Finite Group Databasebeta

← #29575960

Subgroups — #29575960

Order 15616.

548 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 111 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#21455283C211yesmaximal, characteristic
2#21448792C211yesmaximal, characteristic, complemented
3#21444380C211yesmaximal, characteristic, complemented
4#21444315C211yesmaximal, characteristic, complemented
5#21443721C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#1478C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9#16840312C2211yescharacteristic
10#16839977—22no—
11#16839967C412yescomplemented
12#16839957C412yescomplemented
13#16839786C2211yescharacteristic
14#16839761—21no—
15#16839665—21no—
16#16839655—22no—
17#16839628C2211yescharacteristic
18#16839625—21no—
19C4⋊C4×Dic61—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21C4×C22⋊Dic61—21no—
22#16839497—22no—
23#16839497—21no—
24#16839497—21no—
25C2×C4×D4×C61C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C61×D4⋊C8—22no—
29C2×#11758222—21no—
30#14519803—42no—
31#14519803—42no—
32C2×#11758200—22no—
33C2×#11758199—22no—
34C2×#11758194—21no—
35#14519777C2×C411yescharacteristic
36#14519774C2×C411yescharacteristic
37#14519765—21no—
38#14519751—21no—
39#14519751—21no—
40C4×D244—21no—
41C4×D244—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61—22no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—22no—
48C4⋊#92418—21no—
49C4⋊#92418C2×C412yes—
50#14519700—22no—
51#14519697—22no—
52#14519684—22no—
53#14519684—21no—
54#14519684D411yescharacteristic
55#14519684D411yescharacteristic
56#14519684D412yes—
57C22×D4×C61D411yescharacteristic
58C23×C4×C61—21no—
59C4×D4×C61—21no—
60C4×D4×C61—21no—
61C4×D4×C61C2×C412yes—
62C4×D4×C61D412yes—
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70C4×C8×C61—21no—
71C4×C8×C61—21no—
72C24×C61—21no—
73C2×D4×C61—21no—
74C2×D4×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—21no—
77C2×D4×C61—21no—
78C2×D4×C61C22⋊C412yes—
79C22×C4×C61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—82no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—41no—
87C22×Dic61—21no—
88C22×Dic61—22no—
89C22×Dic61—21no—
90#11758222—41no—
91#11758222—41no—
92#11758222—41no—
93#11758222—41no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×C4×D61—41no—
98C2×Dic122—21no—
99C2×C8×C61—41no—
100C2×C8×C61—41no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61C22⋊C412yes—
103C22⋊C4×C61—22no—
104C22⋊C4×C61—22no—
105C42×C61C22×C411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61#4111yescharacteristic
108C42×C61#4111yescharacteristic
109C22⋊Dic61—44no—
110C22⋊Dic61—82no—
111#11758200—44no—
112#11758199—44no—
113#11758198—41no—
114#11758198—41no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—42no—
118C4⋊Dic61—41no—
119C4⋊Dic61—41no—
120C4×Dic61—41no—
121C4×Dic61—41no—
122C4×Dic61—82no—
123#11758194—41no—
124#11758194—41no—
125C2×#92418—22no—
126C23×C61—22no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61#12011yescharacteristic
132C22×D61—41no—
133D4×C61—22no—
134D4×C61—22no—
135D4×C61—41no—
136D4×C61—41no—
137C2×C4×C61—22no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—82no—
142C2×C4×C61—82no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12311yescharacteristic
149C2×C4×C61#13511yescharacteristic
150C2×C4×C61#13511yescharacteristic
151C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C22⋊D61—81no—
154C22⋊D61—81no—
155C2×Dic61—22no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165D244—41no—
166D244—41no—
167D244—81no—
168D244—81no—
169C4×D61—81no—
170C4×D61—81no—
171Dic122—41no—
172Dic122—41no—
173C8×C61—81no—
174C8×C61—81no—
175#92418—42no—
176#9181—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—42no—
182C22×C61—82no—
183C22×C61—82no—
184C22×C61#38611yescharacteristic
185C22×C61#44911yescharacteristic
186C22×C61#44911yescharacteristic
187C2×D61—81no—
188C2×D61—81no—
189C2×D61—81no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—42no—
195C4×C61#41611yescharacteristic
196C4×C61#41811yescharacteristic
197C4×C61#48211yescharacteristic
198C4×C61#48211yescharacteristic
199Dic61—42no—
200Dic61—81no—
201Dic61—82no—
202Dic61—81no—
203#3036—611no—
204#2357—611no—
205#1517—611no—
206#1502—611no—
207#1478D6111yescharacteristic, complemented
208#1303—611no—
209#1286—611no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—82no—
213C2×C61#187511yescharacteristic
214C2×C61#190411yescharacteristic
