Finite Group Databasebeta

← C23×#12781 #19160407

Subgroups — C23×#12781 #19160407

Order 5888.

758 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3545C211yesmaximal, characteristic
2C23×#2133C212yesmaximal, complemented
3C2×C23×#355C211yesmaximal, characteristic, complemented
4C23×#1912C212yesmaximal, complemented
5C23×#1888C211yesmaximal, characteristic, complemented
6C23×#563C411yescharacteristic, complemented
7C23×#562—21no—
8C23×H(3,𝔽4)C411yescharacteristic, complemented
9C23×#555C412yescomplemented
10C23×#551—21no—
11C23×#551—22no—
12C23×#546—22no—
13C23×#542—22no—
14C23×#538—22no—
15C23×#536C2212yes—
16C23×#534—21no—
17C23×#529—21no—
18C2×C23×#124C2211yescharacteristic
19C23×#519C2211yescharacteristic, complemented
20C23×#421—22no—
21C23×#418—21no—
22C2×C23×#101C2211yescharacteristic, complemented
23C2×C23×#101C2212yescomplemented
24C23×#356—22no—
25C23×#355—21no—
26C23×#355—21no—
27C23×#355—22no—
28C23×#339—21no—
29C23×H(3,ℤ/4)—22no—
30C2×#41×C23D412yes—
31C22×Q8×C23C2311yescharacteristic
32C22×D4×C23—21no—
33C22×D4×C23D411yescharacteristic
34C2×M16×C23—22no—
35C2×M16×C23—21no—
36C23×C4⋊Q8C2×C412yes—
37C23×C4⋊Q8D412yes—
38C23×#126—22no—
39C23×#126—22no—
40C23×#126—22no—
41C23×#125—21no—
42C23×#124—22no—
43C23×#124—22no—
44C23×#124—22no—
45C23×#124—21no—
46C23×#124C2×C411yescharacteristic
47C23×#124C2×C412yes—
48C23×#124C2×C411yescharacteristic
49C23×#124D411yescharacteristic
50C23×#124D411yescharacteristic
51C23×#124D412yes—
52C23×#123—22no—
53C23×#123—22no—
54C23×#123—22no—
55C23×#123—21no—
56C23×#123—21no—
57C23×#123—22no—
58C23×#123—22no—
59C23×C22⋊Q8—21no—
60C23×C22⋊Q8—22no—
61C23×C22⋊Q8—22no—
62C23×C22⋊Q8—22no—
63C23×C22⋊Q8—22no—
64C23×#121—22no—
65C23×#121—21no—
66C23×#121—22no—
67C23×#121—22no—
68C23×#121—22no—
69C23×#120—22no—
70C23×#120—21no—
71C23×#120—21no—
72C23×#120—22no—
73C4×Q8×C23—21no—
74C4×Q8×C23—42no—
75C4×D4×C23—22no—
76C4×D4×C23—22no—
77C4×D4×C23—21no—
78C4×D4×C23—22no—
79C23×#117—22no—
80C23×#117—22no—
81C23×#117—22no—
82C23×#117—21no—
83C23×#117D412yes—
84C2×C22⋊C4×C23—21no—
85C2×C22⋊C4×C23—22no—
86C2×C22⋊C4×C23—22no—
87C2×C42×C23D411yescharacteristic
88C23×#104—44no—
89C23×#102—42no—
90C23×#101—22no—
91C23×#101—22no—
92C23×#101—22no—
93C24×C23—21no—
94C24×C23—21no—
95#41×C23—42no—
96#41×C23—42no—
97C2×Q8×C23—21no—
98C2×Q8×C23—41no—
99C2×Q8×C23—41no—
100C2×Q8×C23C22⋊C412yes—
101C2×Q8×C23C22×C411yescharacteristic
102C2×Q8×C23C2×D411yescharacteristic
103C2×Q8×C23C2×D412yes—
104C2×D4×C23—22no—
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—21no—
108C2×D4×C23—22no—
109C2×D4×C23—21no—
110C2×D4×C23—41no—
111C2×D4×C23—41no—
112C2×D4×C23—81no—
113C2×D4×C23—81no—
114C2×D4×C23C22⋊C412yes—
115C2×D4×C23C22⋊C411yescharacteristic
116C2×D4×C23C22⋊C411yescharacteristic
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23—21no—
120C22×C4×C23—21no—
121C22×C4×C23—22no—
122C22×C4×C23C2×D411yescharacteristic
123C22×C4×C23C2×D412yes—
124M16×C23—42no—
125M16×C23—41no—
126M16×C23—41no—
127M16×C23—42no—
128C2×C8×C23—41no—
129C2×C8×C23—42no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—21no—
133C4⋊C4×C23—21no—
