Finite Group Databasebeta

← #29568234

Subgroups — #29568234

Order 15616.

874 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 109 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#390×D61C211yesmaximal, characteristic
2#21445580C212yesmaximal, complemented
3#21443640C211yesmaximal, characteristic, complemented
4#21443639C212yesmaximal, complemented
5C61×#1270C211yesmaximal, characteristic, complemented
6C22×D4×D61C411yescharacteristic
7C2×C4⋊C4×D61C411yescharacteristic
8C2×C22⋊C4×D61—22no—
9C2×C42×D61C2211yescharacteristic
10C2×#14519689—22no—
11C2×#14519687—22no—
12#16839665C411yescharacteristic
13#16839664C2211yescharacteristic
14#16839663C411yescharacteristic
15#16839662—22no—
16#16839517C2212yescomplemented
17C61×#390C2211yescharacteristic
18C61×#336C2212yescomplemented
19C24×D61—21no—
20C22×D244C2×C411yescharacteristic
21C22×C22⋊D61—21no—
22C2×D4×D61—22no—
23C2×D4×D61—22no—
24C2×D4×D61—42no—
25C2×D4×D61—42no—
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29C22×#92418—22no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C4⋊C4×D61—42no—
33C2×C22⋊Dic61—22no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C22⋊C4×D61—44no—
37C22⋊C4×D61—44no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61D412yes—
40C42×D61—42no—
41#14519689—28no—
42#14519687—44no—
43C22×D4×C61C2×C411yescharacteristic
44C22×C8×C61—22no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61—22no—
47C2×C42×C61C2311yescharacteristic
48C24×C61—21no—
49C23×D61—24no—
50C23×D61—21no—
51C23×D61—42no—
52C23×D61—44no—
53C23×D61—42no—
54C23×D61—44no—
55C23×D61C22⋊C411yescharacteristic
56C2×D4×C61—21no—
57C2×D4×C61—41no—
58C2×D4×C61—41no—
59C2×D4×C61M1612yes—
60C22×C4×C61—21no—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D412yes—
63C2×C22⋊D61—24no—
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C22×Dic61—21no—
67C22×Dic61C22⋊C412yes—
68C22×Dic61C22⋊C411yescharacteristic
69D4×D61—48no—
70D4×D61—48no—
71C2×D244—21no—
72C2×D244—41no—
73C2×D244—41no—
74C2×D244M1612yes—
75C2×C4×D61—22no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—22no—
79C2×C4×D61—24no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—42no—
83C2×C4×D61—44no—
84C2×C4×D61—42no—
85C2×C8×C61—24no—
86C2×C8×C61—42no—
87C2×C8×C61—42no—
88C4⋊C4×C61—41no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C42×C61—41no—
92C22⋊Dic61—44no—
93#11758198—44no—
94#11758198—42no—
95#11758198—42no—
96C4⋊Dic61—41no—
97#11758196—42no—
98C4×Dic61—24no—
99C4×Dic61—41no—
100C2×#92418—24no—
101C2×#92418—42no—
102C2×#92418—42no—
103C23×C61—22no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61C2×C22⋊C411yescharacteristic
109C22×D61—21no—
110C22×D61—21no—
111C22×D61—22no—
112C22×D61—22no—
113C22×D61—48no—
114C22×D61—416no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—84no—
120C22×D61—88no—
121C22×D61—84no—
122C22×D61#10012yes—
123C22×D61#10414yes—
124D4×C61—42no—
125D4×C61—42no—
126C2×C4×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—22no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61C2×M1611yescharacteristic
133C2×C4×C61C2×M1611yescharacteristic
134C2×C4×C61AGL(1,ℤ/8)12yes—
135C2×C4×C61AGL(1,ℤ/8)12yes—
136C22⋊D61—48no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—41no—
143C2×Dic61—42no—
144C2×Dic61#9912yes—
145D244—42no—
146D244—42no—
147C4×D61—24no—
148C4×D61—24no—
149C4×D61—44no—
150C4×D61—44no—
151C4×D61—84no—
152C8×C61—44no—
153#92418—44no—
154#7626—611nomaximal, Sylow
155C22×C61—42no—
156C22×C61—44no—
157C22×C61—81no—
158C22×C61—82no—
159C22×C61—81no—
160C22×C61#40711yescharacteristic
161C22×C61C2×#9911yescharacteristic
162C22×C61C2×#10011yescharacteristic
163C22×C61#41812yes—
164C22×C61C2×#10412yes—
165C2×D61—22no—
166C2×D61—22no—
167C2×D61—24no—
168C2×D61—24no—
169C2×D61—42no—
170C2×D61—44no—
171C2×D61—816no—
172C2×D61—81no—
173C2×D61—82no—
174C2×D61—81no—
175C2×D61#32914yes—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—22no—
179C4×C61—22no—
180C4×C61—81no—
181Dic61—22no—
182Dic61—42no—
183Dic61—81no—
184C2×#390—611no—
185#1749—612no—
186#1270—611no—
187#1270—612no—
