Finite Group Databasebeta

← C23×#18433 #19166059

Subgroups — C23×#18433 #19166059

Order 5888.

872 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic
2C23×#3218C211yesmaximal, characteristic
3C23×#3209C211yesmaximal, characteristic, complemented
4C23×#2660C211yesmaximal, characteristic, complemented
5C23×#1605C214yesmaximal, complemented
6C23×#1603C214yesmaximal
7C23×#1506C211yesmaximal, characteristic, complemented
8C23×#1500C211yesmaximal, characteristic
9C23×#1470C211yesmaximal, characteristic, complemented
10C23×#564—21no—
11C23×#563—21no—
12C23×#557—21no—
13C23×#554—21no—
14C23×#552—21no—
15C23×#552—21no—
16C23×#552—22no—
17C23×#544—21no—
18C23×#544C2212yes—
19C23×#541—21no—
20C23×#541—22no—
21C23×#539C2212yes—
22C23×#533C2211yescharacteristic
23C2×C23×C4⋊Q8C2211yescharacteristic
24C23×#529C2211yescharacteristic
25C2×C23×#125C2211yescharacteristic
26C23×#497—21no—
27C23×#494—21no—
28C23×#491—21no—
29C23×#487—21no—
30C23×#462—22no—
31C23×#462—21no—
32C23×#461—21no—
33C23×#451C2214yes—
34C23×#451C2212yes—
35C23×#448C2212yes—
36C23×#432C2211yescharacteristic
37C23×#431C2211yescharacteristic
38C23×#398—22no—
39C23×#395C2214yes—
40C23×#384—21no—
41C23×#382—21no—
42C23×#376C2211yescharacteristic
43C23×#328C2212yes—
44C23×#328C2214yes—
45C23×#327C2212yes—
46C23×#324C2214yes—
47C22×Q8×C23C2312yes—
48C2×Q16×C23—24no—
49C2×Q16×C23—22no—
50C2×SD16×C23—22no—
51C23×C4⋊Q8—21no—
52C23×C4⋊Q8—22no—
53C23×C4⋊Q8—22no—
54C23×C4⋊Q8—21no—
55C23×C4⋊Q8—21no—
56C23×C4⋊Q8—21no—
57C23×C4⋊Q8—21no—
58C23×C4⋊Q8D411yescharacteristic
59C23×C4⋊Q8D411yescharacteristic
60C23×#126—22no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#126—21no—
65C23×#125—22no—
66C23×#125—21no—
67C23×#125—22no—
68C23×#125—22no—
69C23×#125—22no—
70C23×#125—22no—
71C23×#125—22no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—21no—
75C23×#125—21no—
76C23×#125—21no—
77C23×#125—21no—
78C23×#125—21no—
79C23×#125D411yescharacteristic
80C23×#124D411yescharacteristic
81C23×#123—22no—
82C23×#123—21no—
83C23×#123—22no—
84C23×C22⋊Q8—21no—
85C23×C22⋊Q8—21no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8C2311yescharacteristic
92C23×C22⋊Q8C2312yes—
93C23×#121C2311yescharacteristic
94C4×Q8×C23—21no—
95C4×Q8×C23—22no—
96C4×Q8×C23—21no—
97C4×Q8×C23—21no—
98C4×Q8×C23D411yescharacteristic
99C4×Q8×C23D412yes—
100C4×D4×C23—21no—
101C4×D4×C23—22no—
102C4×D4×C23D411yescharacteristic
103C23×#117—22no—
104C23×#117—21no—
105C23×#117—21no—
106C23×#117—22no—
107C23×#117—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23—24no—
112C2×C4⋊C4×C23D411yescharacteristic
113C2×C4⋊C4×C23D412yes—
114C2×C4⋊C4×C23D411yescharacteristic
115C2×C4⋊C4×C23C2312yes—
116C2×C42×C23—21no—
117C2×C42×C23C2311yescharacteristic
118C23×C4⋊C8—21no—
119C23×C4⋊C8—21no—
120C23×#103—22no—
121C23×#103—22no—
122C23×#103—24no—
123C23×C22⋊C8C2312yes—
124C23×C22⋊C8C2312yes—
125C23×#97—21no—
126C4×C8×C23—21no—
127C23×#95—21no—
128C23×#95—24no—
129C23×#95C2312yes—
130C2×Q8×C23—22no—
131C2×Q8×C23—21no—
132C2×Q8×C23—22no—
133C2×Q8×C23—22no—
134C2×Q8×C23—82no—
135C2×Q8×C23C2×D412yes—
136C2×Q8×C23C2×D411yescharacteristic
137C2×D4×C23—21no—
138C2×D4×C23C2×D411yescharacteristic
139C22×C4×C23—22no—
140C22×C4×C23—22no—
141C22×C4×C23—21no—
142C22×C4×C23—22no—
143C22×C4×C23C2×D412yes—
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D411yescharacteristic
146C22×C4×C23C2×D412yes—
147C22×C4×C23C2411yescharacteristic
148Q16×C23—82no—
149Q16×C23—84no—
150SD16×C23—82no—
151C2×C8×C23—22no—
