Finite Group Databasebeta

← #19212335

Subgroups — #19212335

Order 5888.

892 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D23×#390C211yesmaximal, characteristic
2#15673464C211yesmaximal, characteristic, complemented
3#15673089C211yesmaximal, characteristic, complemented
4#15671218C211yesmaximal, characteristic, complemented
5#15671217C211yesmaximal, characteristic, complemented
6#15671217C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic, complemented
8C22×D4×D23C411yescharacteristic
9C22×#67262—21no—
10C2×C4⋊C4×D23C411yescharacteristic
11C2×C22⋊C4×D23—21no—
12C2×C22⋊C4×D23—21no—
13C2×C42×D23C2211yescharacteristic
14C2×(C2×D23)⋊C8—21no—
15#13092320C411yescharacteristic
16#13092319C2211yescharacteristic
17#13092318C411yescharacteristic
18#13092317—21no—
19#13092317—21no—
20#13092172C2211yescharacteristic, complemented
21#13092172C2211yescharacteristic, complemented
22#13092171—21no—
23#13092171—21no—
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—22no—
32C2×D4×D23—42no—
33C2×D4×D23—42no—
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—
58C2×#67262—24no—
59C2×#67262—42no—
60C2×#67262—42no—
61C4⋊C4×D23—42no—
62C22⋊C4×D23—42no—
63C22⋊C4×D23—42no—
64C22⋊C4×D23—42no—
65C22⋊C4×D23—42no—
66C42×D23—42no—
67(C2×D23)⋊C8—42no—
68C24×C23—21no—
69C23×D23—24no—
70C23×D23—21no—
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×C23M1612yescomplemented
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×D92M1612yescomplemented
99C2×C4×D23—22no—
100C2×C4×D23—22no—
101C2×C4×D23—22no—
102C2×C4×D23—21no—
103C2×C4×D23—21no—
104C2×C4×D23—22no—
105C2×C4×D23—22no—
106C2×C4×D23—21no—
107C2×C4×D23—21no—
108C2×C4×D23—41no—
109C2×C4×D23—42no—
110C2×C4×D23—42no—
111C2×C4×D23—42no—
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—41no—
137C4×Dic23—41no—
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#67262—48no—
146#7606—231nomaximal, Sylow
147C23×C23—22no—
148C23×C23—41no—
149C23×C23—41no—
150C23×C23—41no—
151C23×C23—41no—
152C23×C23—41no—
153C23×C23—41no—
154C23×C23C2×C22⋊C411yescharacteristic
155C22×D23—21no—
156C22×D23—21no—
157C22×D23—21no—
158C22×D23—21no—
159C22×D23—22no—
160C22×D23—48no—
161C22×D23—48no—
162C22×D23—48no—
163C22×D23—41no—
164C22×D23—41no—
165C22×D23—41no—
166C22×D23—41no—
167C22×D23—41no—
168C22×D23—41no—
169C22×D23—84no—
170C22×D23—84no—
171C22×D23—84no—
172C22×D23—84no—
173C22×D23#9912yes—
174C22×D23#10212yes—
175C22×D23#10412yes—
176D4×C23—42no—
177D4×C23—42no—
178C2×C4×C23—21no—
179C2×C4×C23—21no—
180C2×C4×C23—21no—
181C2×C4×C23—21no—
182C2×C4×C23—21no—
183C2×C4×C23—21no—
184C2×C4×C23—41no—
185C2×C4×C23—41no—
186C2×C4×C23—41no—
187C2×C4×C23C2×M1611yescharacteristic
188C2×C4×C23C2×M1611yescharacteristic
189C2×C4×C23C2×D811yescharacteristic
190C2×C4×C23C2×SD1611yescharacteristic
191C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
193C22⋊D23—48no—
194C2×Dic23—21no—
195C2×Dic23—21no—
196C2×Dic23—21no—
197C2×Dic23—21no—
198C2×Dic23—21no—
199C2×Dic23—21no—
200C2×Dic23—21no—
201C2×Dic23—41no—
202C2×Dic23—41no—
203C2×Dic23—41no—
204C2×Dic23#10012yes—
205C2×Dic23#10212yes—
206D92—42no—
207D92—42no—
208C4×D23—24no—
209C4×D23—24no—
210C4×D23—24no—
211C4×D23—24no—
212C4×D23—42no—
213C4×D23—42no—
