Finite Group Databasebeta

← #29575956

Subgroups — #29575956

Order 15616.

544 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455293C211yesmaximal, characteristic
2#21448736C211yesmaximal, characteristic, complemented
3#21444378C211yesmaximal, characteristic
4#21444315C211yesmaximal, characteristic
5#21444213C211yesmaximal, characteristic, complemented
6#21443720C211yesmaximal, characteristic, complemented
7C61×#1286C211yesmaximal, characteristic, complemented
8C2×#14519810—21no—
9#16840312C2211yescharacteristic
10C2×C4×Dic122C2211yescharacteristic
11C2×#14519742—21no—
12#16839985—22no—
13#16839973C412yescomplemented
14#16839963C412yescomplemented
15#16839815C2211yescharacteristic
16#16839764—21no—
17#16839659—22no—
18#16839658—21no—
19#16839626—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#16839613—21no—
22#16839497—21no—
23#16839497—21no—
24#16839495—22no—
25#16839492—22no—
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C61×#382C2211yescharacteristic
29C2×#11758222—21no—
30C22×Dic122D411yescharacteristic
31#14519810—42no—
32#14519810—42no—
33C2×Dic244—22no—
34C2×#11758190—22no—
35#14519777C2×C411yescharacteristic
36#14519774C2×C411yescharacteristic
37#14519765—21no—
38#14519751—21no—
39#14519751—21no—
40C4×Dic122—21no—
41C4×Dic122—21no—
42C4×Dic122C2×C412yes—
43C4×Dic122D412yes—
44#14519742—42no—
45#14519742—42no—
46C4×D244—21no—
47C4×D244—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758196—21no—
51C2×#11758196—22no—
52C2×#11758196—22no—
53C2×C4×Dic61—21no—
54C2×C4×Dic61—22no—
55C2×C4×Dic61—21no—
56C4⋊#92418—21no—
57#14519702—22no—
58#14519699—22no—
59C22⋊#92418—21no—
60C4×#92418—21no—
61#14519684—22no—
62#14519684D411yescharacteristic
63#14519684D411yescharacteristic
64C2×M16×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C42×C61C2311yescharacteristic
67C4⋊C8×C61—21no—
68C4⋊C8×C61—21no—
69C4⋊C8×C61C2×C412yes—
70C22⋊C8×C61—21no—
71C4×C8×C61—21no—
72#95×C61—21no—
73#95×C61D412yes—
74C22×C4×C61—22no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—41no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83#11758222—41no—
84#11758222—41no—
85#11758222—41no—
86#11758222—41no—
87C2×D244—21no—
88C2×D244—21no—
89C2×D244—21no—
90C2×C4×D61—41no—
91C2×Dic122—21no—
92C2×Dic122—21no—
93C2×Dic122—21no—
94C2×Dic122—21no—
95C2×Dic122C22⋊C412yes—
96C2×Dic122D812yes—
97C2×Dic122SD1612yes—
98M16×C61—41no—
99M16×C61—41no—
100C2×C8×C61—22no—
101C2×C8×C61—41no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C42×C61C22×C411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61#4111yescharacteristic
107C42×C61#4111yescharacteristic
108#11758198—41no—
109#11758198—41no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—42no—
114C4⋊Dic61C22⋊C412yes—
115#11758196—22no—
116#11758196—22no—
117#11758196—44no—
118#11758196—44no—
119C4×Dic61—22no—
120C4×Dic61—22no—
121C4×Dic61—44no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C2×#92418—41no—
125Dic244—44no—
126#11758190—44no—
127C23×C61#12011yescharacteristic
128C22×D61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12311yescharacteristic
140C2×C4×C61C2×D811yescharacteristic
141C2×C4×C61C2×SD1611yescharacteristic
142C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C22⋊D61—81no—
145C22⋊D61—81no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158D244—41no—
159D244—41no—
160D244—81no—
161D244—81no—
162C4×D61—81no—
163C4×D61—81no—
164Dic122—22no—
165Dic122—22no—
166Dic122—22no—
167Dic122—41no—
168Dic122—41no—
169Dic122#10214yes—
170C8×C61—42no—
171C8×C61—81no—
172#92418—81no—
173#9180—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61#38611yescharacteristic
177C22×C61#44711yescharacteristic
178C22×C61#44811yescharacteristic
179C2×D61—81no—
180C2×D61—81no—
181C2×D61—81no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—82no—
187C4×C61C2×#10211yescharacteristic
188C4×C61#41711yescharacteristic
189C4×C61#48011yescharacteristic
190C4×C61#48111yescharacteristic
191Dic61—24no—
192Dic61—42no—
193Dic61—42no—
194Dic61—81no—
195Dic61—81no—
196#3037—611no—
197#2351—611no—
198#1517—611no—
199#1502—611no—
200#1479—611no—
201#1303—611no—
202#1286D6111yescharacteristic, complemented
203C2×C61—21no—
204C2×C61—21no—
205C2×C61#187411yescharacteristic
206C2×C61#190511yescharacteristic
