Finite Group Databasebeta

← #29576462

Subgroups — #29576462

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C22⋊Dic61C211yesmaximal, characteristic, complemented
2#21446656C211yesmaximal, characteristic, complemented
3#21445750C211yesmaximal, characteristic, complemented
4#21444387C211yesmaximal, characteristic, complemented
5#21443687C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C2×C61×#447C211yesmaximal, characteristic
8C2×D4×Dic61—21no—
9C22×C22⋊Dic61—21no—
10C2×#14519698—22no—
11C2×#14519692—21no—
12C2×#14519692—21no—
13C2×C22⋊#92418C2211yescharacteristic
14#16839665—21no—
15#16839642—22no—
16#16839622C2211yescharacteristic
17#16839622C2211yescharacteristic
18#16839618—21no—
19C4×C22⋊Dic61C2211yescharacteristic
20#16839496—22no—
21C23×D4×C61C2211yescharacteristic
22C22×D8×C61C412yescomplemented
23C2×#121×C61C2211yescharacteristic
24C61×#447—21no—
25C61×#447—22no—
26C61×#447—21no—
27C2×#102×C61C412yescomplemented
28C2×C22⋊C8×C61C2211yescharacteristic
29C23×Dic61—22no—
30C22×#92418—21no—
31D4×Dic61—44no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—21no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—42no—
40C2×C22⋊Dic61—42no—
41C2×C22⋊Dic61D411yescharacteristic
42C2×C22⋊Dic61D411yescharacteristic
43C2×C4×Dic61—21no—
44#14519698—44no—
45#14519692—42no—
46#14519692—42no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684D412yes—
52C25×C61—21no—
53C22×D4×C61—21no—
54C22×D4×C61—21no—
55C22×D4×C61—21no—
56C22×D4×C61—21no—
57C22×D4×C61—44no—
58C22×D4×C61C2×C412yes—
59C22×D4×C61C2×C411yescharacteristic, complemented
60C22×D4×C61D411yescharacteristic
61C22×D4×C61D412yes—
62C22×D4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C2×D8×C61—22no—
65C2×D8×C61—22no—
66C2×D8×C61—44no—
67C22×C8×C61C2×C412yescomplemented
68#121×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71#121×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
73C2×C22⋊C4×C61—21no—
74#102×C61—22no—
75#102×C61—22no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C24×C61—21no—
79C24×C61—21no—
80C24×C61—22no—
81C24×C61—22no—
82C24×C61—21no—
83C24×C61—44no—
84C24×C61—41no—
85C24×C61—41no—
86C24×C61—41no—
87C24×C61—41no—
88C24×C61—82no—
89C24×C61C2×D411yescharacteristic
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—21no—
94C2×D4×C61—22no—
95C2×D4×C61—21no—
96C2×D4×C61—22no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—42no—
102C2×D4×C61—42no—
103C2×D4×C61—44no—
104C2×D4×C61—44no—
105C2×D4×C61—44no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61C22⋊C412yes—
113C2×D4×C61C22⋊C412yes—
114C22×C4×C61—21no—
115C22×C4×C61—42no—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—21no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61—41no—
128C22×Dic61—82no—
129D8×C61—44no—
130D8×C61—44no—
131C2×C8×C61—22no—
132C2×C8×C61—42no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C22⋊Dic61—21no—
138C22⋊Dic61—21no—
139C22⋊Dic61—21no—
140C22⋊Dic61—44no—
141C22⋊Dic61—44no—
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145C22⋊Dic61—41no—
146C22⋊Dic61—82no—
147C22⋊Dic61D812yescomplemented
148C4⋊Dic61—42no—
149C4⋊Dic61—42no—
150C4×Dic61—42no—
151C2×#92418—42no—
152C2×#92418—41no—
153C23×C61—24no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—44no—
159C23×C61—44no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—82no—
175C23×C61—82no—
176C23×C61—81no—
177C23×C61—81no—
178C23×C61—84no—
179C23×C61—84no—
180C23×C61—84no—
181C23×C61—84no—
182C23×C61C4×D411yescharacteristic
183C23×C61#12011yescharacteristic
184C23×C61#12111yescharacteristic
185C23×C61C2×D811yescharacteristic
186C23×C61#13511yescharacteristic
187C23×C61AGL(1,ℤ/8)11yescharacteristic
188C23×C61AGL(1,ℤ/8)11yescharacteristic
189D4×C61—24no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—44no—
194D4×C61—44no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—81no—
198D4×C61—81no—
199D4×C61—81no—
200D4×C61—81no—
201D4×C61—81no—
202D4×C61#9914yescomplemented
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C4×D411yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61#12311yescharacteristic
216C2×Dic61—22no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—88no—
226C2×Dic61—81no—
227C2×Dic61—81no—
228C2×Dic61—81no—
229C8×C61—42no—
230#92418—81no—
231#9259—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—24no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—44no—
244C22×C61—44no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—84no—
251C22×C61—84no—
252C22×C61—84no—
253C22×C61—84no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—161no—
260C22×C61—161no—
261C22×C61#38611yescharacteristic
262C22×C61C4×D811yescharacteristic
263C22×C61#44211yescharacteristic
264C22×C61#44711yescharacteristic
265C22×C61#44911yescharacteristic
266C22×C61#45911yescharacteristic
267C22×C61#46311yescharacteristic
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—81no—
272C4×C61C2×#9912yes—
273Dic61—42no—
274Dic61—81no—
275Dic61—82no—
276Dic61—161no—
277C2×#447D6111yescharacteristic
278D4×C22⋊C4—611no—
279#1964—611no—
280#1786—611no—
281#1519—611no—
282#1289—611no—
