Finite Group Databasebeta

← C23×#61798 #19209424

Subgroups — C23×#61798 #19209424

Order 5888.

1674 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3550C211yesmaximal, characteristic, complemented
2C23×#3512C211yesmaximal, characteristic
3C23×#3350C214yesmaximal, complemented
4C23×#3349C214yesmaximal, complemented
5C23×#3320C214yesmaximal, complemented
6C23×#3312C214yesmaximal, complemented
7C23×#3236C212yesmaximal, complemented
8C23×#3201C214yesmaximal, complemented
9C23×#3116C212yesmaximal, complemented
10C23×#3097C214yesmaximal, complemented
11C23×#2980C211yesmaximal, characteristic, complemented
12C23×#563—22no—
13C23×#559—21no—
14Q82×C23—21no—
15C23×#556C2214yes—
16C23×#553—24no—
17C23×#553C2214yes—
18C23×#552C2214yes—
19D4×Q8×C23C2214yes—
20C23×#542—24no—
21C23×#541—21no—
22C23×#540—21no—
23C23×#537C2212yes—
24C23×#536C2214yes—
25C23×#533C2212yes—
26C23×#532C2211yescharacteristic
27C23×#529—22no—
28C2×C23×#125C2211yescharacteristic
29C23×#520C2212yes—
30C4×#41×C23C2211yescharacteristic
31C23×#501C2214yes—
32C23×#491C2214yes—
33C23×#485C2214yes—
34C23×#485C2214yes—
35C23×#485C2214yes—
36C23×#485C2214yes—
37C23×#484C2214yes—
38C23×#484C2214yes—
39C23×#483C2214yes—
40C23×#483C2214yes—
41C23×#482C2214yes—
42C23×#479C2214yes—
43C23×#479C2214yes—
44C23×#477C2212yes—
45C23×#476C2212yes—
46C23×#470C2214yes—
47C23×#468C2214yes—
48C23×#467C2214yes—
49C23×#465C2214yes—
50C23×#462C2212yes—
51C23×#461C2212yes—
52C23×#452C2214yes—
53C23×#451C2214yes—
54C23×#448C2214yes—
55C23×#441C2214yes—
56C23×#440C2214yes—
57C23×#435C2214yes—
58C23×#435C2214yes—
59C23×#432C2212yes—
60C23×#428C2214yes—
61C23×#427C2214yes—
62C23×#419C2214yes—
63C2×C23×#103C2214yes—
64C23×#408C2214yes—
65C2×#41×C23C2311yescharacteristic
66C2×#41×C23C2312yes—
67C22×Q8×C23C2314yes—
68C2×Q16×C23C2314yes—
69C2×SD16×C23C2314yes—
70C2×M16×C23C2314yes—
71C22×C8×C23C2314yes—
72C23×C4⋊Q8—24no—
73C23×C4⋊Q8—24no—
74C23×C4⋊Q8D411yescharacteristic
75C23×C4⋊Q8C2312yes—
76C23×C4⋊Q8C2312yes—
77C23×#127D411yescharacteristic
78C23×#126—24no—
79C23×#126—24no—
80C23×#126—24no—
81C23×#126—24no—
82C23×#125—24no—
83C23×#125—22no—
84C23×#125—21no—
85C23×#125—21no—
86C23×#125C2314yes—
87C23×#124—24no—
88C23×#124—24no—
89C23×#124D412yes—
90C23×#123—24no—
91C23×#123—24no—
92C23×#123—24no—
93C23×#123—24no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8—24no—
97C23×C22⋊Q8—24no—
98C23×C22⋊Q8C2314yes—
99C23×C22⋊Q8C2314yes—
100C23×C22⋊Q8C2314yes—
101C23×C22⋊Q8C2314yes—
102C23×#121—24no—
103C23×#121—24no—
104C23×#121—22no—
105C23×#121D414yes—
106C23×#121C2314yes—
107C4×Q8×C23—22no—
108C4×Q8×C23—24no—
109C4×Q8×C23—22no—
110C4×Q8×C23C2314yes—
111C4×Q8×C23C2312yes—
112C4×D4×C23—24no—
113C4×D4×C23—24no—
114C4×D4×C23—22no—
115C4×D4×C23—22no—
116C4×D4×C23—24no—
117C4×D4×C23C2312yes—
118C4×D4×C23C2314yes—
119C23×#117—22no—
120C23×#117—24no—
121C23×#117—21no—
122C23×#117C2314yes—
123C23×#117C2314yes—
124C23×#117C2311yescharacteristic
125C2×C4⋊C4×C23—22no—
126C2×C4⋊C4×C23C2314yes—
127C2×C4⋊C4×C23C2312yes—
128C2×C42×C23—21no—
