Finite Group Databasebeta

← #29663292

Subgroups — #29663292

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456970C211yesmaximal, characteristic, complemented
2#21456762C211yesmaximal, characteristic
3C2×#16840110C211yesmaximal, characteristic
4#21454749C211yesmaximal, characteristic, complemented
5#21445362C214yesmaximal, complemented
6#21445238C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic
9C61×#1548C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11C2×#14519774C2211yescharacteristic
12#16840165—21no—
13#16840135—21no—
14#16840114—21no—
15#16840110—21no—
16#16840110—21no—
17#16840110C2212yes—
18#16840110C2212yes—
19#16840109—21no—
20#16840097—22no—
21#16840054C2214yescomplemented
22#16839963C2214yes—
23#16839915—22no—
24#16839889—21no—
25#16839889C2212yescomplemented
26#16839886—21no—
27#16839886C2212yescomplemented
28#16839815C2211yescharacteristic
29#16839790—21no—
30C2×#14519697—21no—
31C2×#14519697—21no—
32#16839720—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#333C2214yescomplemented
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758190—24no—
45C22×#92418D411yescharacteristic
46C22×#92418D411yescharacteristic
47#14519777D411yescharacteristic
48#14519777D411yescharacteristic
49#14519774—21no—
50#14519774—21no—
51#14519774—21no—
52#14519774C2312yes—
53#14519769—21no—
54#14519756—21no—
55#14519756D412yes—
56C4×Dic122—21no—
57C4×Dic122C2312yes—
58#14519736—21no—
59#14519710—24no—
60C4⋊#92418C2312yes—
61#14519706—22no—
62#14519706—22no—
63#14519700—24no—
64#14519699—24no—
65#14519697—22no—
66#14519697—22no—
67#14519697—22no—
68#14519697—22no—
69C22⋊#92418—21no—
70#14519687—22no—
71C4×#92418—21no—
72C4×#92418D411yescharacteristic
73C4×#92418D411yescharacteristic
74C4×#92418C2311yescharacteristic
75C4×#92418C2311yescharacteristic
76C8×Dic61—22no—
77C22×Q8×C61—21no—
78C2×M16×C61—21no—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C42×C61C2311yescharacteristic
86C4⋊C8×C61—21no—
87C4⋊C8×C61C2312yes—
88C4⋊C8×C61C2312yes—
89C22⋊C8×C61—21no—
90C4×C8×C61—21no—
91C23×D61—41no—
92C2×Q8×C61—22no—
93C2×Q8×C61—41no—
94C2×Q8×C61—82no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—42no—
103C2×D244—42no—
104C2×D244—41no—
105C2×D244—41no—
106C2×D244—41no—
107C2×D244—82no—
108C2×D244—82no—
109C2×Dic122—21no—
110C2×Dic122C2×D412yes—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122C22⋊Dic61—22no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61C2×D412yes—
127#11758196—22no—
128#11758196—22no—
129C4×Dic61—22no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137#11758190—84no—
138C23×C61#12111yescharacteristic
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143Q8×C61—82no—
144Q8×C61—82no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—42no—
148C2×C4×C61—82no—
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#13512yes—
153C2×C4×C61#13712yes—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C2×Dic61—22no—
158C2×Dic61—22no—
159D244—42no—
160D244—42no—
161D244—44no—
162D244—82no—
163D244—82no—
164D244—82no—
165D244—82no—
166Dic122—24no—
167Dic122—42no—
168C8×C61—42no—
169C8×C61—42no—
170#92418—21no—
171#92418—21no—
172#92418—21no—
173#92418—21no—
174#92418—42no—
175#19212—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61#46412yes—
179C22×C61C2×#12111yescharacteristic
180C2×D61—82no—
181C2×D61—82no—
182C2×D61—82no—
183C2×D61—164no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—82no—
187C4×C61#46412yes—
188C4×C61#47212yes—
189C4×C61C2×#13511yescharacteristic
190C4×C61C2×#13711yescharacteristic
191C4×C61#57712yes—
192Dic61—42no—
193Dic61—42no—
194#3230—611no—
195#3208—611no—
196C2×#486—611no—
197#2960—611no—
198#1698—614no—
199#1678—614no—
200#1564—611no—
201#1548D6111yescharacteristic, complemented
202#1545—611no—
203C2×C61—22no—
204C2×C61C2×#46411yescharacteristic
205C2×C61#305711yescharacteristic
206C2×C61C2×#47211yescharacteristic
207D61—162no—
208#533—611no—
209C2×C4⋊Q8C2×D6111yescharacteristic
210C2×#127—611no—
211#500—1221no—
212#492—1221no—
213#487—1221no—
214#486—612no—
215#486—612no—
