Finite Group Databasebeta

← #29598682

Subgroups — #29598682

Order 15616.

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

normal subgroups only

This group has 556 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#21450772C211yesmaximal, characteristic
2C4×#14519692C211yesmaximal, characteristic
3#21443943C214yesmaximal
4C61×#1799C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic, complemented
6C2×#14519756—21no—
7C2×#14519756—21no—
8C2×#14519692C2212yes—
9#16839698—21no—
10#16839698—21no—
11#16839665C2212yes—
12#16839631C412yescomplemented
13C4×C4⋊Dic61C411yescharacteristic, complemented
14C4×C22⋊Dic61—21no—
15#16839529—21no—
16C2×C4×D4×C61C2211yescharacteristic
17C2×#99×C61C2212yes—
18C61×#380—21no—
19C61×#342—21no—
20C22×Dic122—21no—
21#14519777—41no—
22#14519777—41no—
23#14519756—44no—
24#14519756—44no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C412yes—
28C2×#11758196—22no—
29C2×#11758196—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C4×Dic61—41no—
34C2×C4×Dic61C2×C412yes—
35#14519692—22no—
36#14519692Q814yescomplemented
37#14519684—21no—
38#14519684—41no—
39C22×D4×C61C2311yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C4×D4×C61—41no—
43C4×D4×C61—41no—
44C2×C4⋊C4×C61—41no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61D411yescharacteristic
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C42×C61C2×C411yescharacteristic
50#99×C61—24no—
51#95×C61—41no—
52C24×C61C2×D411yescharacteristic
53C24×C61C2×D411yescharacteristic
54C2×D4×C61—41no—
55C2×D4×C61—41no—
56C2×D4×C61C2×Q812yes—
57C2×D4×C61#4112yes—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—42no—
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×C61C22×C411yescharacteristic
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—42no—
73C2×Dic122—42no—
74C2×Dic122—42no—
75C2×Dic122—41no—
76C2×Dic122—41no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—82no—
79C22⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C42×C61—21no—
84C42×C61—21no—
85C22⋊Dic61—24no—
86C22⋊Dic61—44no—
87C22⋊Dic61—44no—
88C22⋊Dic61—42no—
89C4⋊Dic61—24no—
90C4⋊Dic61—42no—
91C4⋊Dic61—42no—
92#11758196—44no—
93#11758196—44no—
94C4×Dic61—42no—
95C4×Dic61—82no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C22⋊Q812yes—
106C23×C61#12411yescharacteristic
107C23×C61C4⋊Q811yescharacteristic
108D4×C61—42no—
109D4×C61—42no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—81no—
122C2×C4×C61—81no—
123C2×C4×C61—82no—
124C2×C4×C61—82no—
125C2×C4×C61—82no—
126C2×C4×C61#9911yescharacteristic
127C2×C4×C61#9911yescharacteristic
128C2×C4×C61C4×Q812yes—
129C2×Dic61—24no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—44no—
135C2×Dic61—42no—
136C2×Dic61—82no—
137C2×Dic61—82no—
138Dic122—82no—
139Dic122—82no—
140#13298—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—22no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—42no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61#39111yescharacteristic
156C22×C61C2×#9911yescharacteristic
157C22×C61#41011yescharacteristic
158C4×C61—21no—
159C4×C61—22no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—41no—
163C4×C61—162no—
164C4×C61#35312yes—
165Dic61—44no—
166Dic61—84no—
167Dic61—82no—
168#2542—611no—
169#1799D6111yescharacteristic
170C4×#99—611no—
171#1393—614no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—42no—
175C2×C61—41no—
176C2×C61—41no—
177C2×C61#199611yescharacteristic
178C2×C61C2×#35311yescharacteristic
179C2×C61#212911yescharacteristic
180C2×C4⋊Q8—611no—
181C2×C22⋊Q8—1221no—
182C2×C22⋊Q8—1221no—
183C2×C4×D4C2×D6111yescharacteristic
184C2×#99—612no—
185C2×#99C2×D6112yes—
186#399—1221no—
187#399—1221no—
188#390—612no—
189#384—612no—
190#380—21no—
191C4×C4⋊C4—611no—
192C4×C22⋊C4—1221no—
193#342—21no—
194#342—1221no—
195C61#1329811yescharacteristic, complemented, Sylow
196C22×Q8—1221no—
197C22×D4C22×D6111yescharacteristic
198C23×C4D24411yescharacteristic
199C23×C4C22⋊D6111yescharacteristic
200C4⋊Q8—2441no—
201C4⋊Q8—2441no—
202C22⋊Q8—2444no—
203C22⋊Q8—2444no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—612no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1222no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4C4×D6111yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1222no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4D24411yescharacteristic
218C2×C22⋊C4C22⋊D6111yescharacteristic
219C2×C42—612no—
220C2×C42—2441no—
221C2×C42C4×D6111yescharacteristic
222#99—24no—
223#99—614no—
224#99—1222no—
225#95—41no—
226#95—1221no—
227#95—2441no—
228C24C2×D24411yescharacteristic
229C24C2×C22⋊D6111yescharacteristic
230C2×Q8—2442no—
231C2×Q8—2442no—
232C2×Q8—2441no—
233C2×Q8—2441no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4#1175822412yes—
237C2×D4Q8×D6112yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—2442no—
248C22×C4#1175819811yescharacteristic
249C22×C4#1175819811yescharacteristic
250C22×C4#1175819811yescharacteristic
251C22×C4#1175819811yescharacteristic
252C22×C4C2×C4×D6111yescharacteristic
253C4⋊C4—41no—
254C4⋊C4—82no—
255C4⋊C4—1224no—
256C4⋊C4—2442no—
257C4⋊C4—2442no—
258C4⋊C4—2444no—
259C4⋊C4—2444no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—1224no—
265C22⋊C4—2444no—
266C22⋊C4—2444no—
267C22⋊C4—2442no—
268C42—21no—
269C42—21no—
270C42—4882no—
271C42—2442no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×#1175819811yescharacteristic
281C23#1451975712yes—
282C23#1451976411yescharacteristic
283C23#1451977511yescharacteristic
284Q8—4882no—
285Q8—4882no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—1224no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—2444no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4—2442no—
312C2×C4—2442no—
313C2×C4#1451969111yescharacteristic
314C2×C4#1451969111yescharacteristic
315C2×C4#1451974212yes—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22#1683966711yescharacteristic
331C22C2×#1451969111yescharacteristic
332C22#1683973511yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—162no—
339C4—4884no—
340C4—4882no—
341C4—2444no—
342C4#1683955412yes—
343C2—21no—
344C2—21no—
345C2—41no—
346C2—42no—
347C2—41no—
348C2#2144680311yescharacteristic
349C2C2×#1683955411yescharacteristic
350C2#2144749311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radical, cosocleC22×D4×C61C23
hypercenter#1799D61
hypocenterC61#13298
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C2×#14519777C4semidirect
12. #16839631C4semidirect
13. C4×C4⋊Dic61C4semidirect
36. #14519692Q8semidirect
195. C61#13298semidirect

Maximal subgroups

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

subgroupindex
#214507722
C4×#145196922
#214439432
C61×#17992
#1329861