Finite Group Databasebeta

← C23×#18997 #19166623

Subgroups — C23×#18997 #19166623

Order 5888.

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 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×C23×H(3,𝔽4)C211yesmaximal, characteristic
2C23×#3215C212yesmaximal, complemented
3C23×#3140C211yesmaximal, characteristic, complemented
4C23×#1659C218yesmaximal, complemented
5C23×#1533C211yesmaximal, characteristic, complemented
6C23×#1527C212yesmaximal, complemented
7C23×H(3,𝔽4)—24no—
8C23×H(3,𝔽4)—22no—
9C23×H(3,𝔽4)C2214yes—
10C23×#533C2211yescharacteristic, complemented
11C2×C23×#124—21no—
12C2×C23×#124—22no—
13C2×C23×#124C2212yes—
14C2×C23×#124C2211yescharacteristic
15C2×C23×#120—21no—
16C2×C23×#120—22no—
17C23×#497C2214yescomplemented
18C23×#489C2214yescomplemented
19C23×#489C2214yescomplemented
20C23×#488—22no—
21C23×#464—24no—
22C23×#451—22no—
23C23×#450—22no—
24C23×#432C2212yescomplemented
25C23×#419—21no—
26C23×#418—21no—
27C4×M16×C23C2211yescharacteristic, complemented
28C23×#329C2218yescomplemented
29C23×#329C2214yescomplemented
30C25×C23—21no—
31C22×Q8×C23D411yescharacteristic
32C22×Q8×C23D412yes—
33C22×D4×C23—22no—
34C22×D4×C23—21no—
35C22×D4×C23D411yescharacteristic
36C22×D4×C23D412yes—
37C2×Q16×C23—24no—
38C2×SD16×C23—24no—
39C2×M16×C23—21no—
40C23×C4⋊Q8C2312yescomplemented
41C23×#125—21no—
42C23×#124—24no—
43C23×#124—28no—
44C23×#124—24no—
45C23×#124—28no—
46C23×#124—22no—
47C23×#124—22no—
48C23×#124—22no—
49C23×#124D414yes—
50C23×#124D412yes—
51C23×#124C2312yes—
52C23×#124C2314yes—
53C23×C22⋊Q8—22no—
54C23×#120—24no—
55C23×#120—28no—
56C23×#120—24no—
57C23×#120—28no—
58C4×Q8×C23—22no—
59C23×#117—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—22no—
62C2×C22⋊C4×C23—22no—
63C2×C22⋊C4×C23—22no—
64C2×C22⋊C4×C23—22no—
65C2×C42×C23—21no—
66C2×C42×C23C2311yescharacteristic
67C23×C4⋊C8—22no—
68C23×#103—24no—
69C23×#103—24no—
70C23×#102—24no—
71C23×#102—24no—
72C23×C22⋊C8—22no—
73C23×#97C2312yescomplemented
74C23×#97C2314yescomplemented
75C4×C8×C23—21no—
76C24×C23—24no—
77C24×C23—21no—
78C24×C23—22no—
79C24×C23—84no—
80C24×C23—82no—
81C2×Q8×C23—21no—
82C2×Q8×C23—22no—
83C2×Q8×C23—22no—
84C2×Q8×C23—42no—
85C2×Q8×C23—82no—
86C2×Q8×C23C2×D412yes—
87C2×Q8×C23C2×D414yes—
88C2×D4×C23—24no—
89C2×D4×C23—28no—
90C2×D4×C23—21no—
91C2×D4×C23—22no—
92C2×D4×C23—42no—
93C2×D4×C23—82no—
94C2×D4×C23—84no—
95C2×D4×C23—82no—
96C2×D4×C23C2×D412yes—
97C2×D4×C23C2×D414yes—
98C22×C4×C23—22no—
99C22×C4×C23—21no—
100C22×C4×C23C2×D411yescharacteristic
101C22×C4×C23C2×D412yes—
102Q16×C23—84no—
103SD16×C23—84no—
104M16×C23—42no—
105C2×C8×C23—22no—
106C2×C8×C23—24no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—42no—
110C22⋊C4×C23—28no—
111C22⋊C4×C23—24no—
112C22⋊C4×C23—24no—
113C22⋊C4×C23—24no—
114C22⋊C4×C23—24no—
115C22⋊C4×C23—24no—
116C22⋊C4×C23—24no—
