Finite Group Databasebeta

← C23×#18151 #19165777

Subgroups — C23×#18151 #19165777

Order 5888.

994 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3460C211yesmaximal, characteristic
2C23×#3092C212yesmaximal, complemented
3C23×#2487C211yesmaximal, characteristic, complemented
4C23×#1618C214yesmaximal, complemented
5C23×#1617C212yesmaximal, complemented
6C23×#1616C212yesmaximal, complemented
7C23×#1513C212yesmaximal, complemented
8C23×#1465C211yesmaximal, characteristic, complemented
9C23×#544C2211yescharacteristic
10C23×#542—22no—
11C23×#540—22no—
12C23×#538—21no—
13C23×#538C2212yes—
14C23×#537C2211yescharacteristic
15C23×#535—21no—
16C23×#532—22no—
17C23×#529—22no—
18C23×#525—21no—
19C23×#525C2212yes—
20C4×#41×C23—21no—
21C2×C4×D4×C23C2211yescharacteristic
22C22×C4⋊C4×C23C2211yescharacteristic
23C23×#484C2214yes—
24C23×#483—22no—
25C23×#481C2214yes—
26C23×#480—22no—
27C23×#428—22no—
28C23×#417—22no—
29C23×#407C2212yes—
30C23×#400C2212yes—
31C23×#399C2212yes—
32C23×#389—21no—
33C23×#386—21no—
34C23×#385—21no—
35C23×#380—21no—
36C23×#380C2211yescharacteristic
37C23×#328C2214yes—
38C23×#327C2214yes—
39C23×#324C2212yes—
40C23×#323C2212yes—
41C2×#41×C23—21no—
42C2×#41×C23—21no—
43C2×#41×C23D412yes—
44C22×D4×C23D411yescharacteristic
45C23×C4×C23—21no—
46C23×C4×C23D411yescharacteristic
47C23×C4×C23C2311yescharacteristic
48C2×M16×C23—22no—
49C23×C4⋊Q8—22no—
50C23×#127—22no—
51C23×#126—22no—
52C23×#126—22no—
53C23×#125—22no—
54C23×#124—22no—
55C23×#124—22no—
56C23×#124—22no—
57C23×#123—22no—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×C22⋊Q8—21no—
62C23×C22⋊Q8—21no—
63C23×C22⋊Q8—22no—
64C23×C22⋊Q8D412yes—
65C23×C22⋊Q8C2312yes—
66C23×#121—21no—
67C23×#121—22no—
68C23×#121—22no—
69C23×#121—22no—
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—21no—
73C23×#121D412yes—
74C23×#121C2312yes—
75C23×#120—22no—
76C23×#120—22no—
77C4×Q8×C23—21no—
78C4×Q8×C23—21no—
79C4×D4×C23—21no—
80C4×D4×C23—21no—
81C4×D4×C23—22no—
82C4×D4×C23—22no—
83C4×D4×C23—22no—
84C4×D4×C23—22no—
85C4×D4×C23—22no—
86C4×D4×C23—22no—
87C23×#117—22no—
88C23×#117—21no—
89C23×#117—21no—
90C23×#117D412yes—
91C2×C4⋊C4×C23—22no—
92C2×C4⋊C4×C23—22no—
93C2×C4⋊C4×C23—21no—
94C2×C4⋊C4×C23—21no—
95C2×C4⋊C4×C23—22no—
96C2×C4⋊C4×C23C2312yes—
97C2×C4⋊C4×C23C2311yescharacteristic
98C2×C22⋊C4×C23—21no—
99C2×C22⋊C4×C23—22no—
100C2×C22⋊C4×C23D411yescharacteristic
101C2×C22⋊C4×C23C2311yescharacteristic
102C2×C42×C23—22no—
103C2×C42×C23D411yescharacteristic
104C23×#107—24no—
105C23×#106—24no—
106C23×#103—24no—
107C23×#102—24no—
108C23×C22⋊C8—22no—
109C23×C22⋊C8C2314yes—
110C23×#95—22no—
111C23×#95—22no—
112C23×#95C2312yes—
113C24×C23C2×D411yescharacteristic
114C24×C23C2×D411yescharacteristic
115#41×C23—42no—
116#41×C23—44no—
117#41×C23—42no—
118#41×C23—82no—
119C2×Q8×C23—21no—
120C2×Q8×C23—21no—
121C2×Q8×C23C2×D412yes—
122C2×D4×C23—21no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—21no—
128C2×D4×C23—22no—
129C2×D4×C23—22no—
130C2×D4×C23—22no—
131C2×D4×C23—41no—
132C2×D4×C23C2×D412yes—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—22no—
139C22×C4×C23—22no—
140C22×C4×C23—42no—
141C22×C4×C23—42no—
142C22×C4×C23—42no—
143C22×C4×C23—82no—
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D412yes—
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2411yescharacteristic
151M16×C23—44no—
