Finite Group Databasebeta

← C23×#8463 #19156089

Subgroups — C23×#8463 #19156089

Order 5888.

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C23×#3493C211yesmaximal, characteristic
2C23×#1820C211yesmaximal, characteristic, complemented
3C23×#1818C211yesmaximal, characteristic, complemented
4C23×#1516C212yesmaximal, complemented
5C23×#1516C212yesmaximal, complemented
6C23×#556—22no—
7C23×#555—22no—
8C23×#553—22no—
9C23×#552—22no—
10C23×#544C411yescharacteristic, complemented
11C23×#538—22no—
12C23×#537C411yescharacteristic, complemented
13C23×#529—22no—
14C23×#525C2212yes—
15C23×#520C412yescomplemented
16C2×C23×#117C2211yescharacteristic
17C2×C23×C22⋊C8C2211yescharacteristic
18C2×C23×C22⋊C8C2211yescharacteristic
19C23×#381C2212yes—
20C23×#342—22no—
21C23×#336—22no—
22C23×#328—22no—
23C23×#328—22no—
24C23×#327—22no—
25C23×#327—22no—
26C2×#41×C23C2×C412yes—
27C23×C4×C23C2311yescharacteristic
28C22×C8×C23—21no—
29C22×C8×C23—21no—
30C23×C4⋊Q8—22no—
31C23×#126—24no—
32C23×#126—24no—
33C23×#125—22no—
34C23×#125—22no—
35C23×#124—22no—
36C23×#124—24no—
37C23×#123—24no—
38C23×#123—22no—
39C23×C22⋊Q8—22no—
40C23×C22⋊Q8—22no—
41C23×C22⋊Q8D412yes—
42C23×C22⋊Q8D412yes—
43C23×#121—24no—
44C23×#121—22no—
45C23×#121—22no—
46C23×#121D412yes—
47C23×#121D412yes—
48C23×#120—22no—
49C4×Q8×C23—22no—
50C4×Q8×C23—22no—
51C4×D4×C23—22no—
52C4×D4×C23—22no—
53C4×D4×C23—24no—
54C4×D4×C23—22no—
55C4×D4×C23—22no—
56C23×#117—24no—
57C23×#117—22no—
58C23×#117—22no—
59C23×#117—22no—
60C23×#117C2×C412yes—
61C2×C4⋊C4×C23—22no—
62C2×C4⋊C4×C23C2×C411yescharacteristic
63C2×C4⋊C4×C23C2×C411yescharacteristic
64C2×C22⋊C4×C23—22no—
65C2×C22⋊C4×C23D412yes—
66C2×C42×C23D412yes—
67C23×C22⋊C8—22no—
68C23×C22⋊C8—22no—
69C23×#95—22no—
70C23×#95—22no—
71C24×C23C2×D411yescharacteristic
72#41×C23—44no—
73C2×Q8×C23—22no—
74C2×Q8×C23C22⋊C412yes—
75C2×D4×C23—22no—
76C2×D4×C23—22no—
77C2×D4×C23—22no—
78C2×D4×C23—22no—
79C2×D4×C23C22⋊C412yes—
80C22×C4×C23—24no—
81C22×C4×C23—22no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23C22⋊C412yes—
85C22×C4×C23C22⋊C412yes—
86C22×C4×C23C22×C411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D412yes—
89C22×C4×C23C2×D412yes—
90C2×C8×C23—42no—
91C2×C8×C23—42no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—24no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23C22⋊C412yes—
103C4⋊C4×C23C22⋊C412yes—
104C22⋊C4×C23—24no—
105C22⋊C4×C23—24no—
106C22⋊C4×C23—22no—
107C22⋊C4×C23—24no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—22no—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—22no—
112C22⋊C4×C23—44no—
113C42×C23—24no—
114C42×C23—22no—
115C42×C23—22no—
116C42×C23—22no—
117#8463C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—22no—
119C23×C23—22no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23C2×C22⋊C411yescharacteristic
123C23×C23#12012yes—
124Q8×C23—42no—
125Q8×C23—82no—
126D4×C23—42no—
127D4×C23—44no—
128D4×C23—82no—
129D4×C23—82no—
130D4×C23—82no—
131C2×C4×C23—22no—
132C2×C4×C23—22no—
133C2×C4×C23—22no—
134C2×C4×C23—22no—
135C2×C4×C23—22no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—42no—
139C2×C4×C23—42no—
140C2×C4×C23—44no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23—84no—
144C2×C4×C23—82no—
145C2×C4×C23—82no—
146C2×C4×C23—82no—
147C2×C4×C23C2×C22⋊C412yes—
148C2×C4×C23#12011yescharacteristic
149C2×C4×C23#12011yescharacteristic
150C8×C23—81no—
151C8×C23—81no—
152#3493C2×C2311yescharacteristic
153#1820C2×C2311yescharacteristic, complemented
154#1818C2×C2311yescharacteristic, complemented
155#1516C2×C2312yescomplemented
156#1516C2×C2312yescomplemented
