Finite Group Databasebeta

← C23×#8422 #19156048

Subgroups — C23×#8422 #19156048

Order 5888.

756 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3511C211yesmaximal, characteristic
2C23×#1987C211yesmaximal, characteristic, complemented
3C23×#1806C211yesmaximal, characteristic, complemented
4C23×#1549C211yesmaximal, characteristic, complemented
5C23×#1547C211yesmaximal, characteristic, complemented
6C23×#1502C211yesmaximal, characteristic, complemented
7C23×#1477C211yesmaximal, characteristic, complemented
8C23×#556C412yescomplemented
9C23×#555—21no—
10C23×#554—21no—
11C23×#553—22no—
12C23×#548—22no—
13C23×#547C412yescomplemented
14C23×#542—22no—
15C23×#537—21no—
16C23×#533—21no—
17C23×#532—21no—
18C23×#532C2211yescharacteristic
19C2×C23×C4⋊Q8C2211yescharacteristic
20C23×#520—21no—
21C4×#41×C23C2211yescharacteristic
22C23×#431C2211yescharacteristic
23C23×#423C2211yescharacteristic
24C2×C23×C4⋊C8C2211yescharacteristic
25C2×C23×#104—21no—
26C4×C4⋊C4×C23C2211yescharacteristic
27C23×#340—22no—
28C23×#339—21no—
29C23×#332—21no—
30C23×#332—22no—
31C23×#331—21no—
32C23×#327—22no—
33C23×Q8⋊C8—21no—
34C23×D4⋊C8—21no—
35C2×#41×C23—22no—
36C2×#41×C23—21no—
37C2×#41×C23D411yescharacteristic
38C22×Q8×C23—21no—
39C2×M16×C23—21no—
40C22×C8×C23—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8C2×C412yes—
44C23×C4⋊Q8D412yes—
45C23×C4⋊Q8D411yescharacteristic
46C23×#127D411yescharacteristic
47C23×#126—24no—
48C23×#125—22no—
49C23×#124—24no—
50C23×#124—22no—
51C23×#124—21no—
52C23×#123—24no—
53C23×C22⋊Q8—24no—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—22no—
57C23×#121—24no—
58C23×#121—22no—
59C23×#121—22no—
60C23×#121—22no—
61C23×#121—21no—
62C23×#121D412yes—
63C4×Q8×C23—22no—
64C4×Q8×C23—22no—
65C4×Q8×C23C2×C411yescharacteristic
66C4×Q8×C23D411yescharacteristic
67C4×D4×C23—24no—
68C4×D4×C23—22no—
69C4×D4×C23—22no—
70C4×D4×C23—21no—
71C4×D4×C23—21no—
72C4×D4×C23—21no—
73C4×D4×C23C2×C411yescharacteristic
74C4×D4×C23C2×C412yes—
75C4×D4×C23D411yescharacteristic
76C23×#117—22no—
77C23×#117—22no—
78C23×#117—21no—
79C23×#117D411yescharacteristic
80C2×C4⋊C4×C23—21no—
81C2×C4⋊C4×C23—22no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23D412yes—
84C2×C42×C23—21no—
85C2×C42×C23—21no—
86C2×C42×C23C2311yescharacteristic
87C23×C4⋊C8—21no—
88C23×C4⋊C8—21no—
89C23×C4⋊C8—21no—
90C23×C4⋊C8—21no—
91C23×C4⋊C8—21no—
92C23×#104—42no—
93C23×C22⋊C8—21no—
94C4×C8×C23—21no—
95C23×#95—21no—
96#41×C23—21no—
97#41×C23—21no—
98#41×C23—21no—
99#41×C23—21no—
100#41×C23—41no—
101#41×C23—41no—
102#41×C23—41no—
103#41×C23—41no—
104#41×C23—84no—
105C2×Q8×C23—22no—
106C2×Q8×C23—21no—
107C2×Q8×C23—22no—
108C2×Q8×C23—82no—
109C2×Q8×C23C22⋊C411yescharacteristic
110C2×D4×C23—21no—
111C2×D4×C23—22no—
112C2×D4×C23—24no—
113C2×D4×C23—21no—
114C2×D4×C23—21no—
115C2×D4×C23C22⋊C411yescharacteristic
116C2×D4×C23C22⋊C412yes—
117C22×C4×C23—24no—
118C22×C4×C23—22no—
119C22×C4×C23—21no—
120C22×C4×C23—22no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—41no—
124C22×C4×C23C2×D411yescharacteristic
125C22×C4×C23C2×D411yescharacteristic
126C22×C4×C23C2×D411yescharacteristic
127M16×C23—41no—
128M16×C23—41no—
129C2×C8×C23—42no—
130C2×C8×C23—41no—
131C2×C8×C23—41no—
132C2×C8×C23—41no—
133C4⋊C4×C23—24no—
134C4⋊C4×C23—24no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—21no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—21no—
140C4⋊C4×C23—21no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—41no—
143C4⋊C4×C23C22⋊C411yescharacteristic
144C4⋊C4×C23C22⋊C414yes—
145C4⋊C4×C23C22⋊C411yescharacteristic
146C4⋊C4×C23C22⋊C412yes—
147C22⋊C4×C23—24no—
148C22⋊C4×C23—24no—
149C22⋊C4×C23—24no—
150C22⋊C4×C23—21no—
