Finite Group Databasebeta

← C23×#9018 #19156644

Subgroups — C23×#9018 #19156644

Order 5888.

928 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 150 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×#394C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic, complemented
3C23×#1786C212yesmaximal
4C2×C23×#324C211yesmaximal, characteristic, complemented
5C23×#1294C212yesmaximal, complemented
6C23×D4×C23C2211yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C22×C22⋊C4×C23—21no—
9C22×C22⋊C4×C23—21no—
10C2×C23×#102—24no—
11C2×C23×C22⋊C8C2212yes—
12C23×#394—22no—
13C23×#394C414yescomplemented
14C23×#387—24no—
15C23×#381—21no—
16C23×#380C2212yes—
17C4×C22⋊C4×C23—21no—
18C2×C23×#95C2211yescharacteristic
19C23×#324—22no—
20C25×C23—21no—
21C22×D4×C23—22no—
22C22×D4×C23—21no—
23C22×D4×C23—22no—
24C22×D4×C23—44no—
25C22×D4×C23C2×C412yes—
26C23×C4×C23—21no—
27C23×C4×C23D411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C23×C4×C23C2311yescharacteristic
30C22×C8×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23C2×C412yes—
34C2×C4⋊C4×C23D412yes—
35C2×C4⋊C4×C23D414yes—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—24no—
38C2×C22⋊C4×C23—24no—
39C2×C22⋊C4×C23—44no—
40C2×C22⋊C4×C23—44no—
41C2×C42×C23—22no—
42C23×#102—48no—
43C23×C22⋊C8—22no—
44C23×C22⋊C8—22no—
45C23×#95—22no—
46C23×#95—24no—
47C23×#95—21no—
48C23×#95—22no—
49C23×#95C2×C412yes—
50C24×C23—24no—
51C24×C23—22no—
52C24×C23—42no—
53C24×C23—42no—
54C24×C23—42no—
55C24×C23—41no—
56C24×C23—41no—
57C24×C23—42no—
58C24×C23—42no—
59C24×C23C2×D411yescharacteristic
60C2×D4×C23—28no—
61C2×D4×C23—24no—
62C2×D4×C23—44no—
63C2×D4×C23—44no—
64C2×D4×C23—48no—
65C2×D4×C23—44no—
66C2×D4×C23—42no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—22no—
71C22×C4×C23—22no—
72C22×C4×C23—21no—
73C22×C4×C23—22no—
74C22×C4×C23—24no—
75C22×C4×C23—42no—
76C22×C4×C23—42no—
77C22×C4×C23—42no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
80C22×C4×C23C22⋊C412yes—
81C22×C4×C23C22⋊C412yes—
82C22×C4×C23C22×C411yescharacteristic
83C22×C4×C23C2×D412yes—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23#4112yes—
86C2×C8×C23—44no—
87C2×C8×C23—42no—
88C4⋊C4×C23—24no—
89C4⋊C4×C23—24no—
90C4⋊C4×C23—44no—
91C22⋊C4×C23—48no—
92C22⋊C4×C23—48no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—42no—
95C42×C23—82no—
96#9018C2311yesmaximal, characteristic, direct product factor, Sylow
97C23×C23—22no—
98C23×C23—22no—
99C23×C23—24no—
100C23×C23—42no—
101C23×C23—44no—
102C23×C23—48no—
103C23×C23—48no—
104C23×C23—42no—
105C23×C23—42no—
106C23×C23—42no—
107C23×C23—42no—
108C23×C23—42no—
109C23×C23—44no—
110C23×C23—84no—
111C23×C23—84no—
112C23×C23—82no—
113C23×C23—82no—
114C23×C23—82no—
115C23×C23—82no—
116C23×C23C2×C22⋊C411yescharacteristic
117C23×C23#12011yescharacteristic
118C23×C23#12311yescharacteristic
119C23×C23#13512yes—
120C23×C23AGL(1,ℤ/8)12yes—
121D4×C23—28no—
122D4×C23—48no—
123D4×C23—44no—
124C2×C4×C23—24no—
125C2×C4×C23—22no—
126C2×C4×C23—22no—
127C2×C4×C23—22no—
128C2×C4×C23—22no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—42no—
137C2×C4×C23—42no—
138C2×C4×C23—42no—
139C2×C4×C23—44no—
140C2×C4×C23—44no—
141C2×C4×C23—44no—
142C2×C4×C23—81no—
143C2×C4×C23—84no—
144C2×C4×C23C4×D412yes—
145C2×C4×C23#12112yes—
146C8×C23—82no—
147C2×#394C2×C2311yescharacteristic
148#2313C2×C2311yescharacteristic, complemented
149#1786C2×C2312yes—
150C2×#324C2×C2311yescharacteristic, complemented
151#1294C2×C2312yescomplemented
152C22×C23—24no—
153C22×C23—24no—
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—44no—
