Finite Group Databasebeta

← C23×#2781 #15670400

Subgroups — C23×#2781 #15670400

Order 2944.

434 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#400C211yesmaximal, characteristic, complemented
2C23×#398C211yesmaximal, characteristic, complemented
3C23×#397C211yesmaximal, characteristic, complemented
4C23×#396C211yesmaximal, characteristic, complemented
5C23×#394C211yesmaximal, characteristic
6C23×#394C211yesmaximal, characteristic
7C23×#388C211yesmaximal, characteristic, complemented
8C23×#388C211yesmaximal, characteristic, complemented
9C23×#388C211yesmaximal, characteristic, complemented
10C23×#388C211yesmaximal, characteristic, complemented
11C23×#387C211yesmaximal, characteristic, complemented
12C23×#386C211yesmaximal, characteristic
13C23×#385C211yesmaximal, characteristic, complemented
14C23×#382C211yesmaximal, characteristic, complemented
15C23×#380C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C23×C4×C23C2211yescharacteristic
18C2×C4⋊C4×C23C2211yescharacteristic
19C2×C4⋊C4×C23C2211yescharacteristic
20C2×C4⋊C4×C23C2211yescharacteristic
21C2×C4⋊C4×C23C2211yescharacteristic
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C4⋊C4×C23C2211yescharacteristic
24C2×C4⋊C4×C23C2211yescharacteristic
25C2×C4⋊C4×C23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic
27C2×C22⋊C4×C23C2211yescharacteristic
28C2×C22⋊C4×C23C2211yescharacteristic
29C2×C22⋊C4×C23C2211yescharacteristic
30C2×C22⋊C4×C23C2211yescharacteristic
31C2×C22⋊C4×C23C2211yescharacteristic
32C2×C22⋊C4×C23C2211yescharacteristic
33C2×C22⋊C4×C23C2211yescharacteristic
34C2×C22⋊C4×C23C2211yescharacteristic
35C2×C22⋊C4×C23C2211yescharacteristic
36C2×C22⋊C4×C23C2211yescharacteristic
37C2×C42×C23C2211yescharacteristic
38C2×C42×C23C2211yescharacteristic
39C23×#95C2211yescharacteristic
40C23×#95C2211yescharacteristic
41C23×#95C2211yescharacteristic
42C23×#95C2211yescharacteristic
43C23×#95C2211yescharacteristic
44C23×#95C2211yescharacteristic
45C23×#95C2211yescharacteristic
46C23×#95C2211yescharacteristic
47C23×#95C2211yescharacteristic
48C23×#95C2211yescharacteristic
49C23×#95C2211yescharacteristic
50C23×#95C2211yescharacteristic
51C24×C23C2311yescharacteristic
52C24×C23C2311yescharacteristic
53C2×D4×C23—22no—
54C2×D4×C23—41no—
55C2×D4×C23—41no—
56C22×C4×C23—22no—
57C22×C4×C23—41no—
58C22×C4×C23—41no—
59C22×C4×C23C2311yescharacteristic
60C22×C4×C23C2311yescharacteristic
61C22×C4×C23C2311yescharacteristic
62C22×C4×C23C2311yescharacteristic
63C22×C4×C23C2311yescharacteristic
64C22×C4×C23C2311yescharacteristic
65C22×C4×C23C2311yescharacteristic
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C22×C4×C23C2311yescharacteristic
70C22×C4×C23C2311yescharacteristic
71C22×C4×C23C2311yescharacteristic
72C4⋊C4×C23—22no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—41no—
80C22⋊C4×C23—22no—
81C22⋊C4×C23—22no—
82C22⋊C4×C23—22no—
83C22⋊C4×C23—22no—
84C22⋊C4×C23—22no—
85C22⋊C4×C23—41no—
86C22⋊C4×C23—41no—
87C22⋊C4×C23—41no—
88C22⋊C4×C23—41no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C42×C23—22no—
92C42×C23—41no—
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—21no—
97C23×C23—21no—
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—21no—
105C23×C23—21no—
106C23×C23#4112yes—
107C23×C23C2411yescharacteristic
108D4×C23—44no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
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—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—44no—
147C2×C4×C23#4112yes—
148C2×C4×C23#4112yes—
149#2781C2311yesmaximal, characteristic, direct product factor, Sylow
150C22×C23—22no—
151C22×C23—22no—
152C22×C23—22no—
153C22×C23—41no—
154C22×C23—41no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23C2×#4111yescharacteristic
165C22×C23C2×#4111yescharacteristic
166C22×C23C2×#4111yescharacteristic
167C22×C23H(5,𝔽2)11yescharacteristic
168C22×C23H(5,𝔽2)11yescharacteristic
169C22×C23H(5,𝔽2)11yescharacteristic
170C22×C232−1+411yescharacteristic
171C4×C23—22no—
172C4×C23—22no—
173C4×C23—41no—
174C4×C23—41no—
175C4×C23—41no—
176C4×C23—41no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—41no—
184#400C2×C2311yescharacteristic, complemented
185#398C2×C2311yescharacteristic, complemented
186#397C2×C2311yescharacteristic, complemented
187#396C2×C2311yescharacteristic, complemented
188#394C2×C2311yescharacteristic
189#394C2×C2311yescharacteristic
190#388C2×C2311yescharacteristic, complemented
