Finite Group Databasebeta

← C23×#34973 #19182599

Subgroups — C23×#34973 #19182599

Order 5888.

1716 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2524C214yesmaximal, complemented
2C23×#2519C214yesmaximal
3C23×#2469C214yesmaximal
4C23×#2462C214yesmaximal, complemented
5C23×#2402C211yesmaximal, characteristic, complemented
6C23×#2399C211yesmaximal, characteristic, complemented
7C23×#2396C212yesmaximal
8C4×C23×#125C214yesmaximal, complemented
9C4×C23×#126C214yesmaximal
10C23×#2298C211yesmaximal, characteristic
11C42×Q8×C23C212yesmaximal, complemented
12C2×C23×#126C2214yes—
13C2×C23×#125C2214yes—
14C2×C4×Q8×C23C2212yes—
15C2×C4×Q8×C23C2214yes—
16C2×C23×#117C2211yescharacteristic
17C23×#400C2214yes—
18C23×#400C2212yes—
19C23×#399C2212yes—
20C23×#398C2214yes—
21C23×#398C2214yes—
22C23×#398C2214yes—
23C23×#396C2214yes—
24C23×#395C2214yes—
25C23×#395C2212yes—
26C23×#393C2212yes—
27C23×#389C2214yes—
28C23×#389C2214yes—
29C23×#388C2214yes—
30C23×#388C2214yes—
31C23×#388C2214yes—
32C23×#388C2214yes—
33C23×#387C2214yes—
34C23×#387C2214yes—
35C23×#387C2214yes—
36C23×#387C2214yes—
37C23×#387C2214yes—
38C23×#387C2214yes—
39C23×#387C2214yes—
40C23×#387C2214yes—
41C23×#383C2214yes—
42C23×#382C2214yes—
43C4×C4⋊C4×C23C2214yes—
44C4×C4⋊C4×C23C2212yes—
45C4×C4⋊C4×C23C2214yes—
46C4×C4⋊C4×C23C2214yes—
47C4×C4⋊C4×C23C2214yes—
48C4×C4⋊C4×C23C2214yes—
49C4×C4⋊C4×C23C2214yes—
50C4×C4⋊C4×C23C2212yes—
51C4×C4⋊C4×C23C2214yes—
52C4×C22⋊C4×C23C2212yes—
53C4×C22⋊C4×C23C2214yes—
54C23×#376C2211yescharacteristic
55C43×C23C2211yescharacteristic
56C43×C23C2214yes—
57C22×Q8×C23C2312yes—
58C23×C4×C23C2311yescharacteristic
59C23×#126—216no—
60C23×#125—216no—
61C4×Q8×C23—216no—
62C4×Q8×C23—28no—
63C23×#117—24no—
64C2×C4⋊C4×C23C2314yes—
65C2×C4⋊C4×C23C2314yes—
66C2×C4⋊C4×C23C2314yes—
67C2×C4⋊C4×C23C2312yes—
68C2×C4⋊C4×C23C2314yes—
69C2×C4⋊C4×C23C2314yes—
70C2×C4⋊C4×C23C2314yes—
71C2×C4⋊C4×C23C2314yes—
72C2×C4⋊C4×C23C2314yes—
73C2×C4⋊C4×C23C2314yes—
74C2×C4⋊C4×C23C2314yes—
75C2×C4⋊C4×C23C2312yes—
76C2×C4⋊C4×C23C2314yes—
77C2×C22⋊C4×C23C2314yes—
78C2×C22⋊C4×C23C2314yes—
79C2×C22⋊C4×C23C2314yes—
80C2×C22⋊C4×C23C2312yes—
81C2×C42×C23C2314yes—
82C2×C42×C23C2314yes—
83C2×C42×C23C2314yes—
84C2×C42×C23C2314yes—
85C2×C42×C23C2314yes—
86C2×C42×C23C2312yes—
87C2×C42×C23C2314yes—
88C2×C42×C23C2314yes—
89C2×C42×C23C2312yes—
90C2×C42×C23C2311yescharacteristic
91C2×C42×C23C2311yescharacteristic
92C23×#95C2314yes—
93C23×#95C2314yes—
94C23×#95C2314yes—
95C23×#95C2314yes—
96C23×#95C2314yes—
97C23×#95C2314yes—
98C23×#95C2312yes—
99C23×#95C2312yes—
100C23×#95C2314yes—
101C23×#95C2314yes—
102C23×#95C2314yes—
103C23×#95C2314yes—
104C23×#95C2314yes—
105C23×#95C2314yes—
106C23×#95C2314yes—
107C24×C23C2411yescharacteristic
108C2×Q8×C23—28no—
109C2×Q8×C23—24no—
110C22×C4×C23—22no—
111C22×C4×C23—24no—
112C22×C4×C23C2414yes—
113C22×C4×C23C2414yes—
114C22×C4×C23C2412yes—
115C22×C4×C23C2414yes—
116C22×C4×C23C2411yescharacteristic
117C22×C4×C23C2414yes—
118C22×C4×C23C2414yes—
119C22×C4×C23C2414yes—
120C22×C4×C23C2412yes—
121C22×C4×C23C2411yescharacteristic
122C4⋊C4×C23—24no—
123C4⋊C4×C23—28no—