159C22×C23—44no—
160C22×C23—44no—
161C22×C23—82no—
162C22×C23—82no—
163C22×C23—88no—
164C22×C23—88no—
165C22×C23—82no—
166C22×C23—82no—
167C22×C23#38611yescharacteristic
168C22×C23#41611yescharacteristic
169C22×C23#41811yescharacteristic
170C22×C23#44912yes—
171C22×C23#45412yes—
172C22×C23B(4,𝔽2)12yes—
173C22×C23#48212yes—
174C4×C23—22no—
175C4×C23—22no—
176C4×C23—42no—
177C4×C23—42no—
178C4×C23—82no—
179C4×C23—84no—
180C23×D4C22×C2311yescharacteristic
181C22×C4⋊C4C22×C2311yescharacteristic
182C22×C22⋊C4—21no—
183C22×C22⋊C4—21no—
184C2×#102—24no—
185C2×C22⋊C8C22×C2312yes—
186#394—22no—
187#394C4×C2314yescomplemented
188#387—24no—
189#381—21no—
190#380C22×C2312yes—
191C4×C22⋊C4—21no—
192C2×#95C22×C2311yescharacteristic
193#324—22no—
194C2×C23—21no—
195C2×C23—21no—
196C2×C23—22no—
197C2×C23—84no—
198C2×C23#159714yes—
199C2×C23#187511yescharacteristic
200C2×C23#188811yescharacteristic
201C2×C23#188911yescharacteristic
202C25—21no—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4C2×C4×C2312yes—
208C23×C4—21no—
209C23×C4D4×C2311yescharacteristic
210C23×C4D4×C2311yescharacteristic
211C23×C4C23×C2311yescharacteristic
212C22×C8—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C2×C4×C2312yes—
216C2×C4⋊C4D4×C2312yes—
217C2×C4⋊C4D4×C2314yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C42—22no—
224#102—48no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227#95—22no—
228#95—24no—
229#95—21no—
230#95—22no—
231#95C2×C4×C2312yes—
232C23#901811yescharacteristic, direct product factor, Sylow
233C24—24no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—41no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24C2×D4×C2311yescharacteristic
243C2×D4—28no—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C22⋊C4×C2312yes—
264C22×C4C22⋊C4×C2312yes—
265C22×C4C22×C4×C2311yescharacteristic
266C22×C4C2×D4×C2312yes—
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4#41×C2312yes—
269C2×C8—44no—
270C2×C8—42no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—82no—
279C23—22no—
280C23—22no—
281C23—24no—
282C23—42no—
283C23—44no—
284C23—42no—
285C23—42no—
286C23—48no—
287C23—48no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—44no—
292C23—82no—
293C23—84no—
294C23—84no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23C2×C22⋊C4×C2311yescharacteristic
299C23C23×#12011yescharacteristic
300C23C23×#12311yescharacteristic
301C23C23×#13512yes—
302C23C23×AGL(1,ℤ/8)12yes—
303D4—28no—
304D4—48no—
305D4—44no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4C4×D4×C2312yes—
327C2×C4C23×#12112yes—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—24no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—82no—
339C22—82no—
340C22—88no—
341C22—82no—
342C22—88no—
343C22—82no—
344C22C23×#38611yescharacteristic
345C22C23×#41611yescharacteristic
346C22C23×#41811yescharacteristic
347C22C23×#44912yes—
348C22C23×#45412yes—
349C22C23×B(4,𝔽2)12yes—
350C22C23×#48212yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—84no—
357C2—21no—
358C2—21no—
359C2—22no—
360C2—84no—
361C2C23×#159714yes—
362C2C23×#187511yescharacteristic
363C2C23×#188811yescharacteristic
364C2C23×#188911yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2313C2semidirect
4. C2×C23×#324C2semidirect
5. C23×#1294C2semidirect
13. C23×#394C4semidirect
96. #9018C23direct (N × H)
148. #2313C2×C23semidirect
150. C2×#324C2×C23semidirect
151. #1294C2×C23semidirect
187. #394C4×C23semidirect
232. C23#9018direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3942
C23×#23132
C23×#17862
C2×C23×#3242
C23×#12942
#901823