Finite Group Databasebeta

← #29577365

Subgroups — #29577365

Order 15616.

538 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 117 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#21454842C211yesmaximal, characteristic
2#21450993C211yesmaximal, characteristic
3#21445808C211yesmaximal, characteristic
4#21444500C211yesmaximal, characteristic, complemented
5#21443721C211yesmaximal, characteristic, complemented
6#21443720C211yesmaximal, characteristic, complemented
7C61×#1548C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C8×D244—22no—
10#16839827—22no—
11#16839815C2211yescharacteristic
12#16839786C2211yescharacteristic
13C2×#14519693—21no—
14C22⋊C8×D61—22no—
15#16839673C414yes—
16#16839663—21no—
17#16839627—21no—
18#16839619—22no—
19C4×C4⋊Dic61C2211yescharacteristic
20#16839530—21no—
21#16839511—22no—
22C2×C4⋊Q8×C61C2211yescharacteristic
23C61×#431C2211yescharacteristic
24C61×#423C2211yescharacteristic
25C61×#333—22no—
26C22×D244D411yescharacteristic
27C22×C4×D61D412yes—
28C2×#11758194—21no—
29C2×C8×D61—24no—
30C4×D244—21no—
31C4×D244—21no—
32C4×D244—21no—
33C4×D244—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758198—24no—
37C2×#11758198C2×C412yes—
38C2×C4×Dic61—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418D412yes—
42#14519693—42no—
43C22⋊#92418—21no—
44#14519689—24no—
45C4×#92418—21no—
46#14519684—24no—
47#14519684C2×C412yes—
48C22×Q8×C61C2×C412yes—
49C2×M16×C61—21no—
50C4⋊Q8×C61—21no—
51C4⋊Q8×C61—21no—
52C4⋊Q8×C61—21no—
53C4⋊Q8×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C42×C61C2311yescharacteristic
57C4⋊C8×C61—21no—
58C4⋊C8×C61—21no—
59C4⋊C8×C61D412yes—
60C22⋊C8×C61—21no—
61C4×C8×C61—21no—
62C23×D61C22⋊C412yes—
63C2×Q8×C61—22no—
64C2×Q8×C61—22no—
65C2×Q8×C61—82no—
66C22×C4×C61—22no—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—22no—
71C22×Dic61C22⋊C412yes—
72C2×D244—21no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—21no—
76C2×D244SD1612yes—
77C2×D244Q1612yes—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—44no—
81M16×C61—41no—
82M16×C61—41no—
83C2×C8×C61—22no—
84C2×C8×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C42×C61C2×D411yescharacteristic
90C42×C61C2×D411yescharacteristic
91C42×C61#4111yescharacteristic
92C42×C61#4111yescharacteristic
93#11758198—22no—
94#11758198—22no—
95#11758198—84no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61—42no—
99C4×Dic61—41no—
100C4×Dic61—41no—
101#11758194—41no—
102#11758194—41no—
103C2×#92418—22no—
104C2×#92418—41no—
105C8×D61—84no—
106C23×C61C2×C22⋊C411yescharacteristic
107C22×D61—22no—
108C22×D61—22no—
109C22×D61—42no—
110C22×D61—42no—
111C22×D61#10314yes—
112Q8×C61—82no—
113Q8×C61—82no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—44no—
117C2×C4×C61—42no—
118C2×C4×C61C4×D411yescharacteristic
119C2×C4×C61C4×D411yescharacteristic
120C2×C4×C61#12011yescharacteristic
121C2×C4×C61#12111yescharacteristic
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—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133D244—22no—
134D244—22no—
135D244—22no—
136D244—22no—
137C4×D61—44no—
138C4×D61—82no—
139C8×C61—42no—
140C8×C61—81no—
141#92418—42no—
142#92418—81no—
143#9391—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61#38611yescharacteristic
147C22×C61C2×#10311yescharacteristic
148C22×C61#41611yescharacteristic
149C2×D61—24no—
150C2×D61—42no—
151C2×D61—42no—
152C2×D61—44no—
153C2×D61—82no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—82no—
158C4×C61#45512yes—
159C4×C61#46511yescharacteristic
160C4×C61#46511yescharacteristic
161C4×C61#46711yescharacteristic
162C4×C61#46711yescharacteristic
163Dic61—42no—
164Dic61—82no—
165#2972—611no—
166#2563—611no—
167#1800—611no—
168#1548—611no—
169#1548D6111yescharacteristic, complemented
170#1303—611no—