283#1282—611no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—44no—
291C2×C61—82no—
292C2×C61—161no—
293C2×C61#187911yescharacteristic
294C2×C61#189811yescharacteristic
295C2×C61#190411yescharacteristic
296C23×D4C2×D6111yescharacteristic
297C22×D8Dic6112yescomplemented
298C2×#121C2×D6111yescharacteristic
299C2×C4×D4—1221no—
300C22×C22⋊C4—1221no—
301#447—21no—
302#447—22no—
303#447—21no—
304C2×#102—1222no—
305C2×#102Dic6112yescomplemented
306C2×#99—1221no—
307C2×#99—1221no—
308C2×C22⋊C8—611no—
309C2×C22⋊C8C2×D6111yescharacteristic
310#390—1221no—
311#386—1222no—
312#380—611no—
313#380—611no—
314#379—1221no—
315C4×C22⋊C4—611no—
316#327—1222no—
317C61#925911yescharacteristic, complemented, Sylow
318C25—21no—
319C22×D4—21no—
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4—44no—
324C22×D4C2×Dic6112yes—
325C22×D4C2×Dic6111yescharacteristic, complemented
326C22×D4C22⋊D6111yescharacteristic
327C22×D4C22⋊D6112yes—
328C22×D4C22⋊D6111yescharacteristic
329C23×C4—1222no—
330C23×C4C22×D6111yescharacteristic
331C2×D8—22no—
332C2×D8—22no—
333C2×D8—44no—
334C22×C8—1221no—
335C22×C8C2×Dic6112yescomplemented
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340C4×D4—2444no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C2×Dic6111yescharacteristic, complemented
344C2×C22⋊C4—21no—
345C2×C22⋊C4—611no—
346C2×C22⋊C4—611no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—2442no—
353C2×C22⋊C4—2442no—
354C2×C42—1221no—
355#102—22no—
356#102—22no—
357#102—2444no—
358#99—2442no—
359#99—2442no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364#95—612no—
365#95—1221no—
366#95—1221no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—44no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24—82no—
378C24D4×D6111yescharacteristic
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4C22⋊Dic6112yes—
402C2×D4C22⋊Dic6112yes—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—1221no—
406C22×C4—1222no—
407C22×C4—1221no—
408C22×C4—4882no—
409C22×C4—2444no—
410C22×C4—2441no—
411C22×C4—2444no—
412C22×C4D4×D6111yescharacteristic
413C22×C4#1175822411yescharacteristic
414C22×C4#1175822411yescharacteristic
415C22×C4C22×Dic6111yescharacteristic
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418D8—44no—
419D8—44no—
420C2×C8—22no—
421C2×C8—42no—
422C2×C8—2442no—
423C2×C8—2441no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—2442no—
427C4⋊C4—2442no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—612no—
431C22⋊C4—1221no—
432C22⋊C4—1221no—
433C22⋊C4—1221no—
434C22⋊C4—4882no—
435C22⋊C4—2444no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2444no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C42—2442no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23D4×Dic6111yescharacteristic
472C23#1451974511yescharacteristic
473C23#1451974911yescharacteristic
474C23D8×D6111yescharacteristic
475C23#1451979611yescharacteristic
476C23#1451980111yescharacteristic
477C23#1451980111yescharacteristic
478D4—24no—
479D4—22no—
480D4—22no—
481D4—22no—
482D4—44no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—81no—
487D4—81no—
488D4—81no—
489D4—81no—
490D4—81no—
491D4#1451969214yescomplemented
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—1222no—
502C2×C4—4881no—
503C2×C4—4881no—
504C2×C4—4881no—
505C2×C4—4888no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4C2×C22⋊Dic6111yescharacteristic
515C2×C4D4×Dic6111yescharacteristic
516C2×C4#1451974911yescharacteristic
517C2×C4#1451975811yescharacteristic
518C8—42no—
519C8—4881no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—24no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—161no—
548C22—161no—
549C22#1683964211yescharacteristic
550C22D8×Dic6111yescharacteristic
551C22#1683987411yescharacteristic
552C22#1683989411yescharacteristic
553C22#1683990411yescharacteristic
554C22#1683995511yescharacteristic
555C22#1683997511yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—81no—
560C4—9761no—
561C4—4882no—
562C4—4881no—
563C4—2442no—
564C4C2×#1451969212yes—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—44no—
572C2—82no—
573C2—161no—
574C2#2144616911yescharacteristic
575C2#2144627311yescharacteristic
576C2#2144630811yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC2×C61×#447C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenterC2×#447D61
hypocenterC61#9259
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊Dic61C2semidirect
2. #21446656C2semidirect
3. #21445750C2semidirect
4. #21444387C2semidirect
5. #21443687C2semidirect
6. #21443666C2semidirect
22. C22×D8×C61C4semidirect
27. C2×#102×C61C4semidirect
59. C22×D4×C61C2×C4semidirect
67. C22×C8×C61C2×C4semidirect
72. C2×C4⋊C4×C61C2×C4semidirect
147. C22⋊Dic61D8semidirect
202. D4×C61#99semidirect
297. C22×D8Dic61semidirect
305. C2×#102Dic61semidirect
317. C61#9259semidirect
325. C22×D4C2×Dic61semidirect
335. C22×C8C2×Dic61semidirect
343. C2×C4⋊C4C2×Dic61semidirect
491. D4#14519692semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊Dic612
#214466562
#214457502
#214443872
#214436872
#214436662
C2×C61×#4472
#925961