Finite Group Databasebeta

← #29577897

Subgroups — #29577897

Order 15616.

600 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 107 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#21451385C211yesmaximal, characteristic
2#21444263C212yesmaximal, complemented
3#21444153C211yesmaximal, characteristic, complemented
4#21443684C212yesmaximal, complemented
5C61×#2873C211yesmaximal, characteristic, complemented
6C2×C4×D244C2211yescharacteristic
7#16839819C2212yes—
8#16839701C411yescharacteristic
9#16839698C411yescharacteristic
10#16839679C411yescharacteristic
11#16839671C411yescharacteristic
12#16839653—21no—
13#16839641—21no—
14#16839638—21no—
15#16839633—21no—
16#16839630C2211yescharacteristic
17#16839491—24no—
18#16839488—21no—
19#16839486—21no—
20C22×M16×C61—21no—
21C2×#117×C61C2211yescharacteristic
22C61×#405—24no—
23C4×M16×C61—21no—
24C4×M16×C61C2212yes—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C4×D61—21no—
28C4×D244—22no—
29C4×D244—22no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C4⋊#92418—22no—
38C22⋊#92418—22no—
39#14519688—22no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C2×M16×C61—24no—
48C2×M16×C61—24no—
49C2×M16×C61—22no—
50C2×M16×C61—22no—
51C22×C8×C61—22no—
52#117×C61—22no—
53#117×C61—22no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59C2×C42×C61C2311yescharacteristic
60#97×C61—22no—
61#97×C61—22no—
62C4×C8×C61—22no—
63C4×C8×C61D414yescomplemented
64C24×C61C22⋊C411yescharacteristic
65C23×D61—21no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C2×C22⋊D61—42no—
77C2×C22⋊D61—42no—
78C2×C22⋊D61—81no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C2×D244—22no—
83C2×D244—42no—
84C2×C4×D61—42no—
85C2×C4×D61—81no—
86C2×C4×D61—81no—
87M16×C61—28no—
88M16×C61—24no—
89M16×C61—24no—
90C2×C8×C61—28no—
91C2×C8×C61—24no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—22no—
94C22⋊C4×C61—22no—
95C22⋊C4×C61—22no—
96C42×C61—21no—
97C42×C61—21no—
98C42×C61C2×D412yes—
99C42×C61#4112yes—
100C22⋊Dic61—81no—
101#11758198—42no—
102#11758198—42no—
103#11758198—41no—
104#11758198—41no—
105#11758198—81no—
106C4⋊Dic61—22no—
107#11758196—81no—
108C2×#92418—42no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61#10412yes—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—22no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61—41no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61#10412yes—
136C2×C4×C61C4×D412yes—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12112yes—
139C2×C4×C61#12311yescharacteristic
140C22⋊D61—84no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150D244—24no—
151D244—42no—
152C4×D61—84no—
153C8×C61—28no—
154#92418—82no—
155#9492—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—22no—
159C22×C61—42no—
160C22×C61—41no—
161C22×C61—41no—
162C22×C61—81no—
163C22×C61#38611yescharacteristic
164C22×C61C2×#10411yescharacteristic
165C22×C61C2×#10411yescharacteristic
166C2×D61—42no—
167C2×D61—42no—
168C2×D61—42no—
169C2×D61—81no—
170C2×D61—81no—
171C2×D61—81no—
172C2×D61—81no—
173C4×C61—22no—
174C4×C61—21no—
175C4×C61—22no—
176C4×C61—21no—
177C4×C61—22no—
178C4×C61—41no—
179C4×C61—41no—
180Dic61—81no—
181Dic61—81no—
182Dic61—81no—
183#2873D6111yescharacteristic, complemented
184#2598—611no—
185#1490—612no—
186#1464—611no—
187#1288—612no—
188C2×C61—21no—
189C2×C61—21no—
190C2×C61—42no—
191C2×C61#189111yescharacteristic
192C2×C61#189611yescharacteristic
193C2×C61#191011yescharacteristic
194D61—82no—
195C22×M16—21no—
196C2×C4×D4—611no—
