Finite Group Databasebeta

← #29590676

Subgroups — #29590676

Order 15616.

486 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 79 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#21449055C211yesmaximal, characteristic
2#21444153C211yesmaximal, characteristic, complemented
3#21443818C212yesmaximal, complemented
4#21443815C212yesmaximal
5C61×#1820C211yesmaximal, characteristic, complemented
6C2×C4×D244C411yescharacteristic, complemented
7C2×C4×C22⋊D61—21no—
8#16839819—21no—
9#16839692C411yescharacteristic, complemented
10#16839687C2211yescharacteristic
11#16839684—21no—
12#16839679C411yescharacteristic, complemented
13#16839678—21no—
14#16839671C2211yescharacteristic
15C4×C4⋊Dic61C411yescharacteristic, complemented
16C4×#11758198—21no—
17#16839488—21no—
18#16839486C2212yes—
19C2×#117×C61C2211yescharacteristic
20C61×#381—21no—
21C61×#342—21no—
22C61×#342C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×C4×D61—21no—
26C4×D244—41no—
27C4×D244—41no—
28C4×C22⋊D61—44no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×C4×Dic61—21no—
38C4⋊#92418—41no—
39C22⋊#92418—22no—
40#14519688—41no—
41#14519684—21no—
42#14519684—21no—
43#14519684C2×C411yescharacteristic
44#14519684C2×C411yescharacteristic
45C23×C4×C61D411yescharacteristic
46#117×C61—41no—
47#117×C61—41no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61C2311yescharacteristic
53C2×C42×C61C2×C411yescharacteristic
54C2×C42×C61D411yescharacteristic
55#95×C61—41no—
56#95×C61—41no—
57C24×C61C2×D411yescharacteristic
58C23×D61—21no—
59C22×C4×C61—22no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—41no—
63C22×C4×C61—42no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C2×C22⋊D61—42no—
71C2×C22⋊D61—42no—
72C2×C22⋊D61—42no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C2×D244—41no—
77C2×D244—42no—
78C2×C4×D61—42no—
79C2×C4×D61—42no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—42no—
88C42×C61—21no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61—21no—
92C22⋊Dic61—42no—
93#11758198—42no—
94#11758198—42no—
95#11758198—42no—
96#11758198—42no—
97C4⋊Dic61—42no—
98C4⋊Dic61—41no—
99#11758196—42no—
100C4×Dic61—42no—
101C2×#92418—42no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61AGL(1,ℤ/8)12yes—
108C22×D61—22no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61—41no—
112C22×D61—41no—
113C22×D61—41no—
114C22×D61—41no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—82no—
131C2×C4×C61—82no—
132C2×C4×C61—82no—
133C2×C4×C61#9911yescharacteristic
134C2×C4×C61#9911yescharacteristic
135C22⋊D61—48no—
136C2×Dic61—22no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145D244—42no—
146D244—42no—
147C4×D61—82no—
148C4×D61—82no—
149#92418—82no—
150#11798—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—22no—
154C22×C61—42no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61—81no—
158C22×C61C2×#9911yescharacteristic
159C22×C61#41811yescharacteristic
160C22×C61#42111yescharacteristic
161C2×D61—42no—
162C2×D61—42no—
163C2×D61—42no—
164C2×D61—81no—
165C2×D61—81no—
166C2×D61—81no—
167C2×D61—81no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—41no—
171C4×C61—41no—
172C4×C61—41no—
173C4×C61—41no—
174C4×C61—41no—
175C4×C61—162no—
176Dic61—42no—
177Dic61—81no—
178Dic61—81no—
179#2392—611no—
180#1820D6111yescharacteristic, complemented
181#1464—611no—
182#1344—612no—
183#1343—612no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—42no—
187C2×C61#151011yescharacteristic
188C2×C61#212011yescharacteristic
189C2×C61#212611yescharacteristic
