Finite Group Databasebeta

← C23×#9942 #19157568

Subgroups — C23×#9942 #19157568

Order 5888.

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

normal subgroups only

This group has 894 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×#3007C211yesmaximal, characteristic, complemented
2C23×#2561C211yesmaximal, characteristic, complemented
3C23×#2404C211yesmaximal, characteristic
4C23×#1789C211yesmaximal, characteristic, complemented
5C23×#1511C211yesmaximal, characteristic, complemented
6C23×#1462C211yesmaximal, characteristic, complemented
7C23×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C23C2211yescharacteristic
9C2×C23×#137—21no—
10C2×C23×C22⋊Q8C2211yescharacteristic
11C2×C23×#120—21no—
12C22×C22⋊C4×C23C2211yescharacteristic
13C23×#452—22no—
14C23×#451—21no—
15C23×#450—21no—
16C2×C23×#104—22no—
17C23×#417—21no—
18C23×#407C2211yescharacteristic
19C23×#407C2211yescharacteristic
20C23×#397—21no—
21C23×#393—21no—
22C23×#393—21no—
23C23×#392—21no—
24C23×#388—22no—
25C23×#386—22no—
26C23×#386—21no—
27C23×#381—21no—
28C23×#381C2211yescharacteristic
29C23×#379—21no—
30C4×C22⋊C4×C23C2211yescharacteristic
31C23×#328—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
40C22×Q8×C23D411yescharacteristic, complemented
41C22×D4×C23—21no—
42C22×D4×C23—22no—
43C22×D4×C23D411yescharacteristic
44C23×C4×C23—21no—
45C23×C4×C23D411yescharacteristic
46C23×C4×C23C2311yescharacteristic
47C23×#137—44no—
48C2×Q16×C23—22no—
49C2×SD16×C23—22no—
50C2×M16×C23—21no—
51C2×M16×C23—21no—
52C23×C22⋊Q8—21no—
53C23×C22⋊Q8—21no—
54C23×C22⋊Q8—21no—
55C23×C22⋊Q8—21no—
56C23×#120—44no—
57C2×C4⋊C4×C23—21no—
58C2×C4⋊C4×C23D411yescharacteristic, complemented
59C2×C22⋊C4×C23—22no—
60C2×C22⋊C4×C23—22no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—21no—
65C2×C22⋊C4×C23—21no—
66C2×C22⋊C4×C23—21no—
67C2×C22⋊C4×C23—21no—
68C2×C22⋊C4×C23—21no—
69C2×C22⋊C4×C23—21no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C42×C23D412yescomplemented
73C23×#104—44no—
74C23×#103—22no—
75C23×#102—22no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×C22⋊C8—21no—
79C23×C22⋊C8D412yescomplemented
80C23×#95—22no—
81C23×#95—22no—
82C23×#95—22no—
83C23×#95—21no—
84C23×#95—21no—
85C23×#95D412yes—
86C24×C23—21no—
87C24×C23—21no—
88C24×C23—21no—
89C24×C23—21no—
90C24×C23—42no—
91C24×C23—42no—
92C24×C23—42no—
93C24×C23C2×D411yescharacteristic
94#41×C23—24no—
95#41×C23—24no—
96#41×C23—42no—
97#41×C23—42no—
98#41×C23—42no—
99#41×C23—42no—
100#41×C23—42no—
101#41×C23—42no—
102C2×Q8×C23—21no—
103C2×Q8×C23—21no—
104C2×Q8×C23—21no—
105C2×Q8×C23—21no—
106C2×Q8×C23—42no—
107C2×Q8×C23—42no—
108C2×D4×C23—24no—
109C2×D4×C23—21no—
110C2×D4×C23—21no—
111C2×D4×C23—42no—
112C2×D4×C23—42no—
113C2×D4×C23—41no—
114C2×D4×C23—41no—
115C2×D4×C23—44no—
116C2×D4×C23—44no—
117C2×D4×C23—82no—
118C22×C4×C23—21no—
119C22×C4×C23—22no—
120C22×C4×C23—21no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—22no—
124C22×C4×C23—21no—
125C22×C4×C23—21no—
126C22×C4×C23—42no—
127C22×C4×C23—42no—
128C22×C4×C23—42no—
129C22×C4×C23—41no—
130C22×C4×C23—41no—
131C22×C4×C23—42no—
132C22×C4×C23C2×D411yescharacteristic
133C22×C4×C23C2×D411yescharacteristic
134C22×C4×C23C2×D411yescharacteristic
135C22×C4×C23C2×D411yescharacteristic
136C22×C4×C23C2×D411yescharacteristic
137C22×C4×C23#4111yescharacteristic
138Q16×C23—44no—
139SD16×C23—44no—
140M16×C23—42no—
141M16×C23—42no—
142C2×C8×C23—22no—
143C2×C8×C23—41no—
144C4⋊C4×C23—21no—
