Finite Group Databasebeta

← C23×#18172 #19165798

Subgroups — C23×#18172 #19165798

Order 5888.

998 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic
2C23×#3092C212yesmaximal, complemented
3C2×C23×#400C211yesmaximal, characteristic, complemented
4C23×#1617C212yesmaximal, complemented
5C23×#1616C212yesmaximal, complemented
6C23×#1615C214yesmaximal, complemented
7C23×#1504C212yesmaximal, complemented
8C23×#1463C211yesmaximal, characteristic, complemented
9C23×#554—22no—
10C23×#553—22no—
11C23×#552—22no—
12C23×#547—22no—
13C23×#544C2211yescharacteristic
14C23×#539—21no—
15C23×#539C2212yes—
16C23×#537C2211yescharacteristic
17C23×#525—21no—
18C23×#525—21no—
19C23×#525C2212yes—
20C23×#520—21no—
21C22×C4⋊C4×C23C2211yescharacteristic
22C22×C4⋊C4×C23C2211yescharacteristic
23C23×#484C2214yes—
24C23×#483—22no—
25C23×#481C2214yes—
26C23×#480—22no—
27C23×#428—22no—
28C23×#417—22no—
29C23×#407C2212yes—
30C23×#400—21no—
31C23×#400—21no—
32C23×#400C2212yes—
33C23×#400C2212yes—
34C2×C23×#95—21no—
35C2×C23×#95—21no—
36C2×C23×#95C2211yescharacteristic
37C23×#328C2214yes—
38C23×#327C2214yes—
39C23×#324C2212yes—
40C23×#324C2212yes—
41C2×#41×C23—21no—
42C2×#41×C23—21no—
43C2×#41×C23D412yes—
44C23×C4×C23—21no—
45C23×C4×C23—21no—
46C23×C4×C23D411yescharacteristic
47C23×C4×C23D411yescharacteristic
48C23×C4×C23C2311yescharacteristic
49C2×M16×C23—22no—
50C23×C4⋊Q8—22no—
51C23×#126—22no—
52C23×#126—22no—
53C23×#125—22no—
54C23×#125—22no—
55C23×#125—22no—
56C23×#123—22no—
57C23×#123—22no—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×C22⋊Q8—22no—
62C23×C22⋊Q8—21no—
63C23×C22⋊Q8—21no—
64C23×C22⋊Q8—21no—
65C23×C22⋊Q8—22no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8D412yes—
69C23×C22⋊Q8C2312yes—
70C23×#121—21no—
71C23×#121—21no—
72C23×#121—21no—
73C23×#121—22no—
74C23×#121—21no—
75C23×#121D412yes—
76C23×#121C2312yes—
77C23×#120—22no—
78C4×Q8×C23—22no—
79C4×D4×C23—22no—
80C4×D4×C23—22no—
81C4×D4×C23—22no—
82C4×D4×C23—22no—
83C4×D4×C23—22no—
84C4×D4×C23—22no—
85C4×D4×C23—22no—
86C23×#117—21no—
87C23×#117—22no—
88C23×#117—21no—
89C23×#117D412yes—
90C2×C4⋊C4×C23—22no—
91C2×C4⋊C4×C23—22no—
92C2×C4⋊C4×C23—22no—
93C2×C4⋊C4×C23—22no—
94C2×C4⋊C4×C23—22no—
95C2×C4⋊C4×C23—21no—
96C2×C4⋊C4×C23D411yescharacteristic
97C2×C4⋊C4×C23D411yescharacteristic
98C2×C4⋊C4×C23C2312yes—
99C2×C4⋊C4×C23C2311yescharacteristic
100C2×C4⋊C4×C23C2311yescharacteristic
101C23×#107—24no—
102C23×#106—24no—
103C23×#103—24no—
104C23×#102—24no—
105C23×C22⋊C8—22no—
106C23×C22⋊C8C2314yes—
107C23×#95—22no—
108C23×#95—22no—
109C23×#95—22no—
110C23×#95—22no—
111C23×#95—21no—
112C23×#95C2312yes—
113C24×C23C2×D411yescharacteristic
114#41×C23—42no—
115#41×C23—44no—
116#41×C23—42no—
117#41×C23—82no—
118C2×Q8×C23—21no—
119C2×Q8×C23—21no—
120C2×Q8×C23C2×D412yes—
121C2×D4×C23—21no—
122C2×D4×C23—22no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—21no—
126C2×D4×C23C2×D412yes—
127C22×C4×C23—22no—
128C22×C4×C23—21no—
129C22×C4×C23—21no—
130C22×C4×C23—21no—
131C22×C4×C23—22no—
132C22×C4×C23—21no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—41no—
138C22×C4×C23—42no—
139C22×C4×C23—42no—
140C22×C4×C23—42no—
141C22×C4×C23—42no—
142C22×C4×C23—44no—
143C22×C4×C23—42no—
144C22×C4×C23—41no—
145C22×C4×C23—41no—
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D412yes—
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D411yescharacteristic
