Finite Group Databasebeta

← #29592251

Subgroups — #29592251

Order 15616.

454 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 63 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#21446860C211yesmaximal, characteristic
2#21444855C212yesmaximal, complemented
3#21443846C212yesmaximal, complemented
4#21443751C211yesmaximal, characteristic, complemented
5C61×#2804C211yesmaximal, characteristic, complemented
6C2×#14519803—21no—
7#16840343C2211yescharacteristic
8#16840080—22no—
9C2×#14519705—21no—
10#16839531C2211yescharacteristic
11#16839495—22no—
12#16839488C2212yes—
13C61×#400C2212yes—
14C61×#393—22no—
15C61×#391—21no—
16C61×#390C2211yescharacteristic
17C61×#386—21no—
18C61×#385—22no—
19C2×#11758222—21no—
20#14519803—42no—
21#14519803—42no—
22C2×#11758200—41no—
23C2×#11758199—41no—
24C2×#11758194—21no—
25#14519757—41no—
26#14519752—41no—
27#14519751—22no—
28#14519708—42no—
29#14519705—42no—
30#14519697—42no—
31C22⋊#92418—22no—
32C22×Q8×C61—21no—
33C22×D4×C61D411yescharacteristic
34C23×C4×C61—21no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61D412yes—
37C2×C4⋊C4×C61C2311yescharacteristic
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—21no—
41C2×C42×C61D411yescharacteristic
42#95×C61—22no—
43#95×C61—22no—
44#95×C61—22no—
45#95×C61D412yes—
46C24×C61—21no—
47C2×Q8×C61—42no—
48C2×Q8×C61—81no—
49C2×Q8×C61—81no—
50C2×D4×C61—21no—
51C2×D4×C61—41no—
52C2×D4×C61—41no—
53C2×D4×C61SD1612yes—
54C22×C4×C61—21no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—82no—
58C22×C4×C61—81no—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D412yes—
61C2×C22⋊D61—41no—
62#11758222—41no—
63#11758222—41no—
64#11758222—41no—
65#11758222—41no—
66C2×D244—21no—
67C2×D244—22no—
68C2×C4×D61—41no—
69C2×Dic122—21no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61—42no—
72C4⋊C4×C61—81no—
73C22⋊C4×C61—44no—
74C22⋊C4×C61—82no—
75C22⋊C4×C61—81no—
76C42×C61—41no—
77#11758200—44no—
78#11758199—44no—
79#11758198—42no—
80C4⋊Dic61—22no—
81#11758194—22no—
82#11758194—41no—
83C2×#92418—41no—
84C2×#92418—42no—
85C23×C61—22no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61#12011yescharacteristic
91C22×D61—41no—
92Q8×C61—84no—
93D4×C61—42no—
94D4×C61—81no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—84no—
109C2×C4×C61C2×SD1611yescharacteristic
110C2×C4×C61#13711yescharacteristic
111C22⋊D61—81no—
112C22⋊D61—81no—
113C2×Dic61—41no—
114D244—41no—
115D244—41no—
116D244—82no—
117C4×D61—81no—
118C4×D61—81no—
119Dic122—41no—
120Dic122—41no—
121#92418—42no—
122#92418—82no—
123#12127—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—44no—
127C22×C61—42no—
128C22×C61—82no—
129C22×C61—81no—
130C22×C61—81no—
131C22×C61#44811yescharacteristic
132C22×C61#45311yescharacteristic
133C22×C61B(4,𝔽2)11yescharacteristic
134C2×D61—81no—
135C2×D61—81no—
136C2×D61—81no—
137C4×C61—21no—
138C4×C61—21no—
139C4×C61—42no—
140C4×C61—42no—
141C4×C61—82no—
142C4×C61—82no—
143Dic61—81no—
144#2804D6111yescharacteristic, complemented
145#2006—611no—
146#1616—612no—
147#1353—612no—
148#1318—611no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—82no—
152C2×C61#159611yescharacteristic
153C2×C61#219811yescharacteristic
154C2×C61#220211yescharacteristic
155D61—161no—
156C2×AGL(1,ℤ/8)—1221no—
157#537—611no—
158#481—1222no—
159C2×#104—1221no—
160#400C2×D6112yes—
161#393—22no—
162#391—21no—
163#390C2×D6111yescharacteristic
164#386—21no—
165#385—22no—
166#343—611no—
167#327—1222no—
168#324—612no—
169C61#1212711yescharacteristic, complemented, Sylow
170C2×#41—1221no—
171C22×Q8—21no—
172C22×D4C22⋊D6111yescharacteristic
173C23×C4—21no—
174AGL(1,ℤ/8)—2442no—
175AGL(1,ℤ/8)—2442no—
176C2×SD16—2441no—
177C2×D8—2441no—
178C2×M16—1221no—
179C22⋊Q8—2441no—
180#121—1222no—
181#121—2441no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22⋊D6112yes—
184C2×C4⋊C4C22×D6111yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42C22⋊D6111yescharacteristic
189#106—2442no—
190#104—2442no—
191#102—2442no—
192C22⋊C8—1222no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95C22⋊D6112yes—
197C24—21no—
198#41—2441no—
199#41—2441no—
200#41—2441no—
201#41—2441no—
202C2×Q8—42no—
203C2×Q8—81no—
204C2×Q8—81no—
205C2×Q8—1221no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—1222no—
210C2×D4—1221no—
211C2×D4—2441no—
212C2×D4#1175820112yes—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4—2441no—
219C22×C4C2×C22⋊D6111yescharacteristic
220C22×C4C2×C22⋊D6112yes—
221SD16—2444no—
222D8—2444no—
223M16—1222no—
224M16—2441no—
225C2×C8—2442no—
226C2×C8—2441no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4—1222no—
231C22⋊C4—44no—
232C22⋊C4—82no—
233C22⋊C4—81no—
234C22⋊C4—2442no—
235C42—41no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—2441no—
242C23#1451974611yescharacteristic
243Q8—84no—
244Q8—2441no—
245Q8—2441no—
246D4—42no—
247D4—81no—
248D4—4882no—
249D4—4881no—
250D4—4881no—
251D4—2441no—
252D4—2441no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—84no—
267C2×C4—4881no—
268C2×C4—4881no—
269C2×C4—2441no—
270C2×C4C2×#1175820111yescharacteristic
271C2×C4#1451980811yescharacteristic
272C8—4882no—
273C8—2442no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—42no—
278C22—82no—
279C22—81no—
280C22—81no—
281C22—4881no—
282C22—4881no—
283C22—4881no—
284C22#1683990111yescharacteristic
285C22#1683993011yescharacteristic
286C22#1683994911yescharacteristic
287C4—21no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—82no—
292C4—82no—
293C4—4881no—
294C2—21no—
295C2—21no—
296C2—82no—
297C2—9761no—
298C2#2144474211yescharacteristic
299C2#2144781511yescharacteristic
300C2#2144783611yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2804C2
socleC22×C61B(4,𝔽2)
prime radical#21446860C2
hypercenter#2804D61
hypocenterC61#12127
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #21444855C2semidirect
3. #21443846C2semidirect
4. #21443751C2semidirect
5. C61×#2804C2semidirect
144. #2804D61semidirect
169. C61#12127semidirect

Maximal subgroups

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

subgroupindex
#214468602
#214448552
#214438462
#214437512
C61×#28042
#1212761