Finite Group Databasebeta

← #29582387

Subgroups — #29582387

Order 15616.

658 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 105 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, complemented
2#21455283C211yesmaximal, characteristic
3#21444500C211yesmaximal, characteristic, complemented
4#21444313C211yesmaximal, characteristic, complemented
5#21444210C211yesmaximal, characteristic, complemented
6#21443722C211yesmaximal, characteristic, complemented
7C61×#2542C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9#16840312C2211yescharacteristic
10#16840136—21no—
11#16840112—21no—
12#16839977—22no—
13#16839967—21no—
14#16839964—22no—
15#16839957—21no—
16#16839901—22no—
17#16839815C2211yescharacteristic
18#16839815C2211yescharacteristic
19#16839786C2211yescharacteristic
20#16839761—21no—
21#16839511—22no—
22#16839495—22no—
23D4⋊#92418—22no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×C22⋊Q8×C61—22no—
26C2×C4×D4×C61C2211yescharacteristic
27C61×#399—22no—
28C61×#390—21no—
29C61×#384—21no—
30C4×C4⋊C4×C61C2211yescharacteristic
31C2×#11758222—21no—
32#14519803—42no—
33#14519803—42no—
34C2×#11758200—22no—
35C2×#11758201—24no—
36C2×#11758199—22no—
37C2×#11758194—21no—
38#14519777D411yescharacteristic
39#14519774D411yescharacteristic
40#14519765—21no—
41#14519751—21no—
42#14519751—21no—
43C4×D244—21no—
44C4×D244—21no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418D412yes—
48C4⋊#92418D412yes—
49#14519701—24no—
50#14519700—22no—
51#14519697—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C4×#92418—21no—
55C4×#92418—21no—
56C22×Q8×C61D412yes—
57C22×D4×C61D411yescharacteristic
58C23×C4×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C22⋊Q8×C61—48no—
64C4×D4×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—24no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—21no—
75C2×C42×C61—21no—
76C2×C42×C61C2311yescharacteristic
77#95×C61D412yes—
78C24×C61—21no—
79C2×Q8×C61—22no—
80C2×Q8×C61—22no—
81C2×Q8×C61—44no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61SD1612yescomplemented
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—44no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—41no—
98#11758222—41no—
99#11758222—41no—
100#11758222—41no—
101#11758222—41no—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×C4×D61—41no—
106C2×Dic122—21no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—48no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—48no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61—41no—
119C42×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124#11758201—84no—
125#11758200—44no—
126#11758199—44no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131#11758194—41no—
132#11758194—41no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—41no—
136C23×C61—22no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61#12011yescharacteristic
142C22×D61—41no—
143Q8×C61—44no—
144Q8×C61—82no—
145D4×C61—22no—
146D4×C61—22no—
147D4×C61—41no—
148D4×C61—41no—
149C2×C4×C61—22no—
150C2×C4×C61—24no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—44no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61—82no—
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12711yescharacteristic
166C2×C4×C61C2×SD1611yescharacteristic
167C2×C4×C61#13511yescharacteristic
168C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C61#13711yescharacteristic
170C22⋊D61—81no—
171C22⋊D61—81no—
172C2×Dic61—41no—
173D244—41no—
174D244—41no—
175D244—42no—
176D244—81no—
177C4×D61—81no—
178C4×D61—81no—
179Dic122—41no—
180Dic122—41no—
181#92418—42no—
182#92418—42no—
183#92418—81no—
184#10240—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—44no—
188C22×C61—42no—
189C22×C61—82no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#39211yescharacteristic
193C22×C61#44811yescharacteristic
194C22×C61#44911yescharacteristic
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—44no—
202C4×C61—81no—
203C4×C61—82no—
204C4×C61#45011yescharacteristic
205C4×C61#45211yescharacteristic
206C4×C61#45312yes—
207C4×C61#46511yescharacteristic
208C4×C61#46811yescharacteristic
209Dic61—81no—
210#3206—611no—
211#3036—611no—
212#2542D6111yescharacteristic, complemented
213#1548—611no—
214#1502—611no—
215#1478—611no—
