Finite Group Databasebeta

← C23×#12766 #19160392

Subgroups — C23×#12766 #19160392

Order 5888.

760 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3529C211yesmaximal, characteristic
2C23×#2132C211yesmaximal, characteristic, complemented
3C23×#2120C212yesmaximal, complemented
4C23×#1891C212yesmaximal, complemented
5C23×#1884C211yesmaximal, characteristic, complemented
6C23×#568C2211yescharacteristic
7C23×#563—21no—
8C23×#563C411yescharacteristic, complemented
9C23×#562—21no—
10C23×#562—22no—
11C23×H(3,𝔽4)C411yescharacteristic, complemented
12C23×#560—22no—
13C23×#560C412yescomplemented
14C23×#532—21no—
15C23×#532C2211yescharacteristic
16C23×#529—22no—
17C23×#529—21no—
18C23×#525—22no—
19C23×#525C2212yes—
20C23×#421—21no—
21C23×#413C2212yes—
22C23×#410C2211yescharacteristic
23C23×#407—22no—
24C23×#356—21no—
25C23×#355—21no—
26C23×#352—22no—
27C23×AΣL(1,𝔽16)—22no—
28C23×#344—22no—
29C23×#339—21no—
30C2×#41×C23—21no—
31C2×#41×C23D411yescharacteristic
32C2×#41×C23C2311yescharacteristic
33C23×C4×C23D412yes—
34C23×#131—42no—
35C2×M16×C23—22no—
36C2×M16×C23—21no—
37C2×M16×C23—22no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8D411yescharacteristic
40C23×#127—21no—
41C23×#127C2×C411yescharacteristic
42C23×#126—22no—
43C23×#126—22no—
44C23×#126—22no—
45C23×#125—22no—
46C23×#125—21no—
47C23×#124—21no—
48C23×#124—21no—
49C23×#124—22no—
50C23×#124—21no—
51C23×#124C2×C411yescharacteristic
52C23×#124D411yescharacteristic
53C23×#123—22no—
54C23×#123—22no—
55C23×#123—22no—
56C23×#123—22no—
57C23×#123—22no—
58C23×#123C2×C412yes—
59C23×#123D412yes—
60C23×C22⋊Q8—22no—
61C23×C22⋊Q8—21no—
62C23×C22⋊Q8—22no—
63C23×C22⋊Q8—22no—
64C23×C22⋊Q8—22no—
65C23×#121—21no—
66C23×#121—22no—
67C23×#121—22no—
68C23×#121—22no—
69C23×#121—22no—
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—22no—
73C23×#120—22no—
74C23×#120—22no—
75C23×#120C2×C412yes—
76C23×#120D412yes—
77C4×Q8×C23—21no—
78C4×D4×C23—21no—
79C4×D4×C23—22no—
80C4×D4×C23—22no—
81C4×D4×C23—22no—
82C4×D4×C23—22no—
83C23×#117—22no—
84C23×#117—21no—
85C23×#117D412yes—
86C2×C42×C23—21no—
87C2×C42×C23D411yescharacteristic
88C23×#104—42no—
89C23×#101—22no—
90C23×#100—22no—
91C23×#99—22no—
92C23×C22⋊C8—44no—
93C24×C23C22⋊C412yes—
94#41×C23—41no—
95#41×C23—41no—
96#41×C23—41no—
97#41×C23—41no—
98#41×C23—81no—
99#41×C23—81no—
100C2×Q8×C23—21no—
101C2×Q8×C23—21no—
102C2×Q8×C23C22⋊C411yescharacteristic
103C2×Q8×C23C2×D411yescharacteristic
104C2×D4×C23—21no—
105C2×D4×C23—22no—
106C2×D4×C23—21no—
107C2×D4×C23—22no—
108C2×D4×C23—22no—
109C2×D4×C23—22no—
110C2×D4×C23—22no—
111C2×D4×C23—21no—
112C2×D4×C23C22⋊C411yescharacteristic
113C2×D4×C23C22×C411yescharacteristic
114C2×D4×C23C2×D412yes—
115C22×C4×C23—22no—
116C22×C4×C23—21no—
117C22×C4×C23—21no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—42no—
121C22×C4×C23—42no—
122C22×C4×C23C22⋊C412yes—
123C22×C4×C23C2×D411yescharacteristic
124C22×C4×C23C2×D412yes—
125M16×C23—22no—
126M16×C23—42no—
127M16×C23—41no—
128M16×C23—42no—
129M16×C23—42no—
130C2×C8×C23—24no—
131C2×C8×C23—41no—
