Finite Group Databasebeta

← C23×#12852 #19160478

Subgroups — C23×#12852 #19160478

Order 5888.

992 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×C23×#386C211yesmaximal, characteristic
2C23×#1783C211yesmaximal, characteristic
3C23×#1781C211yesmaximal, characteristic
4C23×#1390C214yesmaximal, complemented
5C23×D4×C23C2211yescharacteristic
6C24×C4×C23C411yescharacteristic, complemented
7C22×C22⋊C4×C23—21no—
8C22×C22⋊C4×C23C411yescharacteristic, complemented
9C2×C23×#100—21no—
10C2×C23×#100C2212yes—
11C2×C23×#99—21no—
12C2×C23×#99C2212yes—
13C23×#407—21no—
14C23×#386—21no—
15C23×#386—21no—
16C23×#386C412yescomplemented
17C23×#386C2212yes—
18C23×#379—21no—
19C2×C23×#95—21no—
20C25×C23C2×C411yescharacteristic
21C25×C23D411yescharacteristic
22C22×D4×C23—24no—
23C22×D4×C23—41no—
24C22×D4×C23—41no—
25C22×D4×C23—42no—
26C22×D4×C23C2×C411yescharacteristic
27C22×D4×C23D411yescharacteristic
28C22×D4×C23D411yescharacteristic
29C22×D4×C23C2311yescharacteristic
30C23×C4×C23—21no—
31C23×C4×C23—21no—
32C23×C4×C23—21no—
33C23×C4×C23—44no—
34C23×C4×C23—41no—
35C23×C4×C23—41no—
36C23×C4×C23C2×C412yes—
37C23×C4×C23D411yescharacteristic
38C2×M16×C23—22no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23—44no—
44C2×C22⋊C4×C23—42no—
45C2×C22⋊C4×C23—41no—
46C2×C22⋊C4×C23—41no—
47C2×C22⋊C4×C23C2×C412yes—
48C23×#100—24no—
49C23×#100—42no—
50C23×#99—24no—
51C23×#99—42no—
52C23×C22⋊C8—41no—
53C23×C22⋊C8—41no—
54C23×#95—22no—
55C23×#95—22no—
56C24×C23—22no—
57C24×C23—21no—
58C24×C23—21no—
59C24×C23—22no—
60C24×C23—22no—
61C24×C23—21no—
62C24×C23—21no—
63C24×C23—41no—
64C24×C23—42no—
65C24×C23—42no—
66C24×C23—42no—
67C24×C23—42no—
68C24×C23C22⋊C411yescharacteristic
69C24×C23C22⋊C411yescharacteristic
70C24×C23C22⋊C411yescharacteristic
71C24×C23C22×C411yescharacteristic
72C24×C23C2×D411yescharacteristic
73C2×D4×C23—22no—
74C2×D4×C23—22no—
75C2×D4×C23—22no—
76C2×D4×C23—22no—
77C2×D4×C23—22no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23—48no—
85C2×D4×C23—48no—
86C2×D4×C23#4114yes—
87C22×C4×C23—24no—
88C22×C4×C23—21no—
89C22×C4×C23—21no—
90C22×C4×C23—22no—
91C22×C4×C23—22no—
92C22×C4×C23—22no—
93C22×C4×C23—41no—
94C22×C4×C23—44no—
95C22×C4×C23—44no—
96C22×C4×C23—42no—
97C22×C4×C23—42no—
98C22×C4×C23—48no—
99C22×C4×C23—44no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23—42no—
103C22×C4×C23—44no—
104C22×C4×C23—42no—
105C22×C4×C23—42no—
106C22×C4×C23—42no—
107C22×C4×C23—42no—
108C22×C4×C23C22⋊C411yescharacteristic
109C22×C4×C23C2×D411yescharacteristic
110M16×C23—42no—
111M16×C23—42no—
112C2×C8×C23—42no—
113C22⋊C4×C23—44no—
114C22⋊C4×C23—44no—
115C22⋊C4×C23—42no—
116C22⋊C4×C23—42no—
117C22⋊C4×C23—42no—
118C22⋊C4×C23—42no—
119C22⋊C4×C23—82no—
120C22⋊C4×C23—82no—
121#12852C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—22no—
