Finite Group Databasebeta

← #29582388

Subgroups — #29582388

Order 15616.

658 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456746C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic
3#21450772C211yesmaximal, characteristic, complemented
4#21444499C211yesmaximal, characteristic, complemented
5#21444315C211yesmaximal, characteristic, complemented
6#21443721C211yesmaximal, characteristic, complemented
7C61×#1478C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9#16840312C2211yescharacteristic
10C2×#14519777C2211yescharacteristic
11C2×#14519756—22no—
12#16840135—21no—
13#16840113—21no—
14#16839977—22no—
15#16839968—22no—
16#16839967—21no—
17#16839957—21no—
18#16839903—22no—
19#16839786C2211yescharacteristic
20#16839761—21no—
21#16839698—22no—
22#16839665—21no—
23#16839631—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839510—22no—
26#16839497—22no—
27C2×C4×D4×C61C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#431C2211yescharacteristic
30C61×D4⋊C8—22no—
31C2×#11758222—21no—
32C22×Dic122D412yes—
33#14519803—42no—
34#14519803—42no—
35C2×#11758190—24no—
36C2×#11758200—22no—
37C2×#11758199—22no—
38C2×#11758194—21no—
39#14519777—21no—
40#14519777—21no—
41#14519777—21no—
42#14519777—21no—
43#14519777D411yescharacteristic
44#14519774D411yescharacteristic
45#14519765—21no—
46#14519756—48no—
47#14519751—21no—
48#14519751—21no—
49C4×D244—21no—
50C4×D244—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×#11758196—24no—
55C2×C22⋊Dic61—24no—
56C2×C4×Dic61—21no—
57C4⋊#92418—21no—
58C4⋊#92418D412yes—
59#14519702—24no—
60#14519700—22no—
61#14519697—22no—
62#14519684D412yes—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—21no—
73C4⋊C8×C61D412yes—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77C4×C8×C61—21no—
78C24×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61SD1612yescomplemented
85C22×C4×C61—21no—
86C22×C4×C61—44no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—41no—
93C22×Dic61—22no—
94C22×Dic61—22no—
95#11758222—41no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—22no—
104C2×Dic122—21no—
105C2×Dic122—22no—
106C2×Dic122—44no—
107C2×C8×C61—22no—
108C2×C8×C61—41no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61#4111yescharacteristic
117C22⋊Dic61—48no—
118#11758200—44no—
119#11758199—44no—
120#11758198—41no—
121#11758198—41no—
122C4⋊Dic61—24no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129#11758196—48no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
135#11758190—84no—
136C23×C61—22no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61#12011yescharacteristic
142C22×D61—41no—
143D4×C61—22no—
144D4×C61—22no—
145D4×C61—41no—
146D4×C61—41no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61—82no—
152C2×C4×C61—82no—
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12711yescharacteristic
159C2×C4×C61C2×SD1611yescharacteristic
160C2×C4×C61#13511yescharacteristic
161C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C61#13711yescharacteristic
163C22⋊D61—81no—
164C22⋊D61—81no—
165C2×Dic61—24no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—44no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171D244—41no—
172D244—41no—
173D244—42no—
174D244—81no—
175C4×D61—81no—
176C4×D61—81no—
177Dic122—41no—
178Dic122—41no—
179Dic122—44no—
180Dic122—82no—
181C8×C61—42no—
182C8×C61—81no—
183#92418—42no—
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—81no—
202C4×C61#45011yescharacteristic
203C4×C61#45211yescharacteristic
204C4×C61#45312yes—
205C4×C61#46511yescharacteristic
206C4×C61#46811yescharacteristic
207Dic61—44no—
208Dic61—81no—
209Dic61—82no—
210#3206—611no—
211#3036—611no—
212#2542—611no—
213#1548—611no—
214#1502—611no—
215#1478D6111yescharacteristic, complemented
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⋊Q8—611no—
227C2×C22⋊Q8—1222no—
228C2×C4×D4C2×D6111yescharacteristic
229#492—1221no—
