Finite Group Databasebeta

← C23×#8483 #19156109

Subgroups — C23×#8483 #19156109

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3558C211yesmaximal, characteristic
2C23×#1967C211yesmaximal, characteristic, complemented
3C23×#1557C211yesmaximal, characteristic, complemented
4C23×#1529C211yesmaximal, characteristic, complemented
5C23×#1495C211yesmaximal, characteristic, complemented
6C23×#1484C211yesmaximal, characteristic, complemented
7C23×#1469C211yesmaximal, characteristic, complemented
8C23×#562—21no—
9C23×H(3,𝔽4)—21no—
10C23×#559—21no—
11C23×#548—21no—
12C23×#546—21no—
13C23×#546—22no—
14D42×C23—21no—
15C23×#542—21no—
16C23×#538—22no—
17C23×#538C412yes—
18C23×#535—21no—
19C23×#534C412yes—
20C23×#532C2211yescharacteristic
21C23×#529C2211yescharacteristic
22C2×C23×#124C2211yescharacteristic
23C23×#432C2211yescharacteristic, complemented
24C23×#431C2211yescharacteristic, complemented
25C23×#423C2211yescharacteristic, complemented
26C2×C23×#101—21no—
27C2×C23×#100—21no—
28C4×M16×C23C2211yescharacteristic, complemented
29C23×#332—21no—
30C23×#331—21no—
31C23×#330—21no—
32C23×#329—21no—
33C23×Q8⋊C8—21no—
34C23×D4⋊C8—22no—
35C23×D4⋊C8—21no—
36C23×#323—22no—
37C2×#41×C23—21no—
38C2×#41×C23—21no—
39C22×Q8×C23D411yescharacteristic
40C22×D4×C23—21no—
41C22×D4×C23—22no—
42C22×D4×C23D411yescharacteristic
43C2×M16×C23—21no—
44C2×M16×C23—21no—
45C23×C4⋊Q8D411yescharacteristic
46C23×#127—21no—
47C23×#127D411yescharacteristic
48C23×#126—22no—
49C23×#126—21no—
50C23×#125D411yescharacteristic
51C23×#124—21no—
52C23×#124—22no—
53C23×#124—22no—
54C23×#124—22no—
55C23×#124—21no—
56C23×#124—21no—
57C23×#124—21no—
58C23×#124—21no—
59C23×#124—21no—
60C23×#124D411yescharacteristic
61C23×#123—22no—
62C23×#123—21no—
63C23×#123—22no—
64C23×#123—22no—
65C23×#123—22no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—21no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8C2×C411yescharacteristic
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—22no—
73C23×#121—21no—
74C23×#121—22no—
75C23×#121—22no—
76C23×#121—21no—
77C23×#121C2×C411yescharacteristic
78C23×#121C2×C412yes—
79C23×#120—22no—
80C23×#120—22no—
81C23×#120—22no—
82C23×#120—22no—
83C23×#120—22no—
84C4×Q8×C23—21no—
85C4×Q8×C23D411yescharacteristic
86C4×D4×C23—22no—
87C4×D4×C23—21no—
88C4×D4×C23—22no—
89C4×D4×C23—21no—
90C4×D4×C23—21no—
91C4×D4×C23D411yescharacteristic
92C4×D4×C23D412yes—
93C23×#117—21no—
94C2×C22⋊C4×C23—22no—
95C2×C22⋊C4×C23C2×C412yes—
96C2×C22⋊C4×C23D412yes—
97C2×C42×C23C2311yescharacteristic
98C23×C4⋊C8—21no—
99C23×C4⋊C8—21no—
100C23×C4⋊C8—21no—
101C23×C4⋊C8—21no—
102C23×#101—42no—
103C23×#100—42no—
104C23×C22⋊C8—21no—
105C23×C22⋊C8—21no—
106C23×#97—21no—
107C23×#97—21no—
108C4×C8×C23—21no—
109C4×C8×C23—21no—
110C24×C23—22no—
111C24×C23C22⋊C412yes—
112#41×C23—82no—
113#41×C23—81no—
114#41×C23—81no—
115C2×Q8×C23—21no—
116C2×Q8×C23—21no—
117C2×Q8×C23—21no—
118C2×Q8×C23—41no—
119C2×Q8×C23—41no—
120C2×Q8×C23C22⋊C411yescharacteristic
121C2×D4×C23—22no—
122C2×D4×C23—21no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—22no—
128C2×D4×C23—21no—
129C2×D4×C23—21no—
130C2×D4×C23—21no—
131C2×D4×C23—21no—
132C2×D4×C23—41no—
133C2×D4×C23—41no—
134C2×D4×C23—81no—
135C2×D4×C23—81no—
136C2×D4×C23—84no—
137C2×D4×C23C22⋊C411yescharacteristic
138C2×D4×C23C22⋊C412yes—
139C22×C4×C23—22no—
140C22×C4×C23—22no—
141C22×C4×C23—21no—
142C22×C4×C23—22no—
143C22×C4×C23C22⋊C412yes—
144C22×C4×C23C22×C411yescharacteristic
145C22×C4×C23C2×D411yescharacteristic
146C22×C4×C23C2×D411yescharacteristic
147M16×C23—22no—
148M16×C23—41no—
149M16×C23—41no—
150M16×C23—41no—
151C2×C8×C23—41no—
152C2×C8×C23—41no—
153C2×C8×C23—41no—
154C2×C8×C23—41no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—22no—
157C4⋊C4×C23—21no—
158C4⋊C4×C23—21no—
159C4⋊C4×C23—21no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—21no—
162C4⋊C4×C23C22⋊C412yes—