215C2×C61#190611yescharacteristic
216D61—161no—
217C2×AGL(1,ℤ/8)—1221no—
218#532—611no—
219C2×C4×D4C2×D6111yescharacteristic
220#463—1222no—
221#461—612no—
222#459—612no—
223#431C2×D6111yescharacteristic
224#431C2×D6111yescharacteristic
225#423—611no—
226#417—1221no—
227#390—1221no—
228#388—1222no—
229#382—611no—
230#381—1221no—
231C4×C4⋊C4—611no—
232C4×C4⋊C4—1221no—
233C4×C22⋊C4—1221no—
234#327—1221no—
235#327—1221no—
236#327—1222no—
237D4⋊C8—22no—
238C61#918111yescharacteristic, complemented, Sylow
239C2×#41—1221no—
240C22×D4D24411yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—2442no—
243AGL(1,ℤ/8)—2442no—
244C2×SD16—1222no—
245C2×D8—1222no—
246C2×M16—1221no—
247C4⋊Q8—611no—
248#127—611no—
249#124—1221no—
250#121—1221no—
251#121—1221no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—1221no—
255C4×D4—1221no—
256C4×D4C4×D6112yes—
257C4×D4C22⋊D6112yes—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4D24411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—1222no—
263C2×C22⋊C4—1222no—
264C2×C42—1222no—
265C2×C42—1221no—
266C2×C42C22×D6111yescharacteristic
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—612no—
270C4⋊C8—1221no—
271#103—1222no—
272#102—1222no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277#95—611no—
278#95—611no—
279#95—612no—
280#95—1222no—
281#95—1221no—
282C24—21no—
283#41—2441no—
284#41—2441no—
285#41—2441no—
286#41—2441no—
287C2×Q8—1221no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—1221no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—2441no—
297C2×D4#1175819812yes—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—82no—
302C22×C4—1221no—
303C22×C4—1222no—
304C22×C4—1221no—
305C22×C4—2441no—
306C22×C4C2×D24411yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309SD16—2444no—
310D8—2444no—
311M16—2441no—
312M16—2441no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—1222no—
316C4⋊C4—21no—
317C4⋊C4—1221no—
318C4⋊C4—1221no—
319C4⋊C4—2442no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4#1175819812yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—4882no—
326C22⋊C4—2444no—
327C22⋊C4—2441no—
328C22⋊C4—2441no—
329C42—4882no—
330C42—2441no—
331C42—2441no—
332C42C2×C4×D6111yescharacteristic
333C42#1175822411yescharacteristic
334C42#1175822411yescharacteristic
335C42C2×C22⋊D6111yescharacteristic
336C23—22no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—2441no—
342C23#1451974711yescharacteristic
343Q8—2441no—
344Q8—2441no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—4881no—
350D4—4881no—
351D4—4881no—
352D4—4881no—
353D4—2441no—
354D4—2441no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—1222no—
362C2×C4—4881no—
363C2×C4—4881no—
364C2×C4—2442no—
365C2×C4—2441no—
366C2×C4—2442no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2442no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4C2×#1175819811yescharacteristic
374C2×C4#1451973711yescharacteristic
375C2×C4#1451973711yescharacteristic
376C2×C4#1451974911yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4#1451976111yescharacteristic
379C2×C4#1451979911yescharacteristic
380C2×C4#1451979911yescharacteristic
381C2×C4#1451980511yescharacteristic
382C2×C4#1451980511yescharacteristic
383C8—81no—
384C8—81no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—4881no—
394C22—4881no—
395C22—4881no—
396C22#1683963911yescharacteristic
397C22#1683990911yescharacteristic
398C22#1683990911yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—4882no—
405C4—4881no—
406C4—4881no—
407C4—2442no—
408C4#1683975911yescharacteristic
409C4#1683976811yescharacteristic
410C4#1684008711yescharacteristic
411C4#1684008711yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—9761no—
416C2#2144613911yescharacteristic
417C2#2144630411yescharacteristic
418C2#2144631811yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1478C2
socleC22×C61#386
prime radical#21455283C2
cosocleC2×C42×C61C23
hypercenter#1478D61
hypocenterC61#9181
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #21448792C2semidirect
3. #21444380C2semidirect
4. #21444315C2semidirect
5. #21443721C2semidirect
6. #21443678C2semidirect
7. C61×#1478C2semidirect
11. #16839967C4semidirect
12. #16839957C4semidirect
207. #1478D61semidirect
238. C61#9181semidirect

Maximal subgroups

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

subgroupindex
#214552832
#214487922
#214443802
#214443152
#214437212
#214436782
C61×#14782
#918161