Finite Group Databasebeta

← C23×#9938 #19157564

Subgroups — C23×#9938 #19157564

Order 5888.

1008 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3006C211yesmaximal, characteristic, complemented
2C23×#2539C211yesmaximal, characteristic, complemented
3C23×#2404C211yesmaximal, characteristic, complemented
4C23×#1789C211yesmaximal, characteristic, complemented
5C23×#1510C211yesmaximal, characteristic, complemented
6C23×#1462C211yesmaximal, characteristic, complemented
7C23×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C23C2211yescharacteristic
9C2×C23×AGL(1,ℤ/8)—21no—
10C2×C23×#123—21no—
11C2×C23×#121C2211yescharacteristic
12C2×C23×#120—21no—
13C2×C23×#120—21no—
14C22×C22⋊C4×C23C2211yescharacteristic
15C23×#449—22no—
16C23×#448—21no—
17C23×#447—21no—
18C2×C23×#104—22no—
19C23×#417—21no—
20C23×#407C2211yescharacteristic
21C23×#407C2211yescharacteristic
22C23×#399—21no—
23C23×#393—21no—
24C23×#392—21no—
25C23×#392—21no—
26C23×#390—22no—
27C23×#386—22no—
28C23×#386—21no—
29C23×#381C2211yescharacteristic
30C4×C22⋊C4×C23C2211yescharacteristic
31C23×#327—22no—
32C23×#323—22no—
33C25×C23D411yescharacteristic
34C2×#41×C23—21no—
35C2×#41×C23—21no—
36C2×#41×C23—21no—
37C2×#41×C23—21no—
38C2×#41×C23—44no—
39C22×Q8×C23D411yescharacteristic, complemented
40C22×D4×C23—21no—
41C22×D4×C23—21no—
42C22×D4×C23—22no—
43C22×D4×C23—22no—
44C22×D4×C23D411yescharacteristic, complemented
45C22×D4×C23D411yescharacteristic, complemented
46C23×C4×C23—21no—
47C23×C4×C23D411yescharacteristic
48C23×C4×C23C2311yescharacteristic
49C23×AGL(1,ℤ/8)—44no—
50C2×SD16×C23—22no—
51C2×D8×C23—22no—
52C2×M16×C23—21no—
53C2×M16×C23—21no—
54C23×#123—44no—
55C23×#121—21no—
56C23×#121—21no—
57C23×#121—21no—
58C23×#121—21no—
59C23×#120—44no—
60C23×#120—44no—
61C2×C4⋊C4×C23—22no—
62C2×C4⋊C4×C23D411yescharacteristic, complemented
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—22no—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—22no—
67C2×C22⋊C4×C23—21no—
68C2×C22⋊C4×C23—21no—
69C2×C22⋊C4×C23—21no—
70C2×C22⋊C4×C23—21no—
71C2×C22⋊C4×C23—21no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—21no—
74C2×C22⋊C4×C23—21no—
75C2×C22⋊C4×C23—22no—
76C2×C42×C23D412yescomplemented
77C23×#104—44no—
78C23×#103—22no—
79C23×#102—22no—
80C23×C22⋊C8—21no—
81C23×C22⋊C8—21no—
82C23×C22⋊C8—21no—
83C23×C22⋊C8D412yescomplemented
84C23×#95—22no—
85C23×#95—21no—
86C23×#95—21no—
87C23×#95D412yescomplemented
88C24×C23—21no—
89C24×C23—21no—
90C24×C23—21no—
91C24×C23—21no—
92C24×C23—21no—
93C24×C23—42no—
94C24×C23—42no—
95C24×C23—42no—
96C24×C23C2×D411yescharacteristic
97#41×C23—24no—
98#41×C23—24no—
99#41×C23—42no—
100#41×C23—42no—
101#41×C23—42no—
102#41×C23—42no—
103#41×C23—42no—
104#41×C23—42no—
105C2×Q8×C23—21no—
106C2×Q8×C23—21no—
107C2×Q8×C23—42no—
108C2×D4×C23—24no—
109C2×D4×C23—21no—
110C2×D4×C23—21no—
111C2×D4×C23—21no—
112C2×D4×C23—21no—
113C2×D4×C23—42no—
114C2×D4×C23—44no—
115C2×D4×C23—42no—
116C2×D4×C23—42no—
117C2×D4×C23—41no—
118C2×D4×C23—41no—
119C2×D4×C23—42no—
120C2×D4×C23—44no—
121C2×D4×C23—42no—
122C2×D4×C23—44no—
123C2×D4×C23—44no—
124C2×D4×C23—41no—
125C2×D4×C23—41no—
126C2×D4×C23—82no—
127C2×D4×C23—82no—
128C22×C4×C23—22no—
129C22×C4×C23—21no—
130C22×C4×C23—21no—
131C22×C4×C23—21no—
132C22×C4×C23—21no—
133C22×C4×C23—22no—
134C22×C4×C23—21no—
135C22×C4×C23—42no—
136C22×C4×C23—42no—
137C22×C4×C23—42no—
138C22×C4×C23—41no—
139C22×C4×C23—41no—
140C22×C4×C23—42no—
141C22×C4×C23C2×D411yescharacteristic
142C22×C4×C23C2×D411yescharacteristic
143C22×C4×C23C2×D411yescharacteristic
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D411yescharacteristic
146C22×C4×C23#4111yescharacteristic
147SD16×C23—44no—
