Finite Group Databasebeta

← #29582417

Subgroups — #29582417

Order 15616.

554 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 101 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#21456744C211yesmaximal, characteristic
2#21455293C211yesmaximal, characteristic
3#21451035C211yesmaximal, characteristic, complemented
4#21444315C211yesmaximal, characteristic
5#21444215C211yesmaximal, characteristic, complemented
6#21443720C211yesmaximal, characteristic, complemented
7C61×#1548C211yesmaximal, characteristic, complemented
8C2×#14519810—21no—
9#16840312C2211yescharacteristic
10C2×C4×Dic122C2211yescharacteristic
11#16840136—21no—
12#16840112—21no—
13#16839985—22no—
14#16839973—21no—
15#16839964—22no—
16#16839963—21no—
17#16839901—22no—
18#16839815C2211yescharacteristic
19#16839764—21no—
20#16839700—22no—
21#16839669—22no—
22#16839667—21no—
23#16839627—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839497—22no—
26#16839494—22no—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C61×#333—22no—
31C2×#11758222—21no—
32C22×Dic122D411yescharacteristic
33#14519810—42no—
34#14519810—42no—
35C2×Dic244—22no—
36C2×#11758190—22no—
37C2×#11758201—24no—
38#14519777D411yescharacteristic
39#14519774D411yescharacteristic
40#14519765—21no—
41#14519751—21no—
42#14519751—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47C4×D244—21no—
48C4×D244—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61D411yescharacteristic
51C2×#11758196—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—21no—
54C4⋊#92418D412yes—
55#14519702—22no—
56#14519701—24no—
57#14519699—22no—
58C22⋊#92418—21no—
59C4×#92418—21no—
60#14519684—24no—
61#14519684—24no—
62#14519684D412yes—
63C22×Q8×C61D412yes—
64C2×M16×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74C4⋊C8×C61D412yes—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77C2×Q8×C61—22no—
78C2×Q8×C61—22no—
79C2×Q8×C61—44no—
80C22×C4×C61—22no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—22no—
87C22×Dic61—21no—
88#11758222—41no—
89#11758222—41no—
90#11758222—41no—
91#11758222—41no—
92C2×D244—21no—
93C2×D244—21no—
94C2×D244—21no—
95C2×C4×D61—41no—
96C2×Dic122—21no—
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122SD1612yescomplemented
103M16×C61—41no—
104M16×C61—41no—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C42×C61C2×D411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115#11758201—84no—
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—42no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129C2×#92418—22no—
130Dic244—44no—
131#11758190—44no—
132C23×C61#12011yescharacteristic
133C22×D61—41no—
134Q8×C61—44no—
135Q8×C61—82no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61—44no—
139C2×C4×C61—42no—
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12711yescharacteristic
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C22⋊D61—81no—
151C22⋊D61—81no—
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161D244—41no—
162D244—41no—
163D244—42no—
164D244—81no—
165C4×D61—81no—
166C4×D61—81no—
167Dic122—22no—
168Dic122—22no—
169Dic122—41no—
170Dic122—41no—
171Dic122—41no—
172Dic122—41no—
173C8×C61—42no—
174C8×C61—81no—
175#92418—42no—
176#10244—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#39211yescharacteristic
180C22×C61#44811yescharacteristic
181C22×C61#44911yescharacteristic
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—82no—
189C4×C61#45011yescharacteristic
190C4×C61#45211yescharacteristic
191C4×C61#45512yes—
192C4×C61#46511yescharacteristic
193C4×C61#46811yescharacteristic
194Dic61—42no—
195Dic61—82no—
196Dic61—81no—
197Dic61—81no—
198#3206—611no—
199#3037—611no—
200#2568—611no—
201#1548D6111yescharacteristic, complemented
202#1502—611no—
203#1479—611no—
204#1303—611no—
205C2×C61—21no—
206C2×C61—21no—
207C2×C61#200011yescharacteristic
208C2×C61#200411yescharacteristic
209C2×C61#200911yescharacteristic
210D61—161no—
211C2×#137—1221no—
212#532—611no—
213C2×C4⋊Q8C2×D6111yescharacteristic
