Finite Group Databasebeta

← C23×#60492 #19208118

Subgroups — C23×#60492 #19208118

Order 5888.

2038 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×#3458C211yesmaximal, characteristic, complemented
2C2×C23×#520C211yesmaximal, characteristic
3C23×#3084C218yesmaximal, complemented
4C23×#3083C218yesmaximal, complemented
5C23×#3078C214yesmaximal, complemented
6C23×#3077C214yesmaximal, complemented
7C2×C23×#428C212yesmaximal, complemented
8C23×#2905C211yesmaximal, characteristic, complemented
9C23×#2896C212yesmaximal
10C22×#41×C23C2211yescharacteristic
11C22×M16×C23C2212yes—
12C23×#544—21no—
13C23×#544—21no—
14C23×#544C2214yescomplemented
15C23×#544C2214yescomplemented
16C23×#543—22no—
17C23×#533—24no—
18C23×#533C2214yes—
19C23×#520—24no—
20C23×#520C2218yes—
21C2×C4×Q8×C23C2212yes—
22C2×C4×D4×C23—22no—
23C2×C4×D4×C23C2212yes—
24C2×C23×#117—21no—
25C2×C23×#117C2212yes—
26C2×C23×#117C2211yescharacteristic
27C22×C4⋊C4×C23—21no—
28C22×C4⋊C4×C23C2211yescharacteristic
29C23×#479C22116yescomplemented
30C23×#478C22116yescomplemented
31C23×#477C2218yescomplemented
32C23×#476C2218yescomplemented
33C23×#475C2218yescomplemented
34C23×#474C2218yescomplemented
35C23×#428C2218yes—
36C23×#428C2218yescomplemented
37C2×C23×#107C2214yescomplemented
38C2×C23×#106C2214yescomplemented
39C23×#424C2218yescomplemented
40C23×#423C2214yes—
41C23×#419C2214yes—
42C23×#418C2214yes—
43C23×#417C2214yes—
44C23×#417C2214yes—
45C23×#416C2218yes—
46C2×#41×C23—22no—
47C2×#41×C23—24no—
48C2×#41×C23Q818yescomplemented
49C2×#41×C23C2314yes—
50C22×Q8×C23C2311yescharacteristic
51C22×D4×C23—21no—
52C22×D4×C23C2311yescharacteristic
53C23×C4×C23—21no—
54C23×C4×C23—21no—
55C23×C4×C23—22no—
56C23×C4×C23C2311yescharacteristic
57C2×M16×C23—28no—
58C2×M16×C23C2314yes—
59C2×M16×C23C2314yes—
60C22×C8×C23C2314yes—
61C23×C4⋊Q8—24no—
62C23×C4⋊Q8—24no—
63C23×C4⋊Q8C2314yes—
64C23×C4⋊Q8C2314yes—
65C23×#125—24no—
66C23×#125—24no—
67C23×#125C2318yes—
68C23×C22⋊Q8—24no—
69C23×C22⋊Q8—24no—
70C23×C22⋊Q8—28no—
71C23×C22⋊Q8—28no—
72C4×Q8×C23—24no—
73C4×Q8×C23—24no—
74C4×Q8×C23—24no—
75C4×Q8×C23C2318yes—
76C4×D4×C23—216no—
77C4×D4×C23—24no—
78C4×D4×C23C2318yes—
79C23×#117—22no—
80C23×#117—22no—
81C23×#117—28no—
82C23×#117D414yes—
83C23×#117C2314yes—
84C23×#117C2314yes—
85C23×#117C2314yes—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—22no—
88C2×C4⋊C4×C23—22no—
89C2×C4⋊C4×C23—28no—
90C2×C4⋊C4×C23D411yescharacteristic
91C2×C4⋊C4×C23D411yescharacteristic
92C2×C4⋊C4×C23C2312yes—
93C2×C4⋊C4×C23C2312yes—
94C2×C4⋊C4×C23C2314yes—
95C2×C4⋊C4×C23C2312yes—
96C2×C22⋊C4×C23—22no—
97C2×C22⋊C4×C23—22no—
98C2×C22⋊C4×C23D412yes—
99C2×C42×C23—22no—
100C2×C42×C23—22no—
101C2×C42×C23C2312yes—
102C23×#107C23116yes—
103C23×#106C23116yes—
104C23×C4⋊C8C2318yes—
105C23×C4⋊C8C2318yes—
106C23×#103C2318yes—
107C23×#103C2318yes—
108C23×#102C2318yes—
109C23×#102C2318yes—
110C24×C23—21no—
111C24×C23C2×D411yescharacteristic
112#41×C23—216no—
113#41×C23—28no—
114#41×C23C2×Q8116yes—
115C2×Q8×C23—21no—
116C2×Q8×C23—24no—
117C2×Q8×C23—24no—
118C2×Q8×C23—22no—
