Finite Group Databasebeta

← #30600293

Subgroups — #30600293

Order 15616.

674 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 124 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#21462666C211yesmaximal, characteristic, complemented
2#21461055C211yesmaximal, characteristic, complemented
3C2×#16840605C212yesmaximal, complemented
4#21460662C212yesmaximal, complemented
5#21448480C211yesmaximal, characteristic
6D4×#92420—24no—
7#16840778—21no—
8C2×C22⋊#92420—21no—
9C2×C22⋊#92420C2212yes—
10C2×C22⋊#92420C2212yes—
11#16840607—21no—
12#16840605—21no—
13#16840605—24no—
14C2×D4×Dic61C2211yescharacteristic
15C22×C22⋊Dic61C2211yescharacteristic
16#16839665—21no—
17#16839655—21no—
18#16839655—21no—
19#16839642—21no—
20#16839642C411yescharacteristic
21#16839637C411yescharacteristic
22#16839618C411yescharacteristic
23C22⋊C4×Dic61C2211yescharacteristic
24C2×#120×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C22×#92420—22no—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33C22⋊#92420D414yescomplemented
34C4×#92420—22no—
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61D411yescharacteristic
37C23×Dic61C2311yescharacteristic
38D4×Dic61—22no—
39D4×Dic61—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61C2×C411yescharacteristic
50C2×C22⋊Dic61C2×C411yescharacteristic
51C2×C22⋊Dic61D411yescharacteristic
52C2×C4×Dic61—21no—
53C2×C4×Dic61D411yescharacteristic
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C25×C61C2×C411yescharacteristic
58C22×D4×C61—21no—
59C22×D4×C61C2×C411yescharacteristic
60#120×C61—22no—
61#120×C61C814yescomplemented
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61C2×C411yescharacteristic
64C24×C61—21no—
65C24×C61—22no—
66C24×C61—21no—
67C24×C61—22no—
68C24×C61—81no—
69C24×C61—81no—
70C24×C61C22⋊C411yescharacteristic
71C24×C61C2×C812yescomplemented
72C24×C61C22×C411yescharacteristic
73C2×D4×C61—22no—
74C2×D4×C61—22no—
75C2×D4×C61—24no—
76C2×D4×C61—81no—
77C2×D4×C61—81no—
78C2×D4×C61C2×C812yes—
79C2×D4×C61M1612yes—
80C22×C4×C61—21no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—22no—
87C22×Dic61—81no—
88C22×Dic61—81no—
89C22×Dic61—81no—
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C2×D412yes—
93C22×Dic61C2×D411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95C22×Dic61#4112yes—
96C2×#92420—24no—
97C2×#92420—42no—
98C2×#92420—82no—
99C2×#92420—82no—
100C2×#92420—82no—
101C22⋊C4×C61—21no—
102C22⋊C4×C61—24no—
103C22⋊C4×C61C2×C812yescomplemented
104C22⋊Dic61—22no—
105C22⋊Dic61—22no—
106C22⋊Dic61—22no—
107C22⋊Dic61—41no—
108C22⋊Dic61—42no—
109C22⋊Dic61—41no—
110C22⋊Dic61—81no—
111C22⋊Dic61—81no—
112C22⋊Dic61—81no—
113C4⋊Dic61—22no—
114C4⋊Dic61—81no—
115C4×Dic61—21no—
116C4×Dic61—21no—
117C4×Dic61—81no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—22no—
121C23×C61—24no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—42no—
133C23×C61—82no—
134C23×C61—82no—
135C23×C61—82no—
136C23×C61—82no—
137C23×C61—84no—
138C23×C61C22⋊C812yes—
139C23×C61C2×C22⋊C411yescharacteristic
140C23×C61C4×D412yes—
141C23×C61C22×C811yescharacteristic
142C23×C61C2×M1611yescharacteristic
143C23×C61#13112yes—
144D4×C61—24no—
145D4×C61—24no—
146D4×C61—84no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61C22⋊C812yes—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—81no—
169C2×Dic61—82no—
170C2×Dic61#12011yescharacteristic
171C2×Dic61#12112yes—
172C2×Dic61#12311yescharacteristic
173#92420—44no—
174#92420—82no—
175#9107—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—21no—
179C22×C61—22no—
180C22×C61—22no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61—42no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—42no—
190C22×C61—81no—
191C22×C61—84no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—84no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61#38611yescharacteristic
198C22×C61C2×C22⋊C811yescharacteristic
199C22×C61#40811yescharacteristic
200C22×C61C8×D412yes—
201C22×C61#43512yes—
202C4×C61—22no—
203C4×C61—42no—
204Dic61—21no—
205Dic61—22no—
206Dic61—21no—
207Dic61—41no—
208Dic61—41no—
209Dic61—81no—
210Dic61—81no—
211#2972—611no—
212#2321—611no—
213#1459—611no—
214C2×#323—612no—
