Finite Group Databasebeta

← C23×#19066 #19166692

Subgroups — C23×#19066 #19166692

Order 5888.

852 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3550C211yesmaximal, characteristic
2C23×#3237C211yesmaximal, characteristic, complemented
3C23×#3216C211yesmaximal, characteristic
4C23×#3162C211yesmaximal, characteristic, complemented
5C23×#1664C214yesmaximal, complemented
6C23×#1660C214yesmaximal, complemented
7C23×#1537C211yesmaximal, characteristic, complemented
8C23×#1531C211yesmaximal, characteristic, complemented
9C23×#1530C211yesmaximal, characteristic
10C23×#563C2214yes—
11C23×#559—21no—
12Q82×C23—21no—
13C23×#556—21no—
14C23×#556—21no—
15D4×Q8×C23—21no—
16D4×Q8×C23—21no—
17C23×#542—22no—
18C23×#542—22no—
19C23×#536—22no—
20C23×#533C2211yescharacteristic
21C23×#533C2211yescharacteristic
22C23×#533C2211yescharacteristic
23C23×#532C2211yescharacteristic
24C23×#501—21no—
25C23×#501—21no—
26C23×#501C2212yes—
27C23×#501C2212yes—
28C23×#501C2212yes—
29C23×#497C2212yes—
30C23×#491C2212yes—
31C23×#489C2212yes—
32C23×#485—22no—
33C23×#477—21no—
34C23×#475—21no—
35C23×#462—21no—
36C23×#460—21no—
37C23×#452—22no—
38C23×#432C2211yescharacteristic
39C23×#432C2211yescharacteristic
40C23×#427—22no—
41C2×C23×#97C2211yescharacteristic
42C23×#330C2214yes—
43C23×#330C2214yes—
44C23×#329C2214yes—
45C2×#41×C23D412yes—
46C22×Q8×C23—22no—
47C2×Q16×C23—22no—
48C2×SD16×C23—22no—
49C22×C8×C23—21no—
50C23×C4⋊Q8—22no—
51C23×C4⋊Q8—22no—
52C23×C4⋊Q8—24no—
53C23×C4⋊Q8D411yescharacteristic
54C23×C4⋊Q8D411yescharacteristic
55C23×C4⋊Q8D411yescharacteristic
56C23×C4⋊Q8D411yescharacteristic
57C23×C4⋊Q8D411yescharacteristic
58C23×C4⋊Q8C2311yescharacteristic
59C23×C4⋊Q8C2311yescharacteristic
60C23×#127D411yescharacteristic
61C23×#126—24no—
62C23×#126—24no—
63C23×#125—21no—
64C23×#125C2312yes—
65C23×#125C2312yes—
66C23×#124—22no—
67C23×#124—22no—
68C23×#124—24no—
69C23×#124C2312yes—
70C23×#123—24no—
71C23×#123—24no—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—22no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—21no—
77C23×C22⋊Q8—21no—
78C23×C22⋊Q8—24no—
79C23×C22⋊Q8—24no—
80C23×C22⋊Q8—21no—
81C23×C22⋊Q8—21no—
82C23×#121—21no—
83C23×#121—21no—
84C4×Q8×C23—22no—
85C4×Q8×C23—22no—
86C4×Q8×C23—21no—
87C4×Q8×C23—21no—
88C4×Q8×C23—21no—
89C4×Q8×C23—21no—
90C4×Q8×C23—21no—
91C4×Q8×C23—21no—
92C4×D4×C23—22no—
93C4×D4×C23—22no—
94C4×D4×C23—21no—
95C4×D4×C23—21no—
96C23×#117—22no—
97C23×#117—21no—
98C23×#117D412yes—
99C23×#117D412yes—
100C2×C42×C23C2311yescharacteristic
101C23×#107—22no—
102C23×#107—22no—
103C23×#107—22no—
104C23×#106—22no—
105C23×#106—22no—
106C23×#106—22no—
107C23×C4⋊C8—21no—
108C23×C4⋊C8—21no—
109C23×#103—22no—
110C23×#103—22no—
111C23×#103—22no—
112C23×#102—22no—
113C23×C22⋊C8—21no—
114C23×C22⋊C8—21no—
115C23×#97—21no—
116C23×#97C2312yes—
117C23×#97C2312yes—
118C23×#97C2312yes—
119#41×C23—82no—
120C2×Q8×C23—24no—
121C2×Q8×C23—21no—
122C2×Q8×C23—21no—
123C2×Q8×C23—21no—
124C2×Q8×C23—21no—
125C2×Q8×C23—21no—
126C2×Q8×C23—21no—
127C2×Q8×C23—84no—
128C2×Q8×C23C2×D412yes—
129C2×D4×C23—22no—
130C2×D4×C23—21no—
131C2×D4×C23—21no—
132C2×D4×C23C2×D412yes—
133C22×C4×C23—24no—
134C22×C4×C23—22no—
135C22×C4×C23C2×D411yescharacteristic
136C22×C4×C23C2×D411yescharacteristic
137C22×C4×C23C2×D411yescharacteristic
138Q16×C23—82no—
139SD16×C23—82no—
140C2×C8×C23—24no—
