Finite Group Databasebeta

← #30599727

Subgroups — #30599727

Order 15616.

438 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462464C211yesmaximal, characteristic
2#21461691C211yesmaximal, characteristic
3#21461387C211yesmaximal, characteristic, complemented
4#21461375C211yesmaximal, characteristic, complemented
5#21460716C211yesmaximal, characteristic, complemented
6#21460654C211yesmaximal, characteristic, complemented
7#21444198C211yesmaximal, characteristic
8C2×#14519864—21no—
9#16840769—21no—
10#16840768—22no—
11#16840766C2211yescharacteristic
12#16840766C2211yescharacteristic
13#16840765—21no—
14D61⋊(C4⋊C8)C2211yescharacteristic
15C2×C4⋊#92420—21no—
16C2×#14519841—21no—
17C4×C4⋊#6923C2211yescharacteristic
18#16840632—21no—
19#16840624—22no—
20#16840624—21no—
21#16840623—21no—
22C4×#11758224C2211yescharacteristic
23#16839814C2211yescharacteristic
24C4×C8×D61C2211yescharacteristic
25#16839494C411yescharacteristic, complemented
26#16839492C411yescharacteristic, complemented
27D4⋊#92418C411yescharacteristic, complemented
28C61×D4⋊C8C411yescharacteristic, complemented
29#14519864—42no—
30#14519864—42no—
31C2×#11758213—21no—
32C2×#11758213—21no—
33C2×#11758217—22no—
34C2×D61⋊C8—21no—
35C22×#92420—22no—
36C2×C4⋊#6923—21no—
37C2×C4×#6923—21no—
38#14519844—22no—
39#14519844—21no—
40C4⋊#92420—22no—
41C4⋊#92420—21no—
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44C4⋊#92420D411yescharacteristic
45C4⋊#92420D411yescharacteristic
46C4⋊#92420D412yes—
47#14519841—42no—
48(C2×#6923)⋊C4—21no—
49C2×#11758224D411yescharacteristic
50C2×C8×D61—21no—
51C4×Dic122C2×C411yescharacteristic
52#14519742Q811yescharacteristic
53D4×Dic61Q811yescharacteristic
54C4×C22⋊D61—21no—
55#14519737—21no—
56#14519735D411yescharacteristic
57#14519733—21no—
58C2×C4×Dic61—21no—
59C42×D61C2311yescharacteristic
60C4⋊#92418C2×C411yescharacteristic
61#14519689—21no—
62C4×#92418C2×C411yescharacteristic
63C8×Dic61—21no—
64C8×Dic61—21no—
65C4×D4×C61C2×C411yescharacteristic
66C4⋊C8×C61C2×C411yescharacteristic
67C4×C8×C61C2×C411yescharacteristic
68C22×#6923—41no—
69C2×D4×C61C4⋊C411yescharacteristic
70C22×C4×C61—21no—
71C2×C22⋊D61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74#11758224—21no—
75#11758224—21no—
76#11758224SD1612yes—
77#11758224SD1612yes—
78C2×C4×D61C2×D411yescharacteristic
79C2×C4×D61#4111yescharacteristic
80C2×C4×D61#4111yescharacteristic
81C2×Dic122C4⋊C411yescharacteristic
82#11758217—44no—
83C2×#92420—21no—
84C2×#92420—22no—
85C2×#92420—21no—
86C2×#92420—44no—
87C2×#92420—41no—
88C4⋊#6923—22no—
89C4⋊#6923—41no—
90C4×#6923—41no—
91C4×#6923—41no—
92#11758213—41no—
93#11758213—41no—
94#11758213—41no—
95#11758213—41no—
96D61⋊C8—41no—
97D61⋊C8—41no—
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C4⋊C4×C61C4⋊C411yescharacteristic
101C22⋊C4×C61—21no—
102C42×C61C22×C411yescharacteristic
103C22⋊Dic61—21no—
104#11758198—21no—
105C4⋊Dic61C4⋊C411yescharacteristic
106#11758196—21no—
107#11758196—21no—
108C4×Dic61—21no—
109C4×Dic61—21no—
110C4×Dic61—21no—
111C4×Dic61C2×D411yescharacteristic
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61C2×Q811yescharacteristic
114C2×#92418—21no—
115C2×#92418—21no—
116C8×D61—42no—
117C23×C61—21no—
118C22×D61#12311yescharacteristic
119C2×#6923—42no—
120C2×#6923—81no—
121C2×#6923—81no—
122D4×C61—21no—
123D4×C61#10612yes—
124C2×C4×C61—21no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61C2×C4⋊C411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C22⋊D61—22no—
131C22⋊D61—22no—
132C2×Dic61—21no—
133C2×Dic61—22no—
134C2×Dic61—22no—
135C2×Dic61—21no—
136C2×Dic61—81no—
137C2×Dic61#12011yescharacteristic
138C2×Dic61C22⋊Q811yescharacteristic
139C2×Dic61C22⋊Q811yescharacteristic
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—41no—
143C4×D61C2×SD1611yescharacteristic
144C4×D61C2×SD1611yescharacteristic
145C4×D61AGL(1,ℤ/8)11yescharacteristic
146C4×D61#13711yescharacteristic
147Dic122—21no—
148Dic122#10612yes—
149#92420—41no—
150#92420—41no—
151#92420—42no—