119C2×Q8×C23C2×Q814yes—
120C2×Q8×C23C2412yes—
121C2×D4×C23—28no—
122C2×D4×C23—24no—
123C2×D4×C23—21no—
124C2×D4×C23—22no—
125C2×D4×C23C2×Q814yes—
126C2×D4×C23C2412yes—
127C22×C4×C23—21no—
128C22×C4×C23—28no—
129C22×C4×C23—24no—
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—22no—
133C22×C4×C23—24no—
134C22×C4×C23—28no—
135C22×C4×C23—84no—
136C22×C4×C23—82no—
137C22×C4×C23C2×D412yes—
138C22×C4×C23C2×D412yes—
139C22×C4×C23C2×D412yes—
140C22×C4×C23C2×D411yescharacteristic
141C22×C4×C23C2×Q814yes—
142C22×C4×C23#4114yes—
143C22×C4×C23C2411yescharacteristic
144C22×C4×C23C2412yes—
145M16×C23—28no—
146M16×C23—28no—
147C2×C8×C23—28no—
148C2×C8×C23C2418yes—
149C4⋊C4×C23—24no—
150C4⋊C4×C23—28no—
151C4⋊C4×C23—22no—
152C4⋊C4×C23—22no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—24no—
155C4⋊C4×C23—24no—
156C4⋊C4×C23—28no—
157C4⋊C4×C23—28no—
158C4⋊C4×C23C2×D414yes—
159C4⋊C4×C23C2×D414yes—
160C4⋊C4×C23C2414yes—
161C4⋊C4×C23C2414yes—
162C4⋊C4×C23C2414yes—
163C22⋊C4×C23—24no—
164C22⋊C4×C23—24no—
165C22⋊C4×C23—28no—
166C22⋊C4×C23C2×D418yes—
167C42×C23—28no—
168C42×C23—24no—
169C42×C23—24no—
170C42×C23C2×D414yes—
171C42×C23C2414yes—
172#60492C2311yesmaximal, characteristic, direct product factor, Sylow
173C23×C23—24no—
174C23×C23—22no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23C22⋊Q814yes—
180C23×C23C22×D411yescharacteristic
181C23×C23C22×D412yes—
182Q8×C23—24no—
183Q8×C23—22no—
184Q8×C23—84no—
185Q8×C23—84no—
186Q8×C23C22×Q814yes—
187D4×C23—216no—
188D4×C23—24no—
189D4×C23—22no—
190D4×C23C22×Q814yes—
191C2×C4×C23—24no—
192C2×C4×C23—22no—
193C2×C4×C23—216no—
194C2×C4×C23—24no—
195C2×C4×C23—24no—
196C2×C4×C23—24no—
197C2×C4×C23—42no—
198C2×C4×C23—42no—
199C2×C4×C23—44no—
200C2×C4×C23—88no—
201C2×C4×C23—82no—
202C2×C4×C23—82no—
203C2×C4×C23—84no—
204C2×C4×C23C22⋊Q814yes—
205C2×C4×C23C22⋊Q818yes—
206C2×C4×C23C22×D414yes—
207C2×C4×C23C22×D414yes—
208C2×C4×C23C22×D411yescharacteristic
209C2×C4×C23C22×D412yes—
210C2×C4×C23C22×Q812yes—
211C2×C4×C23C22×Q814yes—
212C2×C4×C23C2×#4114yes—
213C2×C4×C23C2×#4112yes—
214C2×C4×C23C2511yescharacteristic
215C8×C23—28no—
216#3458C2×C2311yescharacteristic, complemented
217C2×#520C2×C2311yescharacteristic
218#3084C2×C2318yescomplemented
219#3083C2×C2318yescomplemented
220#3078C2×C2314yescomplemented
221#3077C2×C2314yescomplemented
222C2×#428C2×C2312yescomplemented
223#2905C2×C2311yescharacteristic, complemented
224#2896C2×C2312yes—
225C22×C23—21no—
226C22×C23—22no—
227C22×C23—21no—
228C22×C23—22no—
229C22×C23—24no—
230C22×C23—44no—
231C22×C23—48no—
232C22×C23—42no—
233C22×C23—42no—
234C22×C23C2×C22⋊Q812yes—
235C22×C23C2×C22⋊Q814yes—
236C22×C23C23×D411yescharacteristic
237C4×C23—24no—
238C4×C23—24no—
239C4×C23—24no—
240C4×C23—44no—
241C4×C23—44no—
242C4×C23—44no—
243C4×C23C2×C22⋊Q814yes—
244C4×C23C2×C22⋊Q812yes—
245C4×C23C23×Q811yescharacteristic
246C4×C23C22×#4111yescharacteristic
247C22×#41C22×C2311yescharacteristic
248C22×M16C22×C2312yes—
249#544—21no—
250#544—21no—
251#544C22×C2314yescomplemented
252#544C22×C2314yescomplemented