141C2×C8×C23—22no—
142C2×C8×C23—22no—
143C2×C8×C23—22no—
144C4⋊C4×C23—24no—
145C4⋊C4×C23—24no—
146C4⋊C4×C23—21no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—21no—
149C4⋊C4×C23—21no—
150C4⋊C4×C23—21no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—22no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—21no—
157C4⋊C4×C23—21no—
158C4⋊C4×C23—21no—
159C4⋊C4×C23—21no—
160C4⋊C4×C23—41no—
161C4⋊C4×C23—41no—
162C4⋊C4×C23C2×D412yes—
163C4⋊C4×C23C2×D412yes—
164C4⋊C4×C23C2×D412yes—
165C4⋊C4×C23C2×D412yes—
166C22⋊C4×C23—24no—
167C22⋊C4×C23—24no—
168C22⋊C4×C23—24no—
169C22⋊C4×C23—22no—
170C22⋊C4×C23—22no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—41no—
174C42×C23—24no—
175C42×C23—22no—
176C42×C23—22no—
177C42×C23—41no—
178C42×C23C2×D411yescharacteristic
179C42×C23C2×D411yescharacteristic
180C42×C23C2×D411yescharacteristic
181C42×C23C2411yescharacteristic
182#19066C2311yesmaximal, characteristic, direct product factor, Sylow
183C23×C23—22no—
184C23×C23#12011yescharacteristic
185Q8×C23—42no—
186Q8×C23—42no—
187Q8×C23—42no—
188Q8×C23—42no—
189Q8×C23—41no—
190Q8×C23—41no—
191Q8×C23—42no—
192Q8×C23—42no—
193Q8×C23—84no—
194D4×C23—42no—
195D4×C23—42no—
196D4×C23—42no—
197D4×C23—82no—
198C2×C4×C23—22no—
199C2×C4×C23—22no—
200C2×C4×C23—22no—
201C2×C4×C23—22no—
202C2×C4×C23—22no—
203C2×C4×C23—22no—
204C2×C4×C23—41no—
205C2×C4×C23—41no—
206C2×C4×C23—44no—
207C2×C4×C23—41no—
208C2×C4×C23—41no—
209C2×C4×C23—84no—
210C2×C4×C23#12011yescharacteristic
211C2×C4×C23#12011yescharacteristic
212C2×C4×C23#12011yescharacteristic
213C2×C4×C23C22×D411yescharacteristic
214C2×C4×C23C22×D411yescharacteristic
215C2×C4×C23C22×D411yescharacteristic
216C8×C23—24no—
217C8×C23—42no—
218C8×C23—42no—
219#3550C2×C2311yescharacteristic
220#3237C2×C2311yescharacteristic, complemented
221#3216C2×C2311yescharacteristic
222#3162C2×C2311yescharacteristic, complemented
223#1664C2×C2314yescomplemented
224#1660C2×C2314yescomplemented
225#1537C2×C2311yescharacteristic, complemented
226#1531C2×C2311yescharacteristic, complemented
227#1530C2×C2311yescharacteristic
228C22×C23—21no—
229C22×C23—21no—
230C22×C23—21no—
231C22×C23—42no—
232C22×C23—42no—
233C22×C23—42no—
234C22×C23C2×#12011yescharacteristic
235C4×C23—21no—
236C4×C23—21no—
237C4×C23—21no—
238C4×C23—21no—
239C4×C23—21no—
240C4×C23—42no—
241C4×C23—42no—
242C4×C23—42no—
243C4×C23—42no—
244C4×C23—42no—
245C4×C23—81no—
246C4×C23—81no—
247C4×C23#45612yes—
248#563C22×C2314yes—
249#559—21no—
250Q82—21no—
251#556—21no—
252#556—21no—
253D4×Q8—21no—
254D4×Q8—21no—
255#542—22no—
256#542—22no—
257#536—22no—
258#533C22×C2311yescharacteristic
259#533C22×C2311yescharacteristic
260#533C22×C2311yescharacteristic
261#532C22×C2311yescharacteristic
262#501—21no—
263#501—21no—
264#501C22×C2312yes—
265#501C22×C2312yes—
266#501C22×C2312yes—
267#497C22×C2312yes—
268#491C22×C2312yes—
269#489C22×C2312yes—
270#485—22no—
271#477—21no—
272#475—21no—
273#462—21no—
274#460—21no—
275#452—22no—
276#432C22×C2311yescharacteristic
277#432C22×C2311yescharacteristic
278#427—22no—
279C2×#97C22×C2311yescharacteristic
280#330C22×C2314yes—
281#330C22×C2314yes—
282#329C22×C2314yes—
283C2×C23—41no—
284C2×C23—82no—
285C2×C23C2×#45611yescharacteristic
286C2×C23#301911yescharacteristic
287C2×C23#302011yescharacteristic
288C2×#41D4×C2312yes—
289C22×Q8—22no—
290C2×Q16—22no—
