Finite Group Databasebeta

← C23×#19022 #19166648

Subgroups — C23×#19022 #19166648

Order 5888.

892 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3559C211yesmaximal, characteristic
2C23×#3218C211yesmaximal, characteristic, complemented
3C2×C23×#501C211yesmaximal, characteristic, complemented
4C23×#3141C211yesmaximal, characteristic, complemented
5C23×#1664C214yesmaximal, complemented
6C23×#1661C214yesmaximal, complemented
7C23×#1530C211yesmaximal, characteristic, complemented
8C23×#1528C211yesmaximal, characteristic, complemented
9C23×#1524C211yesmaximal, characteristic, complemented
10C23×#563C2212yes—
11C23×#562C2212yes—
12C23×#554—21no—
13C23×#553—21no—
14C23×#553—21no—
15C23×#547—22no—
16C23×#547—21no—
17C23×#544—21no—
18C23×#540—21no—
19C23×#539—22no—
20C23×#539—21no—
21C23×#537—21no—
22C23×#532C2211yescharacteristic
23C2×C23×C4⋊Q8C2211yescharacteristic
24C23×#529C2211yescharacteristic
25C2×C23×#125C2211yescharacteristic
26C23×#501—22no—
27C23×#501C2214yes—
28C23×#497C2212yes—
29C23×#491C2212yes—
30C23×#489C2214yes—
31C23×#487—21no—
32C23×#486—21no—
33C23×#462—21no—
34C23×#461—21no—
35C23×#451—21no—
36C23×#448—21no—
37C23×#432C2211yescharacteristic
38C2×C23×#107—21no—
39C2×C23×#106—21no—
40C23×#417—22no—
41C4×M16×C23C2211yescharacteristic
42C2×C23×#97C2211yescharacteristic
43C23×#330C2214yes—
44C23×#330C2212yes—
45C23×#329C2212yes—
46C23×#329C2214yes—
47C2×#41×C23—22no—
48C2×#41×C23D412yes—
49C22×Q8×C23—21no—
50C2×Q16×C23—22no—
51C2×SD16×C23—22no—
52C2×M16×C23—21no—
53C22×C8×C23—21no—
54C23×C4⋊Q8—21no—
55C23×C4⋊Q8—22no—
56C23×C4⋊Q8—21no—
57C23×C4⋊Q8—21no—
58C23×C4⋊Q8—21no—
59C23×C4⋊Q8D411yescharacteristic
60C23×C4⋊Q8C2312yes—
61C23×#127D411yescharacteristic
62C23×#126—22no—
63C23×#126—22no—
64C23×#126—21no—
65C23×#125—21no—
66C23×#125—22no—
67C23×#125—21no—
68C23×#125—21no—
69C23×#125D412yes—
70C23×#125C2312yes—
71C23×#125C2311yescharacteristic
72C23×#124C2311yescharacteristic
73C23×#124C2312yes—
74C23×#123—22no—
75C23×#123—22no—
76C23×#123—22no—
77C23×#123—22no—
78C23×#123—21no—
79C23×#123—22no—
80C23×#123—22no—
81C23×C22⋊Q8—22no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—22no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8D411yescharacteristic
88C23×#121—22no—
89C23×#121—21no—
90C23×#121—21no—
91C23×#121—22no—
92C23×#121—22no—
93C23×#121—22no—
94C23×#121—21no—
95C23×#121—21no—
96C23×#121D411yescharacteristic
97C4×Q8×C23D411yescharacteristic
98C4×D4×C23—22no—
99C4×D4×C23—22no—
100C4×D4×C23—21no—
101C4×D4×C23—21no—
102C4×D4×C23—21no—
103C4×D4×C23—21no—
104C4×D4×C23—21no—
105C4×D4×C23D411yescharacteristic
106C23×#117—21no—
107C23×#117—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—22no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23—21no—
112C2×C4⋊C4×C23—21no—
113C2×C4⋊C4×C23—21no—
114C2×C4⋊C4×C23D411yescharacteristic
115C2×C4⋊C4×C23D411yescharacteristic
116C2×C42×C23C2311yescharacteristic
117C23×#107—24no—
118C23×#106—24no—
119C23×C4⋊C8—21no—
120C23×#103—24no—
121C23×#103—22no—
122C23×#103—22no—
123C23×#102—24no—
124C23×C22⋊C8—21no—
125C23×#97—21no—
126C23×#97C2312yes—
127C23×#97C2312yes—
128C23×#97C2312yes—
129C4×C8×C23—21no—
130#41×C23—44no—
131#41×C23—84no—
132C2×Q8×C23—22no—
133C2×Q8×C23—22no—
134C2×Q8×C23—41no—
135C2×Q8×C23—82no—
136C2×Q8×C23C2×D412yes—
137C2×Q8×C23C2×D411yescharacteristic
138C2×D4×C23—22no—
139C2×D4×C23—21no—
140C2×D4×C23—22no—
141C2×D4×C23—22no—
142C2×D4×C23—21no—
143C2×D4×C23—21no—
144C2×D4×C23C2×D412yes—
145C2×D4×C23C2×D411yescharacteristic
146C22×C4×C23—21no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—22no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—41no—
