Finite Group Databasebeta

← C23×#18171 #19165797

Subgroups — C23×#18171 #19165797

Order 5888.

1042 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×#3492C211yesmaximal, characteristic
2C23×#3092C212yesmaximal, complemented
3C2×C23×#398C211yesmaximal, characteristic, complemented
4C23×#1611C212yesmaximal, complemented
5C23×#1610C212yesmaximal, complemented
6C23×#1609C214yesmaximal, 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×C23—21no—
22C22×C4⋊C4×C23C2211yescharacteristic
23C22×C4⋊C4×C23C2211yescharacteristic
24C23×#484—22no—
25C23×#483C2214yes—
26C23×#481—22no—
27C23×#480C2214yes—
28C23×#428—22no—
29C23×#417—22no—
30C23×#407C2212yes—
31C23×#398—21no—
32C23×#398—21no—
33C23×#398C2212yes—
34C23×#398C2212yes—
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×C23—22no—
97C2×C4⋊C4×C23—22no—
98C2×C4⋊C4×C23—44no—
99C2×C4⋊C4×C23D411yescharacteristic
100C2×C4⋊C4×C23D411yescharacteristic
101C2×C4⋊C4×C23C2312yes—
102C2×C4⋊C4×C23C2311yescharacteristic
103C2×C4⋊C4×C23C2311yescharacteristic
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×#95—21no—
113C23×#95C2312yes—
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—21no—
127C2×D4×C23C2×D412yes—
128C22×C4×C23—22no—
129C22×C4×C23—24no—
130C22×C4×C23—21no—
131C22×C4×C23—21no—
132C22×C4×C23—21no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—21no—
136C22×C4×C23—22no—
137C22×C4×C23—22no—
138C22×C4×C23—22no—
139C22×C4×C23—41no—
140C22×C4×C23—42no—
141C22×C4×C23—42no—
142C22×C4×C23—42no—
143C22×C4×C23—42no—
144C22×C4×C23—42no—
145C22×C4×C23—42no—
146C22×C4×C23—41no—
147C22×C4×C23—41no—
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D411yescharacteristic
152C22×C4×C23C2×D411yescharacteristic
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2×D412yes—
155C22×C4×C23C2411yescharacteristic
156M16×C23—44no—
157C2×C8×C23—24no—
158C4⋊C4×C23—21no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—21no—
162C4⋊C4×C23—21no—
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—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—24no—
173C4⋊C4×C23—44no—
174C4⋊C4×C23—42no—
175C4⋊C4×C23—44no—
176C4⋊C4×C23C2×D412yes—
177C4⋊C4×C23C2×D412yes—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C22⋊C4×C23—22no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C42×C23—22no—
188C42×C23—22no—
189C42×C23—22no—
190#18171C2311yesmaximal, characteristic, direct product factor, Sylow
191C23×C23—21no—
192C23×C23—21no—
193C23×C23—22no—
194C23×C23—22no—
195C23×C23—22no—
196C23×C23#12011yescharacteristic
197C23×C23#12011yescharacteristic
198C23×C23C22×D411yescharacteristic
199Q8×C23—42no—
200Q8×C23—42no—
201Q8×C23—81no—
202D4×C23—42no—
203D4×C23—42no—
204D4×C23—42no—
205D4×C23—42no—
206D4×C23—44no—
207D4×C23—44no—
208D4×C23—81no—
209C2×C4×C23—21no—
210C2×C4×C23—21no—
211C2×C4×C23—24no—
212C2×C4×C23—24no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—21no—
216C2×C4×C23—22no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—21no—
221C2×C4×C23—42no—
222C2×C4×C23—41no—
223C2×C4×C23—41no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—44no—
227C2×C4×C23—48no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—41no—
232C2×C4×C23—41no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—82no—
237C2×C4×C23—81no—
238C2×C4×C23—82no—
239C2×C4×C23—82no—
240C2×C4×C23—82no—
241C2×C4×C23#12012yes—
242C2×C4×C23AGL(1,ℤ/8)12yes—
243C2×C4×C23C22×D412yes—
244C8×C23—44no—
245#3492C2×C2311yescharacteristic
246#3092C2×C2312yescomplemented
247C2×#398C2×C2311yescharacteristic, complemented
248#1611C2×C2312yescomplemented
249#1610C2×C2312yescomplemented
250#1609C2×C2314yescomplemented
251#1504C2×C2312yescomplemented
252#1463C2×C2311yescharacteristic, complemented
253C22×C23—24no—
254C22×C23—24no—
255C22×C23—21no—
256C22×C23—21no—
257C22×C23—21no—
258C22×C23—21no—
259C22×C23—21no—
260C22×C23—21no—
261C22×C23—42no—
262C22×C23—42no—
263C22×C23—42no—
264C22×C23#45612yes—
265C22×C23#45812yes—
266C22×C23C2×#12011yescharacteristic
267C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C23#57511yescharacteristic