123C23×C23—22no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—24no—
127C23×C23—24no—
128C23×C23—21no—
129C23×C23—21no—
130C23×C23—21no—
131C23×C23—21no—
132C23×C23—21no—
133C23×C23—21no—
134C23×C23—21no—
135C23×C23—21no—
136C23×C23—21no—
137C23×C23—42no—
138C23×C23—42no—
139C23×C23—41no—
140C23×C23—41no—
141C23×C23—41no—
142C23×C23—41no—
143C23×C23—42no—
144C23×C23—48no—
145C23×C23—48no—
146C23×C23—42no—
147C23×C23—41no—
148C23×C23—41no—
149C23×C23—41no—
150C23×C23—41no—
151C23×C23—48no—
152C23×C23—42no—
153C23×C23—42no—
154C23×C23—42no—
155C23×C23—42no—
156C23×C23—81no—
157C23×C23—81no—
158C23×C23—84no—
159C23×C23#9911yescharacteristic
160C23×C23#9911yescharacteristic
161C23×C23#9912yes—
162C23×C23C2×C22⋊C411yescharacteristic
163C23×C23#11712yes—
164C23×C23#12312yes—
165D4×C23—44no—
166D4×C23—44no—
167D4×C23—44no—
168D4×C23—44no—
169C2×C4×C23—21no—
170C2×C4×C23—21no—
171C2×C4×C23—24no—
172C2×C4×C23—21no—
173C2×C4×C23—21no—
174C2×C4×C23—21no—
175C2×C4×C23—42no—
176C2×C4×C23—42no—
177C2×C4×C23—42no—
178C2×C4×C23—44no—
179C2×C4×C23—48no—
180C2×C4×C23—44no—
181C2×C4×C23—44no—
182C2×C4×C23—42no—
183C2×C4×C23—44no—
184C2×C4×C23—42no—
185C2×C4×C23—42no—
186C2×C4×C23—42no—
187C2×C4×C23—84no—
188C2×C4×C23—82no—
189C2×C4×C23—82no—
190C2×C4×C23—82no—
191C2×C4×C23#12312yes—
192C8×C23—82no—
193C2×#386C2×C2311yescharacteristic
194#1783C2×C2311yescharacteristic
195#1781C2×C2311yescharacteristic
196#1390C2×C2314yescomplemented
197C22×C23—24no—
198C22×C23—24no—
199C22×C23—21no—
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—22no—
206C22×C23—22no—
207C22×C23—22no—
208C22×C23—42no—
209C22×C23—42no—
210C22×C23—42no—
211C22×C23—42no—
212C22×C23—42no—
213C22×C23—48no—
214C22×C23—48no—
215C22×C23—42no—
216C22×C23—41no—
217C22×C23—41no—
218C22×C23—41no—
219C22×C23—41no—
220C22×C23—41no—
221C22×C23—41no—
222C22×C23—41no—
223C22×C23—41no—
224C22×C23—41no—
225C22×C23—81no—
226C22×C23—84no—
227C22×C23—84no—
228C22×C23—81no—
229C22×C23AΣL(1,𝔽16)12yes—
230C22×C23#35212yes—
231C22×C23#38111yescharacteristic
232C22×C23C2×#9911yescharacteristic
233C22×C23C2×#9911yescharacteristic
234C4×C23—22no—
235C4×C23—41no—
236C4×C23—44no—
237C4×C23—82no—
238C4×C23—162no—
239C23×D4C22×C2311yescharacteristic
240C24×C4C4×C2311yescharacteristic, complemented
241C22×C22⋊C4—21no—
242C22×C22⋊C4C4×C2311yescharacteristic, complemented
243C2×#100—21no—
244C2×#100C22×C2312yes—
245C2×#99—21no—
246C2×#99C22×C2312yes—
247#407—21no—
248#386—21no—
249#386—21no—
250#386C4×C2312yescomplemented
251#386C22×C2312yes—
252#379—21no—
253C2×#95—21no—
254C2×C23—21no—
255C2×C23—21no—
256C2×C23—22no—
257C2×C23—22no—
258C2×C23—42no—
259C2×C23—42no—
260C2×C23—42no—
261C2×C23—81no—
262C2×C23#181911yescharacteristic
263C2×C23C2×AΣL(1,𝔽16)11yescharacteristic
264C2×C23C2×#35211yescharacteristic
265C25C2×C4×C2311yescharacteristic
266C25D4×C2311yescharacteristic
