Finite Group Databasebeta

← #29577800

Subgroups — #29577800

Order 15616.

556 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 115 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#21454848C211yesmaximal, characteristic
2#21450136C211yesmaximal, characteristic
3#21445907C212yesmaximal, complemented
4#21443686C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic
6C22×C4⋊Dic61C2211yescharacteristic
7#16839834—24no—
8#16839819—21no—
9#16839726—21no—
10C2×C22⋊#92418C2212yes—
11#16839701C412yescomplemented
12#16839698C412yescomplemented
13#16839660—21no—
14#16839642—21no—
15#16839634—21no—
16#16839622—21no—
17#16839622C2211yescharacteristic
18#16839524—24no—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C61×#380C2212yes—
22C61×#342—21no—
23C23×Dic61—21no—
24C2×#11758194—22no—
25C22×#92418D414yes—
26C2×C4⋊Dic61—22no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61C2×C412yes—
31C2×#11758196—22no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—22no—
34C4⋊#92418—22no—
35C22⋊#92418—22no—
36C22⋊#92418—22no—
37C22⋊#92418—22no—
38#14519688—22no—
39#14519684—22no—
40#14519684—22no—
41#14519684C2×C412yes—
42C22×D4×C61D411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C23×C4×C61C2311yescharacteristic
45C4×D4×C61—22no—
46C4×D4×C61—22no—
47C2×C4⋊C4×C61—22no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61—22no—
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C42×C61D411yescharacteristic
53#99×C61—42no—
54#95×C61—22no—
55C24×C61C22⋊C411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—21no—
59C2×D4×C61—21no—
60C2×D4×C61—22no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D412yes—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61#4112yes—
73C22×Dic61—22no—
74C22×Dic61—22no—
75C22×Dic61—42no—
76C22×Dic61—82no—
77C4⋊C4×C61—22no—
78C4⋊C4×C61—44no—
79C22⋊C4×C61—22no—
80C22⋊C4×C61—22no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C42×C61—21no—
84C42×C61—21no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87C22⋊Dic61—82no—
88C4⋊Dic61—24no—
89C4⋊Dic61—42no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92#11758196—82no—
93#11758194—82no—
94C2×#92418—24no—
95C2×#92418—44no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61C2×C22⋊C411yescharacteristic
104C23×C61#12112yes—
105C23×C61AGL(1,ℤ/8)11yescharacteristic
106C23×C61#13711yescharacteristic
107C23×C61#13712yes—
108D4×C61—42no—
109D4×C61—42no—
110C2×C4×C61—24no—
111C2×C4×C61—22no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—81no—
123C2×C4×C61—82no—
124C2×C4×C61—82no—
125C2×C4×C61—82no—
126C2×C4×C61C4×D412yes—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61#12311yescharacteristic
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—84no—
136#92418—44no—
137#9473—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—22no—
140C22×C61—21no—
141C22×C61—22no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—81no—
150C22×C61#38611yescharacteristic
151C22×C61#41711yescharacteristic
152C22×C61#41911yescharacteristic
153C22×C61#46812yes—
154C22×C61#47012yes—
155C4×C61—21no—
156C4×C61—22no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—162no—
161C4×C61#45812yes—
162Dic61—82no—
163Dic61—82no—
164#2973—611no—
165#2487—611no—
166#1824—612no—
167#1799D6111yescharacteristic
168#1289—612no—
169C2×C61—21no—
170C2×C61—22no—
171C2×C61—21no—
172C2×C61—22no—
173C2×C61—81no—
174C2×C61#189011yescharacteristic
175C2×C61#189411yescharacteristic
176C2×C61#191011yescharacteristic
177C2×C4×D4C2×D6111yescharacteristic
178C22×C4⋊C4—611no—
179#435—1224no—
180#432—1221no—
181C2×#99—21no—
182#407—1221no—
183C2×C22⋊C8—612no—
184#400—612no—
185#399—612no—
186#389—1221no—
187#386—1221no—
188#385—1221no—
189#380—611no—
190#380—1221no—
191#380C2×D6112yes—
192#342—21no—
193#340—1224no—
194C61#947311yescharacteristic, complemented, Sylow
195C22×D4C22⋊D6111yescharacteristic
196C23×C4—1221no—
197C23×C4D24411yescharacteristic
198C23×C4C22×D6111yescharacteristic
199C2×M16—1222no—
200C22×C8—614no—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—612no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4—1222no—
207C2×C4⋊C4—1222no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4C4×D6111yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4C4×D6111yescharacteristic
215C2×C22⋊C4C22⋊D6111yescharacteristic
216C2×C42D24411yescharacteristic
217C4⋊C8—1222no—
218#99—42no—
219C22⋊C8—1222no—
220C22⋊C8—1222no—
221C22⋊C8—1222no—
222#97—1222no—
223#95—22no—
224#95—612no—
225#95—1222no—
226#95—1222no—
227C24#1175819811yescharacteristic
228C24C2×C22⋊D6111yescharacteristic
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—21no—
232C2×D4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—1222no—
239C22×C4—1222no—
240C22×C4—4882no—
241C22×C4—2442no—
242C22×C4#1175819811yescharacteristic
243C22×C4#1175819811yescharacteristic
244C22×C4#1175819811yescharacteristic
245C22×C4C2×C4×D6111yescharacteristic
246C22×C4C2×D24411yescharacteristic
247C22×C4D4×D6112yes—
248C22×C4#1175822412yes—
249M16—4882no—
250C2×C8—1224no—
251C2×C8—2444no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—1224no—
255C4⋊C4—4882no—
256C4⋊C4—2442no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—4882no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×#1175819811yescharacteristic
276C23#1451974912yes—
277C23#1451980311yescharacteristic
278C23#1451980712yes—
279C23#1451980811yescharacteristic
280D4—42no—
281D4—42no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—4884no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4#1451973712yes—
306C2×C4#1451974711yescharacteristic
307C2×C4#1451976111yescharacteristic
308C8—2444no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1683963911yescharacteristic
322C22#1683976111yescharacteristic
323C22#1683977011yescharacteristic
324C22#1684000612yes—
325C22#1684001812yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—162no—
332C4—4882no—
333C4—4882no—
334C4#1683995312yes—
335C2—22no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2144623011yescharacteristic
341C2#2144625011yescharacteristic
342C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#9473
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21445907C2semidirect
4. #21443686C2semidirect
11. #16839701C4semidirect
12. #16839698C4semidirect
194. C61#9473semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214501362
#214459072
#214436862
C61×#17992
#947361