145C4⋊C4×C23—21no—
146C4⋊C4×C23—41no—
147C4⋊C4×C23—41no—
148C22⋊C4×C23—44no—
149C22⋊C4×C23—42no—
150C22⋊C4×C23—44no—
151C22⋊C4×C23—42no—
152C22⋊C4×C23—41no—
153C22⋊C4×C23—41no—
154C22⋊C4×C23—41no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23—41no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23—82no—
159C22⋊C4×C23—82no—
160C42×C23—42no—
161#9942C2311yesmaximal, characteristic, direct product factor, Sylow
162C23×C23—22no—
163C23×C23—22no—
164C23×C23—22no—
165C23×C23—22no—
166C23×C23—22no—
167C23×C23—22no—
168C23×C23—41no—
169C23×C23—42no—
170C23×C23—41no—
171C23×C23—42no—
172C23×C23—42no—
173C23×C23—41no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—41no—
178C23×C23—42no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—81no—
182C23×C23—81no—
183C23×C23—88no—
184C23×C23#12011yescharacteristic
185C23×C23#12011yescharacteristic
186C23×C23#12111yescharacteristic
187Q8×C23—22no—
188Q8×C23—22no—
189Q8×C23—41no—
190Q8×C23—41no—
191Q8×C23—42no—
192Q8×C23—81no—
193D4×C23—22no—
194D4×C23—22no—
195D4×C23—41no—
196D4×C23—41no—
197D4×C23—44no—
198D4×C23—44no—
199D4×C23—88no—
200C2×C4×C23—22no—
201C2×C4×C23—22no—
202C2×C4×C23—22no—
203C2×C4×C23—41no—
204C2×C4×C23—41no—
205C2×C4×C23—41no—
206C2×C4×C23—41no—
207C2×C4×C23—44no—
208C2×C4×C23—44no—
209C2×C4×C23—41no—
210C2×C4×C23—41no—
211C2×C4×C23—42no—
212C2×C4×C23—42no—
213C2×C4×C23—42no—
214C2×C4×C23—42no—
215C2×C4×C23—42no—
216C2×C4×C23—42no—
217C2×C4×C23—41no—
218C2×C4×C23—41no—
219C2×C4×C23—41no—
220C2×C4×C23—42no—
221C2×C4×C23—42no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—81no—
225C2×C4×C23#12011yescharacteristic
226C2×C4×C23#12111yescharacteristic
227C2×C4×C23#12111yescharacteristic
228C2×C4×C23#12711yescharacteristic
229C8×C23—42no—
230C8×C23—81no—
231#3007C2×C2311yescharacteristic, complemented
232#2561C2×C2311yescharacteristic, complemented
233#2404C2×C2311yescharacteristic
234#1789C2×C2311yescharacteristic, complemented
235#1511C2×C2311yescharacteristic, complemented
236#1462C2×C2311yescharacteristic, complemented
237#1284C2×C2311yescharacteristic, complemented
238C22×C23—21no—
239C22×C23—21no—
240C22×C23—21no—
241C22×C23—21no—
242C22×C23—21no—
243C22×C23—21no—
244C22×C23—21no—
245C22×C23—44no—
246C22×C23—44no—
247C22×C23—44no—
248C22×C23—44no—
249C22×C23—42no—
250C22×C23—42no—
251C22×C23—82no—
252C22×C23—82no—
253C22×C23—82no—
254C22×C23—82no—
255C22×C23—82no—
256C22×C23—81no—
257C22×C23—81no—
258C22×C23—81no—
259C22×C23—81no—
260C22×C23#39211yescharacteristic
261C22×C23#45512yes—
262C22×C23B(4,𝔽2)12yes—
263C4×C23—21no—
264C4×C23—21no—
265C4×C23—21no—
266C4×C23—21no—
267C4×C23—42no—
268C4×C23—82no—
269C4×C23—82no—
270C4×C23—81no—
271C4×C23—81no—
272C4×C23—81no—
273C22×#41C22×C2311yescharacteristic
274C2×#137—21no—
275C2×C22⋊Q8C22×C2311yescharacteristic
276C2×#120—21no—
277C22×C22⋊C4C22×C2311yescharacteristic
278#452—22no—
279#451—21no—
280#450—21no—
281C2×#104—22no—
282#417—21no—
283#407C22×C2311yescharacteristic
284#407C22×C2311yescharacteristic
285#397—21no—
286#393—21no—
287#393—21no—
288#392—21no—
289#388—22no—
290#386—21no—
291#386—22no—
292#381—21no—
293#381C22×C2311yescharacteristic
294#379—21no—
295C4×C22⋊C4C22×C2311yescharacteristic
296#328—22no—
297#323—22no—
298C2×C23—22no—
299C2×C23—22no—
300C2×C23—41no—
301C2×C23—82no—
302C2×C23—82no—
303C2×C23#200411yescharacteristic
304C2×C23#201011yescharacteristic
305C2×C23#202111yescharacteristic