134C4⋊C4×C23—22no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—22no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—42no—
141C22⋊C4×C23—22no—
142C22⋊C4×C23—22no—
143C22⋊C4×C23—21no—
144C22⋊C4×C23—21no—
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—22no—
156C22⋊C4×C23—41no—
157C42×C23—22no—
158C42×C23—21no—
159C42×C23—24no—
160C42×C23—41no—
161C42×C23C22⋊C411yescharacteristic
162C42×C23C22⋊C411yescharacteristic
163C42×C23C22⋊C412yes—
164C42×C23C22⋊C412yes—
165#12781C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—22no—
169C23×C23—41no—
170C23×C23—41no—
171C23×C23—81no—
172C23×C23—82no—
173C23×C23—82no—
174C23×C23#12011yescharacteristic
175Q8×C23—41no—
176Q8×C23—41no—
177Q8×C23—42no—
178Q8×C23—42no—
179Q8×C23—81no—
180D4×C23—41no—
181D4×C23—41no—
182D4×C23—42no—
183D4×C23—42no—
184D4×C23—42no—
185D4×C23—81no—
186D4×C23—82no—
187D4×C23—82no—
188D4×C23—81no—
189D4×C23—82no—
190C2×C4×C23—22no—
191C2×C4×C23—21no—
192C2×C4×C23—22no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—22no—
196C2×C4×C23—21no—
197C2×C4×C23—22no—
198C2×C4×C23—22no—
199C2×C4×C23—42no—
200C2×C4×C23—82no—
201C2×C4×C23—82no—
202C2×C4×C23—81no—
203C2×C4×C23—82no—
204C2×C4×C23—81no—
205C2×C4×C23C2×C22⋊C412yes—
206C2×C4×C23C2×C22⋊C411yescharacteristic
207C2×C4×C23#12011yescharacteristic
208C2×C4×C23#12012yes—
209C8×C23—82no—
210C8×C23—81no—
211#3545C2×C2311yescharacteristic
212#2133C2×C2312yescomplemented
213C2×#355C2×C2311yescharacteristic, complemented
214#1912C2×C2312yescomplemented
215#1888C2×C2311yescharacteristic, complemented
216C22×C23—41no—
217C22×C23—42no—
218C22×C23—42no—
219C22×C23—42no—
220C22×C23—42no—
221C22×C23—42no—
222C22×C23—41no—
223C22×C23—41no—
224C22×C23—161no—
225C22×C23—161no—
226C22×C23—162no—
227C22×C23#37911yescharacteristic
228C22×C23B(4,𝔽2)11yescharacteristic
229C22×C23#45811yescharacteristic
230C4×C23—22no—
231C4×C23—21no—
232C4×C23—21no—
233C4×C23—22no—
234C4×C23—41no—
235C4×C23—42no—
236C4×C23—42no—
237C4×C23—41no—
238C4×C23—42no—
239C4×C23—82no—
240#563C4×C2311yescharacteristic, complemented
241#562—21no—
242H(3,𝔽4)C4×C2311yescharacteristic, complemented
243#555C4×C2312yescomplemented
244#551—21no—
245#551—22no—
246#546—22no—
247#542—22no—
248#538—22no—
249#536C22×C2312yes—
250#534—21no—
251#529—21no—
252C2×#124C22×C2311yescharacteristic
253#519C22×C2311yescharacteristic, complemented
254#421—22no—
255#418—21no—
256C2×#101C22×C2312yescomplemented
257C2×#101C22×C2311yescharacteristic, complemented
258#356—22no—
259#355—21no—
260#355—21no—
261#355—22no—
262#339—21no—
263H(3,ℤ/4)—22no—
264C2×C23—21no—
265C2×C23—21no—
266C2×C23—21no—
267C2×C23—81no—
268C2×C23—82no—
269C2×C23—81no—
270C2×C23#179011yescharacteristic
271C2×#41D4×C2312yes—
272C22×Q8C23×C2311yescharacteristic
273C22×D4—21no—
274C22×D4D4×C2311yescharacteristic
275C2×M16—22no—
276C2×M16—21no—
277C4⋊Q8C2×C4×C2312yes—
278C4⋊Q8D4×C2312yes—
279#126—22no—
280#126—22no—
281#126—22no—
282#125—21no—
283#124—22no—
284#124—22no—
285#124—21no—
286#124—22no—
287#124C2×C4×C2311yescharacteristic
288#124C2×C4×C2311yescharacteristic
289#124C2×C4×C2312yes—
290#124D4×C2311yescharacteristic
291#124D4×C2311yescharacteristic
