Finite Group Databasebeta

← #30604776

Subgroups — #30604776

Order 15616.

504 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462048C211yesmaximal, characteristic, complemented
2C2×#16840636C211yesmaximal, characteristic, complemented
3#21460807C212yesmaximal, complemented
4#21460805C212yesmaximal, complemented
5#21455396C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C2×#14519851—21no—
8#16840657—21no—
9#16840657C2212yes—
10#16840636—21no—
11#16840636C2212yes—
12C22×C8×D61C2211yescharacteristic
13#16840269—21no—
14#16840000C411yescharacteristic, complemented
15#16840000C2211yescharacteristic
16#16839995C411yescharacteristic, complemented
17#16839995C2211yescharacteristic
18#16839988—21no—
19#16839988—21no—
20#16839800C411yescharacteristic, complemented
21#16839758—21no—
22C2×#135×C61C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×C4⋊#6923—24no—
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923—42no—
27C2×C4⋊#6923—41no—
28#14519851—42no—
29C2×#11758187—22no—
30C22×C4×D61D411yescharacteristic
31C22×#92418D411yescharacteristic
32C2×C8×D61—42no—
33C2×C8×D61C2×C411yescharacteristic
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36C2×C8×D61C2311yescharacteristic
37#14519758—41no—
38#14519753—21no—
39#14519753—21no—
40#14519748—21no—
41#14519748—21no—
42#14519745—41no—
43C4×C22⋊D61—41no—
44C4×C22⋊D61—41no—
45#14519736—21no—
46#14519711C2×C411yescharacteristic, complemented
47#14519711C2×C411yescharacteristic, complemented
48#14519709—21no—
49#14519701—21no—
50#14519701—21no—
51#14519698—21no—
52#14519698—21no—
53C2×#41×C61—21no—
54#135×C61—42no—
55C2×Q16×C61C2×C411yescharacteristic, complemented
56C2×SD16×C61—21no—
57C2×D8×C61C2×C411yescharacteristic, complemented
58C22×C8×C61C2×C411yescharacteristic
59C23×D61D811yescharacteristic
60C22×#6923—44no—
61C22×#6923—42no—
62C22×#6923—82no—
63C22×#6923—82no—
64#41×C61—42no—
65#41×C61—42no—
66C2×Q8×C61—21no—
67C2×D4×C61—21no—
68C2×D4×C61—41no—
69C22×C4×C61—41no—
70C22×C4×C61C22⋊C411yescharacteristic
71C2×C22⋊D61—41no—
72C2×C22⋊D61—41no—
73C22×Dic61SD1611yescharacteristic
74C2×C4×D61—22no—
75C2×C4×D61—22no—
76C2×C4×D61C22⋊C411yescharacteristic
77C2×C4×D61D811yescharacteristic
78C2×C4×D61SD1611yescharacteristic
79C2×C4×D61C2×D411yescharacteristic
80C4⋊#6923—24no—
81C4⋊#6923—44no—
82C4⋊#6923—42no—
83Q16×C61—41no—
84SD16×C61—42no—
85D8×C61—41no—
86C2×C8×C61—22no—
87C2×C8×C61C22⋊C411yescharacteristic
88C2×C8×C61C22×C411yescharacteristic
89C22⋊Dic61—41no—
90C22⋊Dic61—41no—
91C22⋊Dic61—41no—
92#11758198—41no—
93#11758198—41no—
94C4⋊Dic61—21no—
95C4⋊Dic61—21no—
96#11758196—41no—
97#11758196—41no—
98C4×Dic61—41no—
99C2×#92418—22no—
100C2×#92418C22⋊C411yescharacteristic
101C2×#92418C2×D411yescharacteristic
102C8×D61—41no—
103C8×D61—41no—
104C8×D61—41no—
105C8×D61—41no—
106#11758187—42no—
107C23×C61—41no—
108C23×C61#10211yescharacteristic
109C22×D61—22no—
110C22×D61—22no—
111C22×D61—21no—
112C22×D61—21no—
113C22×D61#10211yescharacteristic
114C22×D61C2×D811yescharacteristic
115C2×#6923—44no—
116C2×#6923—88no—
117C2×#6923—82no—
118C2×#6923—82no—
119Q8×C61—41no—
120Q8×C61—41no—
121D4×C61—41no—
122D4×C61—41no—
123D4×C61—82no—
124C2×C4×C61—41no—
125C2×C4×C61—82no—
126C2×C4×C61#9912yes—
127C2×C4×C61#10211yescharacteristic
128C2×C4×C61#10412yes—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C22⋊D61—82no—
131C22⋊D61—81no—
132C22⋊D61—81no—
133C2×Dic61—22no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61#10211yescharacteristic
137C2×Dic61C2×SD1611yescharacteristic
138C4×D61—22no—
139C4×D61—22no—
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C8×C61—21no—
145C8×C61—21no—
146#92418—21no—
147#92418—21no—
148#11704—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61#32712yes—
155C22×C61C2×#10211yescharacteristic
156C2×D61—24no—
157C2×D61—24no—
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—21no—
162C2×D61#36012yes—
163C2×D61#50911yescharacteristic
164C2×D61#51011yescharacteristic
165#6923—84no—
166C4×C61—81no—
167C4×C61#32712yes—
168C4×C61C2×#9911yescharacteristic
169C4×C61C2×#10411yescharacteristic
170Dic61—21no—
171Dic61—21no—
172Dic61—81no—
173Dic61—81no—
174#3050—611no—
175#2112—611no—
