Finite Group Databasebeta

← #29663968

Subgroups — #29663968

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840109C211yesmaximal, characteristic, complemented
2#21455635C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4C8×#11758222C211yesmaximal, characteristic, complemented
5#21445236C214yesmaximal, complemented
6#21445229C214yesmaximal, complemented
7#21444501C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic, complemented
9C61×#1566C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11C2×#14519777C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840109—21no—
15#16840109—21no—
16#16840109C2214yes—
17#16840058C2212yescomplemented
18#16840053C2212yescomplemented
19#16840048—21no—
20#16840043—21no—
21#16839963—21no—
22#16839963C2212yes—
23#16839958—21no—
24#16839958C2212yes—
25C8×Dic122C2212yescomplemented
26C8×D244C2212yescomplemented
27C8×C22⋊D61—22no—
28#16839815C2211yescharacteristic
29#16839813—21no—
30#16839804—21no—
31#16839768—21no—
32#16839764—21no—
33C2×#14519699—22no—
34C4×C8×D61—21no—
35#16839510C2214yescomplemented
36#16839505C2214yes—
37C61×#423C2211yescharacteristic
38C61×#423C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C61×#334C2214yescomplemented
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46#14519805—44no—
47C2×#11758190—22no—
48C2×D488—22no—
49C2×C8×D61—22no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4×Dic122D411yescharacteristic
59C4×Dic122C2311yescharacteristic
60C4×D244D411yescharacteristic
61C4×D244C2311yescharacteristic
62C4×C22⋊D61—21no—
63C4×C22⋊D61—21no—
64#14519736D411yescharacteristic
65#14519734—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C42×D61—21no—
69#14519711—22no—
70#14519709—22no—
71C4⋊#92418C2312yes—
72#14519706—22no—
73#14519702—22no—
74#14519699—22no—
75#14519699—24no—
76#14519699—24no—
77#14519699—22no—
78#14519699—22no—
79C22⋊#92418—21no—
80#14519689—22no—
81C4×#92418—21no—
82C8×Dic61—22no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C22×C8×C61D412yes—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61—21no—
89C4⋊C8×C61C2312yes—
90C4⋊C8×C61C2312yes—
91C4×C8×C61D411yescharacteristic
92C4×C8×C61D411yescharacteristic
93C4×C8×C61C2312yes—
94C23×D61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—21no—
99C22×Dic61—41no—
100#11758222—22no—
101#11758222SD1614yes—
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—44no—
106C2×D244—42no—
107C2×D244—42no—
108C2×D244—41no—
109C2×D244—41no—
110C2×D244—41no—
111C2×D244—82no—
112C2×D244C2×D411yescharacteristic
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×Dic122—22no—
117C2×Dic122—41no—
118C2×Dic122—82no—
119C2×Dic122C2×D411yescharacteristic
120M16×C61—42no—
121M16×C61—42no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61—42no—
125C2×C8×C61C2×D412yes—
126C2×C8×C61C2×D412yes—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131C22⋊Dic61—21no—
132#11758198—21no—
133#11758198—21no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—41no—
137C4⋊Dic61—41no—
138C4⋊Dic61C2×D411yescharacteristic
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—21no—
141#11758196—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61—21no—
145C2×#92418—22no—
146D488—44no—
147#11758190—44no—
148C8×D61—44no—
149C23×C61#12111yescharacteristic
150C22×D61—21no—
151C22×D61—42no—
152C22×D61—82no—
153C22×D61—82no—
154C22×D61—82no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61C2×SD1612yes—
161C2×C4×C61#13512yes—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C22⋊D61—24no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—42no—
170C2×Dic61—82no—
171D244—21no—
172D244—42no—
173D244—42no—
174D244—44no—
175D244—82no—
176D244—82no—
177D244—82no—
178D244—82no—
179D244C2×SD1612yes—
180C4×D61—24no—
181C4×D61—42no—
182C4×D61—42no—
183Dic122—21no—
184Dic122—82no—
185Dic122—82no—
186Dic122C2×SD1612yes—
187C8×C61—22no—
188C8×C61—22no—
189C8×C61—42no—
190C8×C61—42no—
191#92418—42no—
192#19274—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61#46512yes—
196C22×C61C2×#12111yescharacteristic
197C2×D61—22no—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—82no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—164no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61#46512yes—
207C4×C61#47212yes—
208C4×C61C22×SD1611yescharacteristic
209C4×C61C2×#13511yescharacteristic
210C4×C61#57512yes—
211Dic61—22no—
212Dic61—41no—
213Dic61—41no—
214Dic61—82no—
215C2×#486—611no—
216#3079—611no—
217#3038—611no—
218C8×#41—611no—
219#1678—614no—
220#1677—614no—
221#1566D6111yescharacteristic, complemented
222#1548—611no—
223#1545—611no—
224C2×C61—22no—
225C2×C61#304011yescharacteristic
226C2×C61C2×#46511yescharacteristic
227C2×C61C2×#47211yescharacteristic
228D61—42no—
229D61—162no—
230C2×AGL(1,ℤ/8)—1221no—
231C2×C4⋊Q8—611no—
