Finite Group Databasebeta

← #29580472

Subgroups — #29580472

Order 15616.

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

normal subgroups only

This group has 692 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#21456567C211yesmaximal, characteristic
2#21450286C211yesmaximal, characteristic, complemented
3#21446656C211yesmaximal, characteristic, complemented
4#21445905C211yesmaximal, characteristic, complemented
5#21444388C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C2×C61×#480C211yesmaximal, characteristic, complemented
8C2×#14519760—22no—
9C2×#14519751C2211yescharacteristic
10#16840075—21no—
11#16840073—21no—
12#16839996—22no—
13#16839896—21no—
14#16839895—21no—
15C2×#14519692—22no—
16C2×C22⋊#92418C2211yescharacteristic
17#16839697—22no—
18#16839663—21no—
19#16839631—21no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22#16839524—22no—
23#16839497—22no—
24C2×#121×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#480—22no—
27C61×#480—22no—
28C2×#107×C61—21no—
29C2×#102×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758199—22no—
34C22×#92418D412yes—
35#14519760—44no—
36#14519760—44no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—24no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61D412yes—
46C2×#11758198—24no—
47C2×#11758198—21no—
48C2×C4×Dic61—21no—
49#14519710—22no—
50#14519698—22no—
51#14519697—22no—
52#14519692—44no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519684—21no—
56#14519684—21no—
57C22×D4×C61—21no—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60C23×C4×C61C2311yescharacteristic
61C22×C8×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D412yes—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—21no—
74#107×C61—24no—
75#102×C61—24no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61D412yes—
78C24×C61—21no—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—42no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—42no—
87C2×D4×C61—42no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—42no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—42no—
104C2×C22⋊D61—42no—
105C22×Dic61—22no—
106C22×Dic61—21no—
107C22×Dic61—41no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—42no—
111C2×C8×C61—22no—
112C2×C8×C61—42no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊Dic61—22no—
122C22⋊Dic61—42no—
123C22⋊Dic61D814yescomplemented
124#11758199—82no—
125#11758198—48no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—42no—
131#11758196—48no—
132C4×Dic61—42no—
133C2×#92418—22no—
134C2×#92418—42no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—42no—
139C23×C61—81no—
140C23×C61—81no—
141C23×C61#12011yescharacteristic
142C23×C61#12111yescharacteristic
143C23×C61#12711yescharacteristic
144C23×C61C2×D812yes—
145C23×C61AGL(1,ℤ/8)12yes—
146C22×D61—42no—
147C22×D61—42no—
148C22×D61—41no—
149C22×D61—42no—
150D4×C61—42no—
151D4×C61—42no—
152D4×C61—82no—
153D4×C61—81no—
154D4×C61—81no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—44no—
164C2×C4×C61—42no—
165C2×C4×C61—82no—
166C2×C4×C61—81no—
167C2×C4×C61—81no—
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12011yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C22⋊D61—82no—
173C22⋊D61—81no—
174C22⋊D61—81no—
175C2×Dic61—24no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—42no—
180C2×Dic61—81no—
181C2×Dic61—81no—
182C2×Dic61—81no—
183D244—42no—
184D244—81no—
185C8×C61—42no—
186#92418—42no—
187#9952—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—162no—
200C22×C61#39211yescharacteristic
201C22×C61#44712yes—
202C22×C61#45912yes—
203C22×C61#49311yescharacteristic
204C22×C61#49611yescharacteristic
205C2×D61—84no—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—82no—
213C4×C61—81no—
214C4×C61#45812yes—
215Dic61—44no—
216Dic61—81no—
217Dic61—161no—
218#3186—611no—
219C2×#480D6111yescharacteristic, complemented
220#2501—611no—
221#1964—611no—
222#1823—611no—
223#1519—611no—
224#1282—611no—
225C2×C61—22no—
226C2×C61—21no—
227C2×C61—22no—
228C2×C61—21no—
229C2×C61—161no—
230C2×C61#200911yescharacteristic
231C2×C61#201111yescharacteristic