214C2×C4×Q8—611no—
215#492—1221no—
216#487—1221no—
217#464—1222no—
218#462—1221no—
219#461—1222no—
220#460—1221no—
221#448—1222no—
222#431—611no—
223#431C2×D6111yescharacteristic
224#423C2×D6111yescharacteristic
225#417—1221no—
226#400—1222no—
227#391—1221no—
228#391—1222no—
229#382—1221no—
230C4×C4⋊C4—611no—
231#333—22no—
232#327—1222no—
233Q8⋊C8—1222no—
234C61#1024411yescharacteristic, complemented, Sylow
235C2×#41—1221no—
236C22×Q8—611no—
237C22×Q8D24412yes—
238#137—2442no—
239#137—2442no—
240C2×Q16—1222no—
241C2×SD16—1222no—
242C2×SD16—1224no—
243C2×M16—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—611no—
249#127—611no—
250#124—1221no—
251#121—1221no—
252#121—1221no—
253C4×Q8—1221no—
254C4×Q8—1221no—
255C4×Q8—1221no—
256C4×Q8—1221no—
257C4×D4—1221no—
258C4×D4—1221no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C42—1221no—
265C2×C42—1221no—
266C2×C42C22×D6111yescharacteristic
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—612no—
270C4⋊C8C22⋊D6112yes—
271#103—1222no—
272#103—1224no—
273#102—1222no—
274C22⋊C8—21no—
275C22⋊C8—1221no—
276C4×C8—21no—
277C4×C8—1221no—
278#95—612no—
279#95—1224no—
280#95—1224no—
281#41—2441no—
282#41—2441no—
283#41—2441no—
284#41—2441no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—44no—
288C2×Q8—612no—
289C2×Q8—1221no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×Q8—1221no—
293C2×Q8—1221no—
294C2×Q8—1221no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—2441no—
299C22×C4—22no—
300C22×C4—1221no—
301C22×C4—1222no—
302C22×C4—1221no—
303C22×C4—2441no—
304C22×C4C2×D24411yescharacteristic
305C22×C4D4×D6111yescharacteristic
306C22×C4D4×D6111yescharacteristic
307Q16—2444no—
308SD16—4884no—
309SD16—2444no—
310M16—41no—
311M16—41no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—1222no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—2442no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C22⋊C4—2441no—
327C22⋊C4—2441no—
328C42—2441no—
329C42—2441no—
330C42—2441no—
331C42—2441no—
332C42#1175822211yescharacteristic
333C42D4×D6111yescharacteristic
334C42D4×D6111yescharacteristic
335C42C2×C22⋊D6111yescharacteristic
336C23—2441no—
337C23#1451974711yescharacteristic
338Q8—44no—
339Q8—82no—
340Q8—1222no—
341Q8—1222no—
342Q8—2441no—
343Q8—2441no—
344Q8—2441no—
345Q8—2441no—
346D4—4881no—
347D4—4881no—
348D4—4881no—
349D4—2442no—
350D4—2441no—
351D4—2441no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—1222no—
357C2×C4—4881no—
358C2×C4—4881no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4#1451974511yescharacteristic
368C2×C4#1451974711yescharacteristic
369C2×C4#1451975011yescharacteristic
370C2×C4#1451975011yescharacteristic
371C2×C4#1451975111yescharacteristic
372C2×C4#1451977311yescharacteristic
373C2×C4SD16×D6111yescharacteristic
374C2×C4#1451980011yescharacteristic
375C2×C4#1451980411yescharacteristic
376C2×C4#1451980911yescharacteristic
377C8—42no—
378C8—81no—
379C8—2442no—
380C22—21no—
381C22—21no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22#1683967111yescharacteristic
386C22#1683990211yescharacteristic
387C22#1683991011yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—4882no—
393C4—4881no—
394C4—4881no—
395C4—2442no—
396C4#1683991111yescharacteristic
397C4#1683992311yescharacteristic
398C4#1683994212yes—
399C4#1683999411yescharacteristic
400C4#1684001111yescharacteristic
401C2—21no—
402C2—21no—
403C2—9761no—
404C2#2144682111yescharacteristic
405C2#2144685011yescharacteristic
406C2#2144688111yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1548C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#1548D61
hypocenterC61#10244
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. #21451035C2semidirect
5. #21444215C2semidirect
6. #21443720C2semidirect
7. C61×#1548C2semidirect
102. C2×Dic122SD16semidirect
201. #1548D61semidirect
234. C61#10244semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214552932
#214510352
#214443152
#214442152
#214437202
C61×#15482
#1024461