Finite Group Databasebeta

← #29736088

Subgroups — #29736088

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16840172C211yesmaximal, characteristic, complemented
3C2×#16840059C211yesmaximal, characteristic
4#21447157C212yesmaximal, complemented
5#21447157C212yesmaximal, complemented
6#21446771C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446079C212yesmaximal
9#21446065C212yesmaximal
10#21445647C211yesmaximal, characteristic
11C61×#2584C211yesmaximal, characteristic, complemented
12#16840472—24no—
13#16840472—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519769C2212yes—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×D244C2212yes—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840172—24no—
22#16840059—24no—
23C2×#14519710C2212yes—
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2212yes—
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×#125×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#398—21no—
40C61×#395—21no—
41C61×#389—22no—
42C61×#389C2212yes—
43C61×#389C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777D414yes—
54#14519769—24no—
55#14519769—24no—
56#14519755—216no—
57#14519753—28no—
58C4×D244—24no—
59C4×D244—22no—
60C4×D244—24no—
61C4×D244D414yes—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C4⋊C4×D61—28no—
68C2×#11758198—22no—
69C2×#11758198—21no—
70#14519710—22no—
71#14519710—24no—
72#14519708—22no—
73#14519708—24no—
74C4⋊#92418—22no—
75#14519697—22no—
76#14519697—22no—
77#14519697Q818yescomplemented
78#14519688—22no—
79C22×Q8×C61D411yescharacteristic
80#125×C61—22no—
81#125×C61—22no—
82C4×Q8×C61—22no—
83C4×Q8×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61D411yescharacteristic
89C2×C4⋊C4×C61D412yes—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C42×C61—21no—
94C2×C42×C61C2312yes—
95C2×C42×C61C2311yescharacteristic
96#95×C61—21no—
97#95×C61—22no—
98#95×C61—22no—
99C23×D61—21no—
100C2×Q8×C61—22no—
101C2×Q8×C61—42no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—22no—
111C22×Dic61—21no—
112C2×D244—21no—
113C2×D244—22no—
114C2×D244C2×D412yes—
115C2×D244C2×Q814yes—
116C2×C4×D61—24no—
117C2×C4×D61—28no—
118C2×C4×D61—44no—
119C2×C4×D61—82no—
120C2×C4×D61—82no—
121C2×Dic122—24no—
122C2×Dic122—81no—
123C2×Dic122—81no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—44no—
132C4⋊C4×C61—42no—
133C4⋊C4×C61—42no—
134C4⋊C4×C61—82no—
135C4⋊C4×C61C2×Q814yes—
136C42×C61—22no—
137C42×C61—22no—
138C42×C61—81no—
139C42×C61C2×D412yes—
140C42×C61C2×D412yes—
141#11758198—28no—
142#11758198—24no—
143C4⋊Dic61—24no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61C2×D414yes—
147C4⋊Dic61C2×D412yes—
148#11758196—28no—
149#11758196—28no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418C2×Q814yes—
157C23×C61C22×D411yescharacteristic
158C22×D61—22no—
159C22×D61—24no—
160C22×D61—42no—
161C22×D61—41no—
162C22×D61—41no—
163Q8×C61—44no—
164C2×C4×C61—22no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—24no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61#12012yes—
182C2×C4×C61#12012yes—
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×Q812yes—
186C2×C4×C612−1+412yes—
187C2×Dic61—22no—
188C2×Dic61—24no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192D244—22no—
193D244—24no—
194D244C4⋊Q814yes—
195C4×D61—48no—
196C4×D61—88no—
197Dic122—84no—
198#92418—24no—
199#92418—41no—
200#92418—41no—
201#25845—611nomaximal, Sylow
202C22×C61C2×#12011yescharacteristic
203C22×C61D4×Q812yes—
204C22×C61#57512yes—
205C22×C61#57611yescharacteristic
206C22×C61#57711yescharacteristic
207C2×D61—24no—
208C2×D61—44no—
209C2×D61—44no—
210C2×D61—42no—
211C2×D61—42no—
212C4×C61—22no—
213C4×C61—22no—
214C4×C61—44no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61—81no—
218C4×C61—81no—
219C4×C61C2×C4⋊Q811yescharacteristic
220C4×C61#53611yescharacteristic
221C4×C61D4×Q812yes—
222Dic61—42no—
223Dic61—44no—
224C2×#554—611no—
225C2×#501—611no—
226C2×#478—611no—
227#2584D6111yescharacteristic, complemented
228#2059—612no—
229#2059—612no—
230#1988—611no—
231#1985—611no—
232#1864—612no—
233#1862—612no—
234#1762—611no—
235C2×C61#243411yescharacteristic
236C2×C61#300511yescharacteristic
237C2×C61#301211yescharacteristic
238C2×C61#338812yes—
239C2×C61#338912yes—
240D61—44no—
241#554—1224no—
242#554—1224no—
243C2×C4⋊Q8—611no—
