Finite Group Databasebeta

← C23×#18643 #19166269

Subgroups — C23×#18643 #19166269

Order 5888.

906 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3503C211yesmaximal, characteristic
2C23×#3230C211yesmaximal, characteristic, complemented
3C2×C23×#487C211yesmaximal, characteristic, complemented
4C23×#3079C211yesmaximal, characteristic
5C23×#1637C214yesmaximal, complemented
6C23×#1636C214yesmaximal, complemented
7C23×#1549C211yesmaximal, characteristic, complemented
8C23×#1493C211yesmaximal, characteristic, complemented
9C23×#1476C211yesmaximal, characteristic
10Q82×C23—21no—
11C23×#557C2212yes—
12C23×#554—21no—
13C23×#554—22no—
14D4×Q8×C23—22no—
15D4×Q8×C23C2212yes—
16C23×#544—22no—
17C23×#544—21no—
18C23×#533—21no—
19C23×#533C2211yescharacteristic
20C2×C23×C4⋊Q8—21no—
21C2×C23×C4⋊Q8C2211yescharacteristic
22C2×C23×C4⋊Q8C2211yescharacteristic
23C23×#520—21no—
24C4×#41×C23C2211yescharacteristic
25C23×#500—21no—
26C23×#487—21no—
27C23×#487—22no—
28C23×#487C2214yes—
29C23×#484—22no—
30C23×#477C2212yes—
31C23×#477C2214yes—
32C23×#476C2212yes—
33C23×#475—21no—
34C23×#474—21no—
35C23×#431C2211yescharacteristic
36C23×#428—21no—
37C23×#423C2211yescharacteristic
38C23×#417—21no—
39C2×C23×#103—22no—
40C2×C4×C8×C23C2211yescharacteristic
41C23×#332C2214yes—
42C23×Q8⋊C8C2212yes—
43C23×Q8⋊C8C2214yes—
44C23×D4⋊C8C2212yes—
45C2×#41×C23D411yescharacteristic
46C22×Q8×C23—22no—
47C22×Q8×C23—21no—
48C22×Q8×C23—21no—
49C2×M16×C23—21no—
50C22×C8×C23—21no—
51C23×C4⋊Q8—24no—
52C23×C4⋊Q8—22no—
53C23×C4⋊Q8—22no—
54C23×C4⋊Q8—22no—
55C23×C4⋊Q8—22no—
56C23×C4⋊Q8—21no—
57C23×C4⋊Q8—21no—
58C23×C4⋊Q8—21no—
59C23×C4⋊Q8—21no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8D411yescharacteristic
62C23×C4⋊Q8D412yes—
63C23×C4⋊Q8D411yescharacteristic
64C23×C4⋊Q8C2312yes—
65C23×C4⋊Q8C2312yes—
66C23×#125—22no—
67C23×#125—24no—
68C23×#125—22no—
69C23×#125—21no—
70C23×C22⋊Q8—24no—
71C23×C22⋊Q8—24no—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8—24no—
76C23×C22⋊Q8D412yes—
77C4×Q8×C23—22no—
78C4×Q8×C23—22no—
79C4×Q8×C23—22no—
80C4×Q8×C23—21no—
81C4×Q8×C23D411yescharacteristic
82C4×Q8×C23C2312yes—
83C4×Q8×C23C2311yescharacteristic
84C4×D4×C23—24no—
85C4×D4×C23—22no—
86C4×D4×C23—21no—
87C4×D4×C23—21no—
88C4×D4×C23D411yescharacteristic
89C4×D4×C23C2311yescharacteristic
90C23×#117—22no—
91C23×#117—21no—
92C23×#117—21no—
93C23×#117D411yescharacteristic
94C2×C4⋊C4×C23—22no—
95C2×C4⋊C4×C23—22no—
96C2×C4⋊C4×C23—21no—
97C2×C4⋊C4×C23—21no—
98C2×C4⋊C4×C23—21no—
99C2×C4⋊C4×C23D412yes—
100C2×C42×C23—21no—
101C2×C42×C23C2311yescharacteristic
102C23×#107—22no—
103C23×#107—24no—
104C23×#106—22no—
105C23×C4⋊C8—21no—
106C23×C4⋊C8C2312yes—
107C23×C4⋊C8C2312yes—
108C23×#103—22no—
109C23×#103—24no—
110C23×#103—28no—
111C23×#102—22no—
112C23×C22⋊C8—21no—
113C4×C8×C23—21no—
114C4×C8×C23—21no—
115C4×C8×C23C2312yes—
116#41×C23—22no—
117#41×C23Q1614yes—
118C2×Q8×C23—24no—
119C2×Q8×C23—21no—
120C2×Q8×C23—22no—
121C2×Q8×C23—22no—
122C2×Q8×C23—22no—
123C2×Q8×C23—44no—
124C2×Q8×C23—41no—
125C2×Q8×C23—82no—
126C2×Q8×C23—84no—
127C2×Q8×C23C2×D412yes—
128C2×Q8×C23C2×D411yescharacteristic
129C2×D4×C23—21no—
130C2×D4×C23C2×D411yescharacteristic
131C22×C4×C23—21no—
132C22×C4×C23—24no—
133C22×C4×C23—22no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—41no—
137C22×C4×C23C2×D411yescharacteristic
138C22×C4×C23C2×D411yescharacteristic
139C22×C4×C23C2×D411yescharacteristic
140M16×C23—42no—
141C2×C8×C23—22no—
142C2×C8×C23—22no—
143C2×C8×C23—22no—
144C2×C8×C23—42no—
145C4⋊C4×C23—24no—
146C4⋊C4×C23—24no—
147C4⋊C4×C23—24no—
148C4⋊C4×C23—24no—
149C4⋊C4×C23—21no—