152C2×C8×C23—22no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—22no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—21no—
162C4⋊C4×C23—21no—
163C4⋊C4×C23—21no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—21no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—21no—
168C4⋊C4×C23—21no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—41no—
179C4⋊C4×C23—41no—
180C4⋊C4×C23—41no—
181C4⋊C4×C23—41no—
182C4⋊C4×C23—84no—
183C4⋊C4×C23C2×D411yescharacteristic
184C4⋊C4×C23C2×D411yescharacteristic
185C4⋊C4×C23C2×D412yes—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—21no—
191C22⋊C4×C23—21no—
192C22⋊C4×C23—21no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—21no—
195C42×C23—24no—
196C42×C23—22no—
197C42×C23—22no—
198C42×C23—22no—
199C42×C23—22no—
200C42×C23—21no—
201C42×C23C2×D411yescharacteristic
202C42×C23C2×D411yescharacteristic
203C42×C23C2×D411yescharacteristic
204C42×C23C2×D411yescharacteristic
205#18433C2311yesmaximal, characteristic, direct product factor, Sylow
206C23×C23—21no—
207C23×C23C22×D411yescharacteristic
208Q8×C23—41no—
209Q8×C23—42no—
210Q8×C23—42no—
211Q8×C23—42no—
212Q8×C23—44no—
213Q8×C23—42no—
214Q8×C23—82no—
215D4×C23—41no—
216D4×C23—42no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
219C2×C4×C23—24no—
220C2×C4×C23—22no—
221C2×C4×C23—21no—
222C2×C4×C23—21no—
223C2×C4×C23—22no—
224C2×C4×C23—21no—
225C2×C4×C23—22no—
226C2×C4×C23—22no—
227C2×C4×C23—24no—
228C2×C4×C23—41no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—44no—
234C2×C4×C23—42no—
235C2×C4×C23—82no—
236C2×C4×C23#12011yescharacteristic
237C2×C4×C23#12011yescharacteristic
238C2×C4×C23#12011yescharacteristic
239C2×C4×C23#12011yescharacteristic
240C2×C4×C23C22×D411yescharacteristic
241C2×C4×C23C22×D411yescharacteristic
242C8×C23—42no—
243C8×C23—42no—
244#3563C2×C2311yescharacteristic
245#3218C2×C2311yescharacteristic
246#3209C2×C2311yescharacteristic, complemented
247#2660C2×C2311yescharacteristic, complemented
248#1605C2×C2314yescomplemented
249#1603C2×C2314yes—
250#1506C2×C2311yescharacteristic, complemented
251#1500C2×C2311yescharacteristic
252#1470C2×C2311yescharacteristic, complemented
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—41no—
256C22×C23—41no—
257C22×C23—41no—
258C22×C23C2×#12011yescharacteristic
259C22×C23#57812yes—
260C4×C23—21no—
261C4×C23—21no—
262C4×C23—21no—
263C4×C23—21no—
264C4×C23—21no—
265C4×C23—44no—
266C4×C23—42no—
267C4×C23—41no—
268C4×C23—41no—
269C4×C23—42no—
270C4×C23—42no—
271C4×C23—42no—
272C4×C23—41no—
273C4×C23—82no—
274C4×C23#45612yes—
275#564—21no—
276#563—21no—
277#557—21no—
278#554—21no—
279#552—21no—
280#552—21no—
281#552—22no—
282#544—21no—
283#544C22×C2312yes—
284#541—21no—
285#541—22no—
286#539C22×C2312yes—
287#533C22×C2311yescharacteristic
288C2×C4⋊Q8C22×C2311yescharacteristic
289#529C22×C2311yescharacteristic
290C2×#125C22×C2311yescharacteristic
291#497—21no—
292#494—21no—
293#491—21no—
294#487—21no—
295#462—22no—
296#462—21no—
297#461—21no—
298#451C22×C2314yes—
299#451C22×C2312yes—
300#448C22×C2312yes—
301#432C22×C2311yescharacteristic
302#431C22×C2311yescharacteristic
303#398—22no—
304#395C22×C2314yes—
305#384—21no—
306#382—21no—
307#376C22×C2311yescharacteristic
308#328C22×C2314yes—
309#328C22×C2312yes—
310#327C22×C2312yes—
311#324C22×C2314yes—
312C2×C23—22no—
313C2×C23—81no—
314C2×C23#301111yescharacteristic
315C2×C23C2×#45611yescharacteristic
316C2×C23#302711yescharacteristic
317C22×Q8C23×C2312yes—
318C2×Q16—24no—
319C2×Q16—22no—
320C2×SD16—22no—
321C4⋊Q8—21no—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8D4×C2311yescharacteristic
