Finite Group Databasebeta

← #29736070

Subgroups — #29736070

Order 15616.

1224 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic
2C2×#16840059C211yesmaximal, characteristic
3#21451208C211yesmaximal, characteristic, complemented
4#21447150C212yesmaximal, complemented
5#21447145C212yesmaximal, complemented
6#21446768C211yesmaximal, characteristic, complemented
7#21446759C211yesmaximal, characteristic, complemented
8#21446077C212yesmaximal
9#21446063C212yesmaximal
10#21445645C211yesmaximal, characteristic
11C2×C61×#501C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13#16840427—24no—
14#16840427—24no—
15C2×#14519776C2212yes—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739C2212yes—
21#16840059—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28#16839692—21no—
29#16839682—21no—
30#16839658C2212yes—
31#16839658C2212yes—
32#16839657—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#125×C61C2211yescharacteristic
41C61×#501—24no—
42C2×#107×C61C2212yes—
43C2×#106×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C22×Dic122—22no—
46C22×Dic122D411yescharacteristic
47C2×Q8×D61—28no—
48C2×Q8×D61—48no—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61—22no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519776—24no—
55#14519776—24no—
56#14519755—216no—
57#14519753—28no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C4×D244—22no—
61C4×D244D414yes—
62#14519739—24no—
63#14519739—24no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196D412yes—
71C2×#11758198—21no—
72C2×#11758198—22no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61C2312yes—
75#14519710—22no—
76#14519708—22no—
77C4⋊#92418—22no—
78#14519699Q818yescomplemented
79#14519697—22no—
80#14519697—22no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C22×Q8×C61—21no—
85C22×C8×C61C2312yes—
86C4⋊Q8×C61—22no—
87C4⋊Q8×C61D414yes—
88#125×C61—22no—
89#125×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97#107×C61—24no—
98#106×C61—24no—
99#97×C61—22no—
100C23×D61—21no—
101C2×Q8×C61—24no—
102C2×Q8×C61—44no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61C2×D412yes—
112C22×Dic61C2×D412yes—
113Q8×D61—432no—
114C2×D244—21no—
115C2×D244—22no—
116C2×D244C2×D412yes—
117C2×D244C2×Q814yes—
118C2×C4×D61—28no—
119C2×C4×D61—24no—
120C2×C4×D61—44no—
121C2×C4×D61—44no—
122C2×C4×D61—44no—
123C2×Dic122—22no—
124C2×Dic122—24no—
125C2×Dic122—28no—
126C2×Dic122—44no—
127C2×Dic122—44no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61C2×Q814yescomplemented
131C4⋊C4×C61—24no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—42no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61C2×D414yes—
140C42×C61C2×D412yes—
141C42×C61C2×D412yes—
142#11758198—28no—
143#11758198—24no—
144C4⋊Dic61—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61—22no—
147C4⋊Dic61C2×D412yes—
148C4⋊Dic61C2×Q814yescomplemented
149#11758196—28no—
150#11758196—24no—
151#11758196—44no—
152#11758196—82no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—81no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—22no—
159C2×#92418—22no—
160C23×C61C22×D411yescharacteristic
161C22×D61—22no—
162C22×D61—24no—
163C22×D61—42no—
164C22×D61—41no—
165C22×D61—41no—
166Q8×C61—48no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—24no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61#12012yes—
179C2×C4×C61#12012yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×Q812yes—
183C2×C4×C612−1+412yes—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—24no—
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61AGL(1,ℤ/8)14yes—
195D244—22no—
196D244—24no—
197D244C4⋊Q814yes—
198C4×D61—416no—
199C4×D61—48no—
200Dic122—28no—
201Dic122—416no—
202C8×C61—24no—
203#92418—41no—
204#92418—41no—
205#25844—611nomaximal, Sylow
206C22×C61C2×#12011yescharacteristic
207C22×C61D4×Q812yes—
208C22×C61C2×AGL(1,ℤ/8)12yes—
209C22×C61#57711yescharacteristic
210C22×C61#57811yescharacteristic
211C2×D61—24no—
212C2×D61—44no—
213C2×D61—44no—
214C2×D61—42no—
215C2×D61—42no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—44no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61C2×C4⋊Q811yescharacteristic
222C4×C61#53611yescharacteristic
223C4×C61D4×Q812yes—
224Dic61—24no—
225Dic61—44no—
226Dic61—42no—
227Dic61—42no—
228C2×D4×Q8—611no—
229C2×#501D6111yescharacteristic, complemented
230C2×#478—611no—
231#2584—611no—
232#2058—612no—
233#2057—612no—
234#1987—611no—
235#1985—611no—
236#1864—612no—
237#1862—612no—
238#1762—611no—
239C2×C61#243411yescharacteristic
240C2×C61#300311yescharacteristic
241C2×C61#301311yescharacteristic