124C4⋊C4×C23—28no—
125C4⋊C4×C23—28no—
126C4⋊C4×C23—24no—
127C4⋊C4×C23—28no—
128C4⋊C4×C23—28no—
129C4⋊C4×C23—28no—
130C4⋊C4×C23—28no—
131C4⋊C4×C23—28no—
132C4⋊C4×C23—28no—
133C4⋊C4×C23—28no—
134C4⋊C4×C23#41116yes—
135C22⋊C4×C23—24no—
136C22⋊C4×C23—28no—
137C22⋊C4×C23—28no—
138C22⋊C4×C23—28no—
139C42×C23—22no—
140C42×C23—28no—
141C42×C23—28no—
142C42×C23—28no—
143C42×C23—28no—
144C42×C23—28no—
145C42×C23—28no—
146C42×C23—24no—
147C42×C23—28no—
148C42×C23—24no—
149C42×C23#4114yes—
150#34973C2311yesmaximal, characteristic, direct product factor, Sylow
151C23×C23—24no—
152C23×C23—21no—
153C23×C23—22no—
154C23×C23C2511yescharacteristic
155Q8×C23—216no—
156C2×C4×C23—24no—
157C2×C4×C23—24no—
158C2×C4×C23—24no—
159C2×C4×C23—24no—
160C2×C4×C23—24no—
161C2×C4×C23—24no—
162C2×C4×C23—24no—
163C2×C4×C23—22no—
164C2×C4×C23—24no—
165C2×C4×C23—24no—
166C2×C4×C23—24no—
167C2×C4×C23—24no—
168C2×C4×C23—24no—
169C2×C4×C23—24no—
170C2×C4×C23—24no—
171C2×C4×C23—24no—
172C2×C4×C23—21no—
173C2×C4×C23—24no—
174C2×C4×C23—24no—
175C2×C4×C23—22no—
176C2×C4×C23—44no—
177C2×C4×C23C2×#4118yes—
178C2×C4×C23C2×#4118yes—
179C2×C4×C23C2×#4118yes—
180C2×C4×C23C2×#4114yes—
181C2×C4×C23C2×#4112yes—
182C2×C4×C232−1+414yes—
183#2524C2×C2314yescomplemented
184#2519C2×C2314yes—
185#2469C2×C2314yes—
186#2462C2×C2314yescomplemented
187#2402C2×C2311yescharacteristic, complemented
188#2399C2×C2311yescharacteristic, complemented
189#2396C2×C2312yes—
190C4×#125C2×C2314yescomplemented
191C4×#126C2×C2314yes—
192#2298C2×C2311yescharacteristic
193C42×Q8C2×C2312yescomplemented
194C22×C23—42no—
195C22×C23—41no—
196C22×C23—44no—
197C22×C23C22×#4111yescharacteristic
198C22×C23C22×#4114yes—
199C22×C23C2×2−1+412yes—
200C4×C23—28no—
201C4×C23—24no—
202C4×C23—28no—
203C4×C23—28no—
204C4×C23—44no—
205C4×C23—42no—
206C4×C23—41no—
207C4×C23—44no—
208C4×C23—44no—
209C4×C23#54114yes—
210C2×#126C22×C2314yes—
211C2×#125C22×C2314yes—
212C2×C4×Q8C22×C2312yes—
213C2×C4×Q8C22×C2314yes—
214C2×#117C22×C2311yescharacteristic
215#400C22×C2312yes—
216#400C22×C2314yes—
217#399C22×C2312yes—
218#398C22×C2314yes—
219#398C22×C2314yes—
220#398C22×C2314yes—
221#396C22×C2314yes—
222#395C22×C2312yes—
223#395C22×C2314yes—
224#393C22×C2312yes—
225#389C22×C2314yes—
226#389C22×C2314yes—
227#388C22×C2314yes—
228#388C22×C2314yes—
229#388C22×C2314yes—
230#388C22×C2314yes—
231#387C22×C2314yes—
232#387C22×C2314yes—
233#387C22×C2314yes—
234#387C22×C2314yes—
235#387C22×C2314yes—
236#387C22×C2314yes—
237#387C22×C2314yes—
238#387C22×C2314yes—
239#383C22×C2314yes—
240#382C22×C2314yes—
241C4×C4⋊C4C22×C2314yes—
242C4×C4⋊C4C22×C2314yes—
243C4×C4⋊C4C22×C2314yes—
244C4×C4⋊C4C22×C2314yes—
245C4×C4⋊C4C22×C2312yes—
246C4×C4⋊C4C22×C2312yes—
247C4×C4⋊C4C22×C2314yes—
248C4×C4⋊C4C22×C2314yes—
249C4×C4⋊C4C22×C2314yes—
250C4×C22⋊C4C22×C2314yes—
251C4×C22⋊C4C22×C2312yes—
252#376C22×C2311yescharacteristic
253C43C22×C2314yes—
254C43C22×C2311yescharacteristic
255C2×C23—81no—
256C2×C23#344412yes—
257C2×C23C2×#54111yescharacteristic
258C2×C23#348214yes—
259C22×Q8C23×C2312yes—
260C23×C4C23×C2311yescharacteristic
261#126—216no—
262#125—216no—
263C4×Q8—216no—
