Finite Group Databasebeta

← C23×#10284 #19157910

Subgroups — C23×#10284 #19157910

Order 5888.

678 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3226C211yesmaximal, characteristic, complemented
2C23×#3205C211yesmaximal, characteristic
3C23×#2858C211yesmaximal, characteristic, complemented
4C23×#1962C211yesmaximal, characteristic, complemented
5C23×#1546C211yesmaximal, characteristic, complemented
6C23×#1545C211yesmaximal, characteristic, complemented
7C23×#1276C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C23×#127C2211yescharacteristic
11C2×C23×#124—22no—
12C23×#500—21no—
13C23×#493—21no—
14C23×#486—21no—
15C23×#486—21no—
16C23×#480—22no—
17C23×#474—22no—
18C23×#459—22no—
19C23×#447—22no—
20C23×#431C2211yescharacteristic
21C23×#431C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C2×C23×#104—22no—
24C23×#390—22no—
25C23×#390—22no—
26C43×C23C2211yescharacteristic
27C23×#332—22no—
28C23×#331—22no—
29C2×#41×C23—21no—
30C22×Q8×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23D412yes—
34C2×D8×C23—24no—
35C2×M16×C23—21no—
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8D411yescharacteristic
41C23×#127—21no—
42C23×#127—21no—
43C23×#127—21no—
44C23×#127—21no—
45C23×#127D411yescharacteristic
46C23×#124—21no—
47C23×#124—44no—
48C23×#124—44no—
49C23×#121—21no—
50C23×#121—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23D412yes—
55C2×C22⋊C4×C23—28no—
56C2×C42×C23—21no—
57C2×C42×C23—21no—
58C2×C42×C23D412yes—
59C2×C42×C23C2311yescharacteristic
60C23×#107—24no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8D412yes—
64C23×C4⋊C8D412yes—
65C23×#104—44no—
66C23×#102—24no—
67C23×#102—24no—
68C23×C22⋊C8—21no—
69C23×C22⋊C8—21no—
70C4×C8×C23—21no—
71C4×C8×C23—21no—
72C24×C23—22no—
73#41×C23—42no—
74#41×C23—42no—
75C2×Q8×C23—21no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—42no—
79C2×Q8×C23—42no—
80C2×D4×C23—21no—
81C2×D4×C23—22no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—22no—
85C2×D4×C23—44no—
86C2×D4×C23—44no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C2×D4×C23—41no—
91C2×D4×C23—41no—
92C2×D4×C23—42no—
93C2×D4×C23—42no—
94C22×C4×C23—22no—
95C22×C4×C23—22no—
96C22×C4×C23—41no—
97C22×C4×C23C2×D411yescharacteristic
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100D8×C23—84no—
101M16×C23—42no—
102C2×C8×C23—22no—
103C2×C8×C23—22no—
104C2×C8×C23—41no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—22no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C22⋊C4×C23—416no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C42×C23—22no—
115C42×C23—42no—
116C42×C23—42no—
117C42×C23D814yescomplemented
118C42×C23C2×D411yescharacteristic
119C42×C23C2×D411yescharacteristic
120C42×C23C2×D411yescharacteristic
121C42×C23#4111yescharacteristic
122#10284C2311yesmaximal, characteristic, direct product factor, Sylow
123C23×C23—44no—
124C23×C23—44no—
125C23×C23—41no—
126C23×C23—42no—
127C23×C23—44no—
128C23×C23#12011yescharacteristic
129Q8×C23—42no—
130Q8×C23—82no—
131Q8×C23—81no—
132Q8×C23—81no—
133D4×C23—42no—
134D4×C23—42no—
135D4×C23—44no—
136D4×C23—42no—
137D4×C23—82no—
138D4×C23—82no—
139D4×C23—82no—
140D4×C23—82no—
141D4×C23—82no—
142D4×C23—81no—
143D4×C23—81no—
144C2×C4×C23—22no—
145C2×C4×C23—22no—
146C2×C4×C23—24no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—41no—
150C2×C4×C23—42no—
151C2×C4×C23—44no—
152C2×C4×C23—81no—
153C2×C4×C23—81no—
154C2×C4×C23#12011yescharacteristic
155C2×C4×C23#12011yescharacteristic
156C2×C4×C23#12111yescharacteristic
157C2×C4×C23#12111yescharacteristic
158C2×C4×C23#12111yescharacteristic
159C2×C4×C23#12711yescharacteristic
160C2×C4×C23C2×D812yes—
161C2×C4×C23AGL(1,ℤ/8)12yes—
162C8×C23—42no—
163C8×C23—42no—
164C8×C23—81no—
165#3226C2×C2311yescharacteristic, complemented
166#3205C2×C2311yescharacteristic
167#2858C2×C2311yescharacteristic, complemented
168#1962C2×C2311yescharacteristic, complemented
169#1546C2×C2311yescharacteristic, complemented
170#1545C2×C2311yescharacteristic, complemented
171#1276C2×C2311yescharacteristic, complemented