244C2×#125—612no—
245C2×#125C2×D6111yescharacteristic
246C2×C22⋊Q8—1221no—
247C2×C22⋊Q8—1222no—
248C2×C4×Q8C2×D6111yescharacteristic
249C2×C4×D4—611no—
250C2×C4×D4—612no—
251C22×C4⋊C4—1221no—
252#501—1224no—
253#478—1224no—
254C2×#107—611no—
255C2×#107—612no—
256C2×#106—611no—
257C2×#106—612no—
258C2×C4⋊C8—611no—
259C2×#102—612no—
260C2×#102—611no—
261C2×#102—611no—
262C2×#97—611no—
263#398—21no—
264#395—21no—
265#389—22no—
266#389C2×D6112yes—
267#389C2×D6112yes—
268C4×C4⋊C4C2×D6111yescharacteristic
269#340—612no—
270#340—612no—
271#340—612no—
272#336—612no—
273#336—612no—
274C61#2584511yescharacteristic, complemented, Sylow
275C22×Q8—1221no—
276C22×Q8C22⋊D6111yescharacteristic
277C22×D4—611no—
278C23×C4—1222no—
279C23×C4—1221no—
280C22×C8—612no—
281C22×C8—611no—
282C22×C8—611no—
283C4⋊Q8—614no—
284C4⋊Q8—1222no—
285#125—22no—
286#125—22no—
287#125—1224no—
288#125—1224no—
289C22⋊Q8—12216no—
290C22⋊Q8—1228no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×D4—614no—
294C4×D4—1224no—
295C4×D4—1224no—
296C4×D4—1222no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—611no—
302C2×C4⋊C4—612no—
303C2×C4⋊C4—1228no—
304C2×C4⋊C4—1222no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4C22⋊D6111yescharacteristic
308C2×C4⋊C4C22⋊D6112yes—
309C2×C4⋊C4C22×D6111yescharacteristic
310C2×C4⋊C4C22×D6111yescharacteristic
311C2×C4⋊C4C22×D6112yes—
312C2×C22⋊C4—1222no—
313C2×C22⋊C4—1221no—
314C2×C42—21no—
315C2×C42C22×D6112yes—
316C2×C42C22×D6111yescharacteristic
317#107—1222no—
318#107—1224no—
319#106—1222no—
320#106—1224no—
321C4⋊C8—1222no—
322#102—618no—
323#102—1222no—
324#102—1222no—
325#97—1222no—
326#95—21no—
327#95—22no—
328#95—22no—
329C24—1221no—
330C2×Q8—22no—
331C2×Q8—42no—
332C2×Q8—1224no—
333C2×Q8—4881no—
334C2×Q8—4881no—
335C2×D4—612no—
336C2×D4—614no—
337C2×D4—1221no—
338C2×D4—1222no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—1228no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—1224no—
346C22×C4—4882no—
347C22×C4—4882no—
348C22×C4—2444no—
349C22×C4C2×C22⋊D6112yes—
350C22×C4C2×C22⋊D6112yes—
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C23×D6111yescharacteristic
354C2×C8—614no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—1221no—
358C2×C8—1221no—
359C2×C8—1222no—
360C2×C8—1222no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—82no—
372C4⋊C4—612no—
373C4⋊C4—614no—
374C4⋊C4—1228no—
375C4⋊C4—1228no—
376C4⋊C4—1221no—
377C4⋊C4—1222no—
378C4⋊C4—1224no—
379C4⋊C4Q8×D6114yes—
380C22⋊C4—1228no—
381C22⋊C4—1224no—
382C42—22no—
383C42—22no—
384C42—81no—
385C42D4×D6112yes—
386C42D4×D6112yes—
387C23—1222no—
388C23—1224no—
389C23—2442no—
390C23—2441no—
391C23—2441no—
392C23C22×C22⋊D6111yescharacteristic
393Q8—44no—
394Q8—4884no—
395D4—614no—
396D4—1222no—
397D4—1224no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—1222no—
416C2×C4—1224no—
417C2×C4—4888no—
418C2×C4—2448no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4#1451974512yes—
423C2×C4#1451974512yes—
424C2×C4C2×D4×D6111yescharacteristic
425C2×C4C2×D4×D6111yescharacteristic
426C2×C4C2×Q8×D6112yes—
427C2×C4#1451982512yes—
428C8—1224no—
429C8—2441no—
430C8—2441no—
431C22—1224no—
432C22—2442no—
433C22—2444no—
434C22—2444no—
435C22—2442no—
436C22C2×#1451974511yescharacteristic
437C22Q8×C22⋊D6112yes—
438C22#1684056012yes—
439C22#1684056511yescharacteristic
440C22#1684057011yescharacteristic
441C4—22no—
442C4—22no—
443C4—42no—
444C4—44no—
445C4—42no—
446C4—81no—
447C4—81no—
448C4—2444no—
449C4—2442no—
450C4C4⋊Q8×D6111yescharacteristic
451C4#1684033711yescharacteristic
452C4#1684042712yes—
453C2—2444no—
454C2#2144946011yescharacteristic
455C2#2145505911yescharacteristic
456C2#2145512011yescharacteristic
457C2#2145875812yes—
458C2#2145877312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2584C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2584D61
hypocenterC61#25845
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×#16840172C2semidirect
4. #21447157C2semidirect
5. #21447157C2semidirect
6. #21446771C2semidirect
7. #21446760C2semidirect
11. C61×#2584C2semidirect
77. #14519697Q8semidirect
227. #2584D61semidirect
274. C61#25845semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168401722
C2×#168400592
#214471572
#214467712
#214467602
#214460792
#214460652
#214456472
C61×#25842
#2584561