117C22⋊C4×C23—24no—
118C22⋊C4×C23—41no—
119C42×C23—24no—
120C42×C23—41no—
121C42×C23C2×D412yes—
122C42×C23C2×D411yescharacteristic
123C42×C23C2411yescharacteristic
124#18997C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—22no—
126C23×C23—24no—
127C23×C23—44no—
128C23×C23—44no—
129C23×C23—44no—
130C23×C23—42no—
131C23×C23—44no—
132C23×C23—84no—
133C23×C23—88no—
134C23×C23—816no—
135C23×C23#12011yescharacteristic
136Q8×C23—44no—
137Q8×C23—42no—
138Q8×C23—44no—
139Q8×C23—81no—
140Q8×C23—82no—
141D4×C23—44no—
142D4×C23—42no—
143D4×C23—81no—
144D4×C23—82no—
145D4×C23—84no—
146D4×C23—88no—
147C2×C4×C23—22no—
148C2×C4×C23—22no—
149C2×C4×C23—24no—
150C2×C4×C23—42no—
151C2×C4×C23—44no—
152C2×C4×C23—42no—
153C2×C4×C23—42no—
154C2×C4×C23#12011yescharacteristic
155C2×C4×C23#12012yes—
156C2×C4×C23C22×D411yescharacteristic
157C2×C4×C23C22×D412yes—
158C8×C23—42no—
159C8×C23—44no—
160C2×H(3,𝔽4)C2×C2311yescharacteristic
161#3215C2×C2312yescomplemented
162#3140C2×C2311yescharacteristic, complemented
163#1659C2×C2318yescomplemented
164#1533C2×C2311yescharacteristic, complemented
165#1527C2×C2312yescomplemented
166C22×C23—22no—
167C22×C23—42no—
168C22×C23—44no—
169C22×C23—44no—
170C22×C23—44no—
171C22×C23—44no—
172C22×C23—84no—
173C22×C23—88no—
174C22×C23—816no—
175C22×C23#45512yes—
176C22×C23C2×#12011yescharacteristic
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—22no—
180C4×C23—22no—
181C4×C23—42no—
182C4×C23—44no—
183C4×C23—82no—
184H(3,𝔽4)—24no—
185H(3,𝔽4)—22no—
186H(3,𝔽4)C22×C2314yes—
187#533C22×C2311yescharacteristic, complemented
188C2×#124—21no—
189C2×#124—22no—
190C2×#124C22×C2311yescharacteristic
191C2×#124C22×C2312yes—
192C2×#120—21no—
193C2×#120—22no—
194#497C22×C2314yescomplemented
195#489C22×C2314yescomplemented
196#489C22×C2314yescomplemented
197#488—22no—
198#464—24no—
199#451—22no—
200#450—22no—
201#432C22×C2312yescomplemented
202#419—21no—
203#418—21no—
204C4×M16C22×C2311yescharacteristic, complemented
205#329C22×C2318yescomplemented
206#329C22×C2314yescomplemented
207C2×C23—22no—
208C2×C23—82no—
209C2×C23—84no—
210C2×C23C2×#45511yescharacteristic
211C2×C23#301912yes—
212C25—21no—
213C22×Q8D4×C2311yescharacteristic
214C22×Q8D4×C2312yes—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4D4×C2311yescharacteristic
218C22×D4D4×C2312yes—
219C2×Q16—24no—
220C2×SD16—24no—
221C2×M16—21no—
222C4⋊Q8C23×C2312yescomplemented
223#125—21no—
224#124—24no—
225#124—28no—
226#124—24no—
227#124—28no—
228#124—22no—
229#124—22no—
230#124—22no—
231#124D4×C2314yes—
232#124D4×C2312yes—
233#124C23×C2312yes—
234#124C23×C2314yes—
235C22⋊Q8—22no—
236#120—24no—
237#120—28no—
238#120—24no—
239#120—28no—
240C4×Q8—22no—
241#117—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—21no—
248C2×C42C23×C2311yescharacteristic
249C4⋊C8—22no—
250#103—24no—
251#103—24no—
252#102—24no—
