Finite Group Databasebeta

← #29576771

Subgroups — #29576771

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#21455612C211yesmaximal, characteristic
2#21454842C211yesmaximal, characteristic
3#21445830C211yesmaximal, characteristic
4#21444383C211yesmaximal, characteristic, complemented
5#21443716C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#1481C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16840059—22no—
10#16840049C412yescomplemented
11#16840039C412yescomplemented
12C8×D244—22no—
13#16839827—22no—
14#16839815C2211yescharacteristic
15#16839805—21no—
16#16839786C2211yescharacteristic
17#16839766—21no—
18C2×#14519699—21no—
19C22⋊C8×D61—22no—
20#16839628C2211yescharacteristic
21#16839525—21no—
22#16839499—22no—
23C61×#533C2211yescharacteristic
24C61×#431C2211yescharacteristic
25C2×C4⋊C8×C61C2211yescharacteristic
26C61×Q8⋊C8—22no—
27C22×D244D411yescharacteristic
28C22×C4×D61—21no—
29C2×#11758194—21no—
30C2×C8×D61—24no—
31C4×D244—21no—
32C4×D244—21no—
33C4×D244C2×C412yes—
34C4×D244D412yes—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758198—21no—
38#14519710—22no—
39#14519708—22no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418C2×C412yes—
43#14519699—42no—
44#14519697—22no—
45#14519697—22no—
46C22⋊#92418D412yes—
47#14519689—24no—
48C4×#92418—21no—
49#14519684—21no—
50#14519684—21no—
51C22×C8×C61—21no—
52C4⋊Q8×C61C2×C411yescharacteristic
53C4⋊Q8×C61C2×C411yescharacteristic
54#125×C61—21no—
55C22⋊Q8×C61—21no—
56C22⋊Q8×C61—21no—
57C4×Q8×C61—21no—
58C4×Q8×C61—21no—
59#117×C61—21no—
60C2×C42×C61C2311yescharacteristic
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C22⋊C8×C61—21no—
65C4×C8×C61D412yes—
66C23×D61—21no—
67C2×Q8×C61—21no—
68C2×Q8×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4111yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—41no—
74C2×D244—21no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244C22⋊C412yes—
78C2×D244SD1612yes—
79C2×D244Q1612yes—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—82no—
83C2×C8×C61—22no—
84C2×C8×C61—41no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C42×C61—41no—
95C42×C61C22×C411yescharacteristic
96C42×C61C2×D411yescharacteristic
97C42×C61C2×D411yescharacteristic
98C42×C61#4111yescharacteristic
99#11758198—22no—
100#11758198—22no—
101C4⋊Dic61—21no—
102C4⋊Dic61—41no—
103C4⋊Dic61—41no—
104C4⋊Dic61C22⋊C412yes—
105#11758194—41no—
106#11758194—41no—
107C2×#92418—22no—
108C2×#92418—22no—
109C8×D61—84no—
110C23×C61#12111yescharacteristic
111C22×D61—22no—
112C22×D61—22no—
113C22×D61—22no—
114C22×D61—81no—
115C22×D61—81no—
116Q8×C61—42no—
117Q8×C61—81no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61C2×C22⋊C411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×C4×C61#12011yescharacteristic
126C2×C4×C61#12111yescharacteristic
127C2×C4×C61#12311yescharacteristic
128C2×C4×C61C2×SD1611yescharacteristic
129C2×C4×C61C2×Q1611yescharacteristic
130C2×C4×C61#13511yescharacteristic
131C2×C4×C61#13511yescharacteristic
132C2×Dic61—22no—
133C2×Dic61—81no—
134C2×Dic61—81no—
135C2×Dic61—81no—
136C2×Dic61—81no—
137D244—22no—
138D244—22no—
139D244—22no—
140D244#10314yes—
141C4×D61—44no—
142C4×D61—82no—
143C8×C61—42no—
144C8×C61—81no—
145#92418—42no—
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×C61—81no—
163C4×C61—81no—
164C4×C61C2×#10311yescharacteristic
165C4×C61#41611yescharacteristic
166C4×C61#46511yescharacteristic
167C4×C61#46711yescharacteristic
168Dic61—42no—
169Dic61—161no—
170#3077—611no—
171#2972—611no—
172#1804—611no—
173#1518—611no—