132C2×C8×C23—42no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—22no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—21no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—21no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—41no—
144C22⋊C4×C23—22no—
145C22⋊C4×C23—22no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—22no—
149C22⋊C4×C23—22no—
150C22⋊C4×C23—22no—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23C22⋊C412yes—
157C22⋊C4×C23C22⋊C412yes—
158C42×C23—21no—
159C42×C23—21no—
160C42×C23—22no—
161C42×C23—21no—
162C42×C23—41no—
163C42×C23C22⋊C411yescharacteristic
164C42×C23C22⋊C411yescharacteristic
165#12766C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—22no—
167C23×C23—21no—
168C23×C23—42no—
169C23×C23—82no—
170C23×C23C2×C22⋊C412yes—
171C23×C23#12011yescharacteristic
172Q8×C23—41no—
173Q8×C23—41no—
174Q8×C23—81no—
175Q8×C23—81no—
176D4×C23—41no—
177D4×C23—42no—
178D4×C23—41no—
179D4×C23—81no—
180D4×C23—82no—
181D4×C23—82no—
182D4×C23—82no—
183D4×C23—82no—
184D4×C23—81no—
185D4×C23—82no—
186D4×C23—81no—
187C2×C4×C23—21no—
188C2×C4×C23—22no—
189C2×C4×C23—21no—
190C2×C4×C23—22no—
191C2×C4×C23—22no—
192C2×C4×C23—21no—
193C2×C4×C23—22no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—22no—
197C2×C4×C23—42no—
198C2×C4×C23—41no—
199C2×C4×C23—42no—
200C2×C4×C23—41no—
201C2×C4×C23—42no—
202C2×C4×C23—82no—
203C2×C4×C23—81no—
204C2×C4×C23—82no—
205C2×C4×C23—81no—
206C2×C4×C23C2×C22⋊C411yescharacteristic
207C2×C4×C23#12012yes—
208C2×C4×C23#12011yescharacteristic
209C8×C23—42no—
210C8×C23—82no—
211#3529C2×C2311yescharacteristic
212#2132C2×C2311yescharacteristic, complemented
213#2120C2×C2312yescomplemented
214#1891C2×C2312yescomplemented
215#1884C2×C2311yescharacteristic, complemented
216C22×C23—22no—
217C22×C23—21no—
218C22×C23—41no—
219C22×C23—42no—
220C22×C23—42no—
221C22×C23—42no—
222C22×C23—42no—
223C22×C23—44no—
224C22×C23—42no—
225C22×C23—41no—
226C22×C23—82no—
227C22×C23#37911yescharacteristic
228C4×C23—22no—
229C4×C23—21no—
230C4×C23—42no—
231C4×C23—41no—
232C4×C23—42no—
233C4×C23—41no—
234C4×C23—42no—
235C4×C23—41no—
236C4×C23—161no—
237C4×C23B(4,𝔽2)11yescharacteristic
238C4×C23#45811yescharacteristic
239#568C22×C2311yescharacteristic
240#563—21no—
241#563C4×C2311yescharacteristic, complemented
242#562—21no—
243#562—22no—
244H(3,𝔽4)C4×C2311yescharacteristic, complemented
245#560—22no—
246#560C4×C2312yescomplemented
247#532—21no—
248#532C22×C2311yescharacteristic
249#529—22no—
250#529—21no—
251#525—22no—
252#525C22×C2312yes—
253#421—21no—
254#413C22×C2312yes—
255#410C22×C2311yescharacteristic
256#407—22no—
257#356—21no—
258#355—21no—
259#352—22no—
260AΣL(1,𝔽16)—22no—
261#344—22no—
262#339—21no—
263C2×C23—21no—
264C2×C23—41no—
265C2×C23—42no—
266C2×C23—81no—
267C2×C23—82no—
268C2×C23#179011yescharacteristic
269C2×#41—21no—
270C2×#41D4×C2311yescharacteristic
271C2×#41C23×C2311yescharacteristic
272C23×C4D4×C2312yes—
273#131—42no—
274C2×M16—22no—
275C2×M16—21no—
276C2×M16—22no—
277C4⋊Q8—21no—
278C4⋊Q8D4×C2311yescharacteristic
279#127—21no—
280#127C2×C4×C2311yescharacteristic
281#126—22no—
282#126—22no—
283#126—22no—
284#125—22no—
285#125—21no—
286#124—21no—
287#124—21no—
288#124—22no—
289#124—21no—
290#124C2×C4×C2311yescharacteristic
291#124D4×C2311yescharacteristic
