Finite Group Databasebeta

← #30603973

Subgroups — #30603973

Order 15616.

756 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 100 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#21461059C211yesmaximal, characteristic, complemented
2#21461055C211yesmaximal, characteristic, complemented
3#21460763C212yesmaximal, complemented
4#21460743C212yesmaximal, complemented
5#120×Dic61C211yesmaximal, characteristic
6#16840778—21no—
7C2×C22⋊#92420—21no—
8#16840607—21no—
9#16840605—21no—
10#16840605C2212yes—
11#16840605C2212yes—
12C24×Dic61C411yescharacteristic
13C2×D4×Dic61—21no—
14C2×D4×Dic61C2211yescharacteristic
15#16839642—21no—
16#16839642C411yescharacteristic
17#16839637—21no—
18#16839637C2211yescharacteristic
19#16839618C2211yescharacteristic
20C22⋊C4×Dic61—21no—
21C22⋊C4×Dic61C411yescharacteristic
22C2×#120×C61C411yescharacteristic
23C22×#92420—41no—
24C4⋊#92420—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C22⋊#92420—41no—
28C4×#92420—41no—
29C23×Dic61—21no—
30C23×Dic61—21no—
31C23×Dic61—44no—
32C23×Dic61—41no—
33C23×Dic61—41no—
34C23×Dic61C2×C411yescharacteristic
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61D411yescharacteristic
37D4×Dic61—41no—
38D4×Dic61—41no—
39D4×Dic61—44no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61C2×C411yescharacteristic
46C2×C22⋊Dic61D411yescharacteristic
47C2×C22⋊Dic61C2311yescharacteristic
48C2×C4×Dic61—21no—
49C2×C4×Dic61D411yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C25×C61C2×C411yescharacteristic
54C22×D4×C61—21no—
55C22×D4×C61C2×C411yescharacteristic
56#120×C61—22no—
57#120×C61C814yescomplemented
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61C2×C411yescharacteristic
60C24×C61—21no—
61C24×C61—21no—
62C24×C61—42no—
63C24×C61—41no—
64C24×C61—42no—
65C24×C61—41no—
66C24×C61C22⋊C411yescharacteristic
67C24×C61C2×C812yescomplemented
68C24×C61C22×C411yescharacteristic
69C2×D4×C61—24no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—42no—
73C2×D4×C61—42no—
74C2×D4×C61C2×C812yes—
75C2×D4×C61M1612yes—
76C22×C4×C61—21no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×Dic61—24no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88C22×Dic61—42no—
89C22×Dic61—42no—
90C22×Dic61—42no—
91C22×Dic61—42no—
92C22×Dic61—44no—
93C22×Dic61—44no—
94C22×Dic61—44no—
95C22×Dic61—41no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98C22×Dic61—44no—
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C22⋊C411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C2×#92420—42no—
104C2×#92420—42no—
105C2×#92420—82no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61C2×C812yescomplemented
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111C22⋊Dic61—42no—
112C22⋊Dic61—42no—
113C22⋊Dic61—42no—
114C22⋊Dic61—41no—
115C4⋊Dic61—41no—
116C4⋊Dic61—42no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61—42no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—22no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—44no—
130C23×C61—42no—
131C23×C61—41no—
132C23×C61—44no—
133C23×C61—42no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—42no—
138C23×C61—48no—
139C23×C61—42no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—42no—
143C23×C61—42no—
144C23×C61C22⋊C812yes—
145C23×C61C2×C22⋊C411yescharacteristic
146C23×C61C22×C811yescharacteristic
147C23×C61C2×M1611yescharacteristic
148D4×C61—42no—
149D4×C61—42no—
150D4×C61—48no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61C22⋊C812yes—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—44no—
163C2×Dic61—42no—
164C2×Dic61—41no—
165C2×Dic61—48no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—44no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—81no—
179C2×Dic61#9911yescharacteristic
180C2×Dic61#9911yescharacteristic
181#92420—82no—
182#92420—82no—
183#11471—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—24no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—22no—
190C22×C61—42no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—44no—
198C22×C61—44no—
199C22×C61—44no—
200C22×C61—44no—
201C22×C61—42no—
202C22×C61—41no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—42no—
206C22×C61—41no—
207C22×C61—42no—
208C22×C61—81no—
209C22×C61C2×C22⋊C811yescharacteristic
210C22×C61C2×#9911yescharacteristic
211C22×C61#41311yescharacteristic
212C4×C61—22no—