174#1481D6111yescharacteristic, complemented
175#1301—611no—
176#1286—611no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61#188111yescharacteristic
180C2×C61#189211yescharacteristic
181C2×C61#190411yescharacteristic
182D61—44no—
183#533C2×D6111yescharacteristic
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#423—611no—
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—22no—
202C61#930311yescharacteristic, complemented, Sylow
203C22×D4—611no—
204C23×C4—1221no—
205C2×M16—1221no—
206C22×C8—21no—
207C22×C8—1224no—
208C4⋊Q8C4×D6111yescharacteristic
209C4⋊Q8C4×D6111yescharacteristic
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×D4—612no—
216C4×D4—612no—
217C4×D4—1221no—
218C4×D4—1221no—
219#117—21no—
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—612no—
230C4⋊C8—1221no—
231C4⋊C8—1221no—
232#102—1222no—
233#102—1222no—
234#102—2442no—
235C22⋊C8—21no—
236C22⋊C8—612no—
237C22⋊C8—1224no—
238C4×C8—1221no—
239C4×C8C22⋊D6112yes—
240#95—1221no—
241#95—1221no—
242C24—1221no—
243C2×Q8—21no—
244C2×Q8—21no—
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×C4#1175822211yescharacteristic
257C22×C4D4×D6111yescharacteristic
258C22×C4D4×D6111yescharacteristic
259M16—2441no—
260M16—2441no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—1222no—
266C2×C8—1222no—
267C2×C8—4884no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—612no—
275C4⋊C4—1221no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C22⋊C4—41no—
279C22⋊C4—1222no—
280C22⋊C4—1222no—
281C42—41no—
282C42C2×C4×D6111yescharacteristic
283C42D4×D6111yescharacteristic
284C42#1175822411yescharacteristic
285C42C2×C22⋊D6111yescharacteristic
286C23—1222no—
287C23—1222no—
288C23—1222no—
289C23—4881no—
290C23—4881no—
291C23#1451975011yescharacteristic
292Q8—42no—
293Q8—81no—
294D4—614no—
295D4—1222no—
296D4—1222no—
297D4—1222no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—1222no—
303C2×C4—4882no—
304C2×C4—4881no—
305C2×C4—4881no—
306C2×C4—4881no—
307C2×C4—4881no—
308C2×C4—2444no—
309C2×C4C22⋊C4×D6111yescharacteristic
310C2×C4#1451973711yescharacteristic
311C2×C4C4×C22⋊D6111yescharacteristic
312C2×C4#1451974511yescharacteristic
313C2×C4#1451974811yescharacteristic
314C2×C4#1451975911yescharacteristic
315C2×C4SD16×D6111yescharacteristic
316C2×C4Q16×D6111yescharacteristic
317C2×C4#1451979811yescharacteristic
318C2×C4#1451980011yescharacteristic
319C8—42no—
320C8—81no—
321C8—2442no—
322C8—2442no—
323C22—21no—
324C22—21no—
325C22—1224no—
326C22—4882no—
327C22—4882no—
328C22—2442no—
329C22—2442no—
330C22#1683963811yescharacteristic
331C22#1683999411yescharacteristic
332C22#1684000211yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—81no—
338C4—81no—
339C4—9761no—
340C4—2442no—
341C4#103×D6111yescharacteristic
342C4#1683975711yescharacteristic
343C4#1683998811yescharacteristic
344C4#1684000011yescharacteristic
345C2—21no—
346C2—21no—
347C2—2444no—
348C2#2144618411yescharacteristic
349C2#2144623911yescharacteristic
350C2#2144630311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1481C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1481D61
hypocenterC61#9303
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
4. #21444383C2semidirect
5. #21443716C2semidirect
6. #21443678C2semidirect
7. C61×#1481C2semidirect
10. #16840049C4semidirect
11. #16840039C4semidirect
174. #1481D61semidirect
202. C61#9303semidirect

Maximal subgroups

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

subgroupindex
#214556122
#214548422
#214458302
#214443832
#214437162
#214436782
C61×#14812
#930361