Finite Group Databasebeta

← #29576769

Subgroups — #29576769

Order 15616.

488 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 113 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#21455618C211yesmaximal, characteristic
2#21454842C211yesmaximal, characteristic
3#21445830C211yesmaximal, characteristic
4#21444384C211yesmaximal, characteristic, complemented
5#21444221C211yesmaximal, characteristic, complemented
6#21443679C211yesmaximal, characteristic, complemented
7C61×#1301C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16840063—22no—
11#16840053C412yescomplemented
12#16840043C412yescomplemented
13C8×D244—22no—
14#16839827—22no—
15#16839815C2211yescharacteristic
16#16839804—21no—
17#16839768—21no—
18C2×#14519699—21no—
19C22⋊C8×D61—22no—
20#16839628C2211yescharacteristic
21#16839525—21no—
22#16839500—22no—
23#16839492—22no—
24C61×#431C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C2×C4⋊C8×C61C2211yescharacteristic
27C22×D244D411yescharacteristic
28C22×C4×D61—21no—
29C2×C8×D61—24no—
30#14519777C2×C411yescharacteristic
31#14519777C2×C411yescharacteristic
32#14519769—21no—
33#14519756—21no—
34#14519756—21no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244C2×C412yes—
40C4×D244D412yes—
41#14519736—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×#11758198—21no—
45#14519711—22no—
46#14519709—22no—
47C4⋊#92418—21no—
48#14519699—22no—
49#14519699—22no—
50#14519699—42no—
51C22⋊#92418—21no—
52#14519689—24no—
53C4×#92418D412yes—
54#14519684—21no—
55#14519684—21no—
56C2×M16×C61—21no—
57C22×C8×C61—21no—
58C2×C42×C61C2311yescharacteristic
59C4⋊C8×C61—21no—
60C4⋊C8×C61—21no—
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61C2×C412yes—
64C22⋊C8×C61D412yes—
65C4×C8×C61—21no—
66C23×D61—21no—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61#4111yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—41no—
72C2×D244—21no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244C22⋊C412yes—
76C2×D244SD1612yes—
77C2×D244Q1612yes—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—82no—
81C2×Dic122—21no—
82C2×Dic122—21no—
83M16×C61—41no—
84M16×C61—41no—
85C2×C8×C61—22no—
86C2×C8×C61—22no—
87C2×C8×C61—41no—
88C2×C8×C61—41no—
89C2×C8×C61—41no—
90C42×C61C22×C411yescharacteristic
91C42×C61C2×D411yescharacteristic
92C42×C61C2×D411yescharacteristic
93C42×C61#4111yescharacteristic
94C22⋊Dic61—41no—
95#11758198—22no—
96#11758198—22no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99C4⋊Dic61—21no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—41no—
103C4⋊Dic61—41no—
104C4⋊Dic61C22⋊C412yes—
105#11758196—41no—
106#11758196—41no—
107C4×Dic61—41no—
108C2×#92418—22no—
109C8×D61—84no—
110C23×C61#12111yescharacteristic
111C22×D61—22no—
112C22×D61—22no—
113C22×D61—22no—
114C22×D61—81no—
115C22×D61—81no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61C4×D411yescharacteristic
120C2×C4×C61C4×D411yescharacteristic
121C2×C4×C61#12011yescharacteristic
122C2×C4×C61#12111yescharacteristic
123C2×C4×C61#12311yescharacteristic
124C2×C4×C61C2×SD1611yescharacteristic
125C2×C4×C61C2×Q1611yescharacteristic
126C2×C4×C61#13511yescharacteristic
127C2×C4×C61#13511yescharacteristic
128C2×Dic61—22no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—81no—
132C2×Dic61—81no—
133C2×Dic61—81no—
134C2×Dic61—81no—
135D244—22no—
136D244—22no—
137D244—22no—
138D244#10314yes—
139C4×D61—44no—
140C4×D61—82no—
141Dic122—42no—
142Dic122—81no—
143C8×C61—42no—
144C8×C61—42no—
145C8×C61—81no—
146#92418—42no—
147#9303—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#38611yescharacteristic
151C22×C61#46511yescharacteristic
152C22×C61#46711yescharacteristic
153C2×D61—24no—
154C2×D61—42no—
155C2×D61—42no—
156C2×D61—82no—
157C2×D61—82no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61C2×#10311yescharacteristic
163C4×C61#41611yescharacteristic
164C4×C61#46511yescharacteristic
165C4×C61#46711yescharacteristic
166Dic61—42no—
167Dic61—81no—
168Dic61—81no—
169Dic61—161no—
170#3077—611no—
171#2972—611no—
172#1804—611no—
173#1518—611no—
174#1481—611no—