269C4×C23—22no—
270C4×C23—22no—
271C4×C23—22no—
272C4×C23—44no—
273C4×C23—41no—
274C4×C23—42no—
275C4×C23—42no—
276C4×C23—41no—
277C4×C23—41no—
278C4×C23—42no—
279C4×C23—82no—
280#554—22no—
281#553—22no—
282#552—22no—
283#547—22no—
284#544C22×C2311yescharacteristic
285#539—21no—
286#539C22×C2312yes—
287#537C22×C2311yescharacteristic
288#525—21no—
289#525—21no—
290#525C22×C2312yes—
291#520—21no—
292C22×C4⋊C4—21no—
293C22×C4⋊C4C22×C2311yescharacteristic
294C22×C4⋊C4C22×C2311yescharacteristic
295#484—22no—
296#483C22×C2314yes—
297#481—22no—
298#480C22×C2314yes—
299#428—22no—
300#417—22no—
301#407C22×C2312yes—
302#398—21no—
303#398—21no—
304#398C22×C2312yes—
305#398C22×C2312yes—
306C2×#95—21no—
307C2×#95C22×C2311yescharacteristic
308#328C22×C2314yes—
309#327C22×C2314yes—
310#324C22×C2312yes—
311#324C22×C2312yes—
312C2×C23—21no—
313C2×C23—21no—
314C2×C23—22no—
315C2×C23—22no—
316C2×C23—82no—
317C2×C23#300611yescharacteristic
318C2×C23C2×#45611yescharacteristic
319C2×C23C2×#45811yescharacteristic
320C2×#41—21no—
321C2×#41—21no—
322C2×#41D4×C2312yes—
323C23×C4—21no—
324C23×C4—21no—
325C23×C4D4×C2311yescharacteristic
326C23×C4D4×C2311yescharacteristic
327C23×C4C23×C2311yescharacteristic
328C2×M16—22no—
329C4⋊Q8—22no—
330#126—22no—
331#126—22no—
332#125—22no—
333#125—22no—
334#125—22no—
335#123—22no—
336#123—22no—
337#123—22no—
338#123—22no—
339#123—22no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8D4×C2312yes—
348C22⋊Q8C23×C2312yes—
349#121—21no—
350#121—21no—
351#121—21no—
352#121—22no—
353#121—21no—
354#121D4×C2312yes—
355#121C23×C2312yes—
356#120—22no—
357C4×Q8—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365#117—22no—
366#117—21no—
367#117—21no—
368#117D4×C2312yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—44no—
378C2×C4⋊C4D4×C2311yescharacteristic
379C2×C4⋊C4D4×C2311yescharacteristic
380C2×C4⋊C4C23×C2312yes—
381C2×C4⋊C4C23×C2311yescharacteristic
382C2×C4⋊C4C23×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#95—21no—
392#95C23×C2312yes—
393C23#1817111yescharacteristic, direct product factor, Sylow
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—21no—
407C2×D4C2×D4×C2312yes—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C2×D4×C2311yescharacteristic
429C22×C4C2×D4×C2312yes—
430C22×C4C2×D4×C2311yescharacteristic
431C22×C4C2×D4×C2311yescharacteristic
432C22×C4C2×D4×C2311yescharacteristic
433C22×C4C2×D4×C2311yescharacteristic
434C22×C4C2×D4×C2312yes—
435C22×C4C24×C2311yescharacteristic
436M16—44no—
437C2×C8—24no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4C2×D4×C2312yes—
457C4⋊C4C2×D4×C2312yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23C23×#12011yescharacteristic
476C23C23×#12011yescharacteristic
477C23C22×D4×C2311yescharacteristic
478Q8—42no—
479Q8—42no—
480Q8—81no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—44no—
486D4—44no—
487D4—81no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4C23×#12012yes—
521C2×C4C23×AGL(1,ℤ/8)12yes—
522C2×C4C22×D4×C2312yes—
523C8—44no—
524C22—24no—
525C22—24no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22C23×#45612yes—
536C22C23×#45812yes—
537C22C2×C23×#12011yescharacteristic
538C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
539C22C23×#57511yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—82no—
551C2—21no—
552C2—21no—
553C2—22no—
554C2—22no—
555C2—82no—
556C2C23×#300611yescharacteristic
557C2C2×C23×#45611yescharacteristic
558C2C2×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×#398C2semidirect
4. C23×#1611C2semidirect
5. C23×#1610C2semidirect
6. C23×#1609C2semidirect
7. C23×#1504C2semidirect
8. C23×#1463C2semidirect
190. #18171C23direct (N × H)
246. #3092C2×C23semidirect
247. C2×#398C2×C23semidirect
248. #1611C2×C23semidirect
249. #1610C2×C23semidirect
250. #1609C2×C23semidirect
251. #1504C2×C23semidirect
252. #1463C2×C23semidirect
393. C23#18171direct (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×#3982
C23×#16112
C23×#16102
C23×#16092
C23×#15042
C23×#14632
#1817123