190D61—82no—
191C2×C4×D4—611no—
192C2×C4×D4—1221no—
193C2×#117C2×D6111yescharacteristic
194#432—1221no—
195#398—611no—
196#397—611no—
197#396—1221no—
198#394—611no—
199#394—1221no—
200#392—611no—
201#381—21no—
202C4×C4⋊C4—611no—
203C4×C22⋊C4—1221no—
204#342—21no—
205#342C2×D6112yes—
206#324—1221no—
207#323—612no—
208C61#1179811yescharacteristic, complemented, Sylow
209C22×D4—611no—
210C22×D4—1221no—
211C23×C4—1221no—
212C23×C4C22⋊D6111yescharacteristic
213C4×D4—2441no—
214C4×D4—2441no—
215C4×D4—2444no—
216#117—41no—
217#117—41no—
218C2×C4⋊C4—611no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4C4×D6111yescharacteristic
222C2×C4⋊C4D24411yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—1221no—
225C2×C22⋊C4—1221no—
226C2×C22⋊C4—1221no—
227C2×C22⋊C4—1221no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4C22⋊D6111yescharacteristic
230C2×C22⋊C4C22×D6111yescharacteristic
231C2×C42—1221no—
232C2×C42C4×D6111yescharacteristic
233C2×C42D24411yescharacteristic
234C4⋊C8—2441no—
235C22⋊C8—1222no—
236#97—2441no—
237#95—41no—
238#95—41no—
239#95—611no—
240#95—611no—
241#95—1221no—
242#95—1221no—
243C24—1221no—
244C24C2×C22⋊D6111yescharacteristic
245C2×D4—2441no—
246C2×D4—2442no—
247C2×D4—2442no—
248C2×D4—2442no—
249C2×D4—2442no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—2442no—
259C22×C4—2442no—
260C22×C4—2441no—
261C22×C4—2441no—
262C22×C4#1175819811yescharacteristic
263C22×C4#1175819811yescharacteristic
264C22×C4#1175819811yescharacteristic
265C22×C4#1175819811yescharacteristic
266C22×C4C2×C4×D6111yescharacteristic
267C22×C4C2×D24411yescharacteristic
268C2×C8—2442no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—2441no—
272C4⋊C4—2442no—
273C4⋊C4—2442no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—2442no—
279C22⋊C4—2442no—
280C22⋊C4—2442no—
281C22⋊C4—2442no—
282C22⋊C4—2442no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—2442no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—1222no—
293C23—2441no—
294C23—2441no—
295C23—2441no—
296C23—2441no—
297C23—2441no—
298C23—2441no—
299C23C2×#1175819811yescharacteristic
300C23#1451980212yes—
301D4—2442no—
302D4—2442no—
303D4—2448no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—1222no—
323C2×C4—4882no—
324C2×C4—4882no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4#1451969111yescharacteristic
334C2×C4#1451969111yescharacteristic
335C8—4882no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—4881no—
344C22—4881no—
345C22—4881no—
346C22—4881no—
347C22—2442no—
348C22—2442no—
349C22—2442no—
350C22C2×#1451969111yescharacteristic
351C22#1683976611yescharacteristic
352C22#1683978011yescharacteristic
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—162no—
361C4—4881no—
362C4—4881no—
363C4—2442no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—4882no—
368C2#2144434711yescharacteristic
369C2#2144745811yescharacteristic
370C2#2144748211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1820C2
socleC22×C61C2×#99
prime radical#16839671C22
cosocleC2×C22⋊C4×C61C23
hypercenter#1820D61
hypocenterC61#11798
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #21444153C2semidirect
3. #21443818C2semidirect
5. C61×#1820C2semidirect
6. C2×C4×D244C4semidirect
9. #16839692C4semidirect
12. #16839679C4semidirect
15. C4×C4⋊Dic61C4semidirect
180. #1820D61semidirect
208. C61#11798semidirect

Maximal subgroups

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

subgroupindex
#214490552
#214441532
#214438182
#214438152
C61×#18202
#1179861