Finite Group Databasebeta

← C23×#10240 #19157866

Subgroups — C23×#10240 #19157866

Order 5888.

658 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3206C211yesmaximal, characteristic, complemented
2C23×#3036C211yesmaximal, characteristic
3C23×#2542C211yesmaximal, characteristic, complemented
4C23×#1548C211yesmaximal, characteristic, complemented
5C23×#1502C211yesmaximal, characteristic, complemented
6C23×#1478C211yesmaximal, characteristic, complemented
7C23×#1303C211yesmaximal, characteristic, complemented
8C2×C23×AGL(1,ℤ/8)—21no—
9C23×#532C2211yescharacteristic
10C2×C23×C4⋊Q8C2211yescharacteristic
11C2×C23×C22⋊Q8—22no—
12C2×C4×D4×C23C2211yescharacteristic
13C23×#492—21no—
14C23×#487—21no—
15C23×#463—22no—
16C23×#461—21no—
17C23×#461—22no—
18C23×#459—21no—
19C23×#448—22no—
20C23×#431C2211yescharacteristic
21C23×#431C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C23×#417—21no—
24C23×#399—22no—
25C23×#390—21no—
26C23×#384—21no—
27C4×C4⋊C4×C23C2211yescharacteristic
28C23×#333—22no—
29C23×#327—22no—
30C23×D4⋊C8—22no—
31C2×#41×C23—21no—
32C22×Q8×C23D412yes—
33C22×D4×C23D411yescharacteristic
34C23×C4×C23—21no—
35C23×AGL(1,ℤ/8)—42no—
36C23×AGL(1,ℤ/8)—42no—
37C2×SD16×C23—22no—
38C2×SD16×C23—24no—
39C2×D8×C23—22no—
40C2×M16×C23—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8D411yescharacteristic
46C23×#127D411yescharacteristic
47C23×#124—21no—
48C23×C22⋊Q8—48no—
49C23×#121—21no—
50C23×#121—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C4×D4×C23—21no—
57C2×C4⋊C4×C23—21no—
58C2×C4⋊C4×C23—24no—
59C2×C4⋊C4×C23—21no—
60C2×C4⋊C4×C23—21no—
61C2×C4⋊C4×C23D411yescharacteristic
62C2×C22⋊C4×C23—24no—
63C2×C22⋊C4×C23—21no—
64C2×C42×C23—21no—
65C2×C42×C23C2311yescharacteristic
66C23×C4⋊C8—21no—
67C23×C4⋊C8—21no—
68C23×C4⋊C8D412yes—
69C23×C4⋊C8D412yes—
70C23×#103—22no—
71C23×#103—24no—
72C23×#102—22no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C4×C8×C23—21no—
76C4×C8×C23—21no—
77C23×#95D412yes—
78C24×C23—21no—
79#41×C23—41no—
80#41×C23—41no—
81#41×C23—41no—
82#41×C23—41no—
83C2×Q8×C23—21no—
84C2×Q8×C23—22no—
85C2×Q8×C23—22no—
86C2×Q8×C23—44no—
87C2×D4×C23—21no—
88C2×D4×C23—21no—
89C2×D4×C23—21no—
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23—41no—
96C2×D4×C23SD1612yescomplemented
97C22×C4×C23—22no—
98C22×C4×C23—22no—
99C22×C4×C23—21no—
100C22×C4×C23—44no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23C2×D411yescharacteristic
105C22×C4×C23C2×D411yescharacteristic
106C22×C4×C23C2×D411yescharacteristic
107SD16×C23—44no—
108SD16×C23—84no—
109D8×C23—44no—
110M16×C23—41no—
111M16×C23—41no—
112C2×C8×C23—22no—
113C2×C8×C23—22no—
114C2×C8×C23—41no—
115C4⋊C4×C23—24no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—48no—
121C4⋊C4×C23—41no—
122C4⋊C4×C23—41no—
123C4⋊C4×C23—41no—
124C4⋊C4×C23—41no—
125C22⋊C4×C23—48no—
126C22⋊C4×C23—41no—
127C22⋊C4×C23—41no—
128C22⋊C4×C23—41no—
129C22⋊C4×C23—41no—
130C42×C23—41no—
131C42×C23—41no—
132C42×C23C2×D411yescharacteristic
133C42×C23C2×D411yescharacteristic
134C42×C23C2×D411yescharacteristic
135C42×C23#4111yescharacteristic
136#10240C2311yesmaximal, characteristic, direct product factor, Sylow
137C23×C23—22no—
138C23×C23—42no—
139C23×C23—42no—
140C23×C23—41no—
141C23×C23—41no—
142C23×C23—41no—
143C23×C23#12011yescharacteristic
144Q8×C23—41no—
145Q8×C23—41no—
146Q8×C23—44no—
147Q8×C23—82no—
148D4×C23—22no—
149D4×C23—22no—
150D4×C23—41no—
151D4×C23—41no—
152D4×C23—41no—
153D4×C23—42no—
154D4×C23—41no—
155D4×C23—81no—
156D4×C23—81no—
157D4×C23—81no—
158C2×C4×C23—22no—
159C2×C4×C23—24no—
160C2×C4×C23—41no—
161C2×C4×C23—42no—
162C2×C4×C23—42no—
163C2×C4×C23—44no—
164C2×C4×C23—42no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—42no—
168C2×C4×C23—82no—
169C2×C4×C23—82no—
170C2×C4×C23—81no—
171C2×C4×C23—81no—
172C2×C4×C23#12011yescharacteristic
173C2×C4×C23#12011yescharacteristic
174C2×C4×C23#12111yescharacteristic
175C2×C4×C23#12111yescharacteristic
176C2×C4×C23#12111yescharacteristic
177C2×C4×C23#12711yescharacteristic
178C2×C4×C23C2×SD1611yescharacteristic
179C2×C4×C23#13511yescharacteristic
180C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C23#13711yescharacteristic
182C8×C23—42no—
183C8×C23—42no—
184C8×C23—81no—
185#3206C2×C2311yescharacteristic, complemented
186#3036C2×C2311yescharacteristic
187#2542C2×C2311yescharacteristic, complemented
188#1548C2×C2311yescharacteristic, complemented
189#1502C2×C2311yescharacteristic, complemented
190#1478C2×C2311yescharacteristic, complemented
191#1303C2×C2311yescharacteristic, complemented
192C22×C23—21no—
193C22×C23—21no—
194C22×C23—44no—