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D411yescharacteristic
152C22×C4×C23C2×D412yes—
153C22×C4×C23C2411yescharacteristic
154M16×C23—44no—
155C2×C8×C23—24no—
156C4⋊C4×C23—21no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—21no—
159C4⋊C4×C23—21no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—24no—
171C4⋊C4×C23—42no—
172C4⋊C4×C23C2×D412yes—
173C4⋊C4×C23C2×D412yes—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—22no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C42×C23—22no—
184C42×C23—22no—
185C42×C23—22no—
186#18172C2311yesmaximal, characteristic, direct product factor, Sylow
187C23×C23—21no—
188C23×C23—21no—
189C23×C23—22no—
190C23×C23—22no—
191C23×C23—22no—
192C23×C23#12011yescharacteristic
193C23×C23#12011yescharacteristic
194C23×C23C22×D411yescharacteristic
195Q8×C23—42no—
196Q8×C23—42no—
197Q8×C23—81no—
198D4×C23—42no—
199D4×C23—42no—
200D4×C23—42no—
201D4×C23—42no—
202D4×C23—44no—
203D4×C23—44no—
204D4×C23—81no—
205C2×C4×C23—21no—
206C2×C4×C23—21no—
207C2×C4×C23—24no—
208C2×C4×C23—22no—
209C2×C4×C23—22no—
210C2×C4×C23—21no—
211C2×C4×C23—22no—
212C2×C4×C23—21no—
213C2×C4×C23—21no—
214C2×C4×C23—21no—
215C2×C4×C23—21no—
216C2×C4×C23—42no—
217C2×C4×C23—41no—
218C2×C4×C23—41no—
219C2×C4×C23—42no—
220C2×C4×C23—42no—
221C2×C4×C23—44no—
222C2×C4×C23—42no—
223C2×C4×C23—41no—
224C2×C4×C23—41no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—44no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—82no—
231C2×C4×C23—81no—
232C2×C4×C23—82no—
233C2×C4×C23—84no—
234C2×C4×C23—82no—
235C2×C4×C23—82no—
236C2×C4×C23#12012yes—
237C2×C4×C23#13712yes—
238C2×C4×C23C22×D412yes—
239C8×C23—44no—
240#3492C2×C2311yescharacteristic
241#3092C2×C2312yescomplemented
242C2×#400C2×C2311yescharacteristic, complemented
243#1617C2×C2312yescomplemented
244#1616C2×C2312yescomplemented
245#1615C2×C2314yescomplemented
246#1504C2×C2312yescomplemented
247#1463C2×C2311yescharacteristic, complemented
248C22×C23—24no—
249C22×C23—24no—
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—21no—
256C22×C23—42no—
257C22×C23—42no—
258C22×C23—42no—
259C22×C23#45412yes—
260C22×C23#45812yes—
261C22×C23C2×#12011yescharacteristic
262C22×C23C2×#13711yescharacteristic
263C22×C23#57511yescharacteristic
264C4×C23—22no—
265C4×C23—22no—
266C4×C23—22no—
267C4×C23—41no—
268C4×C23—42no—
269C4×C23—42no—
270C4×C23—41no—
271C4×C23—41no—
272C4×C23—42no—
273C4×C23—82no—
274C4×C23—82no—
275#554—22no—
276#553—22no—
277#552—22no—
278#547—22no—
279#544C22×C2311yescharacteristic
280#539—21no—
281#539C22×C2312yes—
282#537C22×C2311yescharacteristic
283#525—21no—
284#525—21no—
285#525C22×C2312yes—
286#520—21no—
287C22×C4⋊C4C22×C2311yescharacteristic
288C22×C4⋊C4C22×C2311yescharacteristic
289#484C22×C2314yes—
290#483—22no—
291#481C22×C2314yes—
292#480—22no—
293#428—22no—
294#417—22no—
295#407C22×C2312yes—
296#400—21no—
297#400—21no—
298#400C22×C2312yes—
299#400C22×C2312yes—
300C2×#95—21no—
301C2×#95—21no—
302C2×#95C22×C2311yescharacteristic
303#328C22×C2314yes—
304#327C22×C2314yes—
305#324C22×C2312yes—
306#324C22×C2312yes—
307C2×C23—21no—
308C2×C23—21no—
309C2×C23—22no—
310C2×C23—22no—
311C2×C23—82no—
312C2×C23#300711yescharacteristic
313C2×C23C2×#45411yescharacteristic
314C2×C23C2×#45811yescharacteristic
315C2×#41—21no—
316C2×#41—21no—
317C2×#41D4×C2312yes—
318C23×C4—21no—
319C23×C4—21no—
