Finite Group Databasebeta

← #29576151

Subgroups — #29576151

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic
2C2×C8×D244C211yesmaximal, characteristic
3#21446780C211yesmaximal, characteristic, complemented
4#21445820C211yesmaximal, characteristic
5C2×#16839510C211yesmaximal, characteristic, complemented
6#21443703C211yesmaximal, characteristic, complemented
7C61×#1277C211yesmaximal, characteristic, complemented
8C22×C8×D61—21no—
9C2×#14519777C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11#16840053—22no—
12#16840053C414yescomplemented
13C8×D244—22no—
14C8×D244—22no—
15C2×#14519709—21no—
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519699—21no—
18C2×#14519699—21no—
19C2×#14519689—21no—
20#16839632C2211yescharacteristic
21#16839525—21no—
22#16839525—21no—
23#16839525—21no—
24#16839510—22no—
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4⋊C8×C61C2211yescharacteristic
27C2×C4×C8×C61C2211yescharacteristic
28C22×Dic122—21no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×#92418—21no—
32C2×C8×D61—24no—
33C2×C8×D61—42no—
34C2×C8×D61—42no—
35#14519777—21no—
36#14519777—21no—
37#14519777—21no—
38#14519777C2×C412yes—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244C2×C412yes—
42C4×D244D412yes—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61D411yescharacteristic
49C2×#11758198—21no—
50#14519709—24no—
51C4⋊#92418—21no—
52C4⋊#92418D412yes—
53#14519699—24no—
54#14519699—42no—
55#14519689—24no—
56C22×C8×C61—21no—
57C22×C8×C61—21no—
58C22×C8×C61D411yescharacteristic
59C22×C8×C61D411yescharacteristic
60C2×C42×C61C2311yescharacteristic
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C4⋊C8×C61C2×C412yes—
65C4×C8×C61—21no—
66C4×C8×C61—21no—
67C23×D61—21no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4111yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C2×D244—21no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244C22⋊C412yes—
78C2×D244SD1612yes—
79C2×D244Q1612yes—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—41no—
83C2×C4×D61—41no—
84C2×C4×D61—41no—
85C2×C4×D61—41no—
86C2×Dic122—22no—
87C2×Dic122—41no—
88C2×Dic122—81no—
89C2×Dic122—81no—
90C2×C8×C61—21no—
91C2×C8×C61—21no—
92C2×C8×C61—22no—
93C2×C8×C61—21no—
94C2×C8×C61—21no—
95C2×C8×C61—21no—
96C2×C8×C61—41no—
97C2×C8×C61—41no—
98C2×C8×C61—41no—
99C2×C8×C61—41no—
100C2×C8×C61—41no—
101C2×C8×C61SD1612yes—
102C42×C61C22×C411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106#11758198—22no—
107#11758198—22no—
108C4⋊Dic61—21no—
109C4⋊Dic61—22no—
110C4⋊Dic61—22no—
111C4⋊Dic61—41no—
112C4⋊Dic61—41no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4⋊Dic61—81no—
118C4⋊Dic61C22⋊C412yes—
119C2×#92418—22no—
120C2×#92418—41no—
121C2×#92418—41no—
122C8×D61—48no—
123C23×C61#12111yescharacteristic
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—22no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61C2×C22⋊C411yescharacteristic
134C2×C4×C61C4×D411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12311yescharacteristic
139C2×C4×C61C2×SD1611yescharacteristic
140C2×C4×C61C2×SD1611yescharacteristic
141C2×C4×C61C2×Q1611yescharacteristic
142C2×C4×C61#13511yescharacteristic
143C2×C4×C61#13511yescharacteristic
144C2×C4×C61#13511yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61#13711yescharacteristic
147C2×Dic61—22no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61—81no—
152C2×Dic61—81no—
153C2×Dic61—81no—
154C2×Dic61—81no—
155C2×Dic61—81no—
156D244—22no—
157D244—22no—
158D244—22no—
159D244#10314yes—
160C4×D61—42no—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—42no—
164Dic122—82no—
165Dic122—82no—
166C8×C61—22no—
167C8×C61—42no—
168C8×C61—41no—
169C8×C61—81no—
170#92418—42no—
171#9215—611nomaximal, Sylow
172C22×C61#38611yescharacteristic
173C22×C61#46111yescharacteristic
174C22×C61#46311yescharacteristic
175C22×C61#46511yescharacteristic
176C22×C61#46711yescharacteristic
177C22×C61#47111yescharacteristic
178C22×C61#47211yescharacteristic
179C2×D61—24no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—42no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61C2×#10311yescharacteristic
189C4×C61#41611yescharacteristic
190C4×C61C4×SD1611yescharacteristic
191C4×C61#44011yescharacteristic
192C4×C61#45011yescharacteristic
193C4×C61#45211yescharacteristic
194C4×C61#46511yescharacteristic
195C4×C61#46711yescharacteristic
196Dic61—42no—
197Dic61—82no—
198Dic61—161no—
199C2×#476—611no—
200C2×C8×D4—611no—
201#1990—611no—
202#1802—611no—
203C2×#333—611no—
204#1296—611no—
205#1277D6111yescharacteristic, complemented
206C2×C61#167111yescharacteristic