197C2×#117C2×D6111yescharacteristic
198#432—612no—
199#405—24no—
200C4×M16—21no—
201C4×M16C2×D6112yes—
202#400—611no—
203#399—611no—
204#394—611no—
205#392—611no—
206#388—1221no—
207#386—1221no—
208#386—1221no—
209#385—1221no—
210#383—611no—
211D4⋊C8—1224no—
212#324—1221no—
213#323—1221no—
214C61#949211yescharacteristic, complemented, Sylow
215C22×D4—611no—
216C22×D4—1221no—
217C23×C4—1221no—
218C23×C4C22⋊D6111yescharacteristic
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—22no—
222C2×M16—22no—
223C22×C8—22no—
224C4×D4—1222no—
225C4×D4—1222no—
226#117—22no—
227#117—22no—
228C2×C4⋊C4—611no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4C4×D6111yescharacteristic
231C2×C4⋊C4D24411yescharacteristic
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4C4×D6111yescharacteristic
238C2×C22⋊C4D24411yescharacteristic
239C2×C42C22⋊D6111yescharacteristic
240C2×C42C22×D6111yescharacteristic
241C4⋊C8—1222no—
242C22⋊C8—1222no—
243#97—22no—
244#97—22no—
245#97—1222no—
246C4×C8—22no—
247C4×C8C22⋊D6114yescomplemented
248#95—611no—
249#95—611no—
250#95—1221no—
251#95—1221no—
252#95—1221no—
253#95—1221no—
254C24—1221no—
255C24#1175819811yescharacteristic
256C2×D4—1222no—
257C2×D4—4881no—
258C2×D4—2442no—
259C2×D4—2442no—
260C2×D4—2442no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—4881no—
269C22×C4—4881no—
270C22×C4—2442no—
271C22×C4#1175819811yescharacteristic
272C22×C4#1175819811yescharacteristic
273C22×C4#1175819811yescharacteristic
274C22×C4C2×C4×D6111yescharacteristic
275C22×C4C2×D24411yescharacteristic
276C22×C4C2×C22⋊D6111yescharacteristic
277M16—28no—
278M16—24no—
279M16—24no—
280C2×C8—28no—
281C2×C8—24no—
282C2×C8—2442no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—1222no—
286C4⋊C4—4881no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—4881no—
290C22⋊C4—4881no—
291C22⋊C4—2442no—
292C22⋊C4—2442no—
293C22⋊C4—2441no—
294C22⋊C4—2441no—
295C42—21no—
296C42—21no—
297C42#1175822212yes—
298C42C2×C22⋊D6112yes—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—1222no—
304C23—2441no—
305C23—2441no—
306C23—2441no—
307C23—2441no—
308C23—2441no—
309C23—2441no—
310C23#1451970512yes—
311C23C2×#1175819811yescharacteristic
312D4—1224no—
313D4—4884no—
314D4—2442no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—4884no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4#1451970512yes—
339C2×C4C4×C22⋊D6112yes—
340C2×C4#1451974611yescharacteristic
341C2×C4#1451975112yes—
342C2×C4#1451976011yescharacteristic
343C8—28no—
344C8—4882no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22—4881no—
353C22—4881no—
354C22—4881no—
355C22—4881no—
356C22—2442no—
357C22—2442no—
358C22—2442no—
359C22#1683964111yescharacteristic
360C22C2×#1451970511yescharacteristic
361C22C2×#1451970511yescharacteristic
362C4—22no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—4881no—
370C4—4881no—
371C4—4881no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—4882no—
376C2#2144623711yescharacteristic
377C2#2144626211yescharacteristic
378C2#2144634611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2873C2
socleC22×C61#386
prime radical#16839671C4
cosocleC2×C42×C61C23
hypercenter#2873D61
hypocenterC61#9492
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
2. #21444263C2semidirect
3. #21444153C2semidirect
4. #21443684C2semidirect
5. C61×#2873C2semidirect
63. C4×C8×C61D4semidirect
183. #2873D61semidirect
214. C61#9492semidirect
247. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214513852
#214442632
#214441532
#214436842
C61×#28732
#949261