291C2×SD16—22no—
292C22×C8—21no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—24no—
296C4⋊Q8D4×C2311yescharacteristic
297C4⋊Q8D4×C2311yescharacteristic
298C4⋊Q8D4×C2311yescharacteristic
299C4⋊Q8D4×C2311yescharacteristic
300C4⋊Q8D4×C2311yescharacteristic
301C4⋊Q8C23×C2311yescharacteristic
302C4⋊Q8C23×C2311yescharacteristic
303#127D4×C2311yescharacteristic
304#126—24no—
305#126—24no—
306#125—21no—
307#125C23×C2312yes—
308#125C23×C2312yes—
309#124—24no—
310#124—22no—
311#124—22no—
312#124C23×C2312yes—
313#123—24no—
314#123—24no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325#121—21no—
326#121—21no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—21no—
338C4×D4—21no—
339#117—22no—
340#117—21no—
341#117D4×C2312yes—
342#117D4×C2312yes—
343C2×C42C23×C2311yescharacteristic
344#107—22no—
345#107—22no—
346#107—22no—
347#106—22no—
348#106—22no—
349#106—22no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352#103—22no—
353#103—22no—
354#103—22no—
355#102—22no—
356C22⋊C8—21no—
357C22⋊C8—21no—
358#97—21no—
359#97C23×C2312yes—
360#97C23×C2312yes—
361#97C23×C2312yes—
362C23#1906611yescharacteristic, direct product factor, Sylow
363#41—82no—
364C2×Q8—24no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—84no—
372C2×Q8C2×D4×C2312yes—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C2×D4×C2312yes—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4C2×D4×C2311yescharacteristic
380C22×C4C2×D4×C2311yescharacteristic
381C22×C4C2×D4×C2311yescharacteristic
382Q16—82no—
383SD16—82no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4C2×D4×C2312yes—
407C4⋊C4C2×D4×C2312yes—
408C4⋊C4C2×D4×C2312yes—
409C4⋊C4C2×D4×C2312yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—41no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—41no—
422C42C2×D4×C2311yescharacteristic
423C42C2×D4×C2311yescharacteristic
424C42C2×D4×C2311yescharacteristic
425C42C24×C2311yescharacteristic
426C23—22no—
427C23C23×#12011yescharacteristic
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—41no—
433Q8—41no—
434Q8—42no—
435Q8—42no—
436Q8—84no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—82no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—84no—
453C2×C4C23×#12011yescharacteristic
454C2×C4C23×#12011yescharacteristic
455C2×C4C23×#12011yescharacteristic
456C2×C4C22×D4×C2311yescharacteristic
457C2×C4C22×D4×C2311yescharacteristic
458C2×C4C22×D4×C2311yescharacteristic
459C8—24no—
460C8—42no—
461C8—42no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22C2×C23×#12011yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4C23×#45612yes—
482C2—41no—
483C2—82no—
484C2C2×C23×#45611yescharacteristic
485C2C23×#301911yescharacteristic
486C2C23×#302011yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ 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×#3237C2semidirect
4. C23×#3162C2semidirect
5. C23×#1664C2semidirect
6. C23×#1660C2semidirect
7. C23×#1537C2semidirect
8. C23×#1531C2semidirect
182. #19066C23direct (N × H)
220. #3237C2×C23semidirect
222. #3162C2×C23semidirect
223. #1664C2×C23semidirect
224. #1660C2×C23semidirect
225. #1537C2×C23semidirect
226. #1531C2×C23semidirect
362. C23#19066direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35502
C23×#32372
C23×#32162
C23×#31622
C23×#16642
C23×#16602
C23×#15372
C23×#15312
C23×#15302
#1906623