253#543—22no—
254#533—24no—
255#533C22×C2314yes—
256#520—24no—
257#520C22×C2318yes—
258C2×C4×Q8C22×C2312yes—
259C2×C4×D4—22no—
260C2×C4×D4C22×C2312yes—
261C2×#117—21no—
262C2×#117C22×C2312yes—
263C2×#117C22×C2311yescharacteristic
264C22×C4⋊C4—21no—
265C22×C4⋊C4C22×C2311yescharacteristic
266#479C22×C23116yescomplemented
267#478C22×C23116yescomplemented
268#477C22×C2318yescomplemented
269#476C22×C2318yescomplemented
270#475C22×C2318yescomplemented
271#474C22×C2318yescomplemented
272#428C22×C2318yes—
273#428C22×C2318yescomplemented
274C2×#107C22×C2314yescomplemented
275C2×#106C22×C2314yescomplemented
276#424C22×C2318yescomplemented
277#423C22×C2314yes—
278#419C22×C2314yes—
279#418C22×C2314yes—
280#417C22×C2314yes—
281#417C22×C2314yes—
282#416C22×C2318yes—
283C2×C23—22no—
284C2×C23—24no—
285C2×C23—44no—
286C2×C23C22×C22⋊Q811yescharacteristic
287C2×C23#358511yescharacteristic
288C2×C23#358611yescharacteristic
289C2×#41—22no—
290C2×#41—24no—
291C2×#41Q8×C2318yescomplemented
292C2×#41C23×C2314yes—
293C22×Q8C23×C2311yescharacteristic
294C22×D4—21no—
295C22×D4C23×C2311yescharacteristic
296C23×C4—21no—
297C23×C4—21no—
298C23×C4—22no—
299C23×C4C23×C2311yescharacteristic
300C2×M16—28no—
301C2×M16C23×C2314yes—
302C2×M16C23×C2314yes—
303C22×C8C23×C2314yes—
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8C23×C2314yes—
307C4⋊Q8C23×C2314yes—
308#125—24no—
309#125—24no—
310#125C23×C2318yes—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—28no—
314C22⋊Q8—28no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8C23×C2318yes—
319C4×D4—216no—
320C4×D4—24no—
321C4×D4C23×C2318yes—
322#117—22no—
323#117—22no—
324#117—28no—
325#117D4×C2314yes—
326#117C23×C2314yes—
327#117C23×C2314yes—
328#117C23×C2314yes—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4D4×C2311yescharacteristic
334C2×C4⋊C4D4×C2311yescharacteristic
335C2×C4⋊C4C23×C2312yes—
336C2×C4⋊C4C23×C2314yes—
337C2×C4⋊C4C23×C2312yes—
338C2×C4⋊C4C23×C2312yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4D4×C2312yes—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42C23×C2312yes—
345#107C23×C23116yes—
346#106C23×C23116yes—
347C4⋊C8C23×C2318yes—
348C4⋊C8C23×C2318yes—
349#103C23×C2318yes—
350#103C23×C2318yes—
351#102C23×C2318yes—
352#102C23×C2318yes—
353C23#6049211yescharacteristic, direct product factor, Sylow
354C24—21no—
355C24C2×D4×C2311yescharacteristic
356#41—216no—
357#41—28no—
358#41C2×Q8×C23116yes—
359C2×Q8—21no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8C2×Q8×C2314yes—
364C2×Q8C24×C2312yes—
365C2×D4—28no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4C2×Q8×C2314yes—
370C2×D4C24×C2312yes—
371C22×C4—21no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—84no—
380C22×C4—82no—
381C22×C4C2×D4×C2312yes—
382C22×C4C2×D4×C2312yes—
383C22×C4C2×D4×C2312yes—
384C22×C4C2×D4×C2311yescharacteristic
385C22×C4C2×Q8×C2314yes—
386C22×C4#41×C2314yes—
387C22×C4C24×C2311yescharacteristic
388C22×C4C24×C2312yes—
389M16—28no—
390M16—28no—
391C2×C8—28no—
392C2×C8C24×C2318yes—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4C2×D4×C2314yes—