154C22×C4×C23C2×D411yescharacteristic
155C22×C4×C23C2×D411yescharacteristic
156C22×C4×C23C2×D411yescharacteristic
157Q16×C23—82no—
158SD16×C23—82no—
159M16×C23—42no—
160C2×C8×C23—24no—
161C2×C8×C23—22no—
162C2×C8×C23—22no—
163C2×C8×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—41no—
183C4⋊C4×C23—41no—
184C4⋊C4×C23C2×D411yescharacteristic
185C4⋊C4×C23C2×D411yescharacteristic
186C4⋊C4×C23C2×D412yes—
187C4⋊C4×C23C2×D412yes—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—21no—
195C22⋊C4×C23—21no—
196C22⋊C4×C23—21no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—21no—
200C42×C23—22no—
201C42×C23—21no—
202C42×C23C2×D411yescharacteristic
203C42×C23C2×D411yescharacteristic
204C42×C23C2×D411yescharacteristic
205C42×C23C2411yescharacteristic
206#19022C2311yesmaximal, characteristic, direct product factor, Sylow
207C23×C23—22no—
208C23×C23—21no—
209C23×C23#12011yescharacteristic
210Q8×C23—41no—
211Q8×C23—42no—
212Q8×C23—42no—
213Q8×C23—42no—
214Q8×C23—82no—
215Q8×C23—82no—
216D4×C23—41no—
217D4×C23—42no—
218D4×C23—44no—
219D4×C23—42no—
220D4×C23—42no—
221D4×C23—81no—
222D4×C23—82no—
223D4×C23—82no—
224C2×C4×C23—22no—
225C2×C4×C23—22no—
226C2×C4×C23—21no—
227C2×C4×C23—21no—
228C2×C4×C23—22no—
229C2×C4×C23—22no—
230C2×C4×C23—21no—
231C2×C4×C23—22no—
232C2×C4×C23—41no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—82no—
237C2×C4×C23—81no—
238C2×C4×C23—82no—
239C2×C4×C23—82no—
240C2×C4×C23—82no—
241C2×C4×C23—82no—
242C2×C4×C23#12011yescharacteristic
243C2×C4×C23#12011yescharacteristic
244C2×C4×C23#12011yescharacteristic
245C2×C4×C23C22×D411yescharacteristic
246C2×C4×C23C22×D411yescharacteristic
247C2×C4×C23C22×D411yescharacteristic
248C8×C23—24no—
249C8×C23—42no—
250C8×C23—42no—
251#3559C2×C2311yescharacteristic
252#3218C2×C2311yescharacteristic, complemented
253C2×#501C2×C2311yescharacteristic, complemented
254#3141C2×C2311yescharacteristic, complemented
255#1664C2×C2314yescomplemented
256#1661C2×C2314yescomplemented
257#1530C2×C2311yescharacteristic, complemented
258#1528C2×C2311yescharacteristic, complemented
259#1524C2×C2311yescharacteristic, complemented
260C22×C23—21no—
261C22×C23—21no—
262C22×C23—42no—
263C22×C23—41no—
264C22×C23—41no—
265C22×C23—42no—
266C22×C23—42no—
267C22×C23—41no—
268C22×C23#45512yes—
269C22×C23C2×#12011yescharacteristic
270C4×C23—21no—
271C4×C23—21no—
272C4×C23—21no—
273C4×C23—21no—
274C4×C23—21no—
275C4×C23—42no—
276C4×C23—42no—
277C4×C23—41no—
278C4×C23—41no—
279C4×C23—42no—
280C4×C23—41no—
281C4×C23—82no—
282C4×C23#45612yes—
283#563C22×C2312yes—
284#562C22×C2312yes—
285#554—21no—
286#553—21no—
287#553—21no—
288#547—22no—
289#547—21no—
290#544—21no—
291#540—21no—
292#539—22no—
293#539—21no—
294#537—21no—
295#532C22×C2311yescharacteristic
296C2×C4⋊Q8C22×C2311yescharacteristic
297#529C22×C2311yescharacteristic
298C2×#125C22×C2311yescharacteristic
299#501—22no—
300#501C22×C2314yes—
301#497C22×C2312yes—
302#491C22×C2312yes—
303#489C22×C2314yes—
304#487—21no—
305#486—21no—
306#462—21no—
307#461—21no—
308#451—21no—
309#448—21no—
310#432C22×C2311yescharacteristic
311C2×#107—21no—
312C2×#106—21no—
313#417—22no—
314C4×M16C22×C2311yescharacteristic
315C2×#97C22×C2311yescharacteristic
316#330C22×C2314yes—
317#330C22×C2312yes—
318#329C22×C2312yes—
319#329C22×C2314yes—
320C2×C23—22no—
321C2×C23—82no—
322C2×C23—81no—
323C2×C23C2×#45511yescharacteristic
324C2×C23C2×#45611yescharacteristic
325C2×C23#302111yescharacteristic
326C2×#41—22no—
327C2×#41D4×C2312yes—
328C22×Q8—21no—
329C2×Q16—22no—
330C2×SD16—22no—
331C2×M16—21no—
332C22×C8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—22no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8D4×C2311yescharacteristic