292#124D4×C2312yes—
293#123—21no—
294#123—22no—
295#123—22no—
296#123—22no—
297#123—21no—
298#123—22no—
299#123—22no—
300C22⋊Q8—21no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305#121—21no—
306#121—22no—
307#121—22no—
308#121—22no—
309#121—22no—
310#120—21no—
311#120—22no—
312#120—21no—
313#120—22no—
314C4×Q8—21no—
315C4×Q8—42no—
316C4×D4—21no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—21no—
324#117D4×C2312yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42D4×C2311yescharacteristic
329#104—44no—
330#102—42no—
331#101—22no—
332#101—22no—
333#101—22no—
334C23#1278111yescharacteristic, direct product factor, Sylow
335C24—21no—
336C24—21no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8C22⋊C4×C2312yes—
343C2×Q8C22×C4×C2311yescharacteristic
344C2×Q8C2×D4×C2311yescharacteristic
345C2×Q8C2×D4×C2312yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4C22⋊C4×C2312yes—
357C2×D4C22⋊C4×C2311yescharacteristic
358C2×D4C22⋊C4×C2311yescharacteristic
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4C2×D4×C2311yescharacteristic
365C22×C4C2×D4×C2312yes—
366M16—42no—
367M16—41no—
368M16—41no—
369M16—42no—
370C2×C8—41no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—21no—
386C22⋊C4—21no—
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—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—41no—
399C42—22no—
400C42—21no—
401C42—24no—
402C42—41no—
403C42C22⋊C4×C2311yescharacteristic
404C42C22⋊C4×C2311yescharacteristic
405C42C22⋊C4×C2312yes—
406C42C22⋊C4×C2312yes—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23C23×#12011yescharacteristic
416Q8—41no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—81no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—81no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—81no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4C2×C22⋊C4×C2312yes—
447C2×C4C2×C22⋊C4×C2311yescharacteristic
448C2×C4C23×#12011yescharacteristic
449C2×C4C23×#12012yes—
450C8—82no—
451C8—81no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—41no—
459C22—41no—
460C22—161no—
461C22—161no—
462C22—162no—
463C22C23×#37911yescharacteristic
464C22C23×B(4,𝔽2)11yescharacteristic
465C22C23×#45811yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—81no—
480C2—81no—
481C2—82no—
482C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×Q8C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC23×#3545C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2133C2semidirect
3. C2×C23×#355C2semidirect
4. C23×#1912C2semidirect
5. C23×#1888C2semidirect
6. C23×#563C4semidirect
8. C23×H(3,𝔽4)C4semidirect
9. C23×#555C4semidirect
19. C23×#519C22semidirect
22. C2×C23×#101C22semidirect
23. C2×C23×#101C22semidirect
165. #12781C23direct (N × H)
212. #2133C2×C23semidirect
213. C2×#355C2×C23semidirect
214. #1912C2×C23semidirect
215. #1888C2×C23semidirect
240. #563C4×C23semidirect
242. H(3,𝔽4)C4×C23semidirect
243. #555C4×C23semidirect
253. #519C22×C23semidirect
256. C2×#101C22×C23semidirect
257. C2×#101C22×C23semidirect
334. C23#12781direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35452
C23×#21332
C2×C23×#3552
C23×#19122
C23×#18882
#1278123