215#1282—612no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—22no—
219C2×C61—22no—
220C2×C61—42no—
221C2×C61—82no—
222C2×C61#186911yescharacteristic
223C2×C61#191311yescharacteristic
224C2×C61#191411yescharacteristic
225C2×#120#692311yescharacteristic
226C2×C4×D4—611no—
227C22×C22⋊C4—611no—
228C8×D4—1224no—
229#431—1221no—
230C2×C22⋊C8—612no—
231C2×C22⋊C8—612no—
232C2×C22⋊C8—1221no—
233#390—1221no—
234#388—1221no—
235#388—1221no—
236#386—611no—
237#386—1221no—
238#385—611no—
239#379—611no—
240C4×C22⋊C4—611no—
241#324—1221no—
242#323—1224no—
243#323—1221no—
244C61#910711yescharacteristic, complemented, Sylow
245C25C2×#692311yescharacteristic
246C22×D4—21no—
247C22×D4C2×#692311yescharacteristic
248C23×C4—611no—
249C23×C4—611no—
250C23×C4—611no—
251C22×C8—1222no—
252C22×C8—1222no—
253C22×C8—1222no—
254#120—22no—
255#120#9242014yescomplemented
256C4×D4—1222no—
257C4×D4—1222no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—1221no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—611no—
262C2×C22⋊C4—611no—
263C2×C22⋊C4—611no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1222no—
271C2×C22⋊C4C2×#692311yescharacteristic
272C2×C42—611no—
273C2×C42—1221no—
274C4⋊C8—1222no—
275C22⋊C8—614no—
276C22⋊C8—1222no—
277C22⋊C8—1222no—
278C22⋊C8—1222no—
279C22⋊C8—1222no—
280C4×C8—1222no—
281#95—1221no—
282#95—1221no—
283#95—1221no—
284C24—21no—
285C24—22no—
286C24—21no—
287C24—22no—
288C24—81no—
289C24—81no—
290C24C2×#9242012yescomplemented
291C24C22⋊#692311yescharacteristic
292C24C22×#692311yescharacteristic
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4C2×#9242012yes—
299C2×D4#1175821712yes—
300C22×C4—21no—
301C22×C4—611no—
302C22×C4—611no—
303C22×C4—612no—
304C22×C4—612no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1222no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—4881no—
313C22×C4—4881no—
314C22×C4—4881no—
315C22×C4C22⋊#692311yescharacteristic
316C2×C8—1224no—
317C2×C8—4882no—
318C2×C8—4882no—
319C2×C8—4882no—
320C2×C8—2442no—
321C4⋊C4—1222no—
322C4⋊C4—4881no—
323C22⋊C4—21no—
324C22⋊C4—24no—
325C22⋊C4—1222no—
326C22⋊C4—1222no—
327C22⋊C4—1222no—
328C22⋊C4—4881no—
329C22⋊C4—4881no—
330C22⋊C4—4881no—
331C22⋊C4—2441no—
332C22⋊C4—2442no—
333C22⋊C4—2441no—
334C22⋊C4C2×#9242012yescomplemented
335C42—1221no—
336C42—1221no—
337C42—4881no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—24no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—84no—
358C23C22⋊#9242012yes—
359C23C2×C22⋊#692311yescharacteristic
360C23D4×#692312yes—
361C23C22×#9242011yescharacteristic
362C23C2×#1175821711yescharacteristic
363C23#1451986412yes—
364D4—24no—
365D4—24no—
366D4—84no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—611no—
373C2×C4—612no—
374C2×C4—611no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—4881no—
378C2×C4—4882no—
379C2×C4—2441no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4C22⋊#9242012yes—
393C8—4882no—
394C8—2444no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—41no—
407C22—41no—
408C22—42no—
409C22—81no—
410C22—84no—
411C22—81no—
412C22—81no—
413C22—84no—
414C22—81no—
415C22—81no—
416C22#1684071011yescharacteristic
417C22C2×C22⋊#9242011yescharacteristic
418C22#1684073511yescharacteristic
419C22D4×#9242012yes—
420C22#1684079312yes—
421C4—22no—
422C4—42no—
423C4—1222no—
424C4—1221no—
425C4—1221no—
426C4—4881no—
427C4—4881no—
428C4—2441no—
429C4—2441no—
430C2—21no—
431C2—21no—
432C2—22no—
433C2—22no—
434C2—42no—
435C2—82no—
436C2#2146179211yescharacteristic
437C2#2146186811yescharacteristic
438C2#2146187011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
Fitting, prime radicalC2×#120×C61C4
socleC22×C61#386
cosocleC23×Dic61C23
hypercenterC2×#120#6923
hypocenterC61#9107
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #21462666C2semidirect
2. #21461055C2semidirect
3. C2×#16840605C2semidirect
4. #21460662C2semidirect
33. C22⋊#92420D4semidirect
61. #120×C61C8semidirect
71. C24×C61C2×C8semidirect
103. C22⋊C4×C61C2×C8semidirect
244. C61#9107semidirect
255. #120#92420semidirect
290. C24C2×#92420semidirect
334. C22⋊C4C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#214626662
#214610552
C2×#168406052
#214606622
#214484802
#910761