129C2×C42×C23C2311yescharacteristic
130C23×#107C2314yes—
131C23×#107C2314yes—
132C23×#107C2314yes—
133C23×#106C2314yes—
134C23×#106C2314yes—
135C23×#106C2314yes—
136C23×C4⋊C8C2312yes—
137C23×C4⋊C8C2312yes—
138C23×#103C2314yes—
139C23×#103C2314yes—
140C23×#103C2314yes—
141C23×#103C2314yes—
142C23×#103C2314yes—
143C23×#103C2314yes—
144C23×#103C2314yes—
145C23×#102C2314yes—
146C23×C22⋊C8C2314yes—
147C23×C22⋊C8C2314yes—
148C23×C22⋊C8C2314yes—
149C23×#97C2314yes—
150#41×C23—42no—
151#41×C23—42no—
152#41×C23—42no—
153C2×Q8×C23—22no—
154C2×Q8×C23—24no—
155C2×Q8×C23—44no—
156C2×Q8×C23—44no—
157C2×Q8×C23C2×D412yes—
158C2×Q8×C23C2414yes—
159C2×Q8×C23C2411yescharacteristic
160C2×D4×C23—21no—
161C2×D4×C23—24no—
162C2×D4×C23C2×D414yes—
163C2×D4×C23C2×D414yes—
164C2×D4×C23C2×D412yes—
165C2×D4×C23C2411yescharacteristic
166C22×C4×C23—22no—
167C22×C4×C23—24no—
168C22×C4×C23—24no—
169C22×C4×C23—21no—
170C22×C4×C23—22no—
171C22×C4×C23C2×D412yes—
172C22×C4×C23C2411yescharacteristic
173C22×C4×C23C2414yes—
174Q16×C23—44no—
175SD16×C23—44no—
176M16×C23—44no—
177C2×C8×C23—44no—
178C2×C8×C23C2414yes—
179C2×C8×C23C2414yes—
180C4⋊C4×C23—24no—
181C4⋊C4×C23—24no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—24no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—24no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—24no—
190C4⋊C4×C23—24no—
191C4⋊C4×C23—24no—
192C4⋊C4×C23—24no—
193C4⋊C4×C23C2×D414yes—
194C4⋊C4×C23C2414yes—
195C4⋊C4×C23C2412yes—
196C4⋊C4×C23C2414yes—
197C22⋊C4×C23—24no—
198C22⋊C4×C23—24no—
199C22⋊C4×C23—21no—
200C22⋊C4×C23—24no—
201C22⋊C4×C23—24no—
202C22⋊C4×C23—24no—
203C22⋊C4×C23—24no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—24no—
206C22⋊C4×C23C2×D414yes—
207C22⋊C4×C23C2×D414yes—
208C42×C23—22no—
209C42×C23—21no—
210C42×C23—24no—
211C42×C23—24no—
212C42×C23—24no—
213C42×C23C2×D411yescharacteristic
214C42×C23C2×D411yescharacteristic
215C42×C23C2412yes—
216#61798C2311yesmaximal, characteristic, direct product factor, Sylow
217C23×C23—21no—
218C23×C23C22×D414yes—
219C23×C23C22×D411yescharacteristic
220Q8×C23—24no—
221Q8×C23—22no—
222Q8×C23—24no—
223Q8×C23—42no—
224Q8×C23—44no—
225Q8×C23—84no—
226D4×C23—22no—
227D4×C23—44no—
228D4×C23—42no—
229D4×C23—44no—
230D4×C23—42no—
231D4×C23—84no—
232C2×C4×C23—24no—
233C2×C4×C23—24no—
234C2×C4×C23—22no—
235C2×C4×C23—22no—
236C2×C4×C23—24no—
237C2×C4×C23—21no—
238C2×C4×C23—24no—
239C2×C4×C23—24no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—82no—
243C2×C4×C23—84no—
244C2×C4×C23—82no—
245C2×C4×C23—84no—
246C2×C4×C23C22×D412yes—
247C2×C4×C23C22×D414yes—
248C2×C4×C23C22×D411yescharacteristic
249C2×C4×C23C22×D412yes—
250C2×C4×C23C2511yescharacteristic
251C8×C23—24no—
252C8×C23—24no—
253#3550C2×C2311yescharacteristic, complemented
254#3512C2×C2311yescharacteristic
255#3350C2×C2314yescomplemented
256#3349C2×C2314yescomplemented
257#3320C2×C2314yescomplemented
258#3312C2×C2314yescomplemented
259#3236C2×C2312yescomplemented
260#3201C2×C2314yescomplemented
261#3116C2×C2312yescomplemented