151C22⋊C4×C23—21no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—21no—
154C22⋊C4×C23—22no—
155C42×C23—24no—
156C42×C23—21no—
157C42×C23—21no—
158C42×C23—21no—
159C42×C23—41no—
160C42×C23—41no—
161C42×C23C22×C411yescharacteristic
162C42×C23C2×D411yescharacteristic
163C42×C23C2×D411yescharacteristic
164C42×C23C2×D411yescharacteristic
165#8422C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—21no—
167C23×C23—22no—
168C23×C23#12011yescharacteristic
169Q8×C23—21no—
170Q8×C23—21no—
171Q8×C23—44no—
172Q8×C23—41no—
173Q8×C23—41no—
174Q8×C23—82no—
175D4×C23—21no—
176D4×C23—21no—
177D4×C23—42no—
178D4×C23—41no—
179D4×C23—42no—
180D4×C23—41no—
181D4×C23—42no—
182D4×C23—41no—
183D4×C23—41no—
184D4×C23—41no—
185D4×C23—41no—
186D4×C23—84no—
187C2×C4×C23—21no—
188C2×C4×C23—21no—
189C2×C4×C23—21no—
190C2×C4×C23—21no—
191C2×C4×C23—22no—
192C2×C4×C23—21no—
193C2×C4×C23—24no—
194C2×C4×C23—41no—
195C2×C4×C23—42no—
196C2×C4×C23—42no—
197C2×C4×C23—42no—
198C2×C4×C23—41no—
199C2×C4×C23—41no—
200C2×C4×C23—41no—
201C2×C4×C23—41no—
202C2×C4×C23—42no—
203C2×C4×C23—82no—
204C2×C4×C23—84no—
205C2×C4×C23—81no—
206C2×C4×C23—81no—
207C2×C4×C23C2×C22⋊C411yescharacteristic
208C2×C4×C23C2×C22⋊C411yescharacteristic
209C2×C4×C23C2×C22⋊C411yescharacteristic
210C2×C4×C23#12011yescharacteristic
211C2×C4×C23#12011yescharacteristic
212C2×C4×C23#12011yescharacteristic
213C2×C4×C23#13511yescharacteristic
214C2×C4×C23#13511yescharacteristic
215C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
216C2×C4×C23#13711yescharacteristic
217C8×C23—81no—
218C8×C23—81no—
219C8×C23—81no—
220#3511C2×C2311yescharacteristic
221#1987C2×C2311yescharacteristic, complemented
222#1806C2×C2311yescharacteristic, complemented
223#1549C2×C2311yescharacteristic, complemented
224#1547C2×C2311yescharacteristic, complemented
225#1502C2×C2311yescharacteristic, complemented
226#1477C2×C2311yescharacteristic, complemented
227C22×C23—21no—
228C22×C23—21no—
229C22×C23—41no—
230C22×C23—42no—
231C22×C23—41no—
232C22×C23—42no—
233C22×C23—41no—
234C22×C23—42no—
235C22×C23#37911yescharacteristic
236C22×C23#44911yescharacteristic
237C22×C23#45211yescharacteristic
238C4×C23—21no—
239C4×C23—21no—
240C4×C23—21no—
241C4×C23—41no—
242C4×C23—41no—
243C4×C23—44no—
244C4×C23—41no—
245C4×C23—42no—
246C4×C23—82no—
247C4×C23#41611yescharacteristic
248C4×C23#41711yescharacteristic
249C4×C23#44911yescharacteristic
250C4×C23#45211yescharacteristic
251C4×C23#45411yescharacteristic
252C4×C23#45611yescharacteristic
253#556C4×C2312yescomplemented
254#555—21no—
255#554—21no—
256#553—22no—
257#548—22no—
258#547C4×C2312yescomplemented
259#542—22no—
260#537—21no—
261#533—21no—
262#532—21no—
263#532C22×C2311yescharacteristic
264C2×C4⋊Q8C22×C2311yescharacteristic
265#520—21no—
266C4×#41C22×C2311yescharacteristic
267#431C22×C2311yescharacteristic
268#423C22×C2311yescharacteristic
269C2×C4⋊C8C22×C2311yescharacteristic
270C2×#104—21no—
271C4×C4⋊C4C22×C2311yescharacteristic
272#340—22no—
273#339—21no—
274#332—21no—
275#332—22no—
276#331—21no—
277#327—22no—
278Q8⋊C8—21no—
279D4⋊C8—21no—
280C2×C23—21no—
281C2×C23—21no—
282C2×C23—81no—
283C2×C23—82no—
284C2×C23#178811yescharacteristic
285C2×C23#179311yescharacteristic
286C2×C23#179511yescharacteristic
287C2×#41—22no—
288C2×#41—21no—
289C2×#41D4×C2311yescharacteristic
290C22×Q8—21no—
291C2×M16—21no—
292C22×C8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8C2×C4×C2312yes—
296C4⋊Q8D4×C2312yes—
297C4⋊Q8D4×C2311yescharacteristic
298#127D4×C2311yescharacteristic
299#126—24no—
300#125—22no—
301#124—24no—
302#124—22no—
303#124—21no—
304#123—24no—
305C22⋊Q8—22no—
306C22⋊Q8—24no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309#121—22no—
310#121—22no—
