Finite Group Databasebeta

← C23×#8486 #19156112

Subgroups — C23×#8486 #19156112

Order 5888.

686 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3559C211yesmaximal, characteristic
2C23×#1998C211yesmaximal, characteristic, complemented
3C23×#1557C211yesmaximal, characteristic, complemented
4C23×#1529C211yesmaximal, characteristic, complemented
5C23×#1495C211yesmaximal, characteristic, complemented
6C23×#1484C211yesmaximal, characteristic, complemented
7C23×#1470C211yesmaximal, characteristic, complemented
8C23×#563—21no—
9C23×#562—21no—
10C23×#554—21no—
11C23×#553—21no—
12C23×#553—21no—
13C23×#547—21no—
14C23×#547—22no—
15C23×#544—21no—
16C23×#540—21no—
17C23×#539—22no—
18C23×#539C412yes—
19C23×#537C412yes—
20C23×#532C2211yescharacteristic
21C23×#529C2211yescharacteristic
22C2×C23×#125C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C23×#431C2211yescharacteristic
25C23×#423C2211yescharacteristic
26C4×M16×C23C2211yescharacteristic
27C23×#343—21no—
28C23×#343—21no—
29C23×#332—21no—
30C23×#331—21no—
31C23×#330—21no—
32C23×#329—21no—
33C23×Q8⋊C8—21no—
34C23×D4⋊C8—22no—
35C23×D4⋊C8—21no—
36C23×#324—22no—
37C2×#41×C23—22no—
38C2×#41×C23—21no—
39C2×M16×C23—21no—
40C2×M16×C23—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—22no—
43C23×C4⋊Q8D411yescharacteristic
44C23×#127D411yescharacteristic
45C23×#126—22no—
46C23×#126—22no—
47C23×#126—21no—
48C23×#125—21no—
49C23×#125—22no—
50C23×#125—21no—
51C23×#125—21no—
52C23×#125—21no—
53C23×#125—21no—
54C23×#125D411yescharacteristic
55C23×#124—21no—
56C23×#124D411yescharacteristic
57C23×#123—22no—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×#123—22no—
63C23×#123—21no—
64C23×C22⋊Q8—22no—
65C23×C22⋊Q8—22no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8—22no—
70C23×C22⋊Q8C2×C411yescharacteristic
71C23×#121—21no—
72C23×#121—22no—
73C23×#121—21no—
74C23×#121—22no—
75C23×#121—22no—
76C23×#121—22no—
77C23×#121—21no—
78C23×#121C2×C411yescharacteristic
79C23×#121C2×C412yes—
80C4×Q8×C23D411yescharacteristic
81C4×D4×C23—22no—
82C4×D4×C23—21no—
83C4×D4×C23—21no—
84C4×D4×C23—22no—
85C4×D4×C23—21no—
86C4×D4×C23—21no—
87C4×D4×C23D411yescharacteristic
88C4×D4×C23D412yes—
89C23×#117—21no—
90C23×#117—21no—
91C2×C4⋊C4×C23—21no—
92C2×C4⋊C4×C23—22no—
93C2×C4⋊C4×C23C2×C412yes—
94C2×C4⋊C4×C23D412yes—
95C2×C4⋊C4×C23D411yescharacteristic
96C2×C4⋊C4×C23D411yescharacteristic
97C2×C42×C23C2311yescharacteristic
98C23×C4⋊C8—21no—
99C23×C4⋊C8—21no—
100C23×C4⋊C8—21no—
101C23×C4⋊C8—21no—
102C23×C22⋊C8—21no—
103C23×C22⋊C8—21no—
104C23×#97—21no—
105C23×#97—21no—
106C4×C8×C23—21no—
107C4×C8×C23—21no—
108#41×C23—82no—
109#41×C23—84no—
110C2×Q8×C23—22no—
111C2×Q8×C23—21no—
112C2×Q8×C23C22⋊C411yescharacteristic
113C2×D4×C23—22no—
114C2×D4×C23—21no—
115C2×D4×C23—22no—
116C2×D4×C23—22no—
117C2×D4×C23—21no—
118C2×D4×C23—21no—
119C2×D4×C23C22⋊C411yescharacteristic
120C2×D4×C23C22⋊C412yes—
121C22×C4×C23—22no—
122C22×C4×C23—22no—
123C22×C4×C23—22no—
124C22×C4×C23—21no—
125C22×C4×C23—22no—
126C22×C4×C23C22⋊C412yes—
127C22×C4×C23C22⋊C412yes—
128C22×C4×C23C22×C411yescharacteristic
129C22×C4×C23C2×D411yescharacteristic
130C22×C4×C23C2×D411yescharacteristic
131M16×C23—22no—
132M16×C23—41no—
133M16×C23—42no—
134C2×C8×C23—41no—
135C2×C8×C23—41no—
136C2×C8×C23—41no—
137C2×C8×C23—41no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—21no—