264C4×Q8—28no—
265#117—24no—
266C2×C4⋊C4C23×C2312yes—
267C2×C4⋊C4C23×C2314yes—
268C2×C4⋊C4C23×C2314yes—
269C2×C4⋊C4C23×C2314yes—
270C2×C4⋊C4C23×C2312yes—
271C2×C4⋊C4C23×C2314yes—
272C2×C4⋊C4C23×C2314yes—
273C2×C4⋊C4C23×C2314yes—
274C2×C4⋊C4C23×C2314yes—
275C2×C4⋊C4C23×C2314yes—
276C2×C4⋊C4C23×C2314yes—
277C2×C4⋊C4C23×C2314yes—
278C2×C4⋊C4C23×C2314yes—
279C2×C22⋊C4C23×C2312yes—
280C2×C22⋊C4C23×C2314yes—
281C2×C22⋊C4C23×C2314yes—
282C2×C22⋊C4C23×C2314yes—
283C2×C42C23×C2311yescharacteristic
284C2×C42C23×C2314yes—
285C2×C42C23×C2314yes—
286C2×C42C23×C2314yes—
287C2×C42C23×C2314yes—
288C2×C42C23×C2314yes—
289C2×C42C23×C2314yes—
290C2×C42C23×C2312yes—
291C2×C42C23×C2314yes—
292C2×C42C23×C2312yes—
293C2×C42C23×C2311yescharacteristic
294#95C23×C2314yes—
295#95C23×C2314yes—
296#95C23×C2314yes—
297#95C23×C2314yes—
298#95C23×C2314yes—
299#95C23×C2314yes—
300#95C23×C2312yes—
301#95C23×C2312yes—
302#95C23×C2314yes—
303#95C23×C2314yes—
304#95C23×C2314yes—
305#95C23×C2314yes—
306#95C23×C2314yes—
307#95C23×C2314yes—
308#95C23×C2314yes—
309C23#3497311yescharacteristic, direct product factor, Sylow
310C24C24×C2311yescharacteristic
311C2×Q8—28no—
312C2×Q8—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4C24×C2311yescharacteristic
316C22×C4C24×C2314yes—
317C22×C4C24×C2314yes—
318C22×C4C24×C2314yes—
319C22×C4C24×C2312yes—
320C22×C4C24×C2314yes—
321C22×C4C24×C2314yes—
322C22×C4C24×C2314yes—
323C22×C4C24×C2312yes—
324C22×C4C24×C2311yescharacteristic
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—24no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—28no—
337C4⋊C4#41×C23116yes—
338C22⋊C4—24no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C42—22no—
343C42—28no—
344C42—28no—
345C42—28no—
346C42—28no—
347C42—28no—
348C42—28no—
349C42—24no—
350C42—28no—
351C42—24no—
352C42#41×C2314yes—
353C23—24no—
354C23—21no—
355C23—22no—
356C23C25×C2311yescharacteristic
357Q8—216no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—44no—
379C2×C4C2×#41×C2318yes—
380C2×C4C2×#41×C2318yes—
381C2×C4C2×#41×C2318yes—
382C2×C4C2×#41×C2314yes—
383C2×C4C2×#41×C2312yes—
384C2×C4C23×2−1+414yes—
385C22—42no—
386C22—41no—
387C22—44no—
388C22C22×#41×C2311yescharacteristic
389C22C22×#41×C2314yes—
390C22C2×C23×2−1+412yes—
391C4—28no—
392C4—24no—
393C4—28no—
394C4—28no—
395C4—44no—
396C4—42no—
397C4—41no—
398C4—44no—
399C4—44no—
400C4C23×#54114yes—
401C2—81no—
402C2C23×#344412yes—
403C2C2×C23×#54111yescharacteristic
404C2C23×#348214yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC24×C23C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2524C2semidirect
4. C23×#2462C2semidirect
5. C23×#2402C2semidirect
6. C23×#2399C2semidirect
8. C4×C23×#125C2semidirect
11. C42×Q8×C23C2semidirect
150. #34973C23direct (N × H)
183. #2524C2×C23semidirect
186. #2462C2×C23semidirect
187. #2402C2×C23semidirect
188. #2399C2×C23semidirect
190. C4×#125C2×C23semidirect
193. C42×Q8C2×C23semidirect
309. C23#34973direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25242
C23×#25192
C23×#24692
C23×#24622
C23×#24022
C23×#23992
C23×#23962
C4×C23×#1252
C4×C23×#1262
C23×#22982
C42×Q8×C232
#3497323