Finite Group Databasebeta

← #30606573

Subgroups — #30606573

Order 15616.

460 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462112C211yesmaximal, characteristic
2#21462092C211yesmaximal, characteristic
3#21461574C212yesmaximal, complemented
4#21460919C212yesmaximal, complemented
5#21454532C211yesmaximal, characteristic
6#16840748C2212yes—
7#16840741C2212yes—
8#16840725—22no—
9#16840673C412yescomplemented
10#16840671C412yescomplemented
11#16840669C412yes—
12#16840667C412yes—
13#16840644—22no—
14C2×#14519783C411yescharacteristic, complemented
15C2×#14519783C2211yescharacteristic
16#16840312C2211yescharacteristic
17#16839841—22no—
18C8×D244—22no—
19#16839786C411yescharacteristic, complemented
20#16839728C411yescharacteristic, complemented
21#16839728C2211yescharacteristic
22C2×C4×C8×C61C411yescharacteristic, complemented
23D61⋊M16—22no—
24#14519856—22no—
25#14519838C2×C414yes—
26#14519837—22no—
27#14519836C2×C414yes—
28#14519835—22no—
29#14519831—42no—
30C8×#6923—42no—
31C2×#11758222C2×C411yescharacteristic
32C2×#11758222C2311yescharacteristic
33#14519783—41no—
34#14519783—41no—
35#14519783—41no—
36#14519783—41no—
37C2×#11758194C2×C411yescharacteristic
38C2×#11758194C2×C411yescharacteristic
39C2×#11758189—21no—
40C2×#11758189D412yes—
41C2×C8×D61—21no—
42C2×C8×D61D412yes—
43#14519777C2×C411yescharacteristic
44#14519774C2×C411yescharacteristic
45#14519765C2×C411yescharacteristic
46#14519765C2×C411yescharacteristic
47#14519751—22no—
48C4×D244—22no—
49C4⋊#92418—22no—
50#14519689—22no—
51#14519689—22no—
52C22×C8×C61C2×C411yescharacteristic
53C22×C8×C61C2×C411yescharacteristic
54C2×C42×C61C2×C411yescharacteristic
55C4×C8×C61—21no—
56C4×C8×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—21no—
60C2×C22⋊D61#4112yes—
61#11758222—41no—
62#11758222—41no—
63#11758222—41no—
64#11758222—41no—
65C2×D244—22no—
66C2×D244C4211yescharacteristic
67C2×D244C22×C411yescharacteristic
68C2×C4×D61—21no—
69C2×C4×D61C2×D412yes—
70C2×Dic122C4211yescharacteristic
71C2×Dic122C22×C411yescharacteristic
72C22⋊#6923—24no—
73#11758217—24no—
74C4⋊#6923—42no—
75C4×#6923—42no—
76#11758213—42no—
77D61⋊C8—42no—
78C2×C8×C61—21no—
79C2×C8×C61—21no—
80C2×C8×C61—21no—
81C2×C8×C61—41no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22⋊C411yescharacteristic
84C42×C61—21no—
85C42×C61C4211yescharacteristic
86C42×C61C4211yescharacteristic
87#11758198—22no—
88#11758198—22no—
89C4⋊Dic61—22no—
90#11758194—41no—
91#11758194—41no—
92C2×#92418—21no—
93C2×#92418C22⋊C412yes—
94#11758189—42no—
95#11758189—42no—
96C8×D61—22no—
97C8×D61—22no—
98C8×D61—41no—
99C8×D61—41no—
100C23×C61#11711yescharacteristic
101C22×D61—21no—
102C22×D61C4×D412yes—
103C2×#6923—44no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—42no—
109C2×C4×C61C2×C4211yescharacteristic
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C22⋊D61—42no—
112C22⋊D61—42no—
113C2×Dic61—21no—
114C2×Dic61C4×D412yes—
115D244—41no—
116D244—41no—
117D244—44no—
118C4×D61—22no—
119C4×D61—22no—
120C4×D61—41no—
121C4×D61—41no—
122Dic122—41no—
123Dic122—41no—
124#92420—44no—
125C8×C61—21no—
126C8×C61—41no—
127C8×C61#9911yescharacteristic
128C8×C61#9911yescharacteristic
129#92418—22no—
130#92418—41no—
131#12633—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—41no—
134C22×C61C4×C22⋊C411yescharacteristic
135C2×D61—22no—
136C2×D61—42no—
137C2×D61—42no—
138C2×D61—41no—
139#6923—84no—
140C4×C61—22no—
141C4×C61—21no—
142C4×C61—41no—
143C4×C61C2×#9911yescharacteristic
144C4×C61#41011yescharacteristic