145C4⋊C4×C23—21no—
146C4⋊C4×C23—21no—
147C4⋊C4×C23—22no—
148C4⋊C4×C23—21no—
149C4⋊C4×C23—21no—
150C4⋊C4×C23—22no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—21no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23C22⋊C412yes—
156C4⋊C4×C23C22⋊C411yescharacteristic
157C4⋊C4×C23C22⋊C411yescharacteristic
158C22⋊C4×C23—22no—
159C22⋊C4×C23—22no—
160C22⋊C4×C23—22no—
161C22⋊C4×C23—22no—
162C22⋊C4×C23—22no—
163C22⋊C4×C23—21no—
164C22⋊C4×C23—21no—
165C22⋊C4×C23—22no—
166C22⋊C4×C23—21no—
167C22⋊C4×C23—21no—
168C22⋊C4×C23—21no—
169C22⋊C4×C23—22no—
170C42×C23—22no—
171C42×C23—21no—
172C42×C23C2×D411yescharacteristic
173C42×C23C2×D411yescharacteristic
174C42×C23C2×D411yescharacteristic
175C42×C23C2×D411yescharacteristic
176#8486C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—21no—
178C23×C23—22no—
179C23×C23C2×C22⋊C411yescharacteristic
180Q8×C23—41no—
181Q8×C23—81no—
182Q8×C23—82no—
183D4×C23—42no—
184D4×C23—41no—
185D4×C23—44no—
186D4×C23—42no—
187D4×C23—41no—
188D4×C23—41no—
189D4×C23—81no—
190D4×C23—82no—
191D4×C23—82no—
192C2×C4×C23—21no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—22no—
197C2×C4×C23—22no—
198C2×C4×C23—21no—
199C2×C4×C23—22no—
200C2×C4×C23—41no—
201C2×C4×C23—41no—
202C2×C4×C23—41no—
203C2×C4×C23—41no—
204C2×C4×C23—42no—
205C2×C4×C23—42no—
206C2×C4×C23—82no—
207C2×C4×C23—82no—
208C2×C4×C23—82no—
209C2×C4×C23—82no—
210C2×C4×C23C2×C22⋊C411yescharacteristic
211C2×C4×C23C2×C22⋊C411yescharacteristic
212C2×C4×C23#12011yescharacteristic
213C2×C4×C23#12011yescharacteristic
214C2×C4×C23#12011yescharacteristic
215C2×C4×C23#12011yescharacteristic
216C8×C23—42no—
217C8×C23—81no—
218C8×C23—81no—
219C8×C23—81no—
220#3559C2×C2311yescharacteristic
221#1998C2×C2311yescharacteristic, complemented
222#1557C2×C2311yescharacteristic, complemented
223#1529C2×C2311yescharacteristic, complemented
224#1495C2×C2311yescharacteristic, complemented
225#1484C2×C2311yescharacteristic, complemented
226#1470C2×C2311yescharacteristic, complemented
227C22×C23—21no—
228C22×C23—21no—
229C22×C23—21no—
230C22×C23—41no—
231C22×C23—42no—
232C22×C23—42no—
233C22×C23—41no—
234C22×C23—42no—
235C22×C23—41no—
236C22×C23#37911yescharacteristic
237C4×C23—21no—
238C4×C23—21no—
239C4×C23—21no—
240C4×C23—21no—
241C4×C23—21no—
242C4×C23—41no—
243C4×C23—42no—
244C4×C23—42no—
245C4×C23—41no—
246C4×C23—42no—
247C4×C23—41no—
248C4×C23#45411yescharacteristic
249C4×C23#45611yescharacteristic
250#563—21no—
251#562—21no—
252#554—21no—
253#553—21no—
254#553—21no—
255#547—22no—
256#547—21no—
257#544—21no—
258#540—21no—
259#539—22no—
260#539C4×C2312yes—
261#537C4×C2312yes—
262#532C22×C2311yescharacteristic
263#529C22×C2311yescharacteristic
264C2×#125C22×C2311yescharacteristic
265#432C22×C2311yescharacteristic
266#431C22×C2311yescharacteristic
267#423C22×C2311yescharacteristic
268C4×M16C22×C2311yescharacteristic
269#343—21no—
270#343—21no—
271#332—21no—
272#331—21no—
273#330—21no—
274#329—21no—
275Q8⋊C8—21no—
276D4⋊C8—22no—
277D4⋊C8—21no—
278#324—22no—
279C2×C23—41no—
280C2×C23—81no—
281C2×C23—82no—
282C2×C23#179111yescharacteristic
283C2×C23#179311yescharacteristic
284C2×C23#179511yescharacteristic
285C2×#41—22no—
286C2×#41—21no—
287C2×M16—21no—
288C2×M16—21no—
289C4⋊Q8—21no—
290C4⋊Q8—22no—
291C4⋊Q8D4×C2311yescharacteristic
292#127D4×C2311yescharacteristic
