Finite Group Databasebeta

← #19212189

Subgroups — #19212189

Order 5888.

884 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D23×#390C211yesmaximal, characteristic
2#15673376C211yesmaximal, characteristic, complemented
3#15673154C211yesmaximal, characteristic, complemented
4#15671218C211yesmaximal, characteristic, complemented
5#15671217C211yesmaximal, characteristic, complemented
6#15671217C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic, complemented
8C22×D4×D23C411yescharacteristic
9C2×C4⋊C4×D23C411yescharacteristic
10C2×C22⋊C4×D23—21no—
11C2×C22⋊C4×D23—21no—
12C2×C42×D23C2211yescharacteristic
13C2×Dic23⋊C8—21no—
14C2×(C2×D23)⋊C8—21no—
15C2×(C2×D23)⋊C8—21no—
16C2×C8×Dic23—21no—
17#13092320C411yescharacteristic
18#13092319C2211yescharacteristic
19#13092318C411yescharacteristic
20#13092317—21no—
21#13092317—21no—
22#13092172C2211yescharacteristic, complemented
23#13092172C2211yescharacteristic, complemented
24C23×#390C2211yescharacteristic
25C23×#336C2211yescharacteristic, complemented
26C23×#336C2211yescharacteristic, complemented
27C24×D23—21no—
28C22×D4×C23C2×C411yescharacteristic
29C22×C22⋊D23—21no—
30C2×D4×D23—22no—
31C2×D4×D23—42no—
32C2×D4×D23—42no—
33C2×D4×D23C814yescomplemented
34C22×D92C2×C411yescharacteristic
35C22×C4×D23—21no—
36C22×C4×D23C2×C411yescharacteristic
37C22×C4×D23D411yescharacteristic
38C22×C4×D23D411yescharacteristic
39C22×C8×C23—21no—
40C22×C8×C23—21no—
41C2×C4⋊C4×C23C2×C411yescharacteristic
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C42×C23C2311yescharacteristic
45C2×C22⋊Dic23—21no—
46C2×C22⋊Dic23—21no—
47C2×(C2×D23)⋊C4—21no—
48C2×(C2×D23)⋊C4—21no—
49C2×(C2×D23)⋊C4—21no—
50C2×(C2×D23)⋊C4—21no—
51C2×C4⋊Dic23C2×C411yescharacteristic
52C2×Dic23⋊C4—21no—
53C2×C4×Dic23C2×C411yescharacteristic
54C2×C4×Dic23D411yescharacteristic
55C2×C4×Dic23D411yescharacteristic
56C22×C23⋊C8—21no—
57C22×C23⋊C8—21no—
58C4⋊C4×D23—42no—
59C22⋊C4×D23—42no—
60C22⋊C4×D23—42no—
61C22⋊C4×D23—42no—
62C22⋊C4×D23—42no—
63C42×D23—42no—
64(C2×D23)⋊C8—24no—
65(C2×D23)⋊C8—42no—
66Dic23⋊C8—42no—
67C8×Dic23—42no—
68C24×C23—21no—
69C23×D23—21no—
70C23×D23—24no—
71C23×D23—42no—
72C23×D23—42no—
73C23×D23—42no—
74C23×D23—42no—
75C23×D23—42no—
76C23×D23—42no—
77C23×D23C22⋊C411yescharacteristic
78C2×D4×C23—21no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23C2×C812yes—
82C22×C4×C23—21no—
83C22×C4×C23C22×C411yescharacteristic
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23C2×D411yescharacteristic
86C2×C22⋊D23—24no—
87C2×C22⋊D23—42no—
88C2×C22⋊D23—42no—
89C22×Dic23—21no—
90C22×Dic23C22⋊C411yescharacteristic
91C22×Dic23C22⋊C411yescharacteristic
92C22×Dic23C22⋊C411yescharacteristic
93D4×D23—48no—
94D4×D23—48no—
95C2×D92—21no—
96C2×D92—41no—
97C2×D92—41no—
98C2×D92C2×C812yes—
99C2×C4×D23—22no—
100C2×C4×D23—22no—
101C2×C4×D23—22no—