171#1303—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#188511yescharacteristic
175C2×C61#189211yescharacteristic
176C2×C61#191411yescharacteristic
177D61—44no—
178C2×C4⋊Q8C2×D6111yescharacteristic
179C2×C4×D4—611no—
180C8×D4—1222no—
181C8×D4—1222no—
182#431—611no—
183#431C2×D6111yescharacteristic
184#423—611no—
185#423C2×D6111yescharacteristic
186C2×#100—1221no—
187C2×C22⋊C8—1222no—
188#393—614no—
189#390—1221no—
190#382—1221no—
191#380—1222no—
192C4×C4⋊C4—611no—
193#343—1221no—
194#333—22no—
195#333—1222no—
196C61#939111yescharacteristic, complemented, Sylow
197C22×Q8C4×D6112yes—
198C22×D4—611no—
199C23×C4—612no—
200C2×M16—21no—
201C2×M16—1221no—
202C22×C8—1224no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4×D4—1221no—
208C4×D4—1221no—
209C4×D4—1221no—
210C4×D4—1221no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4—1221no—
215C2×C22⋊C4—612no—
216C2×C22⋊C4—1224no—
217C2×C42—1221no—
218C2×C42C22×D6111yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—612no—
222C4⋊C8—1221no—
223C4⋊C8—1221no—
224C4⋊C8C22⋊D6112yes—
225#100—2442no—
226C22⋊C8—21no—
227C22⋊C8—1224no—
228C22⋊C8—1221no—
229C4×C8—21no—
230C4×C8—1221no—
231#95—612no—
232#95—1224no—
233C24—612no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—82no—
237C2×D4—612no—
238C2×D4—612no—
239C2×D4—1221no—
240C2×D4—1221no—
241C2×D4—1221no—
242C2×D4—1221no—
243C22×C4—22no—
244C22×C4—612no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—1222no—
248C22×C4—2444no—
249C22×C4C2×C4×D6111yescharacteristic
250C22×C4D4×D6111yescharacteristic
251C22×C4D4×D6111yescharacteristic
252M16—41no—
253M16—41no—
254M16—2441no—
255M16—2441no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—1222no—
259C2×C8—4884no—
260C2×C8—2441no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—2442no—
268C22⋊C4—1222no—
269C22⋊C4—1222no—
270C22⋊C4—4884no—
271C42—2441no—
272C42—2441no—
273C42#1175822211yescharacteristic
274C42D4×D6111yescharacteristic
275C42#1175822411yescharacteristic
276C42C2×C22⋊D6111yescharacteristic
277C23—614no—
278C23—1222no—
279C23—1222no—
280C23—2442no—
281C23—2442no—
282C23C22⋊C4×D6111yescharacteristic
283Q8—82no—
284Q8—82no—
285D4—1222no—
286D4—1222no—
287D4—1222no—
288D4—1222no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—1222no—
294C2×C4—4882no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2444no—
299C2×C4—2442no—
300C2×C4#1451973711yescharacteristic
301C2×C4C4×C22⋊D6111yescharacteristic
302C2×C4#1451974511yescharacteristic
303C2×C4#1451974811yescharacteristic
304C2×C4#1451975011yescharacteristic
305C2×C4#1451975911yescharacteristic
306C2×C4SD16×D6111yescharacteristic
307C2×C4Q16×D6111yescharacteristic
308C2×C4#1451979811yescharacteristic
309C2×C4#1451980011yescharacteristic
310C8—42no—
311C8—81no—
312C8—4881no—
313C8—2442no—
314C22—21no—
315C22—21no—
316C22—1224no—
317C22—4882no—
318C22—2444no—
319C22—2442no—
320C22—2442no—
321C22#1683963811yescharacteristic
322C22#103×D6111yescharacteristic
323C22#1683975711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—4882no—
329C4—2442no—
330C4#1683993812yes—
331C4#1683998811yescharacteristic
332C4#1683999411yescharacteristic
333C4#1684000011yescharacteristic
334C4#1684000211yescharacteristic
335C2—21no—
336C2—21no—
337C2—2444no—
338C2#2144620411yescharacteristic
339C2#2144623911yescharacteristic
340C2#2144636311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1548C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1548D61
hypocenterC61#9391
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
4. #21444500C2semidirect
5. #21443721C2semidirect
6. #21443720C2semidirect
7. C61×#1548C2semidirect
169. #1548D61semidirect
196. C61#9391semidirect

Maximal subgroups

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

subgroupindex
#214548422
#214509932
#214458082
#214445002
#214437212
#214437202
C61×#15482
#939161