293#126—22no—
294#126—22no—
295#126—21no—
296#125—21no—
297#125—22no—
298#125—21no—
299#125—21no—
300#125—21no—
301#125—21no—
302#125D4×C2311yescharacteristic
303#124—21no—
304#124D4×C2311yescharacteristic
305#123—22no—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311#123—21no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8C2×C4×C2311yescharacteristic
319#121—21no—
320#121—22no—
321#121—21no—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—21no—
326#121C2×C4×C2311yescharacteristic
327#121C2×C4×C2312yes—
328C4×Q8D4×C2311yescharacteristic
329C4×D4—22no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—22no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4D4×C2311yescharacteristic
336C4×D4D4×C2312yes—
337#117—21no—
338#117—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C2×C4×C2312yes—
342C2×C4⋊C4D4×C2312yes—
343C2×C4⋊C4D4×C2311yescharacteristic
344C2×C4⋊C4D4×C2311yescharacteristic
345C2×C42C23×C2311yescharacteristic
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352#97—21no—
353#97—21no—
354C4×C8—21no—
355C4×C8—21no—
356C23#848611yescharacteristic, direct product factor, Sylow
357#41—82no—
358#41—84no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8C22⋊C4×C2311yescharacteristic
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C22⋊C4×C2311yescharacteristic
369C2×D4C22⋊C4×C2312yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4C22⋊C4×C2312yes—
376C22×C4C22⋊C4×C2312yes—
377C22×C4C22×C4×C2311yescharacteristic
378C22×C4C2×D4×C2311yescharacteristic
379C22×C4C2×D4×C2311yescharacteristic
380M16—22no—
381M16—41no—
382M16—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4C22⋊C4×C2312yes—
405C4⋊C4C22⋊C4×C2311yescharacteristic
406C4⋊C4C22⋊C4×C2311yescharacteristic
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C42—22no—
420C42—21no—
421C42C2×D4×C2311yescharacteristic
422C42C2×D4×C2311yescharacteristic
423C42C2×D4×C2311yescharacteristic
424C42C2×D4×C2311yescharacteristic
425C23—22no—
426C23—21no—
427C23C2×C22⋊C4×C2311yescharacteristic
428Q8—41no—
429Q8—81no—
430Q8—82no—
431D4—42no—
432D4—41no—
433D4—44no—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4C2×C22⋊C4×C2311yescharacteristic
459C2×C4C2×C22⋊C4×C2311yescharacteristic
460C2×C4C23×#12011yescharacteristic
461C2×C4C23×#12011yescharacteristic
462C2×C4C23×#12011yescharacteristic
463C2×C4C23×#12011yescharacteristic
464C8—42no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22C23×#37911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4C23×#45411yescharacteristic
490C4C23×#45611yescharacteristic
491C2—41no—
492C2—82no—
493C2—81no—
494C2C23×#179111yescharacteristic
495C2C23×#179311yescharacteristic
496C2C23×#179511yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3559C2
LCSG ▷ C22×C4 ▷ 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×#1998C2semidirect
3. C23×#1557C2semidirect
4. C23×#1529C2semidirect
5. C23×#1495C2semidirect
6. C23×#1484C2semidirect
7. C23×#1470C2semidirect
176. #8486C23direct (N × H)
221. #1998C2×C23semidirect
222. #1557C2×C23semidirect
223. #1529C2×C23semidirect
224. #1495C2×C23semidirect
225. #1484C2×C23semidirect
226. #1470C2×C23semidirect
356. C23#8486direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35592
C23×#19982
C23×#15572
C23×#15292
C23×#14952
C23×#14842
C23×#14702
#848623