207C2×C61#190611yescharacteristic
208D61—161no—
209C2×#137—1221no—
210#532—611no—
211C2×C4×Q8—611no—
212C2×C4×Q8—1221no—
213#464—1222no—
214#462—612no—
215#460—612no—
216#431—611no—
217#431C2×D6111yescharacteristic
218#423C2×D6111yescharacteristic
219#417—1221no—
220#389—1221no—
221#389—1222no—
222#382—1221no—
223#382C2×D6111yescharacteristic
224C4×C4⋊C4—611no—
225C4×C4⋊C4—1221no—
226#327—1221no—
227#327—1221no—
228#327—1222no—
229Q8⋊C8—1222no—
230C61#918011yescharacteristic, complemented, Sylow
231C2×#41—1221no—
232C22×Q8—611no—
233#137—2442no—
234#137—2442no—
235C2×Q16—1222no—
236C2×SD16—1222no—
237C2×M16—21no—
238C4⋊Q8—611no—
239#127—611no—
240#124—1221no—
241#121—1221no—
242#121—1221no—
243C4×Q8—612no—
244C4×Q8—612no—
245C4×Q8—1221no—
246C4×Q8—1221no—
247C4×Q8—2442no—
248C4×Q8—2442no—
249C4×D4—1221no—
250C4×D4—1221no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1222no—
255C2×C4⋊C4—1222no—
256C2×C4⋊C4—1221no—
257C2×C42—1221no—
258C2×C42—1222no—
259C2×C42—1221no—
260C2×C42C22×D6111yescharacteristic
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—1221no—
264C4⋊C8C4×D6112yes—
265#103—1222no—
266#102—1222no—
267C22⋊C8—21no—
268C22⋊C8—1221no—
269C4×C8—21no—
270C4×C8—1221no—
271#95—21no—
272#95—611no—
273#95—611no—
274#95—1222no—
275#95D24412yes—
276#41—2441no—
277#41—2441no—
278#41—2441no—
279#41—2441no—
280C2×Q8—612no—
281C2×Q8—612no—
282C2×Q8—612no—
283C2×Q8—1221no—
284C2×Q8—1221no—
285C2×Q8—1221no—
286C2×Q8—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C2×D4—1221no—
290C2×D4—2441no—
291C22×C4—22no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—2441no—
297C22×C4C2×D24411yescharacteristic
298C22×C4D4×D6111yescharacteristic
299C22×C4D4×D6111yescharacteristic
300Q16—2444no—
301SD16—2444no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—2441no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—612no—
310C4⋊C4—1222no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4—2444no—
316C4⋊C4—2444no—
317C4⋊C4—2442no—
318C22⋊C4—2441no—
319C22⋊C4—2441no—
320C42—1222no—
321C42—1222no—
322C42—2444no—
323C42—2441no—
324C42—2441no—
325C42C2×C4×D6111yescharacteristic
326C42#1175822211yescharacteristic
327C42D4×D6111yescharacteristic
328C42#1175822611yescharacteristic
329C23—2441no—
330C23#1451974711yescharacteristic
331Q8—614no—
332Q8—1222no—
333Q8—1222no—
334Q8—1222no—
335Q8—2441no—
336Q8—2441no—
337D4—4881no—
338D4—4881no—
339D4—4881no—
340D4—4881no—
341D4—2441no—
342D4—2441no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—1222no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—1222no—
353C2×C4—4881no—
354C2×C4—4881no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2442no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4C22⋊C4×D6111yescharacteristic
363C2×C4#1451973911yescharacteristic
364C2×C4C4×D24411yescharacteristic
365C2×C4#1451975011yescharacteristic
366C2×C4#1451975211yescharacteristic
367C2×C4#1451976211yescharacteristic
368C2×C4D8×D6111yescharacteristic
369C2×C4SD16×D6111yescharacteristic
370C2×C4#1451980411yescharacteristic
371C2×C4#1451980911yescharacteristic
372C8—42no—
373C8—81no—
374C8—4881no—
375C22—21no—
376C22—21no—
377C22—4881no—
378C22—4881no—
379C22—4881no—
380C22#1683964011yescharacteristic
381C22#1683989711yescharacteristic
382C22#1683990211yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—82no—
388C4—1224no—
389C4—4881no—
390C4—4881no—
391C4—2442no—
392C4—2442no—
393C4#102×D6111yescharacteristic
394C4#1683976211yescharacteristic
395C4#1684007711yescharacteristic
396C4#1684008211yescharacteristic
397C2—21no—
398C2—21no—
399C2—9761no—
400C2#2144613511yescharacteristic
401C2#2144631411yescharacteristic
402C2#2144632111yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1286C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1286D61
hypocenterC61#9180
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
2. #21448736C2semidirect
5. #21444213C2semidirect
6. #21443720C2semidirect
7. C61×#1286C2semidirect
13. #16839973C4semidirect
14. #16839963C4semidirect
202. #1286D61semidirect
230. C61#9180semidirect

Maximal subgroups

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

subgroupindex
#214552932
#214487362
#214443782
#214443152
#214442132
#214437202
C61×#12862
#918061