403C4⋊C4C2×D4×C2314yes—
404C4⋊C4C24×C2314yes—
405C4⋊C4C24×C2314yes—
406C4⋊C4C24×C2314yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—28no—
410C22⋊C4C2×D4×C2318yes—
411C42—28no—
412C42—24no—
413C42—24no—
414C42C2×D4×C2314yes—
415C42C24×C2314yes—
416C23—22no—
417C23—24no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23C23×C22⋊Q814yes—
423C23C22×D4×C2311yescharacteristic
424C23C22×D4×C2312yes—
425Q8—22no—
426Q8—24no—
427Q8—84no—
428Q8—84no—
429Q8C22×Q8×C2314yes—
430D4—216no—
431D4—22no—
432D4—24no—
433D4C22×Q8×C2314yes—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—216no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—88no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4C23×C22⋊Q814yes—
448C2×C4C23×C22⋊Q818yes—
449C2×C4C22×D4×C2314yes—
450C2×C4C22×D4×C2314yes—
451C2×C4C22×D4×C2311yescharacteristic
452C2×C4C22×D4×C2312yes—
453C2×C4C22×Q8×C2312yes—
454C2×C4C22×Q8×C2314yes—
455C2×C4C2×#41×C2314yes—
456C2×C4C2×#41×C2312yes—
457C2×C4C25×C2311yescharacteristic
458C8—28no—
459C22—21no—
460C22—22no—
461C22—21no—
462C22—22no—
463C22—24no—
464C22—44no—
465C22—48no—
466C22—42no—
467C22—42no—
468C22C2×C23×C22⋊Q812yes—
469C22C2×C23×C22⋊Q814yes—
470C22C23×D4×C2311yescharacteristic
471C4—24no—
472C4—24no—
473C4—24no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4C2×C23×C22⋊Q814yes—
478C4C2×C23×C22⋊Q812yes—
479C4C23×Q8×C2311yescharacteristic
480C4C22×#41×C2311yescharacteristic
481C2—22no—
482C2—24no—
483C2—44no—
484C2C22×C23×C22⋊Q811yescharacteristic
485C2C23×#358511yescharacteristic
486C2C23×#358611yescharacteristic
subgroupG/H
center, socleC22×C23C23×D4
commutator, Frattini, cosocleC2×C4C25×C23
prime radicalC22×#41×C23C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3458C2semidirect
3. C23×#3084C2semidirect
4. C23×#3083C2semidirect
5. C23×#3078C2semidirect
6. C23×#3077C2semidirect
7. C2×C23×#428C2semidirect
8. C23×#2905C2semidirect
14. C23×#544C22semidirect
15. C23×#544C22semidirect
29. C23×#479C22semidirect
30. C23×#478C22semidirect
31. C23×#477C22semidirect
32. C23×#476C22semidirect
33. C23×#475C22semidirect
34. C23×#474C22semidirect
36. C23×#428C22semidirect
37. C2×C23×#107C22semidirect
38. C2×C23×#106C22semidirect
39. C23×#424C22semidirect
48. C2×#41×C23Q8semidirect
172. #60492C23direct (N × H)
216. #3458C2×C23semidirect
218. #3084C2×C23semidirect
219. #3083C2×C23semidirect
220. #3078C2×C23semidirect
221. #3077C2×C23semidirect
222. C2×#428C2×C23semidirect
223. #2905C2×C23semidirect
251. #544C22×C23semidirect
252. #544C22×C23semidirect
266. #479C22×C23semidirect
267. #478C22×C23semidirect
268. #477C22×C23semidirect
269. #476C22×C23semidirect
270. #475C22×C23semidirect
271. #474C22×C23semidirect
273. #428C22×C23semidirect
274. C2×#107C22×C23semidirect
275. C2×#106C22×C23semidirect
276. #424C22×C23semidirect
291. C2×#41Q8×C23semidirect
353. C23#60492direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34582
C2×C23×#5202
C23×#30842
C23×#30832
C23×#30782
C23×#30772
C2×C23×#4282
C23×#29052
C23×#28962
#6049223