262#3097C2×C2314yescomplemented
263#2980C2×C2311yescharacteristic, complemented
264C22×C23—22no—
265C22×C23—24no—
266C22×C23—24no—
267C22×C23—24no—
268C22×C23—21no—
269C22×C23—42no—
270C22×C23—41no—
271C22×C23C23×D411yescharacteristic
272C4×C23—22no—
273C4×C23—22no—
274C4×C23—21no—
275C4×C23—24no—
276C4×C23—42no—
277C4×C23—44no—
278C4×C23—44no—
279C4×C23—41no—
280C4×C23—44no—
281C4×C23#58512yes—
282#563—22no—
283#559—21no—
284Q82—21no—
285#556C22×C2314yes—
286#553—24no—
287#553C22×C2314yes—
288#552C22×C2314yes—
289D4×Q8C22×C2314yes—
290#542—24no—
291#541—21no—
292#540—21no—
293#537C22×C2312yes—
294#536C22×C2314yes—
295#533C22×C2312yes—
296#532C22×C2311yescharacteristic
297#529—22no—
298C2×#125C22×C2311yescharacteristic
299#520C22×C2312yes—
300C4×#41C22×C2311yescharacteristic
301#501C22×C2314yes—
302#491C22×C2314yes—
303#485C22×C2314yes—
304#485C22×C2314yes—
305#485C22×C2314yes—
306#485C22×C2314yes—
307#484C22×C2314yes—
308#484C22×C2314yes—
309#483C22×C2314yes—
310#483C22×C2314yes—
311#482C22×C2314yes—
312#479C22×C2314yes—
313#479C22×C2314yes—
314#477C22×C2312yes—
315#476C22×C2312yes—
316#470C22×C2314yes—
317#468C22×C2314yes—
318#467C22×C2314yes—
319#465C22×C2314yes—
320#462C22×C2312yes—
321#461C22×C2312yes—
322#452C22×C2314yes—
323#451C22×C2314yes—
324#448C22×C2314yes—
325#441C22×C2314yes—
326#440C22×C2314yes—
327#435C22×C2314yes—
328#435C22×C2314yes—
329#432C22×C2312yes—
330#428C22×C2314yes—
331#427C22×C2314yes—
332#419C22×C2314yes—
333C2×#103C22×C2314yes—
334#408C22×C2314yes—
335C2×C23—41no—
336C2×C23—44no—
337C2×C23—81no—
338C2×C23#348511yescharacteristic
339C2×C23#358612yes—
340C2×#41C23×C2311yescharacteristic
341C2×#41C23×C2312yes—
342C22×Q8C23×C2314yes—
343C2×Q16C23×C2314yes—
344C2×SD16C23×C2314yes—
345C2×M16C23×C2314yes—
346C22×C8C23×C2314yes—
347C4⋊Q8—24no—
348C4⋊Q8—24no—
349C4⋊Q8D4×C2311yescharacteristic
350C4⋊Q8C23×C2312yes—
351C4⋊Q8C23×C2312yes—
352#127D4×C2311yescharacteristic
353#126—24no—
354#126—24no—
355#126—24no—
356#126—24no—
357#125—24no—
358#125—21no—
359#125—21no—
360#125—22no—
361#125C23×C2314yes—
362#124—24no—
363#124—24no—
364#124D4×C2312yes—
365#123—24no—
366#123—24no—
367#123—24no—
368#123—24no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—24no—
372C22⋊Q8—24no—
373C22⋊Q8C23×C2314yes—
374C22⋊Q8C23×C2314yes—
375C22⋊Q8C23×C2314yes—
376C22⋊Q8C23×C2314yes—
377#121—22no—
378#121—24no—
379#121—24no—
380#121D4×C2314yes—
381#121C23×C2314yes—
382C4×Q8—22no—
383C4×Q8—24no—
384C4×Q8—22no—
385C4×Q8C23×C2312yes—
386C4×Q8C23×C2314yes—
387C4×D4—22no—
388C4×D4—24no—
389C4×D4—24no—
390C4×D4—22no—
391C4×D4—24no—
392C4×D4C23×C2312yes—
393C4×D4C23×C2314yes—
394#117—22no—
395#117—24no—
396#117—21no—
397#117C23×C2311yescharacteristic
398#117C23×C2314yes—
399#117C23×C2314yes—
400C2×C4⋊C4—22no—
401C2×C4⋊C4C23×C2312yes—
402C2×C4⋊C4C23×C2314yes—
403C2×C42—21no—
404C2×C42C23×C2311yescharacteristic
405#107C23×C2314yes—
406#107C23×C2314yes—
407#107C23×C2314yes—
408#106C23×C2314yes—
409#106C23×C2314yes—
410#106C23×C2314yes—
411C4⋊C8C23×C2312yes—