150C4⋊C4×C23—22no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—22no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—22no—
157C4⋊C4×C23—21no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—41no—
162C4⋊C4×C23—41no—
163C4⋊C4×C23C2×D414yes—
164C4⋊C4×C23C2×D411yescharacteristic
165C4⋊C4×C23C2×D411yescharacteristic
166C4⋊C4×C23C2×D412yes—
167C22⋊C4×C23—24no—
168C22⋊C4×C23—22no—
169C22⋊C4×C23—21no—
170C22⋊C4×C23—21no—
171C22⋊C4×C23—21no—
172C42×C23—24no—
173C42×C23—21no—
174C42×C23—21no—
175C42×C23—21no—
176C42×C23—22no—
177C42×C23C2×D411yescharacteristic
178C42×C23C2×D411yescharacteristic
179C42×C23C2×D411yescharacteristic
180C42×C23C2411yescharacteristic
181#18643C2311yesmaximal, characteristic, direct product factor, Sylow
182C23×C23—21no—
183C23×C23#12011yescharacteristic
184Q8×C23—21no—
185Q8×C23—24no—
186Q8×C23—44no—
187Q8×C23—44no—
188Q8×C23—42no—
189Q8×C23—82no—
190Q8×C23—84no—
191Q8×C23—82no—
192Q8×C23C2×Q1612yes—
193D4×C23—24no—
194D4×C23—21no—
195D4×C23C2×Q1612yes—
196C2×C4×C23—21no—
197C2×C4×C23—22no—
198C2×C4×C23—24no—
199C2×C4×C23—22no—
200C2×C4×C23—22no—
201C2×C4×C23—21no—
202C2×C4×C23—21no—
203C2×C4×C23—22no—
204C2×C4×C23—24no—
205C2×C4×C23—42no—
206C2×C4×C23—42no—
207C2×C4×C23—42no—
208C2×C4×C23—82no—
209C2×C4×C23—84no—
210C2×C4×C23—82no—
211C2×C4×C23#12011yescharacteristic
212C2×C4×C23#12011yescharacteristic
213C2×C4×C23#12011yescharacteristic
214C2×C4×C23C2×Q1612yes—
215C2×C4×C23#13712yes—
216C2×C4×C23C22×D411yescharacteristic
217C2×C4×C23C22×D411yescharacteristic
218C2×C4×C23C22×D411yescharacteristic
219C8×C23—42no—
220C8×C23—42no—
221C8×C23—42no—
222#3503C2×C2311yescharacteristic
223#3230C2×C2311yescharacteristic, complemented
224C2×#487C2×C2311yescharacteristic, complemented
225#3079C2×C2311yescharacteristic
226#1637C2×C2314yescomplemented
227#1636C2×C2314yescomplemented
228#1549C2×C2311yescharacteristic, complemented
229#1493C2×C2311yescharacteristic, complemented
230#1476C2×C2311yescharacteristic
231C22×C23—21no—
232C22×C23—21no—
233C22×C23—22no—
234C22×C23—41no—
235C22×C23—41no—
236C22×C23#45112yes—
237C22×C23C2×#12011yescharacteristic
238C4×C23—22no—
239C4×C23—21no—
240C4×C23—21no—
241C4×C23—41no—
242C4×C23—42no—
243C4×C23—44no—
244C4×C23—41no—
245C4×C23—42no—
246C4×C23—82no—
247C4×C23#45112yes—
248C4×C23#45412yes—
249C4×C23C22×Q1611yescharacteristic
250C4×C23C2×#13711yescharacteristic
251C4×C23#57712yes—
252Q82—21no—
253#557C22×C2312yes—
254#554—21no—
255#554—22no—
256D4×Q8—22no—
257D4×Q8C22×C2312yes—
258#544—22no—
259#544—21no—
260#533—21no—
261#533C22×C2311yescharacteristic
262C2×C4⋊Q8—21no—
263C2×C4⋊Q8C22×C2311yescharacteristic
264C2×C4⋊Q8C22×C2311yescharacteristic
265#520—21no—
266C4×#41C22×C2311yescharacteristic
267#500—21no—
268#487—22no—
269#487—21no—
270#487C22×C2314yes—
271#484—22no—
272#477C22×C2312yes—
273#477C22×C2314yes—
274#476C22×C2312yes—
275#475—21no—
276#474—21no—
277#431C22×C2311yescharacteristic
278#428—21no—
279#423C22×C2311yescharacteristic
280#417—21no—
281C2×#103—22no—
282C2×C4×C8C22×C2311yescharacteristic
283#332C22×C2314yes—
284Q8⋊C8C22×C2312yes—
285Q8⋊C8C22×C2314yes—
286D4⋊C8C22×C2312yes—
287C2×C23—22no—
288C2×C23—42no—
289C2×C23C2×#45111yescharacteristic
290C2×C23#300911yescharacteristic
291C2×C23C2×#45411yescharacteristic
292C2×#41D4×C2311yescharacteristic
293C22×Q8—22no—
294C22×Q8—21no—
295C22×Q8—21no—
296C2×M16—21no—
297C22×C8—21no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—24no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8D4×C2311yescharacteristic
309C4⋊Q8D4×C2312yes—
310C4⋊Q8D4×C2311yescharacteristic