320C23×C4D4×C2311yescharacteristic
321C23×C4D4×C2311yescharacteristic
322C23×C4C23×C2311yescharacteristic
323C2×M16—22no—
324C4⋊Q8—22no—
325#126—22no—
326#126—22no—
327#125—22no—
328#125—22no—
329#125—22no—
330#123—22no—
331#123—22no—
332#123—22no—
333#123—22no—
334#123—22no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—21no—
341C22⋊Q8—22no—
342C22⋊Q8D4×C2312yes—
343C22⋊Q8C23×C2312yes—
344#121—21no—
345#121—21no—
346#121—21no—
347#121—22no—
348#121—21no—
349#121D4×C2312yes—
350#121C23×C2312yes—
351#120—22no—
352C4×Q8—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360#117—22no—
361#117—21no—
362#117—21no—
363#117D4×C2312yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4D4×C2311yescharacteristic
371C2×C4⋊C4D4×C2311yescharacteristic
372C2×C4⋊C4C23×C2312yes—
373C2×C4⋊C4C23×C2311yescharacteristic
374C2×C4⋊C4C23×C2311yescharacteristic
375#107—24no—
376#106—24no—
377#103—24no—
378#102—24no—
379C22⋊C8—22no—
380C22⋊C8C23×C2314yes—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95C23×C2312yes—
387C23#1817211yescharacteristic, direct product factor, Sylow
388C24C2×D4×C2311yescharacteristic
389#41—42no—
390#41—44no—
391#41—42no—
392#41—82no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×D4×C2312yes—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4C2×D4×C2312yes—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4C2×D4×C2311yescharacteristic
422C22×C4C2×D4×C2312yes—
423C22×C4C2×D4×C2311yescharacteristic
424C22×C4C2×D4×C2311yescharacteristic
425C22×C4C2×D4×C2311yescharacteristic
426C22×C4C2×D4×C2311yescharacteristic
427C22×C4C2×D4×C2312yes—
428C22×C4C24×C2311yescharacteristic
429M16—44no—
430C2×C8—24no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—42no—
447C4⋊C4C2×D4×C2312yes—
448C4⋊C4C2×D4×C2312yes—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23C23×#12011yescharacteristic
467C23C23×#12011yescharacteristic
468C23C22×D4×C2311yescharacteristic
469Q8—42no—
470Q8—42no—
471Q8—81no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—44no—
477D4—44no—
478D4—81no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—84no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4C23×#12012yes—
511C2×C4C23×#13712yes—
512C2×C4C22×D4×C2312yes—
513C8—44no—
514C22—24no—
515C22—24no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22C23×#45412yes—
526C22C23×#45812yes—
527C22C2×C23×#12011yescharacteristic
528C22C2×C23×#13711yescharacteristic
529C22C23×#57511yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—42no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—82no—
540C4—82no—
541C2—21no—
542C2—21no—
543C2—22no—
544C2—22no—
545C2—82no—
546C2C23×#300711yescharacteristic
547C2C2×C23×#45411yescharacteristic
548C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3492C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3092C2semidirect
3. C2×C23×#400C2semidirect
4. C23×#1617C2semidirect
5. C23×#1616C2semidirect
6. C23×#1615C2semidirect
7. C23×#1504C2semidirect
8. C23×#1463C2semidirect
186. #18172C23direct (N × H)
241. #3092C2×C23semidirect
242. C2×#400C2×C23semidirect
243. #1617C2×C23semidirect
244. #1616C2×C23semidirect
245. #1615C2×C23semidirect
246. #1504C2×C23semidirect
247. #1463C2×C23semidirect
387. C23#18172direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34922
C23×#30922
C2×C23×#4002
C23×#16172
C23×#16162
C23×#16152
C23×#15042
C23×#14632
#1817223