213C4×C61—42no—
214Dic61—21no—
215Dic61—41no—
216Dic61—42no—
217Dic61—41no—
218Dic61—41no—
219Dic61—81no—
220Dic61AΣL(1,𝔽16)12yes—
221C4×#120—611no—
222#1460—611no—
223#1459—611no—
224#1326—612no—
225#1316—612no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—44no—
229C2×C61—41no—
230C2×C61—41no—
231C2×C61—42no—
232C2×C61#145911yescharacteristic
233C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
234C2×C61#212011yescharacteristic
235C24×C4—611no—
236C2×#120#692311yescharacteristic
237C2×C4×D4—611no—
238C2×C4×D4—1221no—
239#431—1221no—
240C2×C22⋊C8—1221no—
241#386—611no—
242#386—1221no—
243#385—611no—
244#385—1221no—
245#379—611no—
246C4×C22⋊C4—611no—
247C4×C22⋊C4—1221no—
248#324—1221no—
249#323—612no—
250#323—612no—
251#323—1221no—
252C61#1147111yescharacteristic, complemented, Sylow
253C25C2×#692311yescharacteristic
254C22×D4—21no—
255C22×D4C2×#692311yescharacteristic
256C23×C4—611no—
257C23×C4—611no—
258C23×C4—611no—
259C23×C4—1221no—
260C23×C4—1221no—
261C23×C4—2441no—
262C23×C4—2441no—
263C23×C4—2444no—
264C22×C8—2441no—
265#120—22no—
266#120#9242014yescomplemented
267C4×D4—2441no—
268C4×D4—2441no—
269C4×D4—2444no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—611no—
275C2×C22⋊C4—611no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4C2×#692311yescharacteristic
280C2×C42—611no—
281C2×C42—1221no—
282C4⋊C8—2441no—
283C22⋊C8—1222no—
284C22⋊C8—1222no—
285C22⋊C8—2441no—
286C4×C8—2441no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290C24—21no—
291C24—21no—
292C24—42no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24C2×#9242012yescomplemented
297C24C22⋊#692311yescharacteristic
298C24C22×#692311yescharacteristic
299C2×D4—24no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4C2×#9242012yes—
305C2×D4#1175821712yes—
306C22×C4—21no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1224no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4—2444no—
328C22×C4—2444no—
329C22×C4—2444no—
330C22×C4—2441no—
331C22×C4—2444no—
332C22×C4C22⋊#692311yescharacteristic
333C2×C8—4882no—
334C2×C8—2442no—
335C2×C8—2442no—
336C4⋊C4—2441no—
337C4⋊C4—2442no—
338C22⋊C4—21no—
339C22⋊C4—24no—
340C22⋊C4—2441no—
341C22⋊C4—2441no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C22⋊C4—2442no—
345C22⋊C4—2441no—
346C22⋊C4C2×#9242012yescomplemented
347C42—1221no—
348C42—1221no—
349C42—2442no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—44no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—44no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—48no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23C22⋊#9242012yes—
375C23C2×C22⋊#692311yescharacteristic
376C23C22×#9242011yescharacteristic
377C23C2×#1175821711yescharacteristic
378D4—42no—
379D4—42no—
380D4—48no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—1222no—
389C2×C4—1222no—
390C2×C4—1221no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—4881no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2444no—
397C2×C4—2442no—
398C2×C4—2444no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2448no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2442no—
410C2×C4C22⋊#9242012yes—
411C8—4882no—
412C8—4882no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—41no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—42no—
437C22—81no—
438C22C2×C22⋊#9242011yescharacteristic
439C22D61⋊#9911yescharacteristic
440C22#1684074911yescharacteristic
441C4—22no—
442C4—42no—
443C4—612no—
444C4—1221no—
445C4—4881no—
446C4—2441no—
447C4—2442no—
448C4—2441no—
449C4—2441no—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—41no—
454C2—41no—
455C2—42no—
456C2#2146105511yescharacteristic
457C2D61⋊AΣL(1,𝔽16)11yescharacteristic
458C2#2146207111yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
Fitting, prime radicalC2×#120×C61C4
socleC22×C61C2×#99
cosocleC2×C22⋊Dic61C23
hypercenterC2×#120#6923
hypocenterC61#11471
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #21461059C2semidirect
2. #21461055C2semidirect
3. #21460763C2semidirect
4. #21460743C2semidirect
57. #120×C61C8semidirect
67. C24×C61C2×C8semidirect
108. C22⋊C4×C61C2×C8semidirect
252. C61#11471semidirect
266. #120#92420semidirect
296. C24C2×#92420semidirect
346. C22⋊C4C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#214610592
#214610552
#214607632
#214607432
#120×Dic612
#1147161