102C2×C4×D23—21no—
103C2×C4×D23—21no—
104C2×C4×D23—41no—
105C2×C4×D23—41no—
106C2×C4×D23—41no—
107C2×C4×D23—42no—
108C2×C4×D23—42no—
109C2×C4×D23—42no—
110C2×C4×D23C2×C812yes—
111C2×C4×D23M1612yes—
112C2×C4×D23D812yes—
113C2×C4×D23SD1612yes—
114C2×C8×C23—22no—
115C2×C8×C23—41no—
116C2×C8×C23—41no—
117C2×C8×C23—41no—
118C2×C8×C23—41no—
119C2×C8×C23—41no—
120C4⋊C4×C23—41no—
121C22⋊C4×C23—41no—
122C22⋊C4×C23—41no—
123C22⋊C4×C23—41no—
124C22⋊C4×C23—41no—
125C42×C23—41no—
126C22⋊Dic23—42no—
127C22⋊Dic23—42no—
128(C2×D23)⋊C4—42no—
129(C2×D23)⋊C4—41no—
130(C2×D23)⋊C4—41no—
131(C2×D23)⋊C4—41no—
132(C2×D23)⋊C4—41no—
133(C2×D23)⋊C4—42no—
134C4⋊Dic23—41no—
135Dic23⋊C4—42no—
136C4×Dic23—22no—
137C4×Dic23—22no—
138C4×Dic23—41no—
139C2×C23⋊C8—22no—
140C2×C23⋊C8—41no—
141C2×C23⋊C8—41no—
142C2×C23⋊C8—41no—
143C2×C23⋊C8—41no—
144C2×C23⋊C8—41no—
145#7572—231nomaximal, Sylow
146C23×C23—22no—
147C23×C23—41no—
148C23×C23—41no—
149C23×C23—41no—
150C23×C23—41no—
151C23×C23—41no—
152C23×C23—41no—
153C23×C23C2×C22⋊C411yescharacteristic
154C22×D23—21no—
155C22×D23—21no—
156C22×D23—21no—
157C22×D23—21no—
158C22×D23—22no—
159C22×D23—48no—
160C22×D23—48no—
161C22×D23—41no—
162C22×D23—41no—
163C22×D23—41no—
164C22×D23—41no—
165C22×D23—41no—
166C22×D23—41no—
167C22×D23—48no—
168C22×D23—84no—
169C22×D23—84no—
170C22×D23—84no—
171C22×D23—84no—
172C22×D23C22⋊C812yes—
173C22×D23#10212yes—
174C22×D23#10412yes—
175D4×C23—42no—
176D4×C23—42no—
177C2×C4×C23—21no—
178C2×C4×C23—21no—
179C2×C4×C23—21no—
180C2×C4×C23—21no—
181C2×C4×C23—21no—
182C2×C4×C23—21no—
183C2×C4×C23—41no—
184C2×C4×C23—41no—
185C2×C4×C23—41no—
186C2×C4×C23C22×C811yescharacteristic
187C2×C4×C23C2×M1611yescharacteristic
188C2×C4×C23C2×D811yescharacteristic
189C2×C4×C23C2×SD1611yescharacteristic
190C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
192C22⋊D23—48no—
193C2×Dic23—21no—
194C2×Dic23—21no—
195C2×Dic23—21no—
196C2×Dic23—21no—
197C2×Dic23—21no—
198C2×Dic23—41no—
199C2×Dic23—41no—
200C2×Dic23—41no—
201C2×Dic23C22⋊C812yes—
202C2×Dic23#9912yes—
203C2×Dic23#10012yes—
204C2×Dic23#10212yes—
205D92—42no—
206D92—42no—
207C4×D23—24no—
208C4×D23—24no—
209C4×D23—42no—
210C4×D23—42no—
211C4×D23—42no—
212C4×D23—42no—
213C4×D23—84no—
214C8×C23—42no—
215C8×C23—81no—
216C23⋊C8—42no—
217C23⋊C8—81no—
218C2×#390—231no—
219#1797—231no—
220C4×C22⋊C8—231no—
221#1270—231no—
222#1270—231no—
223#1270—231no—
224#1270D2311yescharacteristic, complemented
225C22×C23—42no—
226C22×C23—42no—
227C22×C23—42no—
228C22×C23—81no—
229C22×C23—81no—
230C22×C23—81no—
231C22×C23—81no—
232C22×C23C2×C22⋊C811yescharacteristic
233C22×C23C2×#9911yescharacteristic
234C22×C23C2×#10011yescharacteristic
235C22×C23C2×#10211yescharacteristic
236C22×C23#41811yescharacteristic
237C22×C23C2×#10411yescharacteristic
238C22×C23#42111yescharacteristic
239C2×D23—22no—