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—88no—
175C22×C23—81no—
176C22×C23—82no—
177C22×C23—82no—
178C22×C23—81no—
179C22×C23—81no—
180C22×C23—82no—
181C22×C23#39211yescharacteristic
182C22×C23#44712yes—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—44no—
186C4×C23—82no—
187C4×C23—81no—
188C4×C23#45412yes—
189C4×C23#45512yes—
190C4×C23#45912yes—
191C4×C23#49311yescharacteristic
192C4×C23#49611yescharacteristic
193#532C22×C2311yescharacteristic
194C2×C4⋊Q8C22×C2311yescharacteristic
195C2×#127C22×C2311yescharacteristic
196C2×#124—22no—
197#500—21no—
198#493—21no—
199#486—21no—
200#486—21no—
201#480—22no—
202#474—22no—
203#459—22no—
204#447—22no—
205#431C22×C2311yescharacteristic
206#431C22×C2311yescharacteristic
207#423C22×C2311yescharacteristic
208C2×#104—22no—
209#390—22no—
210#390—22no—
211C43C22×C2311yescharacteristic
212#332—22no—
213#331—22no—
214C2×C23—22no—
215C2×C23—161no—
216C2×C23—162no—
217C2×C23#200311yescharacteristic
218C2×C23#200411yescharacteristic
219C2×C23#201111yescharacteristic
220C2×#41—21no—
221C22×Q8—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4D4×C2312yes—
225C2×D8—24no—
226C2×M16—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8D4×C2311yescharacteristic
232#127—21no—
233#127—21no—
234#127—21no—
235#127—21no—
236#127D4×C2311yescharacteristic
237#124—21no—
238#124—44no—
239#124—44no—
240#121—21no—
241#121—21no—
242C4×D4—21no—
243C4×D4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4D4×C2312yes—
246C2×C22⋊C4—28no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42D4×C2312yes—
250C2×C42C23×C2311yescharacteristic
251#107—24no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8D4×C2312yes—
255C4⋊C8D4×C2312yes—
256#104—44no—
257#102—24no—
258#102—24no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C23#1028411yescharacteristic, direct product factor, Sylow
264C24—22no—
265#41—42no—
266#41—42no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4C2×D4×C2311yescharacteristic
290C22×C4C2×D4×C2311yescharacteristic
291C22×C4C2×D4×C2311yescharacteristic
292D8—84no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—416no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C42—22no—
307C42—42no—
308C42—42no—
309C42D8×C2314yescomplemented
310C42C2×D4×C2311yescharacteristic
311C42C2×D4×C2311yescharacteristic
312C42C2×D4×C2311yescharacteristic
313C42#41×C2311yescharacteristic
314C23—44no—
315C23—44no—
316C23—41no—
317C23—42no—
318C23—44no—
319C23C23×#12011yescharacteristic
320Q8—42no—
321Q8—82no—
322Q8—81no—
323Q8—81no—
324D4—42no—
325D4—42no—
326D4—44no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—82no—
332D4—82no—
333D4—81no—
334D4—81no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C23×#12011yescharacteristic
346C2×C4C23×#12011yescharacteristic
347C2×C4C23×#12111yescharacteristic
348C2×C4C23×#12111yescharacteristic
349C2×C4C23×#12111yescharacteristic
350C2×C4C23×#12711yescharacteristic
351C2×C4C2×D8×C2312yes—
352C2×C4C23×AGL(1,ℤ/8)12yes—
353C8—42no—
354C8—42no—
355C8—81no—
356C22—21no—
357C22—21no—
358C22—88no—
359C22—81no—
360C22—82no—
361C22—82no—
362C22—81no—
363C22—81no—
364C22—82no—
365C22C23×#39211yescharacteristic
366C22C23×#44712yes—
367C4—21no—
368C4—21no—
369C4—44no—
370C4—82no—
371C4—81no—
372C4C23×#45412yes—
373C4C23×#45512yes—
374C4C23×#45912yes—
375C4C23×#49311yescharacteristic
376C4C23×#49611yescharacteristic
377C2—22no—
378C2—162no—
379C2—161no—
380C2C23×#200311yescharacteristic
381C2C23×#200411yescharacteristic
382C2C23×#201111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3226C2semidirect
3. C23×#2858C2semidirect
4. C23×#1962C2semidirect
5. C23×#1546C2semidirect
6. C23×#1545C2semidirect
7. C23×#1276C2semidirect
117. C42×C23D8semidirect
122. #10284C23direct (N × H)
165. #3226C2×C23semidirect
167. #2858C2×C23semidirect
168. #1962C2×C23semidirect
169. #1546C2×C23semidirect
170. #1545C2×C23semidirect
171. #1276C2×C23semidirect
263. C23#10284direct (N × H)
309. C42D8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32262
C23×#32052
C23×#28582
C23×#19622
C23×#15462
C23×#15452
C23×#12762
#1028423