242C2×C61#338412yes—
243C2×C61#338612yes—
244D61—44no—
245C23×Q8—1221no—
246D4×Q8—1224no—
247D4×Q8—1224no—
248C2×C4⋊Q8—612no—
249C2×C4⋊Q8C2×D6111yescharacteristic
250C2×#125C2×D6111yescharacteristic
251C2×C22⋊Q8—1222no—
252C2×C22⋊Q8—1221no—
253C2×C4×Q8—611no—
254C2×C4×D4—612no—
255C2×C4×D4—611no—
256#501—24no—
257#478—1224no—
258C2×#107—611no—
259C2×#107C2×D6112yes—
260C2×#106—611no—
261C2×#106C2×D6112yes—
262C2×C4⋊C8—611no—
263C2×#102—612no—
264C2×#102—611no—
265C2×#102—611no—
266C2×#97C2×D6111yescharacteristic
267#398—1221no—
268#395—1221no—
269#389—612no—
270#389—612no—
271#389—1222no—
272C4×C4⋊C4—611no—
273#340—612no—
274#340—612no—
275#340—612no—
276#336—612no—
277#336—612no—
278C61#2584411yescharacteristic, complemented, Sylow
279C22×Q8—21no—
280C22×Q8—611no—
281C22×Q8—1222no—
282C22×Q8—1228no—
283C22×Q8—2448no—
284C22×D4—611no—
285C23×C4—1222no—
286C23×C4—1221no—
287C22×C8—611no—
288C22×C8—611no—
289C22×C8C22×D6112yes—
290C4⋊Q8—22no—
291C4⋊Q8—1224no—
292C4⋊Q8—1224no—
293C4⋊Q8C22⋊D6114yes—
294#125—22no—
295#125—22no—
296C22⋊Q8—12216no—
297C22⋊Q8—1228no—
298C4×Q8—1222no—
299C4×Q8—1222no—
300C4×D4—614no—
301C4×D4—1224no—
302C4×D4—1224no—
303C4×D4—1222no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—612no—
310C2×C4⋊C4—611no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1222no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4C22×D6111yescharacteristic
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C4⋊C4C22×D6112yes—
317C2×C22⋊C4—1222no—
318C2×C22⋊C4—1221no—
319C2×C42—612no—
320C2×C42—1221no—
321C2×C42C22×D6111yescharacteristic
322#107—24no—
323#107—1222no—
324#106—24no—
325#106—1222no—
326C4⋊C8—1222no—
327#102—618no—
328#102—1222no—
329#102—1222no—
330#97—22no—
331#95—1222no—
332#95—1222no—
333#95—1221no—
334C24—1221no—
335C2×Q8—24no—
336C2×Q8—44no—
337C2×Q8—1224no—
338C2×Q8—1228no—
339C2×Q8—1222no—
340C2×Q8—2444no—
341C2×Q8—2444no—
342C2×Q8—24432no—
343C2×D4—612no—
344C2×D4—614no—
345C2×D4—1221no—
346C2×D4—1222no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—1228no—
353C22×C4—1222no—
354C22×C4—1221no—
355C22×C4—1224no—
356C22×C4—2444no—
357C22×C4—2444no—
358C22×C4—2444no—
359C22×C4D4×D6111yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4C23×D6111yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—1221no—
365C2×C8—1221no—
366C2×C8—1222no—
367C2×C8—1222no—
368C2×C8Q8×D6114yescomplemented
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—612no—
378C4⋊C4—614no—
379C4⋊C4—1228no—
380C4⋊C4—1221no—
381C4⋊C4—1222no—
382C4⋊C4—1224no—
383C4⋊C4—1222no—
384C4⋊C4—4882no—
385C4⋊C4—2444no—
386C4⋊C4C2×C22⋊D6114yes—
387C22⋊C4—1228no—
388C22⋊C4—1224no—
389C42—1222no—
390C42—1222no—
391C42—4881no—
392C42D4×D6112yes—
393C42C2×C22⋊D6112yes—
394C23—1222no—
395C23—1224no—
396C23—2442no—
397C23—2441no—
398C23—2441no—
399C23C2×D4×D6111yescharacteristic
400Q8—48no—
401Q8—1228no—
402Q8—24416no—
403D4—614no—
404D4—1222no—
405D4—1224no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—614no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1224no—
425C2×C4—2448no—
426C2×C4—24416no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4#1451974512yes—
431C2×C4#1451974512yes—
432C2×C4C2×D4×D6111yescharacteristic
433C2×C4C22×C22⋊D6111yescharacteristic
434C2×C4C2×Q8×D6112yes—
435C2×C4#1451982512yes—
436C8—24no—
437C8—2441no—
438C8—2441no—
439C22—1224no—
440C22—2444no—
441C22—2442no—
442C22—2444no—
443C22—2442no—
444C22C2×#1451974511yescharacteristic
445C22#1684042712yes—
446C22AGL(1,ℤ/8)×D6112yes—
447C22#1684057111yescharacteristic
448C22#1684057811yescharacteristic
449C4—22no—
450C4—22no—
451C4—44no—
452C4—81no—
453C4—81no—
454C4—1224no—
455C4—2444no—
456C4—2442no—
457C4—2442no—
458C4C4⋊Q8×D6111yescharacteristic
459C4#1684033711yescharacteristic
460C4Q8×C22⋊D6112yes—
461C2—2444no—
462C2#2144946011yescharacteristic
463C2#2145504411yescharacteristic
464C2#2145513311yescharacteristic
465C2#2145871512yes—
466C2#2145873712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#501D61
hypocenterC61#25844
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #21451208C2semidirect
4. #21447150C2semidirect
5. #21447145C2semidirect
6. #21446768C2semidirect
7. #21446759C2semidirect
11. C2×C61×#501C2semidirect
78. #14519699Q8semidirect
130. C2×C8×C61C2×Q8semidirect
148. C4⋊Dic61C2×Q8semidirect
229. C2×#501D61semidirect
278. C61#25844semidirect
368. C2×C8Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
C2×#168400592
#214512082
#214471502
#214471452
#214467682
#214467592
#214460772
#214460632
#214456452
C2×C61×#5012
#2584461