267C22×D4—24no—
268C22×D4—41no—
269C22×D4—41no—
270C22×D4—42no—
271C22×D4C2×C4×C2311yescharacteristic
272C22×D4D4×C2311yescharacteristic
273C22×D4D4×C2311yescharacteristic
274C22×D4C23×C2311yescharacteristic
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—44no—
279C23×C4—41no—
280C23×C4—41no—
281C23×C4C2×C4×C2312yes—
282C23×C4D4×C2311yescharacteristic
283C2×M16—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4C2×C4×C2312yes—
293#100—24no—
294#100—42no—
295#99—24no—
296#99—42no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299#95—22no—
300#95—22no—
301C23#1285211yescharacteristic, direct product factor, Sylow
302C24—22no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—41no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24C22⋊C4×C2311yescharacteristic
315C24C22⋊C4×C2311yescharacteristic
316C24C22⋊C4×C2311yescharacteristic
317C24C22×C4×C2311yescharacteristic
318C24C2×D4×C2311yescharacteristic
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4#41×C2314yes—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—48no—
345C22×C4—44no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C22⋊C4×C2311yescharacteristic
355C22×C4C2×D4×C2311yescharacteristic
356M16—42no—
357M16—42no—
358C2×C8—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—21no—
371C23—24no—
372C23—24no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—48no—
390C23—48no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—48no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—84no—
404C23C23×#9911yescharacteristic
405C23C23×#9911yescharacteristic
406C23C23×#9912yes—
407C23C2×C22⋊C4×C2311yescharacteristic
408C23C23×#11712yes—
409C23C23×#12312yes—
410D4—44no—
411D4—44no—
412D4—44no—
413D4—44no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4C23×#12312yes—
437C8—82no—
438C22—24no—
439C22—24no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—48no—
455C22—48no—
456C22—42no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—84no—
468C22—84no—
469C22—81no—
470C22C23×AΣL(1,𝔽16)12yes—
471C22C23×#35212yes—
472C22C23×#38111yescharacteristic
473C22C2×C23×#9911yescharacteristic
474C22C2×C23×#9911yescharacteristic
475C4—22no—
476C4—41no—
477C4—44no—
478C4—82no—
479C4—162no—
480C2—21no—
481C2—21no—
482C2—22no—
483C2—22no—
484C2—42no—
485C2—42no—
486C2—42no—
487C2—81no—
488C2C23×#181911yescharacteristic
489C2C2×C23×AΣL(1,𝔽16)11yescharacteristic
490C2C2×C23×#35211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1390C2semidirect
6. C24×C4×C23C4semidirect
8. C22×C22⋊C4×C23C4semidirect
16. C23×#386C4semidirect
121. #12852C23direct (N × H)
196. #1390C2×C23semidirect
240. C24×C4C4×C23semidirect
242. C22×C22⋊C4C4×C23semidirect
250. #386C4×C23semidirect
301. C23#12852direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3862
C23×#17832
C23×#17812
C23×#13902
#1285223