Finite Group Databasebeta

← #29665292

Subgroups — #29665292

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#21456757C211yesmaximal, characteristic
2#21455620C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic
4#21454899C211yesmaximal, characteristic, complemented
5#21445238C214yesmaximal, complemented
6#21445170C214yesmaximal, complemented
7#21444589C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic
9C61×#1542C211yesmaximal, characteristic, complemented
10C22×#11758190—21no—
11C2×#14519774C2211yescharacteristic
12C2×C4×Dic122C2211yescharacteristic
13#16840136C2212yes—
14#16840110C2212yes—
15#16840066—22no—
16#16840054C2212yescomplemented
17#16840044C2212yescomplemented
18#16839963—22no—
19#16839963C2214yes—
20#16839962—22no—
21#16839912—22no—
22C8×Dic122—22no—
23#16839886C2214yescomplemented
24#16839815C2211yescharacteristic
25#16839812—22no—
26#16839805—21no—
27#16839786C2211yescharacteristic
28#16839770—21no—
29C2×#14519699—21no—
30#16839512C2214yescomplemented
31#16839505C2214yes—
32C61×#533C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#333C2212yescomplemented
36C61×#332C2212yescomplemented
37C22×Dic122D411yescharacteristic
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41C2×#11758190—24no—
42C2×#11758190—44no—
43C2×#11758201—24no—
44C2×#11758194—21no—
45#14519774—21no—
46#14519774—21no—
47#14519774—21no—
48#14519774C2312yes—
49C4×Dic122—21no—
50C4×Dic122—21no—
51C4×Dic122D412yes—
52C4×Dic122C2312yes—
53C2×C4⋊Dic61D411yescharacteristic
54C2×#11758196—21no—
55C2×C4×Dic61—21no—
56#14519710—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61#14519706—24no—
62#14519700—22no—
63#14519700—22no—
64#14519699—24no—
65#14519697—24no—
66C22⋊#92418D412yes—
67C4×#92418C2312yes—
68C8×Dic61—24no—
69C22×C8×C61—21no—
70C4⋊Q8×C61C2311yescharacteristic
71C4⋊Q8×C61C2311yescharacteristic
72#125×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75C4×Q8×C61—21no—
76C4×Q8×C61—21no—
77#117×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C23×D61—41no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—21no—
92C2×D244—22no—
93C2×D244—22no—
94C2×D244—22no—
95C2×D244—42no—
96C2×D244—42no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—82no—
101C2×D244—82no—
102C2×Dic122—21no—
103C2×Dic122—22no—
104C2×Dic122SD1614yescomplemented
105C2×Dic122C2×D412yes—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C2×C8×C61—42no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C42×C61—41no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61C2411yescharacteristic
121#11758201—84no—
122C4⋊Dic61—21no—
123C4⋊Dic61C2×D412yes—
124#11758196—22no—
125#11758196—22no—
126C4×Dic61—22no—
127C4×Dic61—22no—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418C2×D414yes—
131#11758190—48no—
132C23×C61#12111yescharacteristic
133C22×D61—42no—
134C22×D61—82no—
135C22×D61—82no—
136C22×D61—82no—
137Q8×C61—42no—
138Q8×C61—81no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61C2×SD1612yes—
147C2×C4×C61#13512yes—
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C2×#4111yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—24no—
153C2×Dic61—22no—
154C2×Dic61—81no—
155D244—42no—
156D244—42no—
157D244—44no—
158D244—82no—
159D244—82no—
160D244—82no—
161D244—82no—
162Dic122—22no—
163Dic122—24no—
164Dic122C2×SD1614yes—
165C8×C61—42no—
166C8×C61—42no—
167#92418—24no—
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—81no—
182C4×C61—81no—
183C4×C61#46512yes—
184C4×C61C22×SD1611yescharacteristic
185C4×C61C2×#13511yescharacteristic
186C4×C61#57712yes—
187Dic61—24no—
188Dic61—42no—
189#3208—611no—
190#3078—611no—
191C2×#460—611no—
192#2982—611no—
193#1678—614no—
194#1666—614no—
195#1570—611no—
