Finite Group Databasebeta

← #29568575

Subgroups — #29568575

Order 15616.

640 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 111 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#21454164C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic, complemented
3#21443639C214yesmaximal, complemented
4C61×#1749C211yesmaximal, characteristic, complemented
5C2×#14519774C411yescharacteristic
6C2×#14519751—21no—
7C2×#14519751—21no—
8C2×#14519751C412yes—
9C2×C22⋊#92418—21no—
10#16839664—21no—
11#16839664—21no—
12#16839664C2212yes—
13#16839632C411yescharacteristic
14#16839517—21no—
15#16839517C2212yes—
16C22×C42×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C2×#97×C61—21no—
19C61×#336C2212yes—
20C22×D244—21no—
21C22×D244—21no—
22C22×D244C2×C412yes—
23C22×C22⋊D61—22no—
24C22×#92418—22no—
25#14519774—42no—
26#14519751—44no—
27#14519751—44no—
28#14519751—44no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61C2×C412yes—
32C2×#11758198—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C22⋊#92418—42no—
36C23×C4×C61C2×C411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C22×C8×C61—22no—
40C2×C42×C61—42no—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61D411yescharacteristic
43C2×C42×C61D411yescharacteristic
44C2×C42×C61C2311yescharacteristic
45C22⋊C8×C61—24no—
46#97×C61—42no—
47C24×C61C22⋊C411yescharacteristic
48C23×D61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61—41no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C2×C22⋊D61—44no—
65C2×C22⋊D61—44no—
66C2×C22⋊D61—44no—
67C22×Dic61—22no—
68C2×D244—24no—
69C2×D244—42no—
70C2×D244—42no—
71C2×D244—42no—
72C2×D244—42no—
73C2×D244—42no—
74C2×D244—42no—
75C2×D244—42no—
76C2×D244—42no—
77C2×C8×C61—24no—
78C2×C8×C61—42no—
79C2×C8×C61—42no—
80C42×C61—22no—
81C42×C61—22no—
82C42×C61—41no—
83C42×C61—41no—
84#11758198—44no—
85#11758198—44no—
86#11758198—44no—
87C4⋊Dic61—42no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90C2×#92418—42no—
91C2×#92418—42no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61#10012yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—42no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—42no—
107C22×D61—42no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
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—44no—
122C2×C4×C61—44no—
123C2×C4×C61—44no—
124C2×C4×C61#9912yes—
125C2×C4×C61#10412yes—
126C2×C4×C61#13112yes—
127C2×C4×C61AGL(1,ℤ/8)12yes—
128C2×C4×C61AGL(1,ℤ/8)12yes—
129C22⋊D61—88no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133D244—84no—
134D244—84no—
135D244—84no—
136C8×C61—44no—
137#92418—82no—
138#7704—611nomaximal, Sylow
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61#40811yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#10011yescharacteristic
149C22×C61#41811yescharacteristic
150C22×C61#41811yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C22×C61#42111yescharacteristic
153C2×D61—82no—
154C2×D61—82no—
155C2×D61—82no—
156C2×D61—82no—
157C2×D61—82no—
158C2×D61—82no—
159C2×D61—82no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—22no—
163C4×C61—22no—
164C4×C61—22no—
165C4×C61—41no—
166Dic61—82no—
167#2856—611no—
168#1817—611no—
169#1749D6111yescharacteristic, complemented
170#1270—614no—
171C2×C61—42no—
172C2×C61#147111yescharacteristic
173C2×C61#148411yescharacteristic
174C2×C61#149611yescharacteristic
175C2×C61#150711yescharacteristic
176C2×C61#151011yescharacteristic
177C2×C61#153011yescharacteristic
178C2×C61#155511yescharacteristic
179D61—162no—
180C2×#127—611no—
181C2×#121—612no—
182C2×#121—1221no—
183C2×#121—1221no—
184C22×C42C2×D6111yescharacteristic
185C2×C22⋊C8—21no—
186C2×C22⋊C8—1221no—
187C2×#97—21no—
188#390—612no—
189#390—1221no—
190#390—1221no—
191#384—611no—
192#336—612no—
193#336—1221no—
194#336C2×D6112yes—
195C61#770411yescharacteristic, complemented, Sylow
196C22×D4—612no—
197C22×D4—1222no—
198C22×D4—1221no—
199C22×D4—1221no—
200C23×C4C4×D6111yescharacteristic
201C23×C4D24411yescharacteristic
202C23×C4C22⋊D6111yescharacteristic
203C22×C8—22no—
204C22×C8—1222no—
205#127—2442no—
206#121—2444no—
207#121—2444no—
208#121—2444no—
209C2×C4⋊C4—612no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C42—42no—
216C2×C42C4×D6111yescharacteristic
217C2×C42D24411yescharacteristic
218C2×C42C22⋊D6111yescharacteristic
219C2×C42C22×D6111yescharacteristic
220C22⋊C8—24no—
221C22⋊C8—2442no—
222#97—42no—
223C24—1222no—
224C24#1175819811yescharacteristic
225C2×D4—1224no—
226C2×D4—2442no—
227C2×D4—2442no—
228C2×D4—2442no—
229C2×D4—2442no—
230C2×D4—2442no—
231C2×D4—2442no—
232C2×D4—2444no—
233C2×D4—2444no—
234C2×D4—2444no—
235C2×D4—2442no—
236C2×D4—2442no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—1222no—
247C22×C4#1175819811yescharacteristic
248C22×C4#1175819811yescharacteristic
249C22×C4#1175819811yescharacteristic
250C22×C4C2×C4×D6111yescharacteristic
251C22×C4C2×D24411yescharacteristic
252C22×C4C2×C22⋊D6111yescharacteristic
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2442no—
262C22⋊C4—2444no—
263C22⋊C4—2444no—
264C22⋊C4—2444no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—2442no—
276C23—2442no—
277C23—2442no—
278C23—2442no—
279C23—2442no—
280C23—2442no—
281C23—2442no—
282C23#1451969312yes—
283C23C2×#1175819811yescharacteristic
284D4—4884no—
285D4—4884no—
286D4—4884no—
287D4—4888no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—2442no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4#1451969112yes—
308C2×C4#1451970512yes—
309C2×C4#1451978512yes—
310C2×C4#1451980212yes—
311C2×C4#1451980512yes—
312C8—44no—
313C8—4882no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4882no—
322C22—4882no—
323C22—4882no—
324C22—4882no—
325C22—4882no—
326C22—4882no—
327C22—4882no—
328C22#1683973011yescharacteristic
329C22C2×#1451969111yescharacteristic
330C22C2×#1451969311yescharacteristic
331C22#1683976611yescharacteristic
332C22#1683976811yescharacteristic
333C22C2×#1451970511yescharacteristic
334C22#1683977811yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—4882no—
342C2—42no—
343C2—9762no—
344C2#2144418011yescharacteristic
345C2#2144423711yescharacteristic
346C2#2144428711yescharacteristic
347C2#2144433411yescharacteristic
348C2#2144434911yescharacteristic
349C2#2144442911yescharacteristic
350C2#2144453211yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1749C2
socleC23×C61C2×C22⋊C4
prime radical#21454164C2
cosocleC2×C42×C61C23
hypercenter#1749D61
hypocenterC61#7704
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #21445887C2semidirect
3. #21443639C2semidirect
4. C61×#1749C2semidirect
169. #1749D61semidirect
195. C61#7704semidirect

Maximal subgroups

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

subgroupindex
#214541642
#214458872
#214436392
C61×#17492
#770461