Finite Group Databasebeta

← #29598690

Subgroups — #29598690

Order 15616.

492 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 99 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#21451015C211yesmaximal, characteristic
2C4×#14519692C211yesmaximal, characteristic
3#21443943C212yesmaximal
4#21443943C212yesmaximal
5C61×#1799C211yesmaximal, characteristic
6C2×#14519769C411yescharacteristic, complemented
7C2×#14519692C2212yes—
8#16839685—21no—
9#16839685—21no—
10#16839665C2211yescharacteristic
11#16839665C2211yescharacteristic
12#16839635—21no—
13#16839635—21no—
14#16839631C411yescharacteristic, complemented
15#16839627C411yescharacteristic, complemented
16C4×C4⋊Dic61C411yescharacteristic, complemented
17C4×C22⋊Dic61—21no—
18#16839529—21no—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61C2212yes—
21C61×#380—21no—
22C61×#342—21no—
23#14519769—41no—
24#14519769—41no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C4×Dic61—41no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40#14519692—22no—
41#14519692Q814yes—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—41no—
46C22×D4×C61C2311yescharacteristic
47C23×C4×C61D411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C4×D4×C61—41no—
50C4×D4×C61—41no—
51C2×C4⋊C4×C61—41no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57#99×C61—24no—
58#95×C61—41no—
59C24×C61C2×D411yescharacteristic
60C24×C61C2×D411yescharacteristic
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C2×D4×C61C2×Q812yes—
64C2×D4×C61#4112yes—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—42no—
70C22×C4×C61—81no—
71C22×C4×C61—81no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—42no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—82no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—42no—
88C42×C61—21no—
89C42×C61—21no—
90C22⋊Dic61—24no—
91C22⋊Dic61—42no—
92C22⋊Dic61—42no—
93C22⋊Dic61—42no—
94C22⋊Dic61—81no—
95C22⋊Dic61—81no—
96C4⋊Dic61—42no—
97C4⋊Dic61—41no—
98C4⋊Dic61—41no—
99#11758196—42no—
100#11758196—41no—
101#11758196—41no—
102#11758196—81no—
103#11758196—81no—
104C4×Dic61—41no—
105C4×Dic61—41no—
106C4×Dic61—82no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61C2×C22⋊C411yescharacteristic
116C23×C61C22⋊Q812yes—
117C23×C61#12411yescharacteristic
118C23×C61C4⋊Q811yescharacteristic
119D4×C61—42no—
120D4×C61—42no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—81no—
133C2×C4×C61—81no—
134C2×C4×C61—82no—
135C2×C4×C61—82no—
136C2×C4×C61—82no—
137C2×C4×C61#9911yescharacteristic
138C2×C4×C61#9911yescharacteristic
139C2×C4×C61C4×Q812yes—
140C2×Dic61—22no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—44no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—82no—
154C2×Dic61—82no—
155#13300—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—22no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—42no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61#39111yescharacteristic
171C22×C61C2×#9911yescharacteristic
172C22×C61#41011yescharacteristic
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—22no—
176C4×C61—41no—
177C4×C61—41no—
178C4×C61—41no—
179C4×C61—162no—
180Dic61—42no—
181Dic61—81no—
182Dic61—84no—
183Dic61—81no—
184Dic61—81no—
185#2566—611no—
186#1799D6111yescharacteristic
187C4×#99—611no—
188#1393—612no—
189#1393—612no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61—41no—
194C2×C61—41no—
195C2×C61#199611yescharacteristic
196C2×C61#212611yescharacteristic
197C2×C61#212711yescharacteristic
198C2×#125—611no—
199C2×C4×D4C2×D6111yescharacteristic
200C2×#99—612no—
201C2×#99C2×D6112yes—
202#396—1221no—
203#396—1221no—
204#390—611no—
205#390—611no—
206#385—1221no—
207#385—1221no—
208#384—611no—
209#382—611no—
210#380—21no—
211C4×C4⋊C4—611no—
212C4×C22⋊C4—1221no—
213#342—21no—
214#342—1221no—
215C61#1330011yescharacteristic, complemented, Sylow
216C22×D4C22×D6111yescharacteristic
217C23×C4D24411yescharacteristic
218C23×C4C22⋊D6111yescharacteristic
219#125—2441no—
220#125—2441no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4C4×D6111yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4—1221no—
238C2×C22⋊C4D24411yescharacteristic
239C2×C22⋊C4C22⋊D6111yescharacteristic
240C2×C42—611no—
241C2×C42—611no—
242C2×C42—2441no—
243C2×C42C4×D6111yescharacteristic
244#99—24no—
245#99—614no—
246#99—1222no—
247#95—41no—
248#95—1221no—
249#95—1221no—
250#95—1221no—
251#95—2441no—
252C24C2×D24411yescharacteristic
253C24C2×C22⋊D6111yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4#1175822412yes—
257C2×D4Q8×D6112yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—2442no—
270C22×C4#1175819811yescharacteristic
271C22×C4#1175819811yescharacteristic
272C22×C4#1175819811yescharacteristic
273C22×C4#1175819811yescharacteristic
274C22×C4C2×C4×D6111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—82no—
277C4⋊C4—4881no—
278C4⋊C4—4881no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2442no—
282C4⋊C4—2442no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—1224no—
290C22⋊C4—4881no—
291C22⋊C4—4881no—
292C22⋊C4—2442no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C42—21no—
296C42—21no—
297C42—4882no—
298C42—2441no—
299C42—2441no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×#1175819811yescharacteristic
309C23#1451975712yes—
310C23#1451976411yescharacteristic
311C23#1451977511yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—1222no—
331C2×C4—4882no—
332C2×C4—4882no—
333C2×C4—2444no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4#1451969111yescharacteristic
346C2×C4#1451969111yescharacteristic
347C2×C4#1451974212yes—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22#1683966711yescharacteristic
363C22C2×#1451969111yescharacteristic
364C22#1683973511yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—162no—
372C4—4881no—
373C4—4884no—
374C4—4881no—
375C4—4881no—
376C4—2442no—
377C2—21no—
378C2—21no—
379C2—41no—
380C2—42no—
381C2—41no—
382C2#2144680311yescharacteristic
383C2#2144748211yescharacteristic
384C2#2144748611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radical, cosocleC22×D4×C61C23
hypercenter#1799D61
hypocenterC61#13300
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
6. C2×#14519769C4semidirect
14. #16839631C4semidirect
15. #16839627C4semidirect
16. C4×C4⋊Dic61C4semidirect
215. C61#13300semidirect

Maximal subgroups

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

subgroupindex
#214510152
C4×#145196922
#214439432
C61×#17992
#1330061