196#1545—611no—
197#1542D6111yescharacteristic, complemented
198C2×C61—22no—
199C2×C61#304311yescharacteristic
200C2×C61C2×#46511yescharacteristic
201C2×C61#306811yescharacteristic
202D61—162no—
203C22×SD16—1221no—
204#533C2×D6111yescharacteristic
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#423—611no—
223C2×C4⋊C8C2×D6111yescharacteristic
224#419—1221no—
225C2×#102—1221no—
226#334—614no—
227#333C2×D6112yescomplemented
228#332C2×D6112yescomplemented
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—1221no—
239C22×C8—21no—
240C4⋊Q8C22×D6111yescharacteristic
241C4⋊Q8C22×D6111yescharacteristic
242#127—612no—
243#127—1221no—
244#127—1221no—
245#127—1221no—
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—612no—
252C4×Q8—612no—
253C4×Q8—1221no—
254C4×Q8—1221no—
255#117—21no—
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—612no—
264C4⋊C8—612no—
265C4⋊C8—1224no—
266C4⋊C8—1221no—
267C4⋊C8C22×D6112yes—
268C4⋊C8C22×D6112yes—
269#103—1222no—
270#103—1222no—
271#102—1224no—
272#102—1224no—
273C22⋊C8—21no—
274C22⋊C8—612no—
275C4×C8—21no—
276C4×C8—612no—
277C4×C8—1224no—
278C24—2441no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—612no—
282C2×Q8—614no—
283C2×Q8—1221no—
284C2×Q8—1222no—
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×C4#1175822211yescharacteristic
298C22×C4D4×D6111yescharacteristic
299C22×C4D4×D6111yescharacteristic
300SD16—4884no—
301SD16—2448no—
302M16—2442no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—42no—
306C2×C8—614no—
307C2×C8—1222no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—612no—
315C4⋊C4—1222no—
316C4⋊C4—1221no—
317C4⋊C4—1222no—
318C22⋊C4—41no—
319C42—41no—
320C42—1222no—
321C42—1222no—
322C42#1175822211yescharacteristic
323C42D4×D6111yescharacteristic
324C42D4×D6111yescharacteristic
325C42C23×D6111yescharacteristic
326C23—4882no—
327C23—4882no—
328C23—4882no—
329C23—2442no—
330C23#1451975011yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—614no—
334Q8—1224no—
335Q8—1222no—
336D4—4882no—
337D4—4882no—
338D4—4882no—
339D4—4882no—
340D4—2442no—
341D4—2442no—
342D4—2444no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—1224no—
348C2×C4—1222no—
349C2×C4—1222no—
350C2×C4—4881no—
351C2×C4#1451975011yescharacteristic
352C2×C4#1451975011yescharacteristic
353C2×C4#1451975011yescharacteristic
354C2×C4SD16×D6112yes—
355C2×C4#1451980012yes—
356C2×C4C2×D4×D6111yescharacteristic
357C2×C4C2×D4×D6111yescharacteristic
358C2×C4C2×#1175822211yescharacteristic
359C8—42no—
360C8—42no—
361C8—1224no—
362C8—2442no—
363C22—21no—
364C22—21no—
365C22—9764no—
366C22—4882no—
367C22—4882no—
368C22—4882no—
369C22#1683999412yes—
370C22C2×#1451975011yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—81no—
375C4—81no—
376C4—1224no—
377C4—2442no—
378C4#1683999412yes—
379C4C2×SD16×D6111yescharacteristic
380C4C2×#1451980011yescharacteristic
381C4#1684056912yes—
382C2—22no—
383C2—9762no—
384C2#2145534311yescharacteristic
385C2C2×#1683999411yescharacteristic
386C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1542D61
hypocenterC61#19396
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. #21455620C2semidirect
4. #21454899C2semidirect
5. #21445238C2semidirect
6. #21445170C2semidirect
7. #21444589C2semidirect
9. C61×#1542C2semidirect
16. #16840054C22semidirect
17. #16840044C22semidirect
23. #16839886C22semidirect
30. #16839512C22semidirect
35. C61×#333C22semidirect
36. C61×#332C22semidirect
104. C2×Dic122SD16semidirect
197. #1542D61semidirect
227. #333C2×D61semidirect
228. #332C2×D61semidirect
230. C61#19396semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214556202
C2×#168399632
#214548992
#214452382
#214451702
#214445892
#214444862
C61×#15422
#1939661