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—22no—
160C22×C23—22no—
161C22×C23—22no—
162C22×C23—42no—
163C22×C23—42no—
164C22×C23—42no—
165C22×C23—82no—
166C22×C23#37911yescharacteristic
167C22×C23#41012yes—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—22no—
171C4×C23—42no—
172C4×C23—42no—
173C4×C23—42no—
174C4×C23—42no—
175C4×C23—42no—
176C4×C23—162no—
177#556—22no—
178#555—22no—
179#553—22no—
180#552—22no—
181#544C4×C2311yescharacteristic, complemented
182#538—22no—
183#537C4×C2311yescharacteristic, complemented
184#529—22no—
185#525C22×C2312yes—
186#520C4×C2312yescomplemented
187C2×#117C22×C2311yescharacteristic
188C2×C22⋊C8C22×C2311yescharacteristic
189C2×C22⋊C8C22×C2311yescharacteristic
190#381C22×C2312yes—
191#342—22no—
192#336—22no—
193#328—22no—
194#328—22no—
195#327—22no—
196#327—22no—
197C2×C23—22no—
198C2×C23—42no—
199C2×C23—82no—
200C2×C23#178211yescharacteristic
201C2×C23#179411yescharacteristic
202C2×C23#179411yescharacteristic
203C2×#41C2×C4×C2312yes—
204C23×C4C23×C2311yescharacteristic
205C22×C8—21no—
206C22×C8—21no—
207C4⋊Q8—22no—
208#126—24no—
209#126—24no—
210#125—22no—
211#125—22no—
212#124—22no—
213#124—24no—
214#123—24no—
215#123—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8D4×C2312yes—
219C22⋊Q8D4×C2312yes—
220#121—24no—
221#121—22no—
222#121—22no—
223#121D4×C2312yes—
224#121D4×C2312yes—
225#120—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—24no—
231C4×D4—22no—
232C4×D4—22no—
233#117—24no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117C2×C4×C2312yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C2×C4×C2311yescharacteristic
240C2×C4⋊C4C2×C4×C2311yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4D4×C2312yes—
243C2×C42D4×C2312yes—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#95—22no—
247#95—22no—
248C23#846311yescharacteristic, direct product factor, Sylow
249C24C2×D4×C2311yescharacteristic
250#41—44no—
251C2×Q8—22no—
252C2×Q8C22⋊C4×C2312yes—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊C4×C2312yes—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C22⋊C4×C2312yes—
263C22×C4C22⋊C4×C2312yes—
264C22×C4C22×C4×C2311yescharacteristic
265C22×C4C2×D4×C2311yescharacteristic
266C22×C4C2×D4×C2312yes—
267C22×C4C2×D4×C2312yes—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4C22⋊C4×C2312yes—
281C4⋊C4C22⋊C4×C2312yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—44no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C2311yescharacteristic
300C23C23×#12012yes—
301Q8—42no—
302Q8—82no—
303D4—42no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C2×C22⋊C4×C2312yes—
325C2×C4C23×#12011yescharacteristic
326C2×C4C23×#12011yescharacteristic
327C8—81no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22C23×#37911yescharacteristic
339C22C23×#41012yes—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—162no—
349C2—22no—
350C2—42no—
351C2—82no—
352C2C23×#178211yescharacteristic
353C2C23×#179411yescharacteristic
354C2C23×#179411yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3493C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1820C2semidirect
3. C23×#1818C2semidirect
4. C23×#1516C2semidirect
5. C23×#1516C2semidirect
10. C23×#544C4semidirect
12. C23×#537C4semidirect
15. C23×#520C4semidirect
117. #8463C23direct (N × H)
153. #1820C2×C23semidirect
154. #1818C2×C23semidirect
155. #1516C2×C23semidirect
156. #1516C2×C23semidirect
181. #544C4×C23semidirect
183. #537C4×C23semidirect
186. #520C4×C23semidirect
248. C23#8463direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34932
C23×#18202
C23×#18182
C23×#15162
#846323