152C2×C8×C23—24no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—21no—
157C4⋊C4×C23—21no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—24no—
163C4⋊C4×C23—42no—
164C4⋊C4×C23—41no—
165C4⋊C4×C23—41no—
166C4⋊C4×C23—82no—
167C4⋊C4×C23C2×D412yes—
168C4⋊C4×C23C2×D412yes—
169C22⋊C4×C23—22no—
170C22⋊C4×C23—22no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—22no—
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—42no—
181C22⋊C4×C23—82no—
182C42×C23—22no—
183C42×C23—22no—
184C42×C23—21no—
185C42×C23—22no—
186C42×C23—21no—
187#18151C2311yesmaximal, characteristic, direct product factor, Sylow
188C23×C23—21no—
189C23×C23—21no—
190C23×C23—21no—
191C23×C23—22no—
192C23×C23—21no—
193C23×C23—21no—
194C23×C23—41no—
195C23×C23—41no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23#12012yes—
199C23×C23C22×D411yescharacteristic
200Q8×C23—42no—
201Q8×C23—42no—
202Q8×C23—81no—
203D4×C23—44no—
204D4×C23—42no—
205D4×C23—42no—
206D4×C23—42no—
207D4×C23—42no—
208D4×C23—82no—
209D4×C23—82no—
210D4×C23—82no—
211D4×C23—81no—
212D4×C23—82no—
213C2×C4×C23—21no—
214C2×C4×C23—21no—
215C2×C4×C23—22no—
216C2×C4×C23—24no—
217C2×C4×C23—21no—
218C2×C4×C23—22no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—21no—
222C2×C4×C23—42no—
223C2×C4×C23—41no—
224C2×C4×C23—44no—
225C2×C4×C23—42no—
226C2×C4×C23—44no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—81no—
236C2×C4×C23—82no—
237C2×C4×C23—84no—
238C2×C4×C23#12011yescharacteristic
239C2×C4×C23#12011yescharacteristic
240C2×C4×C23#13712yes—
241C2×C4×C23C22×D412yes—
242C8×C23—44no—
243#3460C2×C2311yescharacteristic
244#3092C2×C2312yescomplemented
245#2487C2×C2311yescharacteristic, complemented
246#1618C2×C2314yescomplemented
247#1617C2×C2312yescomplemented
248#1616C2×C2312yescomplemented
249#1513C2×C2312yescomplemented
250#1465C2×C2311yescharacteristic, complemented
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—22no—
254C22×C23—22no—
255C22×C23—22no—
256C22×C23—42no—
257C22×C23—42no—
258C22×C23—42no—
259C22×C23—42no—
260C22×C23—41no—
261C22×C23—42no—
262C22×C23—41no—
263C22×C23—41no—
264C22×C23—42no—
265C22×C23—81no—
266C22×C23C2×#12011yescharacteristic
267C22×C23C2×#13711yescharacteristic
268C22×C23#57511yescharacteristic
269C4×C23—22no—
270C4×C23—22no—
271C4×C23—21no—
272C4×C23—42no—
273C4×C23—41no—
274C4×C23—42no—
275C4×C23—41no—
276C4×C23—42no—
277C4×C23—82no—
278C4×C23—82no—
279C4×C23#45812yes—
280#544C22×C2311yescharacteristic
281#542—22no—
282#540—22no—
283#538—21no—
284#538C22×C2312yes—
285#537C22×C2311yescharacteristic
286#535—21no—
287#532—22no—
288#529—22no—
289#525—21no—
290#525C22×C2312yes—
291C4×#41—21no—
292C2×C4×D4C22×C2311yescharacteristic
293C22×C4⋊C4C22×C2311yescharacteristic
294#484C22×C2314yes—
295#483—22no—
296#481C22×C2314yes—
297#480—22no—
298#428—22no—
299#417—22no—
300#407C22×C2312yes—
301#400C22×C2312yes—
302#399C22×C2312yes—
303#389—21no—
304#386—21no—
305#385—21no—
306#380—21no—
307#380C22×C2311yescharacteristic
308#328C22×C2314yes—
309#327C22×C2314yes—
310#324C22×C2312yes—
311#323C22×C2312yes—
312C2×C23—21no—
313C2×C23—21no—
314C2×C23—42no—
315C2×C23—82no—
316C2×C23—81no—
317C2×C23#300711yescharacteristic
318C2×C23#302111yescharacteristic
319C2×C23C2×#45811yescharacteristic
320C2×#41—21no—
321C2×#41—21no—
322C2×#41D4×C2312yes—
323C22×D4D4×C2311yescharacteristic
324C23×C4—21no—
325C23×C4D4×C2311yescharacteristic