191#388C2×C2311yescharacteristic, complemented
192#388C2×C2311yescharacteristic, complemented
193#388C2×C2311yescharacteristic, complemented
194#387C2×C2311yescharacteristic, complemented
195#386C2×C2311yescharacteristic
196#385C2×C2311yescharacteristic, complemented
197#382C2×C2311yescharacteristic, complemented
198#380C2×C2311yescharacteristic, complemented
199C2×C23—42no—
200C2×C23—81no—
201C2×C23#53811yescharacteristic
202C2×C23#53811yescharacteristic
203C2×C23#53911yescharacteristic
204C2×C23#55211yescharacteristic
205C2×C23#55311yescharacteristic
206C2×C23#55511yescharacteristic
207C2×C23#56211yescharacteristic
208C22×D4C22×C2311yescharacteristic
209C23×C4C22×C2311yescharacteristic
210C2×C4⋊C4C22×C2311yescharacteristic
211C2×C4⋊C4C22×C2311yescharacteristic
212C2×C4⋊C4C22×C2311yescharacteristic
213C2×C4⋊C4C22×C2311yescharacteristic
214C2×C4⋊C4C22×C2311yescharacteristic
215C2×C4⋊C4C22×C2311yescharacteristic
216C2×C4⋊C4C22×C2311yescharacteristic
217C2×C4⋊C4C22×C2311yescharacteristic
218C2×C22⋊C4C22×C2311yescharacteristic
219C2×C22⋊C4C22×C2311yescharacteristic
220C2×C22⋊C4C22×C2311yescharacteristic
221C2×C22⋊C4C22×C2311yescharacteristic
222C2×C22⋊C4C22×C2311yescharacteristic
223C2×C22⋊C4C22×C2311yescharacteristic
224C2×C22⋊C4C22×C2311yescharacteristic
225C2×C22⋊C4C22×C2311yescharacteristic
226C2×C22⋊C4C22×C2311yescharacteristic
227C2×C22⋊C4C22×C2311yescharacteristic
228C2×C22⋊C4C22×C2311yescharacteristic
229C2×C42C22×C2311yescharacteristic
230C2×C42C22×C2311yescharacteristic
231#95C22×C2311yescharacteristic
232#95C22×C2311yescharacteristic
233#95C22×C2311yescharacteristic
234#95C22×C2311yescharacteristic
235#95C22×C2311yescharacteristic
236#95C22×C2311yescharacteristic
237#95C22×C2311yescharacteristic
238#95C22×C2311yescharacteristic
239#95C22×C2311yescharacteristic
240#95C22×C2311yescharacteristic
241#95C22×C2311yescharacteristic
242#95C22×C2311yescharacteristic
243C23#278111yescharacteristic, direct product factor, Sylow
244C24C23×C2311yescharacteristic
245C24C23×C2311yescharacteristic
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C23×C2311yescharacteristic
253C22×C4C23×C2311yescharacteristic
254C22×C4C23×C2311yescharacteristic
255C22×C4C23×C2311yescharacteristic
256C22×C4C23×C2311yescharacteristic
257C22×C4C23×C2311yescharacteristic
258C22×C4C23×C2311yescharacteristic
259C22×C4C23×C2311yescharacteristic
260C22×C4C23×C2311yescharacteristic
261C22×C4C23×C2311yescharacteristic
262C22×C4C23×C2311yescharacteristic
263C22×C4C23×C2311yescharacteristic
264C22×C4C23×C2311yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—22no—
285C42—41no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#41×C2312yes—
300C23C24×C2311yescharacteristic
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—44no—
340C2×C4#41×C2312yes—
341C2×C4#41×C2312yes—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22C2×#41×C2311yescharacteristic
357C22C2×#41×C2311yescharacteristic
358C22C2×#41×C2311yescharacteristic
359C22C23×H(5,𝔽2)11yescharacteristic
360C22C23×H(5,𝔽2)11yescharacteristic
361C22C23×H(5,𝔽2)11yescharacteristic
362C22C23×2−1+411yescharacteristic
363C4—22no—
364C4—22no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C2—42no—
377C2—81no—
378C2C23×#53811yescharacteristic
379C2C23×#53811yescharacteristic
380C2C23×#53911yescharacteristic
381C2C23×#55211yescharacteristic
382C2C23×#55311yescharacteristic
383C2C23×#55511yescharacteristic
384C2C23×#56211yescharacteristic
subgroupG/H
center, socleC23×C23C24
commutator, Frattini, cosocleC23C24×C23
prime radicalC22×D4×C23C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#400C2semidirect
2. C23×#398C2semidirect
3. C23×#397C2semidirect
4. C23×#396C2semidirect
7. C23×#388C2semidirect
8. C23×#388C2semidirect
9. C23×#388C2semidirect
10. C23×#388C2semidirect
11. C23×#387C2semidirect
13. C23×#385C2semidirect
14. C23×#382C2semidirect
15. C23×#380C2semidirect
149. #2781C23direct (N × H)
184. #400C2×C23semidirect
185. #398C2×C23semidirect
186. #397C2×C23semidirect
187. #396C2×C23semidirect
190. #388C2×C23semidirect
191. #388C2×C23semidirect
192. #388C2×C23semidirect
193. #388C2×C23semidirect
194. #387C2×C23semidirect
196. #385C2×C23semidirect
197. #382C2×C23semidirect
198. #380C2×C23semidirect
243. C23#2781direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#4002
C23×#3982
C23×#3972
C23×#3962
C23×#3942
C23×#3882
C23×#3872
C23×#3862
C23×#3852
C23×#3822
C23×#3802
#278123