Finite Group Databasebeta

← C23×#8415 #19156041

Subgroups — C23×#8415 #19156041

Order 5888.

704 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3515C211yesmaximal, characteristic
2C23×#1759C211yesmaximal, characteristic, complemented
3C23×#1529C211yesmaximal, characteristic, complemented
4C23×#1506C211yesmaximal, characteristic, complemented
5C23×#1500C211yesmaximal, characteristic, complemented
6C23×#1488C211yesmaximal, characteristic, complemented
7C23×#1485C211yesmaximal, characteristic, complemented
8C23×#557C411yescharacteristic, complemented
9C23×#554C411yescharacteristic, complemented
10C23×#554C411yescharacteristic, complemented
11C23×#553—21no—
12C23×#552—21no—
13C23×#552—22no—
14C23×#552—21no—
15C23×#547—21no—
16C23×#547C411yescharacteristic, complemented
17C23×#541—21no—
18C23×#539—22no—
19C23×#533C2211yescharacteristic
20C23×#529—21no—
21C23×#529C2211yescharacteristic
22C2×C23×#125—21no—
23C23×#520—21no—
24C4×#41×C23C2211yescharacteristic
25C23×#432C2211yescharacteristic
26C23×#431C2211yescharacteristic
27C2×C23×#104—21no—
28C4×M16×C23C2211yescharacteristic
29C23×#376C2211yescharacteristic
30C23×#339—21no—
31C23×#330—21no—
32C23×#329—21no—
33C23×#328—21no—
34C23×#328—22no—
35C23×#327—21no—
36C23×Q8⋊C8—22no—
37C23×Q8⋊C8—21no—
38C23×D4⋊C8—21no—
39C2×#41×C23—21no—
40C2×#41×C23D411yescharacteristic
41C2×M16×C23—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8C2×C411yescharacteristic
44C23×C4⋊Q8C2×C411yescharacteristic
45C23×#126—22no—
46C23×#126—22no—
47C23×#126—22no—
48C23×#126—21no—
49C23×#125—22no—
50C23×#125—22no—
51C23×#125—22no—
52C23×#125—21no—
53C23×#125—21no—
54C23×#125—22no—
55C23×#125—21no—
56C23×#125—21no—
57C23×#125D411yescharacteristic
58C23×#124D411yescharacteristic
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×#123—21no—
63C23×#123—22no—
64C23×C22⋊Q8—21no—
65C23×C22⋊Q8—21no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8D411yescharacteristic
71C23×C22⋊Q8D412yes—
72C23×#121—22no—
73C23×#121—21no—
74C23×#121D411yescharacteristic
75C4×Q8×C23—21no—
76C4×Q8×C23—21no—
77C4×Q8×C23—21no—
78C4×Q8×C23C2×C411yescharacteristic
79C4×Q8×C23C2×C411yescharacteristic
80C4×Q8×C23D412yes—
81C4×Q8×C23D411yescharacteristic
82C4×D4×C23—22no—
83C4×D4×C23—22no—
84C4×D4×C23—21no—
85C4×D4×C23—21no—
86C4×D4×C23—21no—
87C4×D4×C23—21no—
88C4×D4×C23—21no—
89C4×D4×C23C2×C411yescharacteristic
90C4×D4×C23C2×C411yescharacteristic
91C4×D4×C23D411yescharacteristic
92C23×#117—22no—
93C23×#117—22no—
94C23×#117—21no—
95C23×#117—21no—
96C23×#117—21no—
97C23×#117D411yescharacteristic
98C2×C4⋊C4×C23—22no—
99C2×C4⋊C4×C23—22no—
100C2×C4⋊C4×C23—21no—
101C2×C4⋊C4×C23—21no—
102C2×C4⋊C4×C23—21no—
103C2×C42×C23—21no—
104C2×C42×C23—21no—
105C2×C42×C23C2311yescharacteristic
106C23×C4⋊C8—21no—
107C23×C4⋊C8—21no—
108C23×#104—24no—
109C23×C22⋊C8—21no—
110C23×C22⋊C8—21no—
111C23×#97—21no—
112C23×#97—21no—
113C4×C8×C23—21no—
114C4×C8×C23—21no—
115C23×#95—21no—
116C23×#95—21no—
117#41×C23—22no—
118#41×C23—41no—
119#41×C23—42no—
120#41×C23—42no—
121C2×Q8×C23—21no—
122C2×Q8×C23C22⋊C411yescharacteristic
123C2×Q8×C23C22⋊C411yescharacteristic
124C2×Q8×C23C22⋊C412yes—
125C2×D4×C23—22no—
126C2×D4×C23—21no—
127C2×D4×C23—21no—
128C2×D4×C23C22⋊C411yescharacteristic
129C2×D4×C23C22⋊C411yescharacteristic
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—22no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—41no—
138C22×C4×C23C2×D411yescharacteristic
139C22×C4×C23C2×D411yescharacteristic
140C22×C4×C23C2×D411yescharacteristic
141M16×C23—22no—
142M16×C23—41no—
143C2×C8×C23—41no—
144C2×C8×C23—41no—
145C2×C8×C23—41no—
146C4⋊C4×C23—22no—
147C4⋊C4×C23—22no—
148C4⋊C4×C23—22no—
149C4⋊C4×C23—22no—
150C4⋊C4×C23—22no—
151C4⋊C4×C23—22no—
