Finite Group Databasebeta

← #29665290

Subgroups — #29665290

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456762C211yesmaximal, characteristic
2#21455626C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic
4#21454899C211yesmaximal, characteristic, complemented
5#21445236C214yesmaximal, complemented
6#21445168C214yesmaximal, complemented
7#21444488C211yesmaximal, characteristic
8#21444477C211yesmaximal, characteristic, complemented
9C61×#1570C211yesmaximal, characteristic, complemented
10C22×#11758190—21no—
11#16840319C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14#16840135C2212yes—
15#16840109C2212yes—
16#16840070—22no—
17#16840058C2212yescomplemented
18#16840048C2212yescomplemented
19#16839963—22no—
20#16839963—22no—
21#16839963C2214yes—
22#16839915—22no—
23C8×Dic122C2214yescomplemented
24#16839886—22no—
25#16839815C2211yescharacteristic
26#16839812—22no—
27#16839804—21no—
28#16839772—21no—
29C2×#14519699—21no—
30#16839510C2212yescomplemented
31#16839507C2212yescomplemented
32#16839505C2214yes—
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#334C2214yescomplemented
37C22×Dic122D411yescharacteristic
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41C2×#11758190—24no—
42C2×#11758190—24no—
43C2×#11758190—44no—
44#14519777C2311yescharacteristic
45#14519777C2311yescharacteristic
46#14519774—21no—
47#14519774—21no—
48#14519774—21no—
49#14519774C2312yes—
50#14519769—21no—
51#14519756—21no—
52#14519756—21no—
53C4×Dic122—21no—
54C4×Dic122—21no—
55C4×Dic122—21no—
56C4×Dic122—21no—
57C4×Dic122D412yes—
58C4×Dic122C2312yes—
59#14519736—21no—
60C2×C4⋊Dic61D411yescharacteristic
61C2×#11758196—21no—
62C2×C4×Dic61—21no—
63#14519711—22no—
64#14519709—22no—
65C4⋊#92418C2312yes—
66#14519706—24no—
67#14519702—22no—
68#14519702—22no—
69#14519699—24no—
70#14519699—24no—
71C22⋊#92418—21no—
72C4×#92418—21no—
73C8×Dic61—24no—
74C2×M16×C61—21no—
75C22×C8×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—21no—
89C22×Dic61—21no—
90C2×D244—22no—
91C2×D244—22no—
92C2×D244—22no—
93C2×D244—42no—
94C2×D244—42no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—82no—
99C2×D244—82no—
100C2×Dic122—21no—
101C2×Dic122—22no—
102C2×Dic122—21no—
103C2×Dic122—21no—
104C2×Dic122SD1614yescomplemented
105C2×Dic122C2×D412yes—
106M16×C61—42no—
107C2×C8×C61—22no—
108C2×C8×C61—22no—
109C2×C8×C61—42no—
110C2×C8×C61C2×D414yes—
111C42×C61C2×D411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61#4111yescharacteristic
114C42×C61C2411yescharacteristic
115C22⋊Dic61—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61C2×D412yes—
122#11758196—22no—
123#11758196—22no—
124#11758196—41no—
125#11758196—41no—
126C4×Dic61—22no—
127C4×Dic61—22no—
128C4×Dic61—41no—
129C2×#92418—22no—
130#11758190—48no—
131#11758190—84no—
132C23×C61#12111yescharacteristic
133C22×D61—42no—
134C22×D61—82no—
135C22×D61—82no—
136C22×D61—82no—
137C2×C4×C61—22no—
138C2×C4×C61—41no—
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61C2×SD1612yes—
143C2×C4×C61#13512yes—
144C2×C4×C61C22×D411yescharacteristic
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C2×#4111yescharacteristic
147C2×Dic61—22no—
148C2×Dic61—24no—
149C2×Dic61—22no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—81no—
153D244—42no—
154D244—42no—
155D244—44no—
156D244—82no—
157D244—82no—
158D244—82no—
159D244—82no—
160Dic122—22no—
161Dic122—24no—
162Dic122—42no—
163Dic122—81no—
164Dic122C2×SD1614yes—
165C8×C61—24no—
166C8×C61—42no—
167C8×C61—42no—
168#92418—42no—
169#19396—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#46512yes—
173C22×C61C2×#12111yescharacteristic
174C2×D61—82no—
175C2×D61—82no—
176C2×D61—82no—
177C2×D61—164no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61#46512yes—
182C4×C61C22×SD1611yescharacteristic
183C4×C61C2×#13511yescharacteristic
184C4×C61#57712yes—
185Dic61—24no—
186Dic61—42no—
187Dic61—81no—
188Dic61—81no—
189#3208—611no—
190#3078—611no—
191C2×#460—611no—
192#2982—611no—
193#1678—614no—
194#1666—614no—
195#1570D6111yescharacteristic, complemented
196#1545—611no—
197#1542—611no—
198C2×C61—22no—
199C2×C61#304311yescharacteristic
200C2×C61C2×#46511yescharacteristic
201C2×C61#306811yescharacteristic
202D61—162no—
203C22×SD16—1221no—
204#533—611no—
205C2×#127—611no—