292#123—22no—
293#123—22no—
294#123—22no—
295#123—22no—
296#123—22no—
297#123C2×C4×C2312yes—
298#123D4×C2312yes—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—21no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304#121—22no—
305#121—21no—
306#121—22no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121—22no—
312#120—22no—
313#120—22no—
314#120C2×C4×C2312yes—
315#120D4×C2312yes—
316C4×Q8—21no—
317C4×D4—21no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322#117—22no—
323#117—21no—
324#117D4×C2312yes—
325C2×C42—21no—
326C2×C42D4×C2311yescharacteristic
327#104—42no—
328#101—22no—
329#100—22no—
330#99—22no—
331C22⋊C8—44no—
332C23#1276611yescharacteristic, direct product factor, Sylow
333C24C22⋊C4×C2312yes—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—81no—
339#41—81no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C22⋊C4×C2311yescharacteristic
343C2×Q8C2×D4×C2311yescharacteristic
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4C22⋊C4×C2311yescharacteristic
353C2×D4C22×C4×C2311yescharacteristic
354C2×D4C2×D4×C2312yes—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C22⋊C4×C2312yes—
363C22×C4C2×D4×C2311yescharacteristic
364C22×C4C2×D4×C2312yes—
365M16—22no—
366M16—42no—
367M16—41no—
368M16—42no—
369M16—42no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—42no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4C22⋊C4×C2312yes—
397C22⋊C4C22⋊C4×C2312yes—
398C42—21no—
399C42—21no—
400C42—22no—
401C42—21no—
402C42—41no—
403C42C22⋊C4×C2311yescharacteristic
404C42C22⋊C4×C2311yescharacteristic
405C23—22no—
406C23—21no—
407C23—42no—
408C23—82no—
409C23C2×C22⋊C4×C2312yes—
410C23C23×#12011yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—81no—
414Q8—81no—
415D4—41no—
416D4—42no—
417D4—41no—
418D4—81no—
419D4—82no—
420D4—81no—
421D4—82no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—81no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4C2×C22⋊C4×C2311yescharacteristic
446C2×C4C23×#12012yes—
447C2×C4C23×#12011yescharacteristic
448C8—42no—
449C8—82no—
450C22—22no—
451C22—21no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—42no—
459C22—41no—
460C22—82no—
461C22C23×#37911yescharacteristic
462C4—22no—
463C4—21no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—41no—
468C4—42no—
469C4—41no—
470C4—161no—
471C4C23×B(4,𝔽2)11yescharacteristic
472C4C23×#45811yescharacteristic
473C2—21no—
474C2—41no—
475C2—42no—
476C2—82no—
477C2—81no—
478C2C23×#179011yescharacteristic
subgroupG/H
centerC4×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
socleC2×C23#1790
prime radicalC23×#3529C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2132C2semidirect
3. C23×#2120C2semidirect
4. C23×#1891C2semidirect
5. C23×#1884C2semidirect
8. C23×#563C4semidirect
11. C23×H(3,𝔽4)C4semidirect
13. C23×#560C4semidirect
165. #12766C23direct (N × H)
212. #2132C2×C23semidirect
213. #2120C2×C23semidirect
214. #1891C2×C23semidirect
215. #1884C2×C23semidirect
241. #563C4×C23semidirect
244. H(3,𝔽4)C4×C23semidirect
246. #560C4×C23semidirect
332. C23#12766direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35292
C23×#21322
C23×#21202
C23×#18912
C23×#18842
#1276623