329C4⋊Q8D4×C2311yescharacteristic
330#126—22no—
331#126—22no—
332#126—22no—
333#126—22no—
334#126—21no—
335#125—22no—
336#125—22no—
337#125—22no—
338#125—22no—
339#125—21no—
340#125—22no—
341#125—22no—
342#125—22no—
343#125—22no—
344#125—21no—
345#125—21no—
346#125—21no—
347#125—21no—
348#125—21no—
349#125D4×C2311yescharacteristic
350#124D4×C2311yescharacteristic
351#123—22no—
352#123—21no—
353#123—22no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—21no—
360C22⋊Q8—22no—
361C22⋊Q8C23×C2311yescharacteristic
362C22⋊Q8C23×C2312yes—
363#121C23×C2311yescharacteristic
364C4×Q8—21no—
365C4×Q8—22no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8D4×C2311yescharacteristic
369C4×Q8D4×C2312yes—
370C4×D4—21no—
371C4×D4—22no—
372C4×D4D4×C2311yescharacteristic
373#117—22no—
374#117—21no—
375#117—21no—
376#117—22no—
377#117—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4D4×C2311yescharacteristic
383C2×C4⋊C4D4×C2312yes—
384C2×C4⋊C4D4×C2311yescharacteristic
385C2×C4⋊C4C23×C2312yes—
386C2×C42—21no—
387C2×C42C23×C2311yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—21no—
390#103—22no—
391#103—22no—
392#103—24no—
393C22⋊C8C23×C2312yes—
394C22⋊C8C23×C2312yes—
395#97—21no—
396C4×C8—21no—
397#95—21no—
398#95—24no—
399#95C23×C2312yes—
400C23#1843311yescharacteristic, direct product factor, Sylow
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—82no—
406C2×Q8C2×D4×C2312yes—
407C2×Q8C2×D4×C2311yescharacteristic
408C2×D4—21no—
409C2×D4C2×D4×C2311yescharacteristic
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4C2×D4×C2312yes—
415C22×C4C2×D4×C2311yescharacteristic
416C22×C4C2×D4×C2311yescharacteristic
417C22×C4C2×D4×C2312yes—
418C22×C4C24×C2311yescharacteristic
419Q16—82no—
420Q16—84no—
421SD16—82no—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—84no—
454C4⋊C4C2×D4×C2311yescharacteristic
455C4⋊C4C2×D4×C2311yescharacteristic
456C4⋊C4C2×D4×C2312yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—21no—
472C42C2×D4×C2311yescharacteristic
473C42C2×D4×C2311yescharacteristic
474C42C2×D4×C2311yescharacteristic
475C42C2×D4×C2311yescharacteristic
476C23—21no—
477C23C22×D4×C2311yescharacteristic
478Q8—41no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—44no—
483Q8—42no—
484Q8—82no—
485D4—41no—
486D4—42no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4C23×#12011yescharacteristic
507C2×C4C23×#12011yescharacteristic
508C2×C4C23×#12011yescharacteristic
509C2×C4C23×#12011yescharacteristic
510C2×C4C22×D4×C2311yescharacteristic
511C2×C4C22×D4×C2311yescharacteristic
512C8—42no—
513C8—42no—
514C22—21no—
515C22—21no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×C23×#12011yescharacteristic
520C22C23×#57812yes—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—44no—
527C4—42no—
528C4—41no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—41no—
534C4—82no—
535C4C23×#45612yes—
536C2—22no—
537C2—81no—
538C2C23×#301111yescharacteristic
539C2C2×C23×#45611yescharacteristic
540C2C23×#302711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#3209C2semidirect
4. C23×#2660C2semidirect
5. C23×#1605C2semidirect
7. C23×#1506C2semidirect
9. C23×#1470C2semidirect
205. #18433C23direct (N × H)
246. #3209C2×C23semidirect
247. #2660C2×C23semidirect
248. #1605C2×C23semidirect
250. #1506C2×C23semidirect
252. #1470C2×C23semidirect
400. C23#18433direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#32182
C23×#32092
C23×#26602
C23×#16052
C23×#16032
C23×#15062
C23×#15002
C23×#14702
#1843323