230#487—1221no—
231#463—1222no—
232#461—1221no—
233#461—1222no—
234#459—1221no—
235#448—1222no—
236#431C2×D6111yescharacteristic
237#431C2×D6111yescharacteristic
238#423—611no—
239#417—1221no—
240#399—1222no—
241#390—1221no—
242#384—1221no—
243C4×C4⋊C4—611no—
244#333—1222no—
245#327—1222no—
246D4⋊C8—22no—
247C61#1024011yescharacteristic, complemented, Sylow
248C2×#41—1221no—
249C22×Q8—612no—
250C22×D4D24411yescharacteristic
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—611no—
259C4⋊Q8—1221no—
260C4⋊Q8—1221no—
261C4⋊Q8—1221no—
262C4⋊Q8—1221no—
263#127—611no—
264#124—1221no—
265C22⋊Q8—2448no—
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—1221no—
275C2×C4⋊C4—1224no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4D24411yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—1224no—
281C2×C42—1221no—
282C2×C42C22×D6111yescharacteristic
283C4⋊C8—21no—
284C4⋊C8—612no—
285C4⋊C8—1221no—
286C4⋊C8C22⋊D6112yes—
287#103—1222no—
288#103—1224no—
289#102—1222no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294#95—612no—
295C24—21no—
296#41—2441no—
297#41—2441no—
298#41—2441no—
299#41—2441no—
300C2×Q8—1222no—
301C2×Q8—1221no—
302C2×Q8—1222no—
303C2×Q8—2444no—
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#1175819012yescomplemented
314C22×C4—21no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—2441no—
321C22×C4C2×D24411yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4D4×D6111yescharacteristic
324SD16—4884no—
325SD16—2444no—
326D8—2444no—
327M16—2441no—
328M16—2441no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—1222no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—1224no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—2448no—
338C4⋊C4—2441no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—2448no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—2441no—
348C42—2441no—
349C42#1175822211yescharacteristic
350C42D4×D6111yescharacteristic
351C42D4×D6111yescharacteristic
352C42C2×C22⋊D6111yescharacteristic
353C23—22no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—2441no—
359C23#1451974711yescharacteristic
360Q8—4882no—
361Q8—2441no—
362Q8—2441no—
363Q8—2444no—
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—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—1224no—
381C2×C4—4881no—
382C2×C4—4881no—
383C2×C4—2444no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974711yescharacteristic
390C2×C4#1451975011yescharacteristic
391C2×C4#1451975011yescharacteristic
392C2×C4#1451975111yescharacteristic
393C2×C4#1451977311yescharacteristic
394C2×C4C2×#1175819011yescharacteristic
395C2×C4#1451979911yescharacteristic
396C2×C4#1451980511yescharacteristic
397C2×C4#1451981011yescharacteristic
398C8—42no—
399C8—81no—
400C8—2442no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—42no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22#1683967111yescharacteristic
412C22#1683990311yescharacteristic
413C22#1683990911yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—4881no—
419C4—4882no—
420C4—2444no—
421C4#1683991511yescharacteristic
422C4#1683992611yescharacteristic
423C4#1683992812yes—
424C4#1683999111yescharacteristic
425C4#1684000811yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—9761no—
430C2#2144682411yescharacteristic
431C2#2144683511yescharacteristic
432C2#2144688011yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1478C2
socleC22×C61#392
prime radical#21455283C2
hypercenter#1478D61
hypocenterC61#10240
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #21456746C2semidirect
3. #21450772C2semidirect
4. #21444499C2semidirect
5. #21444315C2semidirect
6. #21443721C2semidirect
7. C61×#1478C2semidirect
84. C2×D4×C61SD16semidirect
215. #1478D61semidirect
247. C61#10240semidirect
313. C2×D4#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567462
#214552832
#214507722
#214444992
#214443152
#214437212
C61×#14782
#1024061