206C2×C4×Q8—611no—
207#492—612no—
208#486—612no—
209#479—1222no—
210#477—612no—
211#475—612no—
212#460—614no—
213#460—1222no—
214#460—1222no—
215#450—1222no—
216C8×Q8—614no—
217C8×Q8—1222no—
218#431—611no—
219#431—1222no—
220#431C2×D6111yescharacteristic
221#428—1221no—
222#423C2×D6111yescharacteristic
223C2×C4⋊C8C2×D6111yescharacteristic
224#419—1221no—
225C2×#102—1221no—
226#334C2×D6114yescomplemented
227#333—612no—
228#332—612no—
229#331—614no—
230C61#1939611yescharacteristic, complemented, Sylow
231C22×Q8—611no—
232C22×D4—1221no—
233C22×D4—1221no—
234C22×D4—1221no—
235C2×SD16—1224no—
236C2×SD16—1224no—
237C2×SD16—2444no—
238C2×M16—21no—
239C22×C8—21no—
240C4⋊Q8—611no—
241C4⋊Q8—611no—
242#127—612no—
243#127—1221no—
244#127—1221no—
245#127—1221no—
246#125—1221no—
247C22⋊Q8—1221no—
248C22⋊Q8—1221no—
249C4×Q8—612no—
250C4×Q8—612no—
251C4×Q8—1221no—
252C4×Q8—1221no—
253C4×Q8—1221no—
254C4×Q8—1221no—
255#117—1221no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C42—1221no—
259C2×C42C22×D6111yescharacteristic
260#107—1222no—
261#106—1222no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—612no—
265C4⋊C8—1224no—
266C4⋊C8D24412yes—
267C4⋊C8C22×D6112yes—
268C4⋊C8C22×D6112yes—
269#103—1222no—
270#103—1222no—
271#102—1224no—
272#102—1224no—
273C22⋊C8—1221no—
274C22⋊C8D24412yes—
275C4×C8—1224no—
276C4×C8—1221no—
277C4×C8C22×D6112yes—
278C24—2441no—
279C2×Q8—612no—
280C2×Q8—614no—
281C2×Q8—1221no—
282C2×Q8—1222no—
283C2×Q8—1221no—
284C2×Q8—1221no—
285C2×D4—1222no—
286C2×D4—1222no—
287C2×D4—1222no—
288C2×D4—4882no—
289C2×D4—4882no—
290C2×D4—2441no—
291C2×D4—2441no—
292C2×D4—2441no—
293C2×D4—2442no—
294C2×D4—2442no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4C2×D24411yescharacteristic
298C22×C4D4×D6111yescharacteristic
299C22×C4#1175822611yescharacteristic
300SD16—4884no—
301SD16—2448no—
302M16—42no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—42no—
306C2×C8—1222no—
307C2×C8C2×D24414yes—
308C4⋊C4—612no—
309C4⋊C4—1222no—
310C4⋊C4—1221no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4—1221no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C22⋊C4—2441no—
319C42—1222no—
320C42—1222no—
321C42—2441no—
322C42C2×D24411yescharacteristic
323C42D4×D6111yescharacteristic
324C42#1175822611yescharacteristic
325C42C23×D6111yescharacteristic
326C23—4882no—
327C23—4882no—
328C23—4882no—
329C23—2442no—
330C23#1451975211yescharacteristic
331Q8—614no—
332Q8—1224no—
333Q8—1222no—
334Q8—4881no—
335Q8—2442no—
336D4—4882no—
337D4—4882no—
338D4—4882no—
339D4—4882no—
340D4—2442no—
341D4—2442no—
342D4—2444no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—1224no—
346C2×C4—1222no—
347C2×C4—1222no—
348C2×C4—4881no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4#1451975211yescharacteristic
352C2×C4#1451975211yescharacteristic
353C2×C4#1451975211yescharacteristic
354C2×C4SD16×D6112yes—
355C2×C4#1451979812yes—
356C2×C4C2×D4×D6111yescharacteristic
357C2×C4C22×D24411yescharacteristic
358C2×C4C2×#1175822611yescharacteristic
359C8—24no—
360C8—42no—
361C8—42no—
362C8—2442no—
363C22—21no—
364C22—21no—
365C22—9764no—
366C22—4882no—
367C22—4882no—
368C22—4882no—
369C22#1683999212yes—
370C22C2×#1451975211yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—1224no—
375C4—4881no—
376C4—4881no—
377C4—2442no—
378C4#1683999212yes—
379C4C2×SD16×D6111yescharacteristic
380C4C2×#1451979811yescharacteristic
381C4#1684057412yes—
382C2—22no—
383C2—9762no—
384C2#2145534111yescharacteristic
385C2C2×#1683999211yescharacteristic
386C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1570C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1570D61
hypocenterC61#19396
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. #21455626C2semidirect
4. #21454899C2semidirect
5. #21445236C2semidirect
6. #21445168C2semidirect
8. #21444477C2semidirect
9. C61×#1570C2semidirect
17. #16840058C22semidirect
18. #16840048C22semidirect
23. C8×Dic122C22semidirect
30. #16839510C22semidirect
31. #16839507C22semidirect
36. C61×#334C22semidirect
104. C2×Dic122SD16semidirect
195. #1570D61semidirect
226. #334C2×D61semidirect
230. C61#19396semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214556262
C2×#168399632
#214548992
#214452362
#214451682
#214444882
#214444772
C61×#15702
#1939661