207C2×C61#167511yescharacteristic
208C2×C61#167711yescharacteristic
209C2×C61#168311yescharacteristic
210C2×C61#187711yescharacteristic
211C2×C61#189211yescharacteristic
212C2×C61#190711yescharacteristic
213D61—44no—
214C23×C8—1221no—
215C2×C4⋊Q8—611no—
216C2×C4×D4—611no—
217#476—614no—
218#476—1222no—
219C8×D4—1222no—
220C8×D4—1222no—
221C2×#106—1221no—
222C2×C4⋊C8—611no—
223C2×C4⋊C8C2×D6111yescharacteristic
224C2×C4⋊C8C2×D6111yescharacteristic
225C2×#102—1221no—
226C2×#102—1221no—
227C2×C22⋊C8—1221no—
228C2×C4×C8C2×D6111yescharacteristic
229#384—611no—
230#340—1221no—
231#340—1221no—
232#340—1221no—
233#333—1222no—
234C61#921511yescharacteristic, complemented, Sylow
235C22×Q8—1221no—
236C22×D4—611no—
237C23×C4—1221no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—1224no—
241C22×C8—1221no—
242C22×C8—2442no—
243C22×C8—2442no—
244C22×C8D24411yescharacteristic
245C22×C8D24411yescharacteristic
246C4⋊Q8—612no—
247C4⋊Q8—1221no—
248C4⋊Q8—1221no—
249C4⋊Q8—1221no—
250C4×D4—612no—
251C4×D4—612no—
252C4×D4—1221no—
253C4×D4—1221no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C42C22×D6111yescharacteristic
262#106—1224no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—612no—
267C4⋊C8—1221no—
268C4⋊C8C4×D6112yes—
269#102—1224no—
270#102—2442no—
271C22⋊C8—1224no—
272C4×C8—21no—
273C4×C8—21no—
274C24—1221no—
275C2×Q8—1222no—
276C2×Q8—4881no—
277C2×Q8—4881no—
278C2×Q8—2441no—
279C2×D4—612no—
280C2×D4—612no—
281C2×D4—612no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C22×C4—1221no—
286C22×C4—1222no—
287C22×C4—1222no—
288C22×C4—2441no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4C2×D24411yescharacteristic
295C22×C4D4×D6111yescharacteristic
296C22×C4#1175822611yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—1222no—
309C2×C8—2448no—
310C2×C8—2441no—
311C2×C8—2441no—
312C2×C8#1175819012yes—
313C4⋊C4—612no—
314C4⋊C4—1221no—
315C4⋊C4—1222no—
316C4⋊C4—1222no—
317C4⋊C4—4881no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C22⋊C4—1222no—
325C22⋊C4—1222no—
326C42C2×C4×D6111yescharacteristic
327C42#1175822211yescharacteristic
328C42D4×D6111yescharacteristic
329C42D4×D6111yescharacteristic
330C23—1222no—
331C23—1222no—
332C23—1222no—
333C23—2441no—
334C23—2441no—
335C23—2441no—
336C23—2441no—
337C23#1451975211yescharacteristic
338Q8—4882no—
339Q8—4882no—
340D4—614no—
341D4—1222no—
342D4—1222no—
343D4—1222no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—1222no—
347C2×C4—4881no—
348C2×C4—4881no—
349C2×C4—4881no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4C22⋊C4×D6111yescharacteristic
360C2×C4#1451973911yescharacteristic
361C2×C4C4×D24411yescharacteristic
362C2×C4#1451974711yescharacteristic
363C2×C4#1451975011yescharacteristic
364C2×C4#1451976211yescharacteristic
365C2×C4SD16×D6111yescharacteristic
366C2×C4C2×#1175819011yescharacteristic
367C2×C4Q16×D6111yescharacteristic
368C2×C4#1451979611yescharacteristic
369C2×C4#1451979811yescharacteristic
370C2×C4#1451979911yescharacteristic
371C2×C4#1451980511yescharacteristic
372C2×C4#1451981011yescharacteristic
373C8—22no—
374C8—42no—
375C8—41no—
376C8—81no—
377C8—2442no—
378C22—1224no—
379C22—2442no—
380C22—2442no—
381C22—2442no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1683964011yescharacteristic
386C22#1683996811yescharacteristic
387C22#1683997811yescharacteristic
388C22#1683999211yescharacteristic
389C22#1684000411yescharacteristic
390C22#1684002611yescharacteristic
391C22#1684003311yescharacteristic
392C4—21no—
393C4—21no—
394C4—9761no—
395C4—4882no—
396C4—2442no—
397C4#103×D6111yescharacteristic
398C4#1683975511yescharacteristic
399C4C4×#1175819011yescharacteristic
400C4#1683986411yescharacteristic
401C4#1683991511yescharacteristic
402C4#1683992611yescharacteristic
403C4#1683999011yescharacteristic
404C4#1684000211yescharacteristic
405C2—2444no—
406C2#2144519511yescharacteristic
407C2#2144521911yescharacteristic
408C2#2144522911yescharacteristic
409C2#2144527111yescharacteristic
410C2#2144615211yescharacteristic
411C2#2144624111yescharacteristic
412C2#2144632611yescharacteristic
subgroupG/H
centerC23#14519752
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1277C2
socleC23×C61#121
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1277D61
hypocenterC61#9215
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
3. #21446780C2semidirect
5. C2×#16839510C2semidirect
6. #21443703C2semidirect
7. C61×#1277C2semidirect
12. #16840053C4semidirect
205. #1277D61semidirect
234. C61#9215semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
C2×C8×D2442
#214467802
#214458202
C2×#168395102
#214437032
C61×#12772
#921561