216#486—1221no—
217#486—1221no—
218#486—1221no—
219#484—1222no—
220#477—614no—
221#460—614no—
222#450—1222no—
223#446—612no—
224#446—1221no—
225C8×Q8—612no—
226C8×Q8—1221no—
227#431—611no—
228#431C2×D6111yescharacteristic
229#424—1221no—
230#423C2×D6111yescharacteristic
231C2×#102—1221no—
232C2×#102—1221no—
233#405—1221no—
234C2×C4×C8—611no—
235#335—612no—
236#334—612no—
237#333C2×D6114yescomplemented
238#331—614no—
239C61#1921211yescharacteristic, complemented, Sylow
240C22×Q8—21no—
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4—1221no—
244C2×SD16—1224no—
245C2×M16—21no—
246C22×C8—611no—
247C22×C8—611no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—611no—
252C4⋊Q8—611no—
253C4⋊Q8C22×D6112yes—
254#127—612no—
255#127—1221no—
256#127—1221no—
257#127—1221no—
258#125—1221no—
259C22⋊Q8—612no—
260C22⋊Q8—1221no—
261C4×Q8—612no—
262C4×Q8—1221no—
263#117—1221no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42C22×D6111yescharacteristic
267#107—1224no—
268C4⋊C8—21no—
269C4⋊C8—612no—
270C4⋊C8—1222no—
271C4⋊C8—1222no—
272C4⋊C8C22×D6112yes—
273C4⋊C8C22×D6112yes—
274#103—1224no—
275#102—1222no—
276#102—1222no—
277#102—1222no—
278#102—1222no—
279#102—1224no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282#97—1222no—
283C4×C8—21no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—1222no—
289C4×C8—1221no—
290C24—2441no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×Q8—612no—
295C2×Q8—1221no—
296C2×D4—1222no—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—4882no—
300C2×D4—4882no—
301C2×D4—2442no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2442no—
306C22×C4—41no—
307C22×C4#1175822211yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4D4×D6111yescharacteristic
310SD16—4884no—
311M16—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—611no—
315C2×C8—611no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—1222no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—612no—
326C4⋊C4—1222no—
327C4⋊C4—1222no—
328C4⋊C4—1221no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C22⋊C4—1222no—
332C42—1222no—
333C42#1175822211yescharacteristic
334C42D4×D6111yescharacteristic
335C42D4×D6111yescharacteristic
336C42C23×D6111yescharacteristic
337C23—4882no—
338C23—4882no—
339C23—4882no—
340C23—2442no—
341C23#1451975011yescharacteristic
342Q8—82no—
343Q8—82no—
344Q8—1224no—
345Q8—2442no—
346D4—4882no—
347D4—4882no—
348D4—4882no—
349D4—4882no—
350D4—2442no—
351D4—2442no—
352D4—2444no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4#1451975011yescharacteristic
360C2×C4#1451975011yescharacteristic
361C2×C4#1451975011yescharacteristic
362C2×C4#1451979812yes—
363C2×C4#1451980912yes—
364C2×C4C2×D4×D6111yescharacteristic
365C2×C4C2×D4×D6111yescharacteristic
366C2×C4C2×#1175822211yescharacteristic
367C8—42no—
368C8—42no—
369C8—1221no—
370C8—1221no—
371C8—1221no—
372C8—1221no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—9764no—
377C22—4882no—
378C22—4882no—
379C22—4882no—
380C22#1683998712yes—
381C22C2×#1451975011yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—2442no—
386C4—2442no—
387C4#1683998712yes—
388C4#1684003112yes—
389C4C2×#1451979811yescharacteristic
390C4C2×#1451980911yescharacteristic
391C4#1684057112yes—
392C2—22no—
393C2—9762no—
394C2C2×#1683998711yescharacteristic
395C2#2145546111yescharacteristic
396C2C2×#1684003111yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1548D61
hypocenterC61#19212
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #21456970C2semidirect
4. #21454749C2semidirect
5. #21445362C2semidirect
6. #21445238C2semidirect
7. #21444565C2semidirect
9. C61×#1548C2semidirect
21. #16840054C22semidirect
25. #16839889C22semidirect
27. #16839886C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
40. C61×#333C22semidirect
201. #1548D61semidirect
237. #333C2×D61semidirect
239. C61#19212semidirect

Maximal subgroups

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

subgroupindex
#214569702
#214567622
C2×#168401102
#214547492
#214453622
#214452382
#214445652
#214444882
C61×#15482
#1921261