152C4⋊C4×C23—22no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—21no—
156C4⋊C4×C23—21no—
157C4⋊C4×C23—21no—
158C4⋊C4×C23—21no—
159C4⋊C4×C23—21no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—21no—
164C4⋊C4×C23—21no—
165C4⋊C4×C23C22⋊C411yescharacteristic
166C4⋊C4×C23C22⋊C411yescharacteristic
167C4⋊C4×C23C22⋊C411yescharacteristic
168C4⋊C4×C23C22⋊C411yescharacteristic
169C4⋊C4×C23C22⋊C412yes—
170C22⋊C4×C23—22no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—22no—
174C22⋊C4×C23—21no—
175C22⋊C4×C23—21no—
176C22⋊C4×C23—21no—
177C22⋊C4×C23—21no—
178C22⋊C4×C23—21no—
179C22⋊C4×C23—21no—
180C22⋊C4×C23—21no—
181C22⋊C4×C23—21no—
182C42×C23—22no—
183C42×C23—21no—
184C42×C23—21no—
185C42×C23—21no—
186C42×C23—21no—
187C42×C23—22no—
188C42×C23—22no—
189C42×C23—41no—
190C42×C23C22×C411yescharacteristic
191C42×C23C2×D411yescharacteristic
192C42×C23C2×D411yescharacteristic
193C42×C23C2×D411yescharacteristic
194#8415C2311yesmaximal, characteristic, direct product factor, Sylow
195C23×C23—21no—
196C23×C23—21no—
197C23×C23#12011yescharacteristic
198Q8×C23—21no—
199Q8×C23—21no—
200Q8×C23—42no—
201Q8×C23—41no—
202Q8×C23—41no—
203Q8×C23—41no—
204D4×C23—21no—
205D4×C23—41no—
206D4×C23—42no—
207D4×C23—44no—
208D4×C23—81no—
209D4×C23#10412yes—
210C2×C4×C23—21no—
211C2×C4×C23—21no—
212C2×C4×C23—21no—
213C2×C4×C23—21no—
214C2×C4×C23—21no—
215C2×C4×C23—21no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—44no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—42no—
228C2×C4×C23—81no—
229C2×C4×C23—82no—
230C2×C4×C23—82no—
231C2×C4×C23—81no—
232C2×C4×C23—81no—
233C2×C4×C23C2×C22⋊C411yescharacteristic
234C2×C4×C23C2×C22⋊C411yescharacteristic
235C2×C4×C23C2×C22⋊C411yescharacteristic
236C2×C4×C23#12011yescharacteristic
237C2×C4×C23#12011yescharacteristic
238C2×C4×C23#12011yescharacteristic
239C8×C23—42no—
240C8×C23—81no—
241C8×C23—81no—
242#3515C2×C2311yescharacteristic
243#1759C2×C2311yescharacteristic, complemented
244#1529C2×C2311yescharacteristic, complemented
245#1506C2×C2311yescharacteristic, complemented
246#1500C2×C2311yescharacteristic, complemented
247#1488C2×C2311yescharacteristic, complemented
248#1485C2×C2311yescharacteristic, complemented
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—22no—
253C22×C23—41no—
254C22×C23—41no—
255C22×C23—41no—
256C22×C23—41no—
257C22×C23—41no—
258C22×C23#37911yescharacteristic
259C4×C23—21no—
260C4×C23—21no—
261C4×C23—21no—
262C4×C23—21no—
263C4×C23—21no—
264C4×C23—41no—
265C4×C23—41no—
266C4×C23—42no—
267C4×C23—41no—
268C4×C23—41no—
269C4×C23—41no—
270C4×C23—41no—
271C4×C23—42no—
272C4×C23—82no—
273C4×C23C2×#10411yescharacteristic
274C4×C23#42111yescharacteristic
275#557C4×C2311yescharacteristic, complemented
276#554C4×C2311yescharacteristic, complemented
277#554C4×C2311yescharacteristic, complemented
278#553—21no—
279#552—21no—
280#552—22no—
281#552—21no—
282#547—21no—
283#547C4×C2311yescharacteristic, complemented
284#541—21no—
285#539—22no—
286#533C22×C2311yescharacteristic
287#529—21no—
288#529C22×C2311yescharacteristic
289C2×#125—21no—
290#520—21no—
291C4×#41C22×C2311yescharacteristic
292#432C22×C2311yescharacteristic
293#431C22×C2311yescharacteristic
294C2×#104—21no—
295C4×M16C22×C2311yescharacteristic
296#376C22×C2311yescharacteristic
297#339—21no—
298#330—21no—
299#329—21no—
300#328—21no—
301#328—22no—
302#327—21no—
303Q8⋊C8—22no—
304Q8⋊C8—21no—
305D4⋊C8—21no—
306C2×C23—41no—
307C2×C23—42no—
308C2×C23—81no—
309C2×C23#178911yescharacteristic
310C2×C23#179411yescharacteristic
311C2×C23#179611yescharacteristic
312C2×#41—21no—
313C2×#41D4×C2311yescharacteristic
314C2×M16—21no—
315C4⋊Q8—21no—
316C4⋊Q8C2×C4×C2311yescharacteristic
317C4⋊Q8C2×C4×C2311yescharacteristic
318#126—22no—
319#126—22no—
320#126—22no—
321#126—21no—
322#125—22no—