175#1301D6111yescharacteristic, complemented
176#1286—611no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61#188111yescharacteristic
180C2×C61#189211yescharacteristic
181C2×C61#190411yescharacteristic
182D61—44no—
183#533—611no—
184C2×C4×D4—611no—
185#478—1222no—
186#476—612no—
187#474—612no—
188C8×D4—1222no—
189C8×D4—1222no—
190#431—611no—
191#431C2×D6111yescharacteristic
192#428—1221no—
193#423C2×D6111yescharacteristic
194C2×C4⋊C8C2×D6111yescharacteristic
195#418—1221no—
196C2×#102—1221no—
197C2×C22⋊C8—1222no—
198#382—611no—
199#340—1221no—
200#328—1222no—
201Q8⋊C8—1222no—
202C61#930311yescharacteristic, complemented, Sylow
203C22×D4—611no—
204C23×C4—1221no—
205C2×M16—21no—
206C22×C8—21no—
207C22×C8—1224no—
208C4⋊Q8—611no—
209C4⋊Q8—611no—
210#125—1221no—
211C22⋊Q8—1221no—
212C22⋊Q8—1221no—
213C4×Q8—1221no—
214C4×Q8—1221no—
215C4×D4—612no—
216C4×D4—612no—
217C4×D4—1221no—
218C4×D4—1221no—
219#117—1221no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—1221no—
222C2×C22⋊C4—1221no—
223C2×C42C22×D6111yescharacteristic
224#107—1222no—
225#106—1222no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—1221no—
231C4⋊C8C4×D6112yes—
232#102—1222no—
233#102—1222no—
234#102—2442no—
235C22⋊C8—1224no—
236C22⋊C8—1221no—
237C22⋊C8D24412yes—
238C4×C8—21no—
239C4×C8—612no—
240#95—1221no—
241#95—1221no—
242C24—1221no—
243C2×Q8—1221no—
244C2×Q8—1221no—
245C2×D4—612no—
246C2×D4—612no—
247C2×D4—612no—
248C2×D4—1221no—
249C2×D4—1221no—
250C2×D4—1221no—
251C22×C4—1221no—
252C22×C4—1222no—
253C22×C4—1222no—
254C22×C4—4882no—
255C22×C4—2441no—
256C22×C4C2×D24411yescharacteristic
257C22×C4D4×D6111yescharacteristic
258C22×C4#1175822611yescharacteristic
259M16—41no—
260M16—41no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—1222no—
267C2×C8—4884no—
268C4⋊C4—612no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—1221no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C22⋊C4—1222no—
279C22⋊C4—1222no—
280C22⋊C4—2441no—
281C42—2441no—
282C42C2×C4×D6111yescharacteristic
283C42#1175822211yescharacteristic
284C42D4×D6111yescharacteristic
285C42D4×D6111yescharacteristic
286C23—1222no—
287C23—1222no—
288C23—1222no—
289C23—4881no—
290C23—4881no—
291C23#1451975211yescharacteristic
292Q8—4881no—
293Q8—2442no—
294D4—614no—
295D4—1222no—
296D4—1222no—
297D4—1222no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—1222no—
301C2×C4—4882no—
302C2×C4—4881no—
303C2×C4—4881no—
304C2×C4—4881no—
305C2×C4—4881no—
306C2×C4—2444no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4C22⋊C4×D6111yescharacteristic
310C2×C4#1451973911yescharacteristic
311C2×C4C4×D24411yescharacteristic
312C2×C4#1451974711yescharacteristic
313C2×C4#1451975011yescharacteristic
314C2×C4#1451976211yescharacteristic
315C2×C4SD16×D6111yescharacteristic
316C2×C4Q16×D6111yescharacteristic
317C2×C4#1451979611yescharacteristic
318C2×C4#1451979811yescharacteristic
319C8—42no—
320C8—42no—
321C8—81no—
322C8—2442no—
323C22—21no—
324C22—21no—
325C22—1224no—
326C22—4882no—
327C22—4882no—
328C22—2442no—
329C22—2442no—
330C22#1683964011yescharacteristic
331C22#1683999211yescharacteristic
332C22#1684000411yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—9761no—
338C4—4881no—
339C4—4881no—
340C4—2442no—
341C4#103×D6111yescharacteristic
342C4#1683975511yescharacteristic
343C4#1683999011yescharacteristic
344C4#1684000211yescharacteristic
345C2—21no—
346C2—21no—
347C2—2444no—
348C2#2144618211yescharacteristic
349C2#2144624111yescharacteristic
350C2#2144630511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1301C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1301D61
hypocenterC61#9303
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
4. #21444384C2semidirect
5. #21444221C2semidirect
6. #21443679C2semidirect
7. C61×#1301C2semidirect
11. #16840053C4semidirect
12. #16840043C4semidirect
175. #1301D61semidirect
202. C61#9303semidirect

Maximal subgroups

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

subgroupindex
#214556182
#214548422
#214458302
#214443842
#214442212
#214436792
C61×#13012
#930361