311C4⋊Q8C23×C2312yes—
312C4⋊Q8C23×C2312yes—
313#125—22no—
314#125—24no—
315#125—22no—
316#125—21no—
317C22⋊Q8—22no—
318C22⋊Q8—24no—
319C22⋊Q8—24no—
320C22⋊Q8—21no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8D4×C2312yes—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—21no—
328C4×Q8D4×C2311yescharacteristic
329C4×Q8C23×C2311yescharacteristic
330C4×Q8C23×C2312yes—
331C4×D4—24no—
332C4×D4—22no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4D4×C2311yescharacteristic
336C4×D4C23×C2311yescharacteristic
337#117—22no—
338#117—21no—
339#117—21no—
340#117D4×C2311yescharacteristic
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4D4×C2312yes—
347C2×C42—21no—
348C2×C42C23×C2311yescharacteristic
349#107—22no—
350#107—24no—
351#106—22no—
352C4⋊C8—21no—
353C4⋊C8C23×C2312yes—
354C4⋊C8C23×C2312yes—
355#103—22no—
356#103—24no—
357#103—28no—
358#102—22no—
359C22⋊C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8C23×C2312yes—
363C23#1864311yescharacteristic, direct product factor, Sylow
364#41—22no—
365#41Q16×C2314yes—
366C2×Q8—24no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—82no—
374C2×Q8—84no—
375C2×Q8C2×D4×C2312yes—
376C2×Q8C2×D4×C2311yescharacteristic
377C2×D4—21no—
378C2×D4C2×D4×C2311yescharacteristic
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4C2×D4×C2311yescharacteristic
386C22×C4C2×D4×C2311yescharacteristic
387C22×C4C2×D4×C2311yescharacteristic
388M16—42no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—42no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4C2×D4×C2314yes—
412C4⋊C4C2×D4×C2311yescharacteristic
413C4⋊C4C2×D4×C2311yescharacteristic
414C4⋊C4C2×D4×C2312yes—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C42—24no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—22no—
425C42C2×D4×C2311yescharacteristic
426C42C2×D4×C2311yescharacteristic
427C42C2×D4×C2311yescharacteristic
428C42C24×C2311yescharacteristic
429C23—21no—
430C23C23×#12011yescharacteristic
431Q8—21no—
432Q8—24no—
433Q8—44no—
434Q8—44no—
435Q8—42no—
436Q8—82no—
437Q8—84no—
438Q8—82no—
439Q8C2×Q16×C2312yes—
440D4—24no—
441D4—21no—
442D4C2×Q16×C2312yes—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4C23×#12011yescharacteristic
459C2×C4C23×#12011yescharacteristic
460C2×C4C23×#12011yescharacteristic
461C2×C4C2×Q16×C2312yes—
462C2×C4C23×#13712yes—
463C2×C4C22×D4×C2311yescharacteristic
464C2×C4C22×D4×C2311yescharacteristic
465C2×C4C22×D4×C2311yescharacteristic
466C8—42no—
467C8—42no—
468C8—42no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22C23×#45112yes—
475C22C2×C23×#12011yescharacteristic
476C4—22no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—42no—
481C4—44no—
482C4—41no—
483C4—42no—
484C4—82no—
485C4C23×#45112yes—
486C4C23×#45412yes—
487C4C22×Q16×C2311yescharacteristic
488C4C2×C23×#13711yescharacteristic
489C4C23×#57712yes—
490C2—22no—
491C2—42no—
492C2C2×C23×#45111yescharacteristic
493C2C23×#300911yescharacteristic
494C2C2×C23×#45411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC2×#41×C23D4
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×#3230C2semidirect
3. C2×C23×#487C2semidirect
5. C23×#1637C2semidirect
6. C23×#1636C2semidirect
7. C23×#1549C2semidirect
8. C23×#1493C2semidirect
181. #18643C23direct (N × H)
223. #3230C2×C23semidirect
224. C2×#487C2×C23semidirect
226. #1637C2×C23semidirect
227. #1636C2×C23semidirect
228. #1549C2×C23semidirect
229. #1493C2×C23semidirect
363. C23#18643direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35032
C23×#32302
C2×C23×#4872
C23×#30792
C23×#16372
C23×#16362
C23×#15492
C23×#14932
C23×#14762
#1864323