195C22×C23—42no—
196C22×C23—82no—
197C22×C23—81no—
198C22×C23—81no—
199C22×C23—81no—
200C22×C23—81no—
201C22×C23—81no—
202C22×C23#39211yescharacteristic
203C22×C23#44811yescharacteristic
204C22×C23#44911yescharacteristic
205C4×C23—21no—
206C4×C23—21no—
207C4×C23—21no—
208C4×C23—44no—
209C4×C23—81no—
210C4×C23—81no—
211C4×C23—82no—
212C4×C23#45011yescharacteristic
213C4×C23#45211yescharacteristic
214C4×C23#45312yes—
215C4×C23#46511yescharacteristic
216C4×C23#46811yescharacteristic
217C2×AGL(1,ℤ/8)—21no—
218#532C22×C2311yescharacteristic
219C2×C4⋊Q8C22×C2311yescharacteristic
220C2×C22⋊Q8—22no—
221C2×C4×D4C22×C2311yescharacteristic
222#492—21no—
223#487—21no—
224#463—22no—
225#461—21no—
226#461—22no—
227#459—21no—
228#448—22no—
229#431C22×C2311yescharacteristic
230#431C22×C2311yescharacteristic
231#423C22×C2311yescharacteristic
232#417—21no—
233#399—22no—
234#390—21no—
235#384—21no—
236C4×C4⋊C4C22×C2311yescharacteristic
237#333—22no—
238#327—22no—
239D4⋊C8—22no—
240C2×C23—21no—
241C2×C23—21no—
242C2×C23—82no—
243C2×C23—161no—
244C2×C23#200011yescharacteristic
245C2×C23#200211yescharacteristic
246C2×C23#200911yescharacteristic
247C2×#41—21no—
248C22×Q8D4×C2312yes—
249C22×D4D4×C2311yescharacteristic
250C23×C4—21no—
251AGL(1,ℤ/8)—42no—
252AGL(1,ℤ/8)—42no—
253C2×SD16—22no—
254C2×SD16—24no—
255C2×D8—22no—
256C2×M16—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8D4×C2311yescharacteristic
262#127D4×C2311yescharacteristic
263#124—21no—
264C22⋊Q8—48no—
265#121—21no—
266#121—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4D4×C2311yescharacteristic
278C2×C22⋊C4—24no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42C23×C2311yescharacteristic
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8D4×C2312yes—
285C4⋊C8D4×C2312yes—
286#103—22no—
287#103—24no—
288#102—22no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293#95D4×C2312yes—
294C23#1024011yescharacteristic, direct product factor, Sylow
295C24—21no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—44no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4SD16×C2312yescomplemented
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4C2×D4×C2311yescharacteristic
322C22×C4C2×D4×C2311yescharacteristic
323C22×C4C2×D4×C2311yescharacteristic
324SD16—44no—
325SD16—84no—
326D8—44no—
327M16—41no—
328M16—41no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—48no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C22⋊C4—48no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—41no—
348C42—41no—
349C42C2×D4×C2311yescharacteristic
350C42C2×D4×C2311yescharacteristic
351C42C2×D4×C2311yescharacteristic
352C42#41×C2311yescharacteristic
353C23—22no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23C23×#12011yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—44no—
363Q8—82no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—81no—
372D4—81no—
373D4—81no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4C23×#12011yescharacteristic
389C2×C4C23×#12011yescharacteristic
390C2×C4C23×#12111yescharacteristic
391C2×C4C23×#12111yescharacteristic
392C2×C4C23×#12111yescharacteristic
393C2×C4C23×#12711yescharacteristic
394C2×C4C2×SD16×C2311yescharacteristic
395C2×C4C23×#13511yescharacteristic
396C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
397C2×C4C23×#13711yescharacteristic
398C8—42no—
399C8—42no—
400C8—81no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—42no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C23×#39211yescharacteristic
412C22C23×#44811yescharacteristic
413C22C23×#44911yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—81no—
419C4—81no—
420C4—82no—
421C4C23×#45011yescharacteristic
422C4C23×#45211yescharacteristic
423C4C23×#45312yes—
424C4C23×#46511yescharacteristic
425C4C23×#46811yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2C23×#200011yescharacteristic
431C2C23×#200211yescharacteristic
432C2C23×#200911yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3036C2
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×#3206C2semidirect
3. C23×#2542C2semidirect
4. C23×#1548C2semidirect
5. C23×#1502C2semidirect
6. C23×#1478C2semidirect
7. C23×#1303C2semidirect
96. C2×D4×C23SD16semidirect
136. #10240C23direct (N × H)
185. #3206C2×C23semidirect
187. #2542C2×C23semidirect
188. #1548C2×C23semidirect
189. #1502C2×C23semidirect
190. #1478C2×C23semidirect
191. #1303C2×C23semidirect
294. C23#10240direct (N × H)
313. C2×D4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32062
C23×#30362
C23×#25422
C23×#15482
C23×#15022
C23×#14782
C23×#13032
#1024023