412C4⋊C8C23×C2312yes—
413#103C23×C2314yes—
414#103C23×C2314yes—
415#103C23×C2314yes—
416#103C23×C2314yes—
417#103C23×C2314yes—
418#103C23×C2314yes—
419#103C23×C2314yes—
420#102C23×C2314yes—
421C22⋊C8C23×C2314yes—
422C22⋊C8C23×C2314yes—
423C22⋊C8C23×C2314yes—
424#97C23×C2314yes—
425C23#6179811yescharacteristic, direct product factor, Sylow
426#41—42no—
427#41—42no—
428#41—42no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8C2×D4×C2312yes—
434C2×Q8C24×C2314yes—
435C2×Q8C24×C2311yescharacteristic
436C2×D4—21no—
437C2×D4—24no—
438C2×D4C2×D4×C2314yes—
439C2×D4C2×D4×C2314yes—
440C2×D4C2×D4×C2312yes—
441C2×D4C24×C2311yescharacteristic
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4C2×D4×C2312yes—
448C22×C4C24×C2311yescharacteristic
449C22×C4C24×C2314yes—
450Q16—44no—
451SD16—44no—
452M16—44no—
453C2×C8—44no—
454C2×C8C24×C2314yes—
455C2×C8C24×C2314yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4C2×D4×C2314yes—
470C4⋊C4C24×C2314yes—
471C4⋊C4C24×C2312yes—
472C4⋊C4C24×C2314yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—24no—
482C22⋊C4C2×D4×C2314yes—
483C22⋊C4C2×D4×C2314yes—
484C42—22no—
485C42—21no—
486C42—24no—
487C42—24no—
488C42—24no—
489C42C2×D4×C2311yescharacteristic
490C42C2×D4×C2311yescharacteristic
491C42C24×C2312yes—
492C23—21no—
493C23C22×D4×C2314yes—
494C23C22×D4×C2311yescharacteristic
495Q8—24no—
496Q8—22no—
497Q8—24no—
498Q8—42no—
499Q8—44no—
500Q8—84no—
501D4—22no—
502D4—42no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—84no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4C22×D4×C2312yes—
522C2×C4C22×D4×C2314yes—
523C2×C4C22×D4×C2311yescharacteristic
524C2×C4C22×D4×C2312yes—
525C2×C4C25×C2311yescharacteristic
526C8—24no—
527C8—24no—
528C22—24no—
529C22—21no—
530C22—22no—
531C22—24no—
532C22—24no—
533C22—41no—
534C22—42no—
535C22C23×D4×C2311yescharacteristic
536C4—21no—
537C4—24no—
538C4—22no—
539C4—22no—
540C4—41no—
541C4—44no—
542C4—42no—
543C4—44no—
544C4—44no—
545C4C23×#58512yes—
546C2—44no—
547C2—41no—
548C2—81no—
549C2C23×#348511yescharacteristic
550C2C23×#358612yes—
subgroupG/H
center, socleC22×C23C23×D4
commutator, Frattini, cosocleC2×C4C25×C23
prime radicalC23×#3512C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×#532 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3550C2semidirect
3. C23×#3350C2semidirect
4. C23×#3349C2semidirect
5. C23×#3320C2semidirect
6. C23×#3312C2semidirect
7. C23×#3236C2semidirect
8. C23×#3201C2semidirect
9. C23×#3116C2semidirect
10. C23×#3097C2semidirect
11. C23×#2980C2semidirect
216. #61798C23direct (N × H)
253. #3550C2×C23semidirect
255. #3350C2×C23semidirect
256. #3349C2×C23semidirect
257. #3320C2×C23semidirect
258. #3312C2×C23semidirect
259. #3236C2×C23semidirect
260. #3201C2×C23semidirect
261. #3116C2×C23semidirect
262. #3097C2×C23semidirect
263. #2980C2×C23semidirect
425. C23#61798direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35502
C23×#35122
C23×#33502
C23×#33492
C23×#33202
C23×#33122
C23×#32362
C23×#32012
C23×#31162
C23×#30972
C23×#29802
#6179823