Finite Group Databasebeta

← C23×#9281 #19156907

Subgroups — C23×#9281 #19156907

Order 5888.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3207C211yesmaximal, characteristic
2C23×#2981C211yesmaximal, characteristic, complemented
3C23×#1505C211yesmaximal, characteristic, complemented
4C23×#1505C211yesmaximal, characteristic, complemented
5C23×#1496C211yesmaximal, characteristic, complemented
6C23×#1287C211yesmaximal, characteristic, complemented
7C23×#1274C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#493C411yescharacteristic, complemented
11C23×#492C411yescharacteristic, complemented
12C23×#487C411yescharacteristic, complemented
13C23×#486C411yescharacteristic, complemented
14C23×#463—22no—
15C23×#449—22no—
16C23×#436—21no—
17C23×#435—22no—
18C8×D4×C23—21no—
19C23×#432C2211yescharacteristic
20C23×#432C2211yescharacteristic
21C23×#431C2211yescharacteristic
22C23×#431C2211yescharacteristic
23C23×#408—22no—
24C23×#376C2211yescharacteristic
25C23×#327—22no—
26C23×#327—22no—
27C23×D4⋊C8—21no—
28C23×D4⋊C8—21no—
29C2×#41×C23—21no—
30C2×#41×C23—21no—
31C2×SD16×C23—22no—
32C2×D8×C23—22no—
33C2×M16×C23—22no—
34C22×C8×C23—22no—
35C23×C4⋊Q8C2×C411yescharacteristic
36C23×C4⋊Q8C2×C411yescharacteristic
37C23×#127C2×C411yescharacteristic
38C23×#127C2×C411yescharacteristic
39C23×#124—21no—
40C23×#124—21no—
41C23×#121—21no—
42C23×#121—21no—
43C23×#121—21no—
44C23×#121D412yes—
45C4×D4×C23—21no—
46C4×D4×C23—21no—
47C4×D4×C23—21no—
48C4×D4×C23D412yes—
49C2×C42×C23—21no—
50C2×C42×C23C2311yescharacteristic
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×#103—22no—
56C23×#102—22no—
57C23×C22⋊C8—22no—
58C23×C22⋊C8—22no—
59C23×C22⋊C8—21no—
60C23×C22⋊C8—21no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8D412yes—
63C23×#97—21no—
64C23×#97—21no—
65C4×C8×C23C2×C411yescharacteristic
66C4×C8×C23C2×C411yescharacteristic
67C4×C8×C23D411yescharacteristic
68C4×C8×C23D411yescharacteristic
69C23×#95—21no—
70C23×#95—21no—
71#41×C23—82no—
72#41×C23—82no—
73C2×Q8×C23—21no—
74C2×Q8×C23—21no—
75C2×D4×C23—22no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C2×D4×C23C22⋊C412yes—
83C22×C4×C23—22no—
84C22×C4×C23—41no—
85C22×C4×C23—41no—
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23#4111yescharacteristic
89SD16×C23—82no—
90D8×C23—82no—
91M16×C23—82no—
92C2×C8×C23—21no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—22no—
96C2×C8×C23—21no—
97C2×C8×C23—41no—
98C2×C8×C23—82no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23C22⋊C412yes—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—22no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C42×C23—22no—
108C42×C23—41no—
109C42×C23C22×C411yescharacteristic
110C42×C23C2×D411yescharacteristic
111C42×C23C2×D411yescharacteristic
112C42×C23#4111yescharacteristic
113#9281C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—22no—
115C23×C23—41no—
116C23×C23#12111yescharacteristic
117Q8×C23—42no—
118Q8×C23—81no—
119D4×C23—42no—
120D4×C23—42no—
121D4×C23—41no—
122D4×C23—41no—
123D4×C23—41no—
124D4×C23—41no—
125D4×C23—81no—
126D4×C23—81no—
127D4×C23—82no—
128D4×C23—82no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—22no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—42no—
137C2×C4×C23—41no—
138C2×C4×C23—82no—
139C2×C4×C23—81no—
140C2×C4×C23—81no—
141C2×C4×C23C2×C22⋊C411yescharacteristic
142C2×C4×C23C4×D411yescharacteristic
143C2×C4×C23C4×D411yescharacteristic
144C2×C4×C23#12011yescharacteristic
145C2×C4×C23#12111yescharacteristic
146C2×C4×C23#12311yescharacteristic
147C8×C23—41no—
148C8×C23—42no—
149C8×C23—41no—
150C8×C23—41no—
151C8×C23—41no—
152C8×C23—81no—
153#3207C2×C2311yescharacteristic
154#2981C2×C2311yescharacteristic, complemented
155#1505C2×C2311yescharacteristic, complemented
156#1505C2×C2311yescharacteristic, complemented
157#1496C2×C2311yescharacteristic, complemented
158#1287C2×C2311yescharacteristic, complemented
159#1274C2×C2311yescharacteristic, complemented
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—42no—
164C22×C23—42no—
165C22×C23—42no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23—81no—
