Finite Group Databasebeta

← #29598308

Subgroups — #29598308

Order 15616.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C2×#16839553C214yesmaximal
3C4×#14519692C211yesmaximal, characteristic
4C61×#1800C211yesmaximal, characteristic
5C2×#14519758C2212yes—
6C22×C4⋊Dic61C411yescharacteristic, complemented
7C2×#14519692C2212yes—
8#16839692—21no—
9#16839687—21no—
10#16839660—21no—
11#16839623C412yescomplemented
12#16839622—21no—
13#16839622C411yescharacteristic, complemented
14C4×C22⋊Dic61—21no—
15#16839553—28no—
16#16839529—21no—
17C2×C4×D4×C61C2211yescharacteristic
18C61×#423—21no—
19C2×#100×C61C2212yes—
20C61×#343—21no—
21C23×Dic61C2×C412yes—
22#14519758—24no—
23#14519758—24no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—41no—
28C2×C4⋊Dic61—41no—
29C2×#11758196—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61C2×C412yes—
33C2×C4×Dic61—41no—
34#14519692—22no—
35#14519692D414yes—
36#14519684—21no—
37#14519684—22no—
38#14519684—22no—
39#14519684—41no—
40C22×D4×C61C2311yescharacteristic
41C23×C4×C61C2×C411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C2×M16×C61—22no—
44C4×D4×C61—41no—
45C4×D4×C61—41no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C4⋊C8×C61—41no—
51C4⋊C8×C61—41no—
52#100×C61—24no—
53C24×C61C22×C411yescharacteristic
54C24×C61C2×D411yescharacteristic
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61C2×D412yes—
58C2×D4×C61#4112yes—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—24no—
73C22×Dic61—42no—
74C22×Dic61—42no—
75C22×Dic61—42no—
76M16×C61—42no—
77M16×C61—42no—
78C2×C8×C61—42no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C42×C61—21no—
83C42×C61—21no—
84C22⋊Dic61—24no—
85C22⋊Dic61—24no—
86C22⋊Dic61—42no—
87C22⋊Dic61—44no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92#11758196—42no—
93#11758196—42no—
94C4×Dic61—82no—
95C23×C61—21no—
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×C61C4×D412yes—
105C23×C61#12112yes—
106D4×C61—42no—
107D4×C61—42no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—81no—
121C2×C4×C61#9911yescharacteristic
122C2×C4×C61#9911yescharacteristic
123C2×C4×C61#12411yescharacteristic
124C2×C4×C61#12711yescharacteristic
125C2×Dic61—42no—
126C2×Dic61—44no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—82no—
133C2×Dic61—82no—
134C2×Dic61—84no—
135C8×C61—82no—
136#13220—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—81no—
149C22×C61#39011yescharacteristic
150C22×C61C2×#9911yescharacteristic
151C22×C61#41011yescharacteristic
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—41no—
158C4×C61#35212yescomplemented
159Dic61—82no—
160Dic61—84no—
161Dic61—82no—
162#2486—611no—
163C2×#352—614no—
164#1800D6111yescharacteristic
165C4×#99—611no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—42no—
169C2×C61—42no—
170C2×C61#196411yescharacteristic
171C2×C61C2×#35211yescharacteristic
172C2×C61#212311yescharacteristic
173C2×#123—612no—
174C2×C4×D4C2×D6111yescharacteristic
175C22×C4⋊C4—611no—
176#423—21no—
177C2×#100C2×D6112yes—
178C2×#99—612no—
179#398—1221no—
180#397—1221no—
181#389—1221no—
182#381—612no—
183#380—611no—
184#380—1221no—
185C4×C22⋊C4—1221no—
186#352—1228no—
187#343—21no—
188#342—1221no—
189C61#1322011yescharacteristic, complemented, Sylow
190C22×D4C22×D6111yescharacteristic
191C23×C4—612no—
192C23×C4C4×D6111yescharacteristic
193C23×C4D24411yescharacteristic
194C2×M16—22no—
195#123—1224no—
196#123—1224no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—2441no—
204C2×C4⋊C4—2441no—
205C2×C4⋊C4C22⋊D6111yescharacteristic
206C2×C22⋊C4—612no—
207C2×C22⋊C4—1222no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4C4×D6111yescharacteristic
210C2×C22⋊C4D24411yescharacteristic
211C2×C42—2441no—
212C2×C42C22⋊D6111yescharacteristic
213C4⋊C8—41no—
214C4⋊C8—41no—
215#100—24no—
216#99—614no—
217#99—1222no—
218#95—1221no—
219#95—1222no—
220#95—1222no—
221#95—2441no—
222C24C2×C4×D6111yescharacteristic
223C24C2×D24411yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D6112yes—
227C2×D4#1175822612yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1224no—
235C22×C4—1222no—
236C22×C4—1222no—
237C22×C4—2442no—
238C22×C4—2442no—
239C22×C4—2442no—
240C22×C4#1175819811yescharacteristic
241C22×C4#1175819811yescharacteristic
242C22×C4#1175819811yescharacteristic
243C22×C4#1175819811yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C4⋊C4—41no—
249C4⋊C4—2442no—
250C4⋊C4—2442no—
251C4⋊C4—2441no—
252C4⋊C4—2441no—
253C4⋊C4—2442no—
254C4⋊C4—2442no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—1224no—
258C22⋊C4—1224no—
259C22⋊C4—2442no—
260C22⋊C4—2444no—
261C42—21no—
262C42—21no—
263C42—4882no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×#1175819811yescharacteristic
273C23#1451973912yes—
274C23#1451975212yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—4882no—
291C2×C4—4882no—
292C2×C4—4884no—
293C2×C4—2444no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4#1451969111yescharacteristic
301C2×C4#1451969111yescharacteristic
302C2×C4#1451976611yescharacteristic
303C2×C4#1451977311yescharacteristic
304C8—82no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1683966311yescharacteristic
318C22C2×#1451969111yescharacteristic
319C22#1683973511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—4882no—
327C4—4884no—
328C4—4882no—
329C4#1683955312yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2144665411yescharacteristic
335C2C2×#1683955311yescharacteristic
336C2#2144747411yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1800C2
socleC22×C61C2×#99
prime radical, cosocleC22×D4×C61C23
hypercenter#1800D61
hypocenterC61#13220
LCSG ▷ C24×C61 ▷ C22×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
6. C22×C4⋊Dic61C4semidirect
11. #16839623C4semidirect
13. #16839622C4semidirect
158. C4×C61#352semidirect
189. C61#13220semidirect
329. C4#16839553semidirect

Maximal subgroups

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

subgroupindex
#214501272
C2×#168395532
C4×#145196922
C61×#18002
#1322061