176C2×#340—611no—
177#1348—612no—
178#1347—612no—
179C2×C61—22no—
180C2×C61—161no—
181C2×C61C2×#32711yescharacteristic
182C2×C61#213911yescharacteristic
183C2×C61C2×#36011yescharacteristic
184D61—21no—
185D61—22no—
186D61—21no—
187C2×#135#692311yescharacteristic, complemented
188C23×C8—611no—
189#525—1221no—
190C22×C4⋊C4—1221no—
191C2×M32—1221no—
192#467—611no—
193#467—611no—
194#466—611no—
195#466—611no—
196#465—1221no—
197#465—1221no—
198#427—611no—
199#416—1221no—
200C22⋊C16—612no—
201C22⋊C16—1221no—
202#340—612no—
203#340—1221no—
204C61#1170411yescharacteristic, complemented, Sylow
205C2×#41—21no—
206C23×C4—611no—
207C23×C4—2441no—
208#135—42no—
209C2×Q16C2×#692311yescharacteristic, complemented
210C2×SD16—21no—
211C2×D8C2×#692311yescharacteristic, complemented
212C22×C8—611no—
213C22×C8—611no—
214C22×C8—611no—
215C22×C8—611no—
216C22×C8—611no—
217C22×C8—2442no—
218C22×C8C2×#692311yescharacteristic
219#123—2441no—
220C22⋊Q8—1221no—
221C22⋊Q8—1221no—
222#121—1221no—
223#121—1221no—
224#120—2441no—
225C4×D4—2441no—
226C4×D4—2441no—
227#117—1221no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1224no—
230C2×C4⋊C4—2441no—
231C2×C4⋊C4—2442no—
232M32—2442no—
233C2×C16—1222no—
234#107—611no—
235#107—611no—
236#106—1221no—
237#103—1221no—
238#103—1221no—
239#102—1221no—
240#102—1221no—
241C24—611no—
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—2441no—
248C2×D4—2441no—
249C22×C4—41no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—611no—
254C22×C4—611no—
255C22×C4—1222no—
256C22×C4—1222no—
257C22×C4—4882no—
258C22×C4—4882no—
259C22×C4—2442no—
260C22×C4—2444no—
261C22×C4C22⋊#692311yescharacteristic
262Q16—41no—
263SD16—42no—
264D8—41no—
265C2×C8—22no—
266C2×C8—611no—
267C2×C8—611no—
268C2×C8—1222no—
269C2×C8—2441no—
270C2×C8—2441no—
271C2×C8—2441no—
272C2×C8—2441no—
273C2×C8C22⋊#692311yescharacteristic
274C2×C8C22×#692311yescharacteristic
275C4⋊C4—1224no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C4⋊C4—2442no—
281C4⋊C4—2444no—
282C22⋊C4—2441no—
283C22⋊C4—2441no—
284C22⋊C4—2441no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—2441no—
288C16—2442no—
289C23—41no—
290C23—611no—
291C23—611no—
292C23—1221no—
293C23—1222no—
294C23—1221no—
295C23—1222no—
296C23#1451983911yescharacteristic
297Q8—41no—
298Q8—41no—
299D4—41no—
300D4—41no—
301D4—82no—
302D4—4881no—
303D4—4881no—
304D4—4882no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—1222no—
310C2×C4—1221no—
311C2×C4—1221no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1222no—
315C2×C4—1222no—
316C2×C4—4882no—
317C2×C4—4888no—
318C2×C4—4882no—
319C2×C4—2444no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4#1451983512yes—
323C2×C4#1451983911yescharacteristic
324C2×C4#1451984112yes—
325C2×C4C2×C22⋊#692311yescharacteristic
326C8—21no—
327C8—21no—
328C8—1221no—
329C8—1221no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—611no—
336C22—611no—
337C22—612no—
338C22—1224no—
339C22—1224no—
340C22—1221no—
341C22—1221no—
342C22—1221no—
343C22—1221no—
344C22#1684061312yes—
345C22C2×#1451983911yescharacteristic
346C4—81no—
347C4—1221no—
348C4—1221no—
349C4—4884no—
350C4—4881no—
351C4—4881no—
352C4#1684061312yes—
353C4C2×#1451983511yescharacteristic
354C4C2×#1451984111yescharacteristic
355C2—22no—
356C2—161no—
357C2—1221no—
358C2—1221no—
359C2—1222no—
360C2C2×#1684061311yescharacteristic
361C2#2146212711yescharacteristic
362C2D61⋊#36011yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC2×#135×C61C4
socleC22×C61C2×#102
prime radical#21455396C2
cosocleC2×C8×D61C23
hypercenterC2×#135#6923
hypocenterC61#11704
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #21462048C2semidirect
2. C2×#16840636C2semidirect
3. #21460807C2semidirect
4. #21460805C2semidirect
14. #16840000C4semidirect
16. #16839995C4semidirect
20. #16839800C4semidirect
22. C2×#135×C61C4semidirect
46. #14519711C2×C4semidirect
47. #14519711C2×C4semidirect
55. C2×Q16×C61C2×C4semidirect
57. C2×D8×C61C2×C4semidirect
187. C2×#135#6923semidirect
204. C61#11704semidirect
209. C2×Q16C2×#6923semidirect
211. C2×D8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214620482
C2×#168406362
#214608072
#214608052
#214553962
#1170461