240C2×D23—22no—
241C2×D23—22no—
242C2×D23—22no—
243C2×D23—22no—
244C2×D23—22no—
245C2×D23—42no—
246C2×D23—42no—
247C2×D23—42no—
248C2×D23—816no—
249C2×D23—81no—
250C2×D23—81no—
251C2×D23—81no—
252C2×D23—81no—
253C2×D23D4⋊C814yes—
254C4×C23—21no—
255C4×C23—21no—
256C4×C23—21no—
257C4×C23—21no—
258C4×C23—21no—
259C4×C23—21no—
260C4×C23—81no—
261Dic23—22no—
262Dic23—41no—
263Dic23—81no—
264Dic23#32314yes—
265C23×D4—231no—
266C22×C4⋊C4—231no—
267C22×C22⋊C4—461no—
268C22×C22⋊C4—461no—
269C22×C42—231no—
270C2×C4⋊C8—461no—
271C2×C22⋊C8—461no—
272C2×C22⋊C8—461no—
273C2×C4×C8—461no—
274#390—231no—
275#390—231no—
276#390—231no—
277#390—461no—
278#390—461no—
279#390C2×D2311yescharacteristic
280#336—231no—
281#336—231no—
282#336C2×D2311yescharacteristic, complemented
283#336C2×D2311yescharacteristic, complemented
284C2×C23—82no—
285C2×C23C2×#32311yescharacteristic
286C2×C23C2×D4⋊C811yescharacteristic
287C2×C23#147811yescharacteristic
288C2×C23#150711yescharacteristic
289C2×C23#151911yescharacteristic
290C2×C23#152711yescharacteristic
291C2×C23#154511yescharacteristic
292D23—24no—
293D23—82no—
294C25—461no—
295C22×D4—234no—
296C22×D4—231no—
297C22×D4—462no—
298C22×D4—461no—
299C22×D4—922no—
300C22×D4—922no—
301C22×D4C4×D2311yescharacteristic
302C23×C4—231no—
303C23×C4—231no—
304C23×C4—231no—
305C23×C4—461no—
306C22×C8—21no—
307C22×C8—21no—
308C22×C8—461no—
309C22×C8—461no—
310C2×C4⋊C4—231no—
311C2×C4⋊C4—461no—
312C2×C4⋊C4—922no—
313C2×C4⋊C4C4×D2311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—461no—
317C2×C22⋊C4—461no—
318C2×C22⋊C4—461no—
319C2×C22⋊C4—461no—
320C2×C22⋊C4—461no—
321C2×C22⋊C4—461no—
322C2×C22⋊C4—922no—
323C2×C22⋊C4—922no—
324C2×C22⋊C4—922no—
325C2×C22⋊C4—922no—
326C2×C42—231no—
327C2×C42—231no—
328C2×C42—231no—
329C2×C42—922no—
330C2×C42C22×D2311yescharacteristic
331C4⋊C8—922no—
332C22⋊C8—464no—
333C22⋊C8—922no—
334C4×C8—922no—
335C23#757211yescharacteristic, complemented, Sylow
336C24—21no—
337C24—231no—
338C24—464no—
339C24—461no—
340C24—922no—
341C24—922no—
342C24—922no—
343C24—922no—
344C24—922no—
345C24—922no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—232no—
350C2×D4—464no—
351C2×D4—461no—
352C2×D4—921no—
353C2×D4—921no—
354C2×D4—922no—
355C2×D4—922no—
356C2×D4—928no—
357C2×D4—928no—
358C2×D4C8×D2312yes—
359C22×C4—21no—
360C22×C4—231no—
361C22×C4—231no—
362C22×C4—232no—
363C22×C4—232no—
364C22×C4—232no—
365C22×C4—232no—
366C22×C4—231no—
367C22×C4—462no—
368C22×C4—462no—
369C22×C4—462no—
370C22×C4—461no—
371C22×C4—461no—
372C22×C4—461no—
373C22×C4—921no—
374C22×C4—921no—
375C22×C4—921no—
376C22×C4—922no—
377C22×C4—922no—
378C22×C4—922no—
379C22×C4C2×C4×D2311yescharacteristic
380C22×C4D4×D2311yescharacteristic
381C22×C4D4×D2311yescharacteristic
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—462no—
389C2×C8—921no—
390C2×C8—921no—