188#1270D6111yescharacteristic, complemented
189C2×C61—82no—
190C2×C61#146211yescharacteristic
191C2×C61#149012yes—
192C2×C61#150712yes—
193C2×C61C2×#32911yescharacteristic
194C2×C61#155511yescharacteristic
195D61—24no—
196D61—82no—
197C23×D4—611no—
198C22×C4⋊C4—611no—
199C22×C22⋊C4—1222no—
200C22×C42—611no—
201C2×C22⋊C8—1222no—
202C2×#97—1222no—
203#390—611no—
204#390—611no—
205#390—611no—
206#390—1222no—
207#390C2×D6111yescharacteristic
208#336—612no—
209#336C2×D6112yescomplemented
210C61#762611yescharacteristic, complemented, Sylow
211C25—1221no—
212C22×D4—611no—
213C22×D4—1221no—
214C22×D4—1222no—
215C22×D4—1222no—
216C22×D4—2442no—
217C22×D4—2442no—
218C22×D4C4×D6111yescharacteristic
219C23×C4—611no—
220C23×C4—612no—
221C23×C4—1221no—
222C22×C8—22no—
223C22×C8—1222no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—2442no—
227C2×C4⋊C4C4×D6111yescharacteristic
228C2×C22⋊C4—22no—
229C2×C22⋊C4—1222no—
230C2×C22⋊C4—1222no—
231C2×C22⋊C4—1222no—
232C2×C22⋊C4—2444no—
233C2×C22⋊C4—2444no—
234C2×C42—612no—
235C2×C42—611no—
236C2×C42—2442no—
237C2×C42C22×D6111yescharacteristic
238C22⋊C8—1228no—
239#97—2444no—
240C24—21no—
241C24—611no—
242C24—1224no—
243C24—1221no—
244C24—2442no—
245C24—2444no—
246C24—2444no—
247C24—2442no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—612no—
252C2×D4—1224no—
253C2×D4—1221no—
254C2×D4—2441no—
255C2×D4—2441no—
256C2×D4—2442no—
257C2×D4—2448no—
258C2×D4—2448no—
259C2×D4—2442no—
260C2×D4#1175818912yes—
261C22×C4—21no—
262C22×C4—611no—
263C22×C4—612no—
264C22×C4—1222no—
265C22×C4—1222no—
266C22×C4—1224no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1222no—
270C22×C4—1222no—
271C22×C4—1221no—
272C22×C4—2442no—
273C22×C4—2442no—
274C22×C4—2444no—
275C22×C4C2×C4×D6111yescharacteristic
276C22×C4D4×D6112yes—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—1224no—
281C2×C8—2442no—
282C2×C8—2442no—
283C4⋊C4—41no—
284C4⋊C4—2441no—
285C4⋊C4—2442no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—2444no—
289C22⋊C4—2444no—
290C22⋊C4—2442no—
291C22⋊C4—2442no—
292C42—41no—
293C42—1224no—
294C42—2441no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—612no—
301C23—614no—
302C23—1221no—
303C23—1221no—
304C23—1222no—
305C23—1222no—
306C23—4884no—
307C23—4884no—
308C23—4888no—
309C23—2448no—
310C23—24416no—
311C23—2441no—
312C23—2441no—
313C23—2442no—
314C23—2442no—
315C23C22⋊C4×D6111yescharacteristic
316D4—42no—
317D4—42no—
318D4—2442no—
319D4—2442no—
320D4—2448no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—612no—
328C2×C4—1224no—
329C2×C4—1224no—
330C2×C4—1221no—
331C2×C4—1221no—
332C2×C4—1222no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—4884no—
336C2×C4—2444no—
337C2×C4—2444no—
338C2×C4—2441no—
339C2×C4—2442no—
340C2×C4M16×D6111yescharacteristic
341C2×C4C2×#1175818911yescharacteristic
342C2×C4#1451980112yes—
343C2×C4#1451980412yes—
344C8—44no—
345C8—2444no—
346C22—42no—
347C22—44no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22—614no—
352C22—1222no—
353C22—1224no—
354C22—1222no—
355C22—1224no—
356C22—48816no—
357C22—4881no—
358C22—4881no—
359C22—4882no—
360C22—2442no—
361C22—2444no—
362C22#1683972411yescharacteristic
363C22#99×D6111yescharacteristic
364C22#100×D6111yescharacteristic
365C22#1683976512yes—
366C22#104×D6112yes—
367C4—21no—
368C4—21no—
369C4—22no—
370C4—22no—
371C4—81no—
372C4—1222no—
373C4—4881no—
374C4—2442no—
375C2—82no—
376C2—1224no—
377C2—4882no—
378C2#2144414411yescharacteristic
379C2#2144426012yes—
380C2#2144433312yes—
381C2#329×D6111yescharacteristic
382C2#2144453111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×D61C4
cosocleC2×C42×C61C23
hypercenter#1270D61
hypocenterC61#7626
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21445580C2semidirect
3. #21443640C2semidirect
4. #21443639C2semidirect
5. C61×#1270C2semidirect
16. #16839517C22semidirect
18. C61×#336C22semidirect
188. #1270D61semidirect
209. #336C2×D61semidirect
210. C61#7626semidirect

Maximal subgroups

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

subgroupindex
#390×D612
#214455802
#214436402
#214436392
C61×#12702
#762661