145Dic61—22no—
146Dic61—41no—
147#2931—611no—
148#2132—611no—
149#2126—611no—
150#1757—612no—
151#1395—612no—
152C2×C61—21no—
153C2×C61—41no—
154C2×C61C4×#9911yescharacteristic
155D61—42no—
156D61—81no—
157C2×#131—611no—
158C2×#131—611no—
159#532—611no—
160#436—1222no—
161C8×D4—1222no—
162#423—611no—
163#413—612no—
164#410—612no—
165#408—611no—
166#408—611no—
167#405—1222no—
168C2×C4×C8#692311yescharacteristic, complemented
169#356—612no—
170#355—612no—
171#354—612no—
172#353—612no—
173#344—1222no—
174C61#1263311yescharacteristic, complemented, Sylow
175C2×#41—611no—
176C2×#41—611no—
177#131—2441no—
178#131—2441no—
179#131—2441no—
180#131—2441no—
181C2×M16—611no—
182C2×M16—612no—
183C2×M16—611no—
184C2×M16—1221no—
185C2×M16—1222no—
186C22×C8—612no—
187C22×C8—1221no—
188C22×C8C2×#692311yescharacteristic
189C22×C8C2×#692311yescharacteristic
190C4⋊Q8—611no—
191#127—611no—
192#124—611no—
193#124—611no—
194#121—1222no—
195C4×D4—1222no—
196#117—1222no—
197C2×C42C2×#692311yescharacteristic
198C4⋊C8—1222no—
199#101—614no—
200#100—1222no—
201#99—614no—
202#99—1222no—
203C22⋊C8—1222no—
204C22⋊C8—1222no—
205#97—2442no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—2442no—
209#41—2441no—
210#41—2441no—
211#41—2441no—
212#41—2441no—
213C2×Q8—611no—
214C2×Q8—611no—
215C2×D4—612no—
216C2×D4—611no—
217C2×D4—611no—
218C2×D4—1222no—
219C2×D4—1221no—
220C22×C4—21no—
221C22×C4—612no—
222C22×C4—1221no—
223C22×C4C22×#692311yescharacteristic
224M16—1224no—
225M16—2442no—
226M16—2442no—
227M16—2441no—
228M16—2442no—
229M16—2441no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—41no—
234C2×C8—612no—
235C2×C8—1222no—
236C2×C8—1222no—
237C2×C8—1221no—
238C2×C8—2442no—
239C2×C8—2441no—
240C2×C8—2441no—
241C2×C8C22⋊#692311yescharacteristic
242C2×C8C22⋊#692311yescharacteristic
243C4⋊C4—1222no—
244C4⋊C4—2442no—
245C22⋊C4—1222no—
246C22⋊C4—1222no—
247C22⋊C4—1224no—
248C42—21no—
249C42—2442no—
250C42C4×#692311yescharacteristic
251C42C4×#692311yescharacteristic
252C23—612no—
253C23—1221no—
254C23#1451985611yescharacteristic
255Q8—2441no—
256Q8—2441no—
257D4—2441no—
258D4—2441no—
259D4—2444no—
260D4—2442no—
261D4—2442no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—612no—
268C2×C4—1222no—
269C2×C4—1222no—
270C2×C4—1221no—
271C2×C4—2444no—
272C2×C4—2441no—
273C2×C4—2441no—
274C2×C4C2×C4×#692311yescharacteristic
275C2×C4C2×C22⋊#692311yescharacteristic
276C8—21no—
277C8—41no—
278C8—1222no—
279C8—2444no—
280C8—2441no—
281C8#1451983611yescharacteristic
282C8#1451983611yescharacteristic
283C22—21no—
284C22—41no—
285C22—1222no—
286C22—2442no—
287C22—2442no—
288C22—2441no—
289C22C4×C22⋊#692311yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—1222no—
294C4—4884no—
295C4—2441no—
296C4C2×#1451983611yescharacteristic
297C4#1684074111yescharacteristic
298C2—21no—
299C2—41no—
300C2—4881no—
301C2—2442no—
302C2C4×#1451983611yescharacteristic
subgroupG/H
centerC8#14519836
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C4×C8×C61C4
socleC2×C61C4×#99
prime radical#16840312C22
cosocleC2×#11758222C23
hypercenterC2×C4×C8#6923
hypocenterC61#12633
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
3. #21461574C2semidirect
4. #21460919C2semidirect
9. #16840673C4semidirect
10. #16840671C4semidirect
14. C2×#14519783C4semidirect
19. #16839786C4semidirect
20. #16839728C4semidirect
22. C2×C4×C8×C61C4semidirect
168. C2×C4×C8#6923semidirect
174. C61#12633semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214620922
#214615742
#214609192
#214545322
#1263361