391C2×C8—921no—
392C2×C8—921no—
393C2×C8—921no—
394C4⋊C4—41no—
395C4⋊C4—921no—
396C4⋊C4—922no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—922no—
402C22⋊C4—922no—
403C22⋊C4—922no—
404C22⋊C4—922no—
405C22⋊C4—921no—
406C22⋊C4—921no—
407C22⋊C4—921no—
408C22⋊C4—921no—
409C42—41no—
410C42—462no—
411C42—462no—
412C42—921no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—232no—
421C23—232no—
422C23—232no—
423C23—461no—
424C23—461no—
425C23—461no—
426C23—461no—
427C23—462no—
428C23—928no—
429C23—928no—
430C23—928no—
431C23—921no—
432C23—921no—
433C23—921no—
434C23—921no—
435C23—921no—
436C23—921no—
437C23—1844no—
438C23—1844no—
439C23—1844no—
440C23—1844no—
441C23C22⋊C4×D2311yescharacteristic
442D4—42no—
443D4—42no—
444D4—922no—
445D4—922no—
446D4—928no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—232no—
457C2×C4—232no—
458C2×C4—232no—
459C2×C4—232no—
460C2×C4—464no—
461C2×C4—464no—
462C2×C4—461no—
463C2×C4—461no—
464C2×C4—461no—
465C2×C4—461no—
466C2×C4—461no—
467C2×C4—922no—
468C2×C4—922no—
469C2×C4—922no—
470C2×C4—922no—
471C2×C4—921no—
472C2×C4—921no—
473C2×C4—921no—
474C2×C4—1844no—
475C2×C4C2×C8×D2311yescharacteristic
476C2×C4M16×D2311yescharacteristic
477C2×C4D8×D2311yescharacteristic
478C2×C4#1062508211yescharacteristic
479C2×C4SD16×D2311yescharacteristic
480C2×C4#1062508511yescharacteristic
481C8—42no—
482C8—81no—
483C8—922no—
484C8—1841no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—234no—
493C22—462no—
494C22—462no—
495C22—462no—
496C22—462no—
497C22—462no—
498C22—462no—
499C22—922no—
500C22—922no—
501C22—922no—
502C22—18416no—
503C22—1841no—
504C22—1841no—
505C22—1841no—
506C22—1841no—
507C22C22⋊C8×D2311yescharacteristic
508C22#99×D2311yescharacteristic
509C22#100×D2311yescharacteristic
510C22#102×D2311yescharacteristic
511C22#1309242011yescharacteristic
512C22#104×D2311yescharacteristic
513C22#1309243211yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—81no—
521C4—234no—
522C4—462no—
523C4—921no—
524C4—1841no—
525C2—82no—
526C2—464no—
527C2—1842no—
528C2D23×#32311yescharacteristic
529C2D23×D4⋊C811yescharacteristic
530C2#1567178611yescharacteristic
531C2#1567191111yescharacteristic
532C2#1567196311yescharacteristic
533C2#1567199311yescharacteristic
534C2#1567206311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#1270C2
socleC23×C23C2×C22⋊C4
hypercenter#1270D23
hypocenterC23#7572
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #15673376C2semidirect
3. #15673154C2semidirect
4. #15671218C2semidirect
5. #15671217C2semidirect
6. #15671217C2semidirect
7. C23×#1270C2semidirect
22. #13092172C22semidirect
23. #13092172C22semidirect
25. C23×#336C22semidirect
26. C23×#336C22semidirect
33. C2×D4×D23C8semidirect
224. #1270D23semidirect
282. #336C2×D23semidirect
283. #336C2×D23semidirect
335. C23#7572semidirect