148D8×C23—44no—
149M16×C23—42no—
150M16×C23—42no—
151C2×C8×C23—22no—
152C2×C8×C23—41no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—44no—
156C22⋊C4×C23—24no—
157C22⋊C4×C23—44no—
158C22⋊C4×C23—44no—
159C22⋊C4×C23—44no—
160C22⋊C4×C23—42no—
161C22⋊C4×C23—41no—
162C22⋊C4×C23—41no—
163C22⋊C4×C23—41no—
164C22⋊C4×C23—41no—
165C22⋊C4×C23—41no—
166C22⋊C4×C23—41no—
167C22⋊C4×C23—82no—
168C42×C23—42no—
169#9938C2311yesmaximal, characteristic, direct product factor, Sylow
170C23×C23—22no—
171C23×C23—22no—
172C23×C23—22no—
173C23×C23—22no—
174C23×C23—22no—
175C23×C23—22no—
176C23×C23—41no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23—41no—
180C23×C23—42no—
181C23×C23—41no—
182C23×C23—41no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—42no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—42no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—41no—
193C23×C23—42no—
194C23×C23—81no—
195C23×C23—81no—
196C23×C23—88no—
197C23×C23#12011yescharacteristic
198C23×C23#12011yescharacteristic
199C23×C23#12111yescharacteristic
200Q8×C23—22no—
201Q8×C23—22no—
202Q8×C23—41no—
203Q8×C23—41no—
204D4×C23—22no—
205D4×C23—22no—
206D4×C23—41no—
207D4×C23—41no—
208D4×C23—42no—
209D4×C23—44no—
210D4×C23—44no—
211D4×C23—81no—
212D4×C23—88no—
213D4×C23—82no—
214D4×C23—88no—
215D4×C23—82no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—24no—
219C2×C4×C23—22no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—41no—
223C2×C4×C23—41no—
224C2×C4×C23—44no—
225C2×C4×C23—44no—
226C2×C4×C23—41no—
227C2×C4×C23—42no—
228C2×C4×C23—41no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—41no—
232C2×C4×C23—42no—
233C2×C4×C23—41no—
234C2×C4×C23—42no—
235C2×C4×C23—41no—
236C2×C4×C23—41no—
237C2×C4×C23—42no—
238C2×C4×C23—42no—
239C2×C4×C23—81no—
240C2×C4×C23#12011yescharacteristic
241C2×C4×C23#12111yescharacteristic
242C2×C4×C23#12111yescharacteristic
243C2×C4×C23#12711yescharacteristic
244C8×C23—42no—
245C8×C23—81no—
246#3006C2×C2311yescharacteristic, complemented
247#2539C2×C2311yescharacteristic, complemented
248#2404C2×C2311yescharacteristic, complemented
249#1789C2×C2311yescharacteristic, complemented
250#1510C2×C2311yescharacteristic, complemented
251#1462C2×C2311yescharacteristic, complemented
252#1284C2×C2311yescharacteristic, complemented
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—21no—
256C22×C23—21no—
257C22×C23—21no—
258C22×C23—21no—
259C22×C23—21no—
260C22×C23—44no—
261C22×C23—44no—
262C22×C23—44no—
263C22×C23—44no—
264C22×C23—42no—
265C22×C23—42no—
266C22×C23—82no—
267C22×C23—82no—
268C22×C23—82no—
269C22×C23—82no—
270C22×C23—82no—
271C22×C23—82no—
272C22×C23—82no—
273C22×C23—81no—
274C22×C23—81no—
275C22×C23—81no—
276C22×C23—81no—
277C22×C23—81no—
278C22×C23—81no—
279C22×C23—81no—
280C22×C23#39211yescharacteristic
281C22×C23#45312yes—
282C22×C23B(4,𝔽2)12yes—
283C4×C23—21no—
284C4×C23—21no—
285C4×C23—21no—
286C4×C23—21no—
287C4×C23—44no—
288C4×C23—42no—
289C4×C23—82no—
290C4×C23—81no—
291C4×C23—81no—
292C22×#41C22×C2311yescharacteristic
293C2×AGL(1,ℤ/8)—21no—
294C2×#123—21no—
295C2×#121C22×C2311yescharacteristic
296C2×#120—21no—
297C2×#120—21no—
298C22×C22⋊C4C22×C2311yescharacteristic
299#449—22no—
300#448—21no—
301#447—21no—
302C2×#104—22no—
303#417—21no—
304#407C22×C2311yescharacteristic
305#407C22×C2311yescharacteristic
306#399—21no—
307#393—21no—
308#392—21no—
309#392—21no—
310#390—22no—
311#386—21no—
312#386—22no—
313#381C22×C2311yescharacteristic
314C4×C22⋊C4C22×C2311yescharacteristic
315#327—22no—
316#323—22no—
317C2×C23—22no—
318C2×C23—22no—
319C2×C23—41no—
320C2×C23—82no—
321C2×C23—82no—
322C2×C23—161no—
323C2×C23#200211yescharacteristic