216#1303—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—82no—
220C2×C61#200011yescharacteristic
221C2×C61#200211yescharacteristic
222C2×C61#200911yescharacteristic
223D61—161no—
224C2×AGL(1,ℤ/8)—1221no—
225#532—611no—
226C2×C4⋊Q8C2×D6111yescharacteristic
227C2×C22⋊Q8—22no—
228C2×C4×D4C2×D6111yescharacteristic
229#492—1221no—
230#487—1221no—
231#463—1222no—
232#461—1222no—
233#461—1221no—
234#459—1221no—
235#448—1222no—
236#431—611no—
237#431—611no—
238#423—611no—
239#417—1221no—
240#399—22no—
241#390—21no—
242#384—21no—
243C4×C4⋊C4C2×D6111yescharacteristic
244#333—1222no—
245#327—1222no—
246D4⋊C8—1222no—
247C61#1024011yescharacteristic, complemented, Sylow
248C2×#41—1221no—
249C22×Q8C22⋊D6112yes—
250C22×D4C22⋊D6111yescharacteristic
251C23×C4—21no—
252AGL(1,ℤ/8)—2442no—
253AGL(1,ℤ/8)—2442no—
254C2×SD16—1224no—
255C2×SD16—1222no—
256C2×D8—1222no—
257C2×M16—1221no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—611no—
263#127—611no—
264#124—1221no—
265C22⋊Q8—48no—
266#121—1221no—
267#121—1221no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—1221no—
273C4×D4—1221no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C22⋊D6111yescharacteristic
279C2×C22⋊C4—24no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42C22×D6111yescharacteristic
283C4⋊C8—612no—
284C4⋊C8—612no—
285C4⋊C8—1221no—
286C4⋊C8—1221no—
287#103—1222no—
288#103—1224no—
289#102—1222no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C4×C8—1221no—
293C4×C8—1221no—
294#95C22⋊D6112yes—
295C24—21no—
296#41—2441no—
297#41—2441no—
298#41—2441no—
299#41—2441no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8—1221no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—2441no—
313C2×D4#1175820112yescomplemented
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—2441no—
321C22×C4C2×C22⋊D6111yescharacteristic
322C22×C4C2×C22⋊D6111yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324SD16—4884no—
325SD16—2444no—
326D8—2444no—
327M16—2441no—
328M16—2441no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—2441no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—48no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C22⋊C4—48no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—41no—
348C42—41no—
349C42D4×D6111yescharacteristic
350C42D4×D6111yescharacteristic
351C42D4×D6111yescharacteristic
352C42#1175822411yescharacteristic
353C23—22no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—2441no—
359C23#1451974611yescharacteristic
360Q8—44no—
361Q8—82no—
362Q8—2441no—
363Q8—2441no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368D4—4881no—
369D4—4881no—
370D4—4881no—
371D4—2441no—
372D4—2441no—
373D4—2442no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—2441no—
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974511yescharacteristic
390C2×C4#1451974811yescharacteristic
391C2×C4#1451974911yescharacteristic
392C2×C4#1451974911yescharacteristic
393C2×C4#1451977311yescharacteristic
394C2×C4C2×#1175820111yescharacteristic
395C2×C4#1451979711yescharacteristic
396C2×C4#1451980211yescharacteristic
397C2×C4#1451980811yescharacteristic
398C8—4881no—
399C8—2442no—
400C8—2442no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—42no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22#1683967011yescharacteristic
412C22#1683990111yescharacteristic
413C22#1683990711yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—81no—
419C4—82no—
420C4—4881no—
421C4#1683991211yescharacteristic
422C4#1683992211yescharacteristic
423C4#1683992912yes—
424C4#1683999311yescharacteristic
425C4#1684001011yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—9761no—
430C2#2144682011yescharacteristic
431C2#2144683411yescharacteristic
432C2#2144687911yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2542C2
socleC22×C61#392
prime radical#21455283C2
hypercenter#2542D61
hypocenterC61#10240
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. #21456744C2semidirect
3. #21444500C2semidirect
4. #21444313C2semidirect
5. #21444210C2semidirect
6. #21443722C2semidirect
7. C61×#2542C2semidirect
87. C2×D4×C61SD16semidirect
212. #2542D61semidirect
247. C61#10240semidirect
313. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214552832
#214445002
#214443132
#214442102
#214437222
C61×#25422
#1024061