339C4⋊Q8C23×C2312yes—
340#127D4×C2311yescharacteristic
341#126—22no—
342#126—22no—
343#126—21no—
344#125—22no—
345#125—21no—
346#125—21no—
347#125—21no—
348#125D4×C2312yes—
349#125C23×C2312yes—
350#125C23×C2311yescharacteristic
351#124C23×C2311yescharacteristic
352#124C23×C2312yes—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—21no—
358#123—22no—
359#123—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8D4×C2311yescharacteristic
367#121—22no—
368#121—22no—
369#121—21no—
370#121—21no—
371#121—22no—
372#121—22no—
373#121—21no—
374#121—21no—
375#121D4×C2311yescharacteristic
376C4×Q8D4×C2311yescharacteristic
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4D4×C2311yescharacteristic
385#117—21no—
386#117—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4D4×C2311yescharacteristic
394C2×C4⋊C4D4×C2311yescharacteristic
395C2×C42C23×C2311yescharacteristic
396#107—24no—
397#106—24no—
398C4⋊C8—21no—
399#103—24no—
400#103—22no—
401#103—22no—
402#102—24no—
403C22⋊C8—21no—
404#97—21no—
405#97C23×C2312yes—
406#97C23×C2312yes—
407#97C23×C2312yes—
408C4×C8—21no—
409C23#1902211yescharacteristic, direct product factor, Sylow
410#41—44no—
411#41—84no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—41no—
415C2×Q8—82no—
416C2×Q8C2×D4×C2312yes—
417C2×Q8C2×D4×C2311yescharacteristic
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×D4×C2312yes—
425C2×D4C2×D4×C2311yescharacteristic
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4C2×D4×C2311yescharacteristic
435C22×C4C2×D4×C2311yescharacteristic
436C22×C4C2×D4×C2311yescharacteristic
437Q16—82no—
438SD16—82no—
439M16—42no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4C2×D4×C2311yescharacteristic
465C4⋊C4C2×D4×C2311yescharacteristic
466C4⋊C4C2×D4×C2312yes—
467C4⋊C4C2×D4×C2312yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C42—22no—
481C42—21no—
482C42C2×D4×C2311yescharacteristic
483C42C2×D4×C2311yescharacteristic
484C42C2×D4×C2311yescharacteristic
485C42C24×C2311yescharacteristic
486C23—22no—
487C23—21no—
488C23C23×#12011yescharacteristic
489Q8—41no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—82no—
494Q8—82no—
495D4—41no—
496D4—42no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—81no—
501D4—82no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4C23×#12011yescharacteristic
522C2×C4C23×#12011yescharacteristic
523C2×C4C23×#12011yescharacteristic
524C2×C4C22×D4×C2311yescharacteristic
525C2×C4C22×D4×C2311yescharacteristic
526C2×C4C22×D4×C2311yescharacteristic
527C8—24no—
528C8—42no—
529C8—42no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22C23×#45512yes—
539C22C2×C23×#12011yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—82no—
552C4C23×#45612yes—
553C2—22no—
554C2—82no—
555C2—81no—
556C2C2×C23×#45511yescharacteristic
557C2C2×C23×#45611yescharacteristic
558C2C23×#302111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3559C2
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×#3218C2semidirect
3. C2×C23×#501C2semidirect
4. C23×#3141C2semidirect
5. C23×#1664C2semidirect
6. C23×#1661C2semidirect
7. C23×#1530C2semidirect
8. C23×#1528C2semidirect
9. C23×#1524C2semidirect
206. #19022C23direct (N × H)
252. #3218C2×C23semidirect
253. C2×#501C2×C23semidirect
254. #3141C2×C23semidirect
255. #1664C2×C23semidirect
256. #1661C2×C23semidirect
257. #1530C2×C23semidirect
258. #1528C2×C23semidirect
259. #1524C2×C23semidirect
409. C23#19022direct (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×#32182
C2×C23×#5012
C23×#31412
C23×#16642
C23×#16612
C23×#15302
C23×#15282
C23×#15242
#1902223