163C4⋊C4×C23C22⋊C411yescharacteristic
164C4⋊C4×C23C22⋊C411yescharacteristic
165C22⋊C4×C23—22no—
166C22⋊C4×C23—22no—
167C22⋊C4×C23—22no—
168C22⋊C4×C23—22no—
169C22⋊C4×C23—22no—
170C22⋊C4×C23—21no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—21no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—21no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—21no—
179C22⋊C4×C23—21no—
180C22⋊C4×C23—22no—
181C42×C23—22no—
182C42×C23—21no—
183C42×C23C2×D411yescharacteristic
184C42×C23C2×D411yescharacteristic
185C42×C23C2×D411yescharacteristic
186C42×C23C2×D411yescharacteristic
187#8483C2311yesmaximal, characteristic, direct product factor, Sylow
188C23×C23—22no—
189C23×C23—21no—
190C23×C23—22no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—82no—
195C23×C23—82no—
196C23×C23—82no—
197C23×C23C2×C22⋊C411yescharacteristic
198Q8×C23—41no—
199Q8×C23—41no—
200Q8×C23—41no—
201Q8×C23—81no—
202Q8×C23—81no—
203D4×C23—42no—
204D4×C23—41no—
205D4×C23—42no—
206D4×C23—44no—
207D4×C23—41no—
208D4×C23—41no—
209D4×C23—41no—
210D4×C23—41no—
211D4×C23—81no—
212D4×C23—81no—
213D4×C23—82no—
214D4×C23—82no—
215D4×C23—82no—
216D4×C23—82no—
217D4×C23—82no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—21no—
221C2×C4×C23—22no—
222C2×C4×C23—22no—
223C2×C4×C23—21no—
224C2×C4×C23—21no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—42no—
230C2×C4×C23—82no—
231C2×C4×C23—82no—
232C2×C4×C23—82no—
233C2×C4×C23C2×C22⋊C411yescharacteristic
234C2×C4×C23C2×C22⋊C411yescharacteristic
235C2×C4×C23#12011yescharacteristic
236C2×C4×C23#12011yescharacteristic
237C2×C4×C23#12011yescharacteristic
238C2×C4×C23#12011yescharacteristic
239C8×C23—42no—
240C8×C23—81no—
241C8×C23—81no—
242C8×C23—81no—
243#3558C2×C2311yescharacteristic
244#1967C2×C2311yescharacteristic, complemented
245#1557C2×C2311yescharacteristic, complemented
246#1529C2×C2311yescharacteristic, complemented
247#1495C2×C2311yescharacteristic, complemented
248#1484C2×C2311yescharacteristic, complemented
249#1469C2×C2311yescharacteristic, complemented
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—41no—
254C22×C23—42no—
255C22×C23—42no—
256C22×C23—42no—
257C22×C23—42no—
258C22×C23—41no—
259C22×C23—42no—
260C22×C23—41no—
261C22×C23—42no—
262C22×C23—162no—
263C22×C23—164no—
264C22×C23#37911yescharacteristic
265C4×C23—21no—
266C4×C23—21no—
267C4×C23—21no—
268C4×C23—21no—
269C4×C23—21no—
270C4×C23—41no—
271C4×C23—42no—
272C4×C23—41no—
273C4×C23—42no—
274C4×C23—41no—
275C4×C23#45311yescharacteristic
276C4×C23#45511yescharacteristic
277#562—21no—
278H(3,𝔽4)—21no—
279#559—21no—
280#548—21no—
281#546—21no—
282#546—22no—
283D42—21no—
284#542—21no—
285#538—22no—
286#538C4×C2312yes—
287#535—21no—
288#534C4×C2312yes—
289#532C22×C2311yescharacteristic
290#529C22×C2311yescharacteristic
291C2×#124C22×C2311yescharacteristic
292#432C22×C2311yescharacteristic, complemented
293#431C22×C2311yescharacteristic, complemented
294#423C22×C2311yescharacteristic, complemented
295C2×#101—21no—
296C2×#100—21no—
297C4×M16C22×C2311yescharacteristic, complemented
298#332—21no—
299#331—21no—
300#330—21no—
301#329—21no—
302Q8⋊C8—21no—
303D4⋊C8—22no—
304D4⋊C8—21no—
305#323—22no—
306C2×C23—41no—
307C2×C23—82no—
308C2×C23—82no—
309C2×C23—81no—
310C2×C23#179111yescharacteristic
311C2×C23#179211yescharacteristic
312C2×C23#179511yescharacteristic
313C2×#41—21no—
314C2×#41—21no—
315C22×Q8D4×C2311yescharacteristic
316C22×D4—21no—
317C22×D4—22no—
318C22×D4D4×C2311yescharacteristic
319C2×M16—21no—
320C2×M16—21no—
321C4⋊Q8D4×C2311yescharacteristic
322#127—21no—
323#127D4×C2311yescharacteristic
324#126—22no—
325#126—21no—
326#125D4×C2311yescharacteristic
327#124—21no—
328#124—22no—
329#124—22no—
330#124—22no—
331#124—21no—
332#124—21no—
333#124—21no—
334#124—21no—
335#124—21no—
336#124D4×C2311yescharacteristic
337#123—22no—
338#123—21no—
339#123—22no—
340#123—22no—
341#123—22no—