232C2×#127—611no—
233C4×#41—611no—
234#486—614no—
235#486—1221no—
236#486—1221no—
237#477—612no—
238#476—612no—
239#475—1221no—
240#474—1221no—
241#460—612no—
242#460—1221no—
243#459—612no—
244#459—1221no—
245C8×Q8—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#431—611no—
249#431—1221no—
250#428—1221no—
251#423C2×D6111yescharacteristic
252#423C2×D6111yescharacteristic
253#418—1221no—
254#417—1221no—
255C2×#102—1222no—
256C2×C4×C8—1221no—
257C2×C4×C8C2×D6111yescharacteristic
258#334C2×D6114yescomplemented
259#333—614no—
260#331—614no—
261C61#1927411yescharacteristic, complemented, Sylow
262C2×#41—611no—
263C22×Q8—1221no—
264C22×D4—1221no—
265C22×D4—1221no—
266C22×D4—1221no—
267AGL(1,ℤ/8)—2444no—
268C2×SD16—1222no—
269C2×D8—1222no—
270C2×M16—21no—
271C2×M16—21no—
272C22×C8—1222no—
273C22×C8D24412yes—
274C4⋊Q8—612no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278#127—612no—
279#127—1221no—
280#127—1221no—
281#127—1221no—
282C4×Q8—611no—
283C4×Q8—611no—
284C4×D4—611no—
285C4×D4—611no—
286C4×D4—1221no—
287C4×D4—1221no—
288#117—611no—
289#117—1221no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C42—1221no—
293C2×C42C22×D6111yescharacteristic
294#107—1222no—
295#106—1222no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—612no—
299C4⋊C8—1222no—
300C4⋊C8C22×D6112yes—
301C4⋊C8C22×D6112yes—
302#103—1222no—
303#102—1222no—
304#102—1222no—
305#102—1222no—
306#102—1224no—
307#102—1224no—
308C22⋊C8—1221no—
309C22⋊C8—1222no—
310C4×C8—1222no—
311C4×C8—1221no—
312C4×C8D24411yescharacteristic
313C4×C8D24411yescharacteristic
314C4×C8C22×D6112yes—
315C24—2441no—
316#41—614no—
317#41—1222no—
318C2×Q8—611no—
319C2×Q8—1222no—
320C2×Q8—4882no—
321C2×Q8—2441no—
322C2×D4—611no—
323C2×D4—1221no—
324C2×D4—1222no—
325C2×D4—1222no—
326C2×D4—1222no—
327C2×D4—4882no—
328C2×D4—2444no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2441no—
332C2×D4—2442no—
333C2×D4—2442no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—2441no—
338C22×C4C2×D24411yescharacteristic
339C22×C4D4×D6111yescharacteristic
340C22×C4#1175822611yescharacteristic
341SD16—2444no—
342D8—2444no—
343M16—42no—
344M16—42no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—42no—
348C2×C8—1222no—
349C2×C8—2444no—
350C2×C8C2×D24412yes—
351C2×C8C2×D24412yes—
352C4⋊C4—611no—
353C4⋊C4—611no—
354C4⋊C4—1221no—
355C4⋊C4—1221no—
356C4⋊C4—1222no—
357C4⋊C4—1222no—
358C4⋊C4—2441no—
359C4⋊C4—2441no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C42—1221no—
364C42—1221no—
365C42—1221no—
366C42C2×D24411yescharacteristic
367C42D4×D6111yescharacteristic
368C42#1175822611yescharacteristic
369C42C23×D6111yescharacteristic
370C23—1221no—
371C23—4882no—
372C23—4882no—
373C23—4882no—
374C23—2442no—
375C23#1451975211yescharacteristic
376Q8—612no—
377Q8—1221no—
378Q8—4882no—
379Q8—4882no—
380D4—612no—
381D4—1224no—
382D4—1221no—
383D4—4882no—
384D4—4882no—
385D4—4882no—
386D4—4882no—
387D4—2442no—
388D4—2442no—
389D4—2444no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—1221no—
393C2×C4—1224no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—4882no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4#1451975211yescharacteristic
401C2×C4#1451975211yescharacteristic
402C2×C4#1451975211yescharacteristic
403C2×C4SD16×D6112yes—
404C2×C4#1451979812yes—
405C2×C4C2×D4×D6111yescharacteristic
406C2×C4C22×D24411yescharacteristic
407C2×C4C2×#1175822611yescharacteristic
408C8—22no—
409C8—22no—
410C8—42no—
411C8—42no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—1222no—
416C22—9764no—
417C22—4882no—
418C22—4882no—
419C22—4882no—
420C22—2441no—
421C22—2441no—
422C22#1683999212yes—
423C22C2×#1451975211yescharacteristic
424C4—21no—
425C4—21no—
426C4—1222no—
427C4—4882no—
428C4—2441no—
429C4—2441no—
430C4#1683999212yes—
431C4#1684003312yes—
432C4C2×SD16×D6111yescharacteristic
433C4C2×#1451979811yescharacteristic
434C4#1684056212yes—
435C2—22no—
436C2—9762no—
437C2—2442no—
438C2#2145531411yescharacteristic
439C2C2×#1683999211yescharacteristic
440C2C2×#1684003311yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1566C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1566D61
hypocenterC61#19274
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
1. C2×#16840109C2semidirect
2. #21455635C2semidirect
4. C8×#11758222C2semidirect
5. #21445236C2semidirect
6. #21445229C2semidirect
7. #21444501C2semidirect
8. #21444488C2semidirect
9. C61×#1566C2semidirect
17. #16840058C22semidirect
18. #16840053C22semidirect
25. C8×Dic122C22semidirect
26. C8×D244C22semidirect
35. #16839510C22semidirect
40. C61×#334C22semidirect
221. #1566D61semidirect
258. #334C2×D61semidirect
261. C61#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
#214556352
#214553002
C8×#117582222
#214452362
#214452292
#214445012
#214444882
C61×#15662
#1927461