323#125—22no—
324#125—22no—
325#125—21no—
326#125—21no—
327#125—22no—
328#125—21no—
329#125—21no—
330#125D4×C2311yescharacteristic
331#124D4×C2311yescharacteristic
332#123—22no—
333#123—22no—
334#123—22no—
335#123—21no—
336#123—22no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8D4×C2311yescharacteristic
344C22⋊Q8D4×C2312yes—
345#121—22no—
346#121—21no—
347#121D4×C2311yescharacteristic
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C2×C4×C2311yescharacteristic
352C4×Q8C2×C4×C2311yescharacteristic
353C4×Q8D4×C2311yescharacteristic
354C4×Q8D4×C2312yes—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4C2×C4×C2311yescharacteristic
363C4×D4C2×C4×C2311yescharacteristic
364C4×D4D4×C2311yescharacteristic
365#117—22no—
366#117—22no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117D4×C2311yescharacteristic
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42C23×C2311yescharacteristic
379C4⋊C8—21no—
380C4⋊C8—21no—
381#104—24no—
382C22⋊C8—21no—
383C22⋊C8—21no—
384#97—21no—
385#97—21no—
386C4×C8—21no—
387C4×C8—21no—
388#95—21no—
389#95—21no—
390C23#841511yescharacteristic, direct product factor, Sylow
391#41—22no—
392#41—41no—
393#41—42no—
394#41—42no—
395C2×Q8—21no—
396C2×Q8C22⋊C4×C2311yescharacteristic
397C2×Q8C22⋊C4×C2311yescharacteristic
398C2×Q8C22⋊C4×C2312yes—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C22⋊C4×C2311yescharacteristic
403C2×D4C22⋊C4×C2311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4C2×D4×C2311yescharacteristic
413C22×C4C2×D4×C2311yescharacteristic
414C22×C4C2×D4×C2311yescharacteristic
415M16—22no—
416M16—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4C22⋊C4×C2311yescharacteristic
440C4⋊C4C22⋊C4×C2311yescharacteristic
441C4⋊C4C22⋊C4×C2311yescharacteristic
442C4⋊C4C22⋊C4×C2311yescharacteristic
443C4⋊C4C22⋊C4×C2312yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42C22×C4×C2311yescharacteristic
465C42C2×D4×C2311yescharacteristic
466C42C2×D4×C2311yescharacteristic
467C42C2×D4×C2311yescharacteristic
468C23—21no—
469C23—21no—
470C23C23×#12011yescharacteristic
471Q8—21no—
472Q8—21no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477D4—21no—
478D4—41no—
479D4—42no—
480D4—44no—
481D4—81no—
482D4C23×#10412yes—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4C2×C22⋊C4×C2311yescharacteristic
507C2×C4C2×C22⋊C4×C2311yescharacteristic
508C2×C4C2×C22⋊C4×C2311yescharacteristic
509C2×C4C23×#12011yescharacteristic
510C2×C4C23×#12011yescharacteristic
511C2×C4C23×#12011yescharacteristic
512C8—42no—
513C8—81no—
514C8—81no—
515C22—21no—
516C22—22no—
517C22—21no—
518C22—21no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22C23×#37911yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—41no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4C2×C23×#10411yescharacteristic
540C4C23×#42111yescharacteristic
541C2—41no—
542C2—42no—
543C2—81no—
544C2C23×#178911yescharacteristic
545C2C23×#179411yescharacteristic
546C2C23×#179611yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3515C2
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×#1759C2semidirect
3. C23×#1529C2semidirect
4. C23×#1506C2semidirect
5. C23×#1500C2semidirect
6. C23×#1488C2semidirect
7. C23×#1485C2semidirect
8. C23×#557C4semidirect
9. C23×#554C4semidirect
10. C23×#554C4semidirect
16. C23×#547C4semidirect
194. #8415C23direct (N × H)
243. #1759C2×C23semidirect
244. #1529C2×C23semidirect
245. #1506C2×C23semidirect
246. #1500C2×C23semidirect
247. #1488C2×C23semidirect
248. #1485C2×C23semidirect
275. #557C4×C23semidirect
276. #554C4×C23semidirect
277. #554C4×C23semidirect
283. #547C4×C23semidirect
390. C23#8415direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35152
C23×#17592
C23×#15292
C23×#15062
C23×#15002
C23×#14882
C23×#14852
#841523