342C22⋊Q8—22no—
343C22⋊Q8—21no—
344C22⋊Q8—22no—
345C22⋊Q8C2×C4×C2311yescharacteristic
346#121—22no—
347#121—22no—
348#121—22no—
349#121—21no—
350#121—22no—
351#121—22no—
352#121—21no—
353#121C2×C4×C2311yescharacteristic
354#121C2×C4×C2312yes—
355#120—22no—
356#120—22no—
357#120—22no—
358#120—22no—
359#120—22no—
360C4×Q8—21no—
361C4×Q8D4×C2311yescharacteristic
362C4×D4—22no—
363C4×D4—21no—
364C4×D4—22no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4D4×C2311yescharacteristic
368C4×D4D4×C2312yes—
369#117—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C2×C4×C2312yes—
372C2×C22⋊C4D4×C2312yes—
373C2×C42C23×C2311yescharacteristic
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378#101—42no—
379#100—42no—
380C22⋊C8—21no—
381C22⋊C8—21no—
382#97—21no—
383#97—21no—
384C4×C8—21no—
385C4×C8—21no—
386C23#848311yescharacteristic, direct product factor, Sylow
387C24—22no—
388C24C22⋊C4×C2312yes—
389#41—82no—
390#41—81no—
391#41—81no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8C22⋊C4×C2311yescharacteristic
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—84no—
414C2×D4C22⋊C4×C2311yescharacteristic
415C2×D4C22⋊C4×C2312yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4C22⋊C4×C2312yes—
421C22×C4C22×C4×C2311yescharacteristic
422C22×C4C2×D4×C2311yescharacteristic
423C22×C4C2×D4×C2311yescharacteristic
424M16—22no—
425M16—41no—
426M16—41no—
427M16—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4C22⋊C4×C2312yes—
440C4⋊C4C22⋊C4×C2311yescharacteristic
441C4⋊C4C22⋊C4×C2311yescharacteristic
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—21no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—21no—
451C22⋊C4—22no—
452C22⋊C4—21no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C42—22no—
459C42—21no—
460C42C2×D4×C2311yescharacteristic
461C42C2×D4×C2311yescharacteristic
462C42C2×D4×C2311yescharacteristic
463C42C2×D4×C2311yescharacteristic
464C23—22no—
465C23—21no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23C2×C22⋊C4×C2311yescharacteristic
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—81no—
478Q8—81no—
479D4—42no—
480D4—41no—
481D4—42no—
482D4—44no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—81no—
488D4—81no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—82no—
493D4—82no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4C2×C22⋊C4×C2311yescharacteristic
510C2×C4C2×C22⋊C4×C2311yescharacteristic
511C2×C4C23×#12011yescharacteristic
512C2×C4C23×#12011yescharacteristic
513C2×C4C23×#12011yescharacteristic
514C2×C4C23×#12011yescharacteristic
515C8—42no—
516C8—81no—
517C8—81no—
518C8—81no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—42no—
523C22—41no—
524C22—42no—
525C22—41no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—162no—
532C22—164no—
533C22C23×#37911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—42no—
544C4C23×#45311yescharacteristic
545C4C23×#45511yescharacteristic
546C2—41no—
547C2—81no—
548C2—82no—
549C2—82no—
550C2C23×#179111yescharacteristic
551C2C23×#179211yescharacteristic
552C2C23×#179511yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3558C2
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
2. C23×#1967C2semidirect
3. C23×#1557C2semidirect
4. C23×#1529C2semidirect
5. C23×#1495C2semidirect
6. C23×#1484C2semidirect
7. C23×#1469C2semidirect
23. C23×#432C22semidirect
24. C23×#431C22semidirect
25. C23×#423C22semidirect
28. C4×M16×C23C22semidirect
187. #8483C23direct (N × H)
244. #1967C2×C23semidirect
245. #1557C2×C23semidirect
246. #1529C2×C23semidirect
247. #1495C2×C23semidirect
248. #1484C2×C23semidirect
249. #1469C2×C23semidirect
292. #432C22×C23semidirect
293. #431C22×C23semidirect
294. #423C22×C23semidirect
297. C4×M16C22×C23semidirect
386. C23#8483direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35582
C23×#19672
C23×#15572
C23×#15292
C23×#14952
C23×#14842
C23×#14692
#848323