214C4×D23—84no—
215C8×C23—42no—
216C8×C23—81no—
217C23⋊C8—42no—
218C23⋊C8—81no—
219C2×#390—231no—
220#1817—231no—
221#1728—231no—
222#1270—231no—
223#1270—231no—
224#1270—231no—
225#1270D2311yescharacteristic, complemented
226C22×C23—42no—
227C22×C23—42no—
228C22×C23—42no—
229C22×C23—81no—
230C22×C23—81no—
231C22×C23—81no—
232C22×C23—81no—
233C22×C23#40711yescharacteristic
234C22×C23C2×#9911yescharacteristic
235C22×C23C2×#10011yescharacteristic
236C22×C23C2×#10211yescharacteristic
237C22×C23#41811yescharacteristic
238C22×C23C2×#10411yescharacteristic
239C22×C23#42111yescharacteristic
240C2×D23—22no—
241C2×D23—22no—
242C2×D23—22no—
243C2×D23—22no—
244C2×D23—22no—
245C2×D23—22no—
246C2×D23—42no—
247C2×D23—42no—
248C2×D23—42no—
249C2×D23—816no—
250C2×D23—81no—
251C2×D23—81no—
252C2×D23—81no—
253C2×D23—81no—
254C2×D23#32714yes—
255C4×C23—21no—
256C4×C23—21no—
257C4×C23—21no—
258C4×C23—21no—
259C4×C23—21no—
260C4×C23—21no—
261C4×C23—81no—
262Dic23—22no—
263Dic23—22no—
264Dic23—41no—
265Dic23—81no—
266C23×D4—231no—
267C22×M16—461no—
268C22×C4⋊C4—231no—
269C22×C22⋊C4—461no—
270C22×C22⋊C4—461no—
271C22×C42—231no—
272C2×C22⋊C8—461no—
273#390—231no—
274#390—231no—
275#390—231no—
276#390—461no—
277#390—461no—
278#390C2×D2311yescharacteristic
279#336—231no—
280#336—231no—
281#336—461no—
282#336—461no—
283#336C2×D2311yescharacteristic, complemented
284#336C2×D2311yescharacteristic, complemented
285C2×C23—82no—
286C2×C23#146211yescharacteristic
287C2×C23#148611yescharacteristic
288C2×C23#149011yescharacteristic
289C2×C23C2×#32711yescharacteristic
290C2×C23#151011yescharacteristic
291C2×C23#152711yescharacteristic
292C2×C23#154511yescharacteristic
293D23—24no—
294D23—82no—
295C25—461no—
296C22×D4—231no—
297C22×D4—461no—
298C22×D4—462no—
299C22×D4—462no—
300C22×D4—922no—
301C22×D4—922no—
302C22×D4C4×D2311yescharacteristic
303C23×C4—231no—
304C23×C4—231no—
305C23×C4—231no—
306C23×C4—461no—
307C2×M16—464no—
308C2×M16—922no—
309C2×M16—922no—
310C22×C8—21no—
311C22×C8—21no—
312C22×C8—461no—
313C22×C8—461no—
314C2×C4⋊C4—231no—
315C2×C4⋊C4—461no—
316C2×C4⋊C4—922no—
317C2×C4⋊C4C4×D2311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—461no—
321C2×C22⋊C4—461no—
322C2×C22⋊C4—461no—
323C2×C22⋊C4—461no—
324C2×C22⋊C4—461no—
325C2×C22⋊C4—461no—
326C2×C22⋊C4—922no—
327C2×C22⋊C4—922no—
328C2×C22⋊C4—922no—
329C2×C22⋊C4—922no—
330C2×C42—231no—
331C2×C42—231no—
332C2×C42—231no—
333C2×C42—922no—
334C2×C42C22×D2311yescharacteristic
335C22⋊C8—922no—
336C23#760611yescharacteristic, complemented, Sylow
337C24—21no—
338C24—231no—
339C24—464no—
340C24—461no—
341C24—922no—
342C24—922no—
343C24—922no—
344C24—922no—
345C24—922no—
346C24—922no—
347C2×D4—21no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—232no—
351C2×D4—464no—
352C2×D4—461no—
353C2×D4—921no—
354C2×D4—921no—
355C2×D4—922no—
356C2×D4—928no—
357C2×D4—928no—
358C2×D4—922no—
359C2×D4#6726212yescomplemented
360C22×C4—21no—
361C22×C4—231no—
362C22×C4—231no—
363C22×C4—232no—
364C22×C4—232no—
365C22×C4—231no—
366C22×C4—462no—
367C22×C4—462no—
368C22×C4—462no—
369C22×C4—462no—
370C22×C4—462no—
371C22×C4—461no—
372C22×C4—461no—
373C22×C4—461no—
374C22×C4—461no—
375C22×C4—461no—
376C22×C4—921no—
377C22×C4—922no—
