Finite Group Databasebeta

← #29591062

Subgroups — #29591062

Order 15616.

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

normal subgroups only

This group has 460 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#21450127C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic
3#21443824C214yesmaximal, complemented
4C61×#1800C211yesmaximal, characteristic
5C2×#14519758—21no—
6C22×C4⋊Dic61C411yescharacteristic
7C2×C22⋊#92418—21no—
8#16839692C2212yes—
9#16839687C412yes—
10#16839660—21no—
11#16839623—21no—
12#16839622—21no—
13#16839622C411yescharacteristic
14#16839488C2212yes—
15#16839486—21no—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#423—21no—
18C2×#100×C61—21no—
19C61×#343C2212yes—
20C23×Dic61—21no—
21C22×#92418—41no—
22#14519758—42no—
23#14519758—42no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—41no—
27C2×C4⋊Dic61—41no—
28C2×C4⋊Dic61C2×C412yes—
29C2×#11758196—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—21no—
32C22⋊#92418—22no—
33C22⋊#92418—41no—
34#14519684—22no—
35#14519684—22no—
36#14519684C2×C412yes—
37C22×D4×C61D411yescharacteristic
38C23×C4×C61C2×C411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C2×M16×C61—22no—
41C4×D4×C61—41no—
42C4×D4×C61—41no—
43C2×C4⋊C4×C61C2311yescharacteristic
44C2×C22⋊C4×C61C2×C411yescharacteristic
45C2×C22⋊C4×C61D411yescharacteristic
46C2×C42×C61D411yescharacteristic
47C4⋊C8×C61—41no—
48C4⋊C8×C61—41no—
49#100×C61—42no—
50C24×C61C22⋊C411yescharacteristic
51C24×C61C22⋊C411yescharacteristic
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61D812yes—
55C2×D4×C61SD1612yes—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72M16×C61—42no—
73M16×C61—42no—
74C2×C8×C61—42no—
75C4⋊C4×C61—41no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C42×C61—21no—
79C42×C61—21no—
80C22⋊Dic61—42no—
81C22⋊Dic61—44no—
82C4⋊Dic61—42no—
83C4⋊Dic61—41no—
84C4⋊Dic61—41no—
85C4⋊Dic61—42no—
86#11758196—42no—
87#11758196—42no—
88C2×#92418—42no—
89C2×#92418—82no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61#10212yes—
99C23×C61#10212yes—
100C23×C61C2×C22⋊C411yescharacteristic
101D4×C61—42no—
102D4×C61—42no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—81no—
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61#9911yescharacteristic
118C2×C4×C61C2×D811yescharacteristic
119C2×C4×C61C2×SD1611yescharacteristic
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—84no—
127C8×C61—82no—
128#92418—82no—
129#11881—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—42no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—81no—
142C22×C61C2×#9911yescharacteristic
143C22×C61C2×#10211yescharacteristic
144C22×C61#42111yescharacteristic
145C4×C61—21no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—41no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61#35212yes—
152Dic61—82no—
153Dic61—82no—
154#2486—611no—
155#1800D6111yescharacteristic
156#1459—611no—
157#1346—614no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61—42no—
161C2×C61—42no—
162C2×C61#151911yescharacteristic
163C2×C61C2×#35211yescharacteristic
164C2×C61#213511yescharacteristic
165C2×#123—1221no—
166C2×C4×D4C2×D6111yescharacteristic
167C22×C4⋊C4—611no—
168#423—21no—
169C2×#100—21no—
170C2×C22⋊C8—1221no—
171#398—612no—
172#397—612no—
173#389—1221no—
174#381—1221no—
175#380—611no—
176#380—1221no—
177#343C2×D6112yes—
178#324—612no—
179#323—1221no—
180C61#1188111yescharacteristic, complemented, Sylow
181C22×D4C22⋊D6111yescharacteristic
182C23×C4—1221no—
183C23×C4C4×D6111yescharacteristic
184C23×C4D24411yescharacteristic
185C2×M16—22no—
186C22×C8—2441no—
187#123—2442no—
188#123—2442no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—612no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—2441no—
196C2×C4⋊C4—2441no—
197C2×C4⋊C4C22×D6111yescharacteristic
198C2×C22⋊C4—1221no—
199C2×C22⋊C4—1222no—
200C2×C22⋊C4C4×D6111yescharacteristic
201C2×C22⋊C4D24411yescharacteristic
202C2×C42C22⋊D6111yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205#100—42no—
206C22⋊C8—1222no—
207C22⋊C8—2441no—
208#95—612no—
209#95—1222no—
210#95—1222no—
211C24#1175819811yescharacteristic
212C24#1175819811yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1175819912yes—
216C2×D4#1175820112yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—2442no—
226C22×C4—2442no—
227C22×C4—2442no—
228C22×C4#1175819811yescharacteristic
229C22×C4#1175819811yescharacteristic
230C22×C4C2×C4×D6111yescharacteristic
231C22×C4C2×D24411yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—4882no—
237C2×C8—2442no—
238C4⋊C4—41no—
239C4⋊C4—2442no—
240C4⋊C4—2442no—
241C4⋊C4—2441no—
242C4⋊C4—2441no—
243C4⋊C4—2442no—
244C4⋊C4—2442no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2442no—
248C22⋊C4—2444no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1451969712yes—
260C23#1451969712yes—
261C23C2×#1175819811yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—4884no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4#1451969111yescharacteristic
285C2×C4#1451969111yescharacteristic
286C2×C4C2×#1175819911yescharacteristic
287C2×C4C2×#1175820111yescharacteristic
288C8—82no—
289C8—4882no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1451969111yescharacteristic
303C22C2×#1451969711yescharacteristic
304C22#1683978011yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—4882no—
312C4—4882no—
313C4#1683955312yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2144438611yescharacteristic
319C2C2×#1683955311yescharacteristic
320C2#2144751311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1800C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C4⋊C4×C61C23
hypercenter#1800D61
hypocenterC61#11881
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #21443824C2semidirect
180. C61#11881semidirect

Maximal subgroups

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

subgroupindex
#214501272
#214441372
#214438242
C61×#18002
#1188161