232C2×C61#202211yescharacteristic
233D61—161no—
234C2×#123—1222no—
235C2×#121—611no—
236C2×#121C2×D6111yescharacteristic
237C22×C4⋊C4C2×D6111yescharacteristic
238#480—22no—
239#480—22no—
240#480—1221no—
241#480—1221no—
242#466—1222no—
243#447—1221no—
244#447—1221no—
245C2×#107—21no—
246C2×#102—21no—
247C2×#99—1222no—
248C2×C22⋊C8—611no—
249C2×C22⋊C8C2×D6111yescharacteristic
250#399—1222no—
251#390—1221no—
252#384—1221no—
253#380—611no—
254C4×C22⋊C4—611no—
255#340—1222no—
256#327—1222no—
257C61#995211yescharacteristic, complemented, Sylow
258C22×D4—21no—
259C22×D4—611no—
260C22×D4—1221no—
261C22×D4C22⋊D6111yescharacteristic
262C23×C4—21no—
263C23×C4C22×D6111yescharacteristic
264C2×D8—1222no—
265C22×C8—21no—
266C22×C8—612no—
267#123—2444no—
268#123—2444no—
269#121—21no—
270#121—21no—
271#121—21no—
272#121—21no—
273#121—1221no—
274#121—1221no—
275#121—1221no—
276#121—1221no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—611no—
283C2×C4⋊C4—1224no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4C22⋊D6112yes—
286C2×C4⋊C4C22⋊D6111yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—612no—
289C2×C22⋊C4—1224no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C42—1221no—
293#107—24no—
294#107—1222no—
295#102—24no—
296#102—1222no—
297#102—1222no—
298#99—2444no—
299C22⋊C8—21no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302C22⋊C8C22⋊D6112yes—
303#95—1221no—
304#95—1221no—
305C24—21no—
306C24—1221no—
307C24D4×D6111yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—1221no—
316C2×D4—1221no—
317C2×D4—2442no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2442no—
321C2×D4—2442no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—1222no—
330C22×C4—1221no—
331C22×C4—2441no—
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334C22×C4#1175822411yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337C22×C4C2×C22⋊D6111yescharacteristic
338D8—4882no—
339C2×C8—22no—
340C2×C8—42no—
341C2×C8—1222no—
342C2×C8—2442no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—2442no—
352C4⋊C4—2448no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—614no—
356C22⋊C4—1222no—
357C22⋊C4—2448no—
358C22⋊C4—2442no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C42—2442no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—81no—
367C23—81no—
368C23—2442no—
369C23—2442no—
370C23—2442no—
371C23—2441no—
372C23#1451974511yescharacteristic
373C23#1451974911yescharacteristic
374C23#1451977311yescharacteristic
375C23D8×D6112yes—
376C23#1451980112yes—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—82no—
381D4—81no—
382D4—4881no—
383D4—4882no—
384D4—4881no—
385D4—4881no—
386D4—2442no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—1224no—
401C2×C4—4881no—
402C2×C4—4881no—
403C2×C4—4881no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4#1451974511yescharacteristic
409C2×C4#1451974611yescharacteristic
410C2×C4#1451974811yescharacteristic
411C2×C4#1451974911yescharacteristic
412C8—42no—
413C8—2442no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—162no—
426C22—4884no—
427C22—4881no—
428C22—4881no—
429C22—4881no—
430C22#1683967011yescharacteristic
431C22#1683989412yes—
432C22#1683995512yes—
433C22#1684013711yescharacteristic
434C22#1684015011yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—82no—
439C4—81no—
440C4—9761no—
441C4—4881no—
442C4—2444no—
443C4#1683995112yes—
444C2—21no—
445C2—22no—
446C2—22no—
447C2—21no—
448C2—161no—
449C2—9761no—
450C2#2144687711yescharacteristic
451C2#2144688811yescharacteristic
452C2#2144694711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#480C2
socleC22×C61#392
prime radical#21456567C2
hypercenterC2×#480D61
hypocenterC61#9952
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. #21450286C2semidirect
3. #21446656C2semidirect
4. #21445905C2semidirect
5. #21444388C2semidirect
6. #21443665C2semidirect
7. C2×C61×#480C2semidirect
123. C22⋊Dic61D8semidirect
219. C2×#480D61semidirect
257. C61#9952semidirect

Maximal subgroups

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

subgroupindex
#214565672
#214502862
#214466562
#214459052
#214443882
#214436652
C2×C61×#4802
#995261