324C2×C23#201011yescharacteristic
325C2×C23#202111yescharacteristic
326C25D4×C2311yescharacteristic
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—21no—
330C2×#41—21no—
331C2×#41—44no—
332C22×Q8D4×C2311yescharacteristic, complemented
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4D4×C2311yescharacteristic, complemented
338C22×D4D4×C2311yescharacteristic, complemented
339C23×C4—21no—
340C23×C4D4×C2311yescharacteristic
341C23×C4C23×C2311yescharacteristic
342AGL(1,ℤ/8)—44no—
343C2×SD16—22no—
344C2×D8—22no—
345C2×M16—21no—
346C2×M16—21no—
347#123—44no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—21no—
352#120—44no—
353#120—44no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4D4×C2311yescharacteristic, complemented
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C42D4×C2312yescomplemented
370#104—44no—
371#103—22no—
372#102—22no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8D4×C2312yescomplemented
377#95—22no—
378#95—21no—
379#95—21no—
380#95D4×C2312yescomplemented
381C23#993811yescharacteristic, direct product factor, Sylow
382C24—21no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—21no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24C2×D4×C2311yescharacteristic
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—82no—
421C2×D4—82no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4C2×D4×C2311yescharacteristic
436C22×C4C2×D4×C2311yescharacteristic
437C22×C4C2×D4×C2311yescharacteristic
438C22×C4C2×D4×C2311yescharacteristic
439C22×C4C2×D4×C2311yescharacteristic
440C22×C4#41×C2311yescharacteristic
441SD16—44no—
442D8—44no—
443M16—42no—
444M16—42no—
445C2×C8—22no—
446C2×C8—41no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—44no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—82no—
462C42—42no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—81no—
488C23—81no—
489C23—88no—
490C23C23×#12011yescharacteristic
491C23C23×#12011yescharacteristic
492C23C23×#12111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—81no—
505D4—88no—
506D4—82no—
507D4—88no—
508D4—82no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4C23×#12011yescharacteristic
534C2×C4C23×#12111yescharacteristic
535C2×C4C23×#12111yescharacteristic
536C2×C4C23×#12711yescharacteristic
537C8—42no—
538C8—81no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22C23×#39211yescharacteristic
567C22C23×#45312yes—
568C22C23×B(4,𝔽2)12yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—44no—
574C4—42no—
575C4—82no—
576C4—81no—
577C4—81no—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—82no—
583C2—161no—
584C2C23×#200211yescharacteristic
585C2C23×#201011yescharacteristic
586C2C23×#202111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3006C2semidirect
2. C23×#2539C2semidirect
3. C23×#2404C2semidirect
4. C23×#1789C2semidirect
5. C23×#1510C2semidirect
6. C23×#1462C2semidirect
7. C23×#1284C2semidirect
39. C22×Q8×C23D4semidirect
44. C22×D4×C23D4semidirect
45. C22×D4×C23D4semidirect
62. C2×C4⋊C4×C23D4semidirect
76. C2×C42×C23D4semidirect
83. C23×C22⋊C8D4semidirect
87. C23×#95D4semidirect
169. #9938C23direct (N × H)
246. #3006C2×C23semidirect
247. #2539C2×C23semidirect
248. #2404C2×C23semidirect
249. #1789C2×C23semidirect
250. #1510C2×C23semidirect
251. #1462C2×C23semidirect
252. #1284C2×C23semidirect
332. C22×Q8D4×C23semidirect
337. C22×D4D4×C23semidirect
338. C22×D4D4×C23semidirect
355. C2×C4⋊C4D4×C23semidirect
369. C2×C42D4×C23semidirect
376. C22⋊C8D4×C23semidirect
380. #95D4×C23semidirect
381. C23#9938direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30062
C23×#25392
C23×#24042
C23×#17892
C23×#15102
C23×#14622
C23×#12842
#993823