169C22×C23#38611yescharacteristic
170C4×C23—21no—
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—42no—
176C4×C23—82no—
177C4×C23—81no—
178C4×C23#44311yescharacteristic
179C4×C23#44411yescharacteristic
180#532C22×C2311yescharacteristic
181#532C22×C2311yescharacteristic
182#493C4×C2311yescharacteristic, complemented
183#492C4×C2311yescharacteristic, complemented
184#487C4×C2311yescharacteristic, complemented
185#486C4×C2311yescharacteristic, complemented
186#463—22no—
187#449—22no—
188#436—21no—
189#435—22no—
190C8×D4—21no—
191#432C22×C2311yescharacteristic
192#432C22×C2311yescharacteristic
193#431C22×C2311yescharacteristic
194#431C22×C2311yescharacteristic
195#408—22no—
196#376C22×C2311yescharacteristic
197#327—22no—
198#327—22no—
199D4⋊C8—21no—
200D4⋊C8—21no—
201C2×C23—41no—
202C2×C23—82no—
203C2×C23—161no—
204C2×C23#187611yescharacteristic
205C2×C23#190411yescharacteristic
206C2×C23#191011yescharacteristic
207C2×#41—21no—
208C2×#41—21no—
209C2×SD16—22no—
210C2×D8—22no—
211C2×M16—22no—
212C22×C8—22no—
213C4⋊Q8C2×C4×C2311yescharacteristic
214C4⋊Q8C2×C4×C2311yescharacteristic
215#127C2×C4×C2311yescharacteristic
216#127C2×C4×C2311yescharacteristic
217#124—21no—
218#124—21no—
219#121—21no—
220#121—21no—
221#121—21no—
222#121D4×C2312yes—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4D4×C2312yes—
227C2×C42—21no—
228C2×C42C23×C2311yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233#103—22no—
234#102—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8D4×C2312yes—
241#97—21no—
242#97—21no—
243C4×C8C2×C4×C2311yescharacteristic
244C4×C8C2×C4×C2311yescharacteristic
245C4×C8D4×C2311yescharacteristic
246C4×C8D4×C2311yescharacteristic
247#95—21no—
248#95—21no—
249C23#928111yescharacteristic, direct product factor, Sylow
250#41—82no—
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4C22⋊C4×C2312yes—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×D4×C2311yescharacteristic
266C22×C4C2×D4×C2311yescharacteristic
267C22×C4#41×C2311yescharacteristic
268SD16—82no—
269D8—82no—
270M16—82no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—82no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4C22⋊C4×C2312yes—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C42—22no—
287C42—41no—
288C42C22×C4×C2311yescharacteristic
289C42C2×D4×C2311yescharacteristic
290C42C2×D4×C2311yescharacteristic
291C42#41×C2311yescharacteristic
292C23—22no—
293C23—41no—
294C23C23×#12111yescharacteristic
295Q8—42no—
296Q8—81no—
297D4—42no—
298D4—42no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—82no—
306D4—82no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4C2×C22⋊C4×C2311yescharacteristic
320C2×C4C4×D4×C2311yescharacteristic
321C2×C4C4×D4×C2311yescharacteristic
322C2×C4C23×#12011yescharacteristic
323C2×C4C23×#12111yescharacteristic
324C2×C4C23×#12311yescharacteristic
325C8—41no—
326C8—42no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—81no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C23×#38611yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—82no—
348C4—81no—
349C4C23×#44311yescharacteristic
350C4C23×#44411yescharacteristic
351C2—41no—
352C2—82no—
353C2—161no—
354C2C23×#187611yescharacteristic
355C2C23×#190411yescharacteristic
356C2C23×#191011yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3207C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2981C2semidirect
3. C23×#1505C2semidirect
4. C23×#1505C2semidirect
5. C23×#1496C2semidirect
6. C23×#1287C2semidirect
7. C23×#1274C2semidirect
10. C23×#493C4semidirect
11. C23×#492C4semidirect
12. C23×#487C4semidirect
13. C23×#486C4semidirect
113. #9281C23direct (N × H)
154. #2981C2×C23semidirect
155. #1505C2×C23semidirect
156. #1505C2×C23semidirect
157. #1496C2×C23semidirect
158. #1287C2×C23semidirect
159. #1274C2×C23semidirect
182. #493C4×C23semidirect
183. #492C4×C23semidirect
184. #487C4×C23semidirect
185. #486C4×C23semidirect
249. C23#9281direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32072
C23×#29812
C23×#15052
C23×#14962
C23×#12872
C23×#12742
#928123