311#121—24no—
312#121—22no—
313#121—21no—
314#121D4×C2312yes—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8C2×C4×C2311yescharacteristic
318C4×Q8D4×C2311yescharacteristic
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4C2×C4×C2311yescharacteristic
326C4×D4C2×C4×C2312yes—
327C4×D4D4×C2311yescharacteristic
328#117—22no—
329#117—22no—
330#117—21no—
331#117D4×C2311yescharacteristic
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4D4×C2312yes—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C23×C2311yescharacteristic
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344#104—42no—
345C22⋊C8—21no—
346C4×C8—21no—
347#95—21no—
348C23#842211yescharacteristic, direct product factor, Sylow
349#41—21no—
350#41—21no—
351#41—21no—
352#41—21no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—84no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—82no—
362C2×Q8C22⋊C4×C2311yescharacteristic
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C22⋊C4×C2311yescharacteristic
369C2×D4C22⋊C4×C2312yes—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4C2×D4×C2311yescharacteristic
378C22×C4C2×D4×C2311yescharacteristic
379C22×C4C2×D4×C2311yescharacteristic
380M16—41no—
381M16—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4C22⋊C4×C2311yescharacteristic
397C4⋊C4C22⋊C4×C2314yes—
398C4⋊C4C22⋊C4×C2311yescharacteristic
399C4⋊C4C22⋊C4×C2312yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—22no—
406C22⋊C4—21no—
407C22⋊C4—22no—
408C42—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—41no—
413C42—41no—
414C42C22×C4×C2311yescharacteristic
415C42C2×D4×C2311yescharacteristic
416C42C2×D4×C2311yescharacteristic
417C42C2×D4×C2311yescharacteristic
418C23—21no—
419C23—22no—
420C23C23×#12011yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—44no—
424Q8—41no—
425Q8—41no—
426Q8—82no—
427D4—21no—
428D4—21no—
429D4—42no—
430D4—41no—
431D4—42no—
432D4—41no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4C2×C22⋊C4×C2311yescharacteristic
460C2×C4C2×C22⋊C4×C2311yescharacteristic
461C2×C4C2×C22⋊C4×C2311yescharacteristic
462C2×C4C23×#12011yescharacteristic
463C2×C4C23×#12011yescharacteristic
464C2×C4C23×#12011yescharacteristic
465C2×C4C23×#13511yescharacteristic
466C2×C4C23×#13511yescharacteristic
467C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
468C2×C4C23×#13711yescharacteristic
469C8—81no—
470C8—81no—
471C8—81no—
472C22—21no—
473C22—21no—
474C22—41no—
475C22—42no—
476C22—41no—
477C22—42no—
478C22—41no—
479C22—42no—
480C22C23×#37911yescharacteristic
481C22C23×#44911yescharacteristic
482C22C23×#45211yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—44no—
491C4—82no—
492C4C23×#41611yescharacteristic
493C4C23×#41711yescharacteristic
494C4C23×#44911yescharacteristic
495C4C23×#45211yescharacteristic
496C4C23×#45411yescharacteristic
497C4C23×#45611yescharacteristic
498C2—21no—
499C2—21no—
500C2—81no—
501C2—82no—
502C2C23×#178811yescharacteristic
503C2C23×#179311yescharacteristic
504C2C23×#179511yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3511C2
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×#1987C2semidirect
3. C23×#1806C2semidirect
4. C23×#1549C2semidirect
5. C23×#1547C2semidirect
6. C23×#1502C2semidirect
7. C23×#1477C2semidirect
8. C23×#556C4semidirect
13. C23×#547C4semidirect
165. #8422C23direct (N × H)
221. #1987C2×C23semidirect
222. #1806C2×C23semidirect
223. #1549C2×C23semidirect
224. #1547C2×C23semidirect
225. #1502C2×C23semidirect
226. #1477C2×C23semidirect
253. #556C4×C23semidirect
258. #547C4×C23semidirect
348. C23#8422direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35112
C23×#19872
C23×#18062
C23×#15492
C23×#15472
C23×#15022
C23×#14772
#842223