306C25D4×C2311yescharacteristic
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—21no—
310C2×#41—21no—
311C2×#41—44no—
312C22×Q8D4×C2311yescharacteristic
313C22×Q8D4×C2311yescharacteristic, complemented
314C22×D4—21no—
315C22×D4—22no—
316C22×D4D4×C2311yescharacteristic
317C23×C4—21no—
318C23×C4D4×C2311yescharacteristic
319C23×C4C23×C2311yescharacteristic
320#137—44no—
321C2×Q16—22no—
322C2×SD16—22no—
323C2×M16—21no—
324C2×M16—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329#120—44no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4D4×C2311yescharacteristic, complemented
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C42D4×C2312yescomplemented
346#104—44no—
347#103—22no—
348#102—22no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8D4×C2312yescomplemented
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95D4×C2312yes—
359C23#994211yescharacteristic, direct product factor, Sylow
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24C2×D4×C2311yescharacteristic
368#41—24no—
369#41—24no—
370#41—42no—
371#41—42no—
372#41—42no—
373#41—42no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×D4—24no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—82no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4C2×D4×C2311yescharacteristic
407C22×C4C2×D4×C2311yescharacteristic
408C22×C4C2×D4×C2311yescharacteristic
409C22×C4C2×D4×C2311yescharacteristic
410C22×C4C2×D4×C2311yescharacteristic
411C22×C4#41×C2311yescharacteristic
412Q16—44no—
413SD16—44no—
414M16—42no—
415M16—42no—
416C2×C8—22no—
417C2×C8—41no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C42—42no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—81no—
455C23—81no—
456C23—88no—
457C23C23×#12011yescharacteristic
458C23C23×#12011yescharacteristic
459C23C23×#12111yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—41no—
463Q8—41no—
464Q8—42no—
465Q8—81no—
466D4—22no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—44no—
471D4—44no—
472D4—88no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—81no—
498C2×C4C23×#12011yescharacteristic
499C2×C4C23×#12111yescharacteristic
500C2×C4C23×#12111yescharacteristic
501C2×C4C23×#12711yescharacteristic
502C8—42no—
503C8—81no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22C23×#39211yescharacteristic
527C22C23×#45512yes—
528C22C23×B(4,𝔽2)12yes—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—42no—
534C4—82no—
535C4—82no—
536C4—81no—
537C4—81no—
538C4—81no—
539C2—22no—
540C2—22no—
541C2—41no—
542C2—82no—
543C2—82no—
544C2C23×#200411yescharacteristic
545C2C23×#201011yescharacteristic
546C2C23×#202111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2404C2
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×#3007C2semidirect
2. C23×#2561C2semidirect
4. C23×#1789C2semidirect
5. C23×#1511C2semidirect
6. C23×#1462C2semidirect
7. C23×#1284C2semidirect
40. C22×Q8×C23D4semidirect
58. C2×C4⋊C4×C23D4semidirect
72. C2×C42×C23D4semidirect
79. C23×C22⋊C8D4semidirect
161. #9942C23direct (N × H)
231. #3007C2×C23semidirect
232. #2561C2×C23semidirect
234. #1789C2×C23semidirect
235. #1511C2×C23semidirect
236. #1462C2×C23semidirect
237. #1284C2×C23semidirect
313. C22×Q8D4×C23semidirect
331. C2×C4⋊C4D4×C23semidirect
345. C2×C42D4×C23semidirect
352. C22⋊C8D4×C23semidirect
359. C23#9942direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30072
C23×#25612
C23×#24042
C23×#17892
C23×#15112
C23×#14622
C23×#12842
#994223