378C22×C4—922no—
379C22×C4—922no—
380C22×C4C2×C4×D2311yescharacteristic
381C22×C4D4×D2311yescharacteristic
382C22×C4D4×D2311yescharacteristic
383M16—928no—
384C2×C8—22no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—462no—
391C2×C8—921no—
392C2×C8—921no—
393C2×C8—921no—
394C2×C8—921no—
395C2×C8—921no—
396C4⋊C4—41no—
397C4⋊C4—921no—
398C4⋊C4—922no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—922no—
404C22⋊C4—922no—
405C22⋊C4—922no—
406C22⋊C4—922no—
407C22⋊C4—921no—
408C22⋊C4—921no—
409C22⋊C4—921no—
410C22⋊C4—921no—
411C42—41no—
412C42—921no—
413C42—921no—
414C42—921no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—232no—
423C23—232no—
424C23—232no—
425C23—461no—
426C23—461no—
427C23—461no—
428C23—461no—
429C23—462no—
430C23—928no—
431C23—928no—
432C23—928no—
433C23—921no—
434C23—921no—
435C23—921no—
436C23—921no—
437C23—921no—
438C23—921no—
439C23—1844no—
440C23—1844no—
441C23—1844no—
442C23—1844no—
443C23C22⋊C4×D2311yescharacteristic
444D4—42no—
445D4—42no—
446D4—922no—
447D4—922no—
448D4—928no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—232no—
459C2×C4—232no—
460C2×C4—464no—
461C2×C4—464no—
462C2×C4—464no—
463C2×C4—464no—
464C2×C4—461no—
465C2×C4—461no—
466C2×C4—461no—
467C2×C4—461no—
468C2×C4—461no—
469C2×C4—461no—
470C2×C4—461no—
471C2×C4—922no—
472C2×C4—922no—
473C2×C4—921no—
474C2×C4—921no—
475C2×C4—921no—
476C2×C4—1844no—
477C2×C4C2×#6726211yescharacteristic
478C2×C4M16×D2311yescharacteristic
479C2×C4D8×D2311yescharacteristic
480C2×C4#1062508211yescharacteristic
481C2×C4SD16×D2311yescharacteristic
482C2×C4#1062508511yescharacteristic
483C8—42no—
484C8—81no—
485C8—922no—
486C8—1841no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—234no—
495C22—462no—
496C22—462no—
497C22—462no—
498C22—462no—
499C22—462no—
500C22—462no—
501C22—922no—
502C22—922no—
503C22—922no—
504C22—18416no—
505C22—1841no—
506C22—1841no—
507C22—1841no—
508C22—1841no—
509C22#1309237911yescharacteristic
510C22#99×D2311yescharacteristic
511C22#100×D2311yescharacteristic
512C22#102×D2311yescharacteristic
513C22#1309242011yescharacteristic
514C22#104×D2311yescharacteristic
515C22#1309243211yescharacteristic
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—81no—
523C4—462no—
524C4—462no—
525C4—921no—
526C4—1841no—
527C2—82no—
528C2—464no—
529C2—1842no—
530C2#1567172211yescharacteristic
531C2#1567182011yescharacteristic
532C2#1567183811yescharacteristic
533C2D23×#32711yescharacteristic
534C2#1567192411yescharacteristic
535C2#1567199311yescharacteristic
536C2#1567206311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC22×C4×C23C22×C4
FrattiniC2×C42C22×D23
FittingC23×#1270C2
socleC23×C23C2×C22⋊C4
hypercenter#1270D23
hypocenterC23#7606
LCSG ▷ C22×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. #15673464C2semidirect
3. #15673089C2semidirect
4. #15671218C2semidirect
5. #15671217C2semidirect
6. #15671217C2semidirect
7. C23×#1270C2semidirect
20. #13092172C22semidirect
21. #13092172C22semidirect
25. C23×#336C22semidirect
26. C23×#336C22semidirect
81. C2×D4×C23M16semidirect
98. C2×D92M16semidirect
225. #1270D23semidirect
283. #336C2×D23semidirect
284. #336C2×D23semidirect
336. C23#7606semidirect
359. C2×D4#67262semidirect