253#102—24no—
254C22⋊C8—22no—
255#97C23×C2312yescomplemented
256#97C23×C2314yescomplemented
257C4×C8—21no—
258C23#1899711yescharacteristic, direct product factor, Sylow
259C24—24no—
260C24—21no—
261C24—22no—
262C24—84no—
263C24—82no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—82no—
269C2×Q8C2×D4×C2312yes—
270C2×Q8C2×D4×C2314yes—
271C2×D4—24no—
272C2×D4—28no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—82no—
277C2×D4—84no—
278C2×D4—82no—
279C2×D4C2×D4×C2312yes—
280C2×D4C2×D4×C2314yes—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4C2×D4×C2311yescharacteristic
284C22×C4C2×D4×C2312yes—
285Q16—84no—
286SD16—84no—
287M16—42no—
288C2×C8—22no—
289C2×C8—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C22⋊C4—28no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C42—24no—
303C42—41no—
304C42C2×D4×C2311yescharacteristic
305C42C2×D4×C2312yes—
306C42C24×C2311yescharacteristic
307C23—22no—
308C23—24no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—44no—
314C23—84no—
315C23—88no—
316C23—816no—
317C23C23×#12011yescharacteristic
318Q8—44no—
319Q8—42no—
320Q8—44no—
321Q8—81no—
322Q8—82no—
323D4—44no—
324D4—42no—
325D4—81no—
326D4—82no—
327D4—84no—
328D4—88no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4C23×#12011yescharacteristic
337C2×C4C23×#12012yes—
338C2×C4C22×D4×C2311yescharacteristic
339C2×C4C22×D4×C2312yes—
340C8—42no—
341C8—44no—
342C22—22no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—88no—
350C22—816no—
351C22C23×#45512yes—
352C22C2×C23×#12011yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—44no—
359C4—82no—
360C2—22no—
361C2—82no—
362C2—84no—
363C2C2×C23×#45511yescharacteristic
364C2C23×#301912yes—
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC2×C23×H(3,𝔽4)C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3215C2semidirect
3. C23×#3140C2semidirect
4. C23×#1659C2semidirect
5. C23×#1533C2semidirect
6. C23×#1527C2semidirect
10. C23×#533C22semidirect
17. C23×#497C22semidirect
18. C23×#489C22semidirect
19. C23×#489C22semidirect
24. C23×#432C22semidirect
27. C4×M16×C23C22semidirect
28. C23×#329C22semidirect
29. C23×#329C22semidirect
40. C23×C4⋊Q8C23semidirect
73. C23×#97C23semidirect
74. C23×#97C23semidirect
124. #18997C23direct (N × H)
161. #3215C2×C23semidirect
162. #3140C2×C23semidirect
163. #1659C2×C23semidirect
164. #1533C2×C23semidirect
165. #1527C2×C23semidirect
187. #533C22×C23semidirect
194. #497C22×C23semidirect
195. #489C22×C23semidirect
196. #489C22×C23semidirect
201. #432C22×C23semidirect
204. C4×M16C22×C23semidirect
205. #329C22×C23semidirect
206. #329C22×C23semidirect
222. C4⋊Q8C23×C23semidirect
255. #97C23×C23semidirect
256. #97C23×C23semidirect
258. C23#18997direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×H(3,𝔽4)2
C23×#32152
C23×#31402
C23×#16592
C23×#15332
C23×#15272
#1899723