #124C22×C23semidirect
210. C4×Q8C22×C23semidirect
217. C4×D4C22×C23semidirect
222. #107C22×C23semidirect
223. #107C22×C23semidirect
224. #106C22×C23semidirect
225. #106C22×C23semidirect
226. C4⋊C8C22×C23semidirect
227. C4⋊C8C22×C23semidirect
228. #103C22×C23semidirect
229. #103C22×C23semidirect
234. #102C22×C23semidirect
235. #102C22×C23semidirect
238. #97C22×C23semidirect
239. C4×C8C22×C23semidirect
240. C23#3118direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5422
C23×#5182
C23×#4852
C23×#4822
C23×#4762
C23×#4742
C23×#4632
C23×#4502
C23×#4472
C23×#4422
C23×#4402
C4×SD16×C232
C4×D8×C232
C23×#4332
#311823