326C23×C4C23×C2311yescharacteristic
327C2×M16—22no—
328C4⋊Q8—22no—
329#127—22no—
330#126—22no—
331#126—22no—
332#125—22no—
333#124—22no—
334#124—22no—
335#124—22no—
336#123—22no—
337#123—22no—
338#123—22no—
339#123—22no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—22no—
343C22⋊Q8D4×C2312yes—
344C22⋊Q8C23×C2312yes—
345#121—21no—
346#121—22no—
347#121—22no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121—21no—
352#121D4×C2312yes—
353#121C23×C2312yes—
354#120—22no—
355#120—22no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366#117—22no—
367#117—21no—
368#117—21no—
369#117D4×C2312yes—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4C23×C2312yes—
376C2×C4⋊C4C23×C2311yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4D4×C2311yescharacteristic
380C2×C22⋊C4C23×C2311yescharacteristic
381C2×C42—22no—
382C2×C42D4×C2311yescharacteristic
383#107—24no—
384#106—24no—
385#103—24no—
386#102—24no—
387C22⋊C8—22no—
388C22⋊C8C23×C2314yes—
389#95—22no—
390#95—22no—
391#95C23×C2312yes—
392C23#1815111yescharacteristic, direct product factor, Sylow
393C24C2×D4×C2311yescharacteristic
394C24C2×D4×C2311yescharacteristic
395#41—42no—
396#41—44no—
397#41—42no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C2×D4×C2312yes—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—41no—
412C2×D4C2×D4×C2312yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4C2×D4×C2311yescharacteristic
425C22×C4C2×D4×C2312yes—
426C22×C4C2×D4×C2311yescharacteristic
427C22×C4C2×D4×C2311yescharacteristic
428C22×C4C2×D4×C2311yescharacteristic
429C22×C4C2×D4×C2312yes—
430C22×C4C24×C2311yescharacteristic
431M16—44no—
432C2×C8—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—82no—
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—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—42no—
461C22⋊C4—82no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—22no—
466C42—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C23×#12012yes—
478C23C22×D4×C2311yescharacteristic
479Q8—42no—
480Q8—42no—
481Q8—81no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—82no—
488D4—82no—
489D4—81no—
490D4—82no—
491D4—82no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4C23×#12011yescharacteristic
518C2×C4C23×#12011yescharacteristic
519C2×C4C23×#13712yes—
520C2×C4C22×D4×C2312yes—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—81no—
537C22C2×C23×#12011yescharacteristic
538C22C2×C23×#13711yescharacteristic
539C22C23×#57511yescharacteristic
540C4—22no—
541C4—22no—
542C4—21no—
543C4—42no—
544C4—41no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4C23×#45812yes—
551C2—21no—
552C2—21no—
553C2—42no—
554C2—81no—
555C2—82no—
556C2C23×#300711yescharacteristic
557C2C23×#302111yescharacteristic
558C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3460C2
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. C23×#2487C2semidirect
4. C23×#1618C2semidirect
5. C23×#1617C2semidirect
6. C23×#1616C2semidirect
7. C23×#1513C2semidirect
8. C23×#1465C2semidirect
187. #18151C23direct (N × H)
244. #3092C2×C23semidirect
245. #2487C2×C23semidirect
246. #1618C2×C23semidirect
247. #1617C2×C23semidirect
248. #1616C2×C23semidirect
249. #1513C2×C23semidirect
250. #1465C2×C23semidirect
392. C23#18151direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34602
C23×#30922
C23×#24872
C23×#16182
C23×#16172
C23×#16162
C23×#15132
C23×#14652
#1815123