152#92420—81no—
153C8×C61—41no—
154C8×C61—41no—
155#92418—22no—
156#92418—41no—
157#8904—611nomaximal, Sylow
158C22×C61—22no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61#38511yescharacteristic
162C2×D61—21no—
163C2×D61—21no—
164C2×D61#48111yescharacteristic
165C2×D61#48311yescharacteristic
166#6923—82no—
167C4×C61—21no—
168C4×C61—41no—
169C4×C61C2×#10611yescharacteristic
170C4×C61#42811yescharacteristic
171C4×C61#44311yescharacteristic
172C4×C61#44411yescharacteristic
173Dic61—22no—
174Dic61—21no—
175Dic61—21no—
176Dic61—41no—
177Dic61#44811yescharacteristic
178Dic61#45011yescharacteristic
179#2906—611no—
180#1806—611no—
181#1570—611no—
182#1566—611no—
183#1476—611no—
184#1303—611no—
185#1279—611no—
186C2×C61—42no—
187C2×C61#185311yescharacteristic
188C2×C61#186111yescharacteristic
189C2×C61#186611yescharacteristic
190D61—21no—
191D61—21no—
192C2×#131—1221no—
193C4×#41—611no—
194#431—611no—
195#424—1222no—
196#424—1221no—
197#423—611no—
198#423—611no—
199#423—1221no—
200C2×C4⋊C8—611no—
201C2×C4⋊C8—1221no—
202C2×#104—1221no—
203C2×C4×C8—611no—
204C4×C4⋊C4—611no—
205#339—1221no—
206#334—1222no—
207#334—1221no—
208#334—1221no—
209Q8⋊C8—611no—
210Q8⋊C8—611no—
211D4⋊C8—611no—
212D4⋊C8#692311yescharacteristic, complemented
213C61#890411yescharacteristic, complemented, Sylow
214C2×#41—611no—
215#131—2442no—
216#131—2442no—
217C2×M16—1221no—
218C2×M16—1222no—
219C2×M16—1221no—
220C22×C8—1221no—
221C22×C8—1221no—
222C22×C8—1222no—
223C4×Q8—611no—
224C4×Q8—611no—
225C4×D4—611no—
226C4×D4—1221no—
227C4×D4—1221no—
228C4×D4C2×#692311yescharacteristic
229#117—611no—
230#117—1221no—
231C2×C4⋊C4—1221no—
232C2×C42—611no—
233C2×C42—1221no—
234C2×C42—1221no—
235C4⋊C8—611no—
236C4⋊C8—611no—
237C4⋊C8—612no—
238C4⋊C8—611no—
239C4⋊C8—1221no—
240C4⋊C8—1221no—
241C4⋊C8—1221no—
242C4⋊C8—1221no—
243C4⋊C8—1222no—
244C4⋊C8—1222no—
245C4⋊C8C2×#692311yescharacteristic
246#104—2442no—
247C22⋊C8—1221no—
248C4×C8—611no—
249C4×C8—1221no—
250C4×C8—1221no—
251C4×C8C2×#692311yescharacteristic
252#95—1221no—
253#41—612no—
254#41—612no—
255#41—1221no—
256#41—1221no—
257C2×Q8—611no—
258C2×D4—1221no—
259C2×D4C4⋊#692311yescharacteristic
260C22×C4—21no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—2441no—
267M16—2441no—
268M16—2441no—
269M16—2441no—
270M16—2441no—
271M16—2444no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—1221no—
275C2×C8—1221no—
276C2×C8—1221no—
277C2×C8—1222no—
278C2×C8—1221no—
279C2×C8—2442no—
280C2×C8—2441no—
281C2×C8—2441no—
282C2×C8—2441no—
283C2×C8—2444no—
284C4⋊C4—611no—
285C4⋊C4—1221no—
286C4⋊C4—1221no—
287C4⋊C4—1222no—
288C4⋊C4—2441no—
289C4⋊C4C4⋊#692311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—1221no—
292C22⋊C4—1221no—
293C42—611no—
294C42—611no—
295C42—611no—
296C42—1221no—
297C42—1221no—
298C42—1221no—
299C42—2441no—
300C42—2441no—
301C42C22×#692311yescharacteristic
302C23—21no—
303C23—611no—
304Q8—612no—
305Q8—1221no—
306D4—21no—
307D4—1222no—
308D4—1222no—
309D4#1451984512yes—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—611no—
316C2×C4—611no—
317C2×C4—611no—
318C2×C4—611no—
319C2×C4—611no—
320C2×C4—1222no—
321C2×C4—1222no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—4881no—
327C2×C4—4881no—
328C2×C4—4881no—
329C2×C4—2441no—
330C2×C4—2442no—
331C2×C4C2×C4⋊#692311yescharacteristic
332C2×C4D4×#692311yescharacteristic
333C2×C4D4×#692311yescharacteristic
334C8—41no—
335C8—41no—
336C8—1222no—
337C8—4881no—
338C8—2441no—
339C8—2441no—
340C8—2442no—
341C8—2441no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—611no—
346C22—611no—
347C22—1221no—
348C22—1221no—
349C22#1684070911yescharacteristic
350C4—21no—
351C4—41no—
352C4—611no—
353C4—611no—
354C4—1222no—
355C4—1221no—
356C4—1221no—
357C4—4882no—
358C4—2441no—
359C4C2×#1451984511yescharacteristic
360C4#1684077311yescharacteristic
361C4#1684080411yescharacteristic
362C4#1684080711yescharacteristic
363C2—42no—
364C2—1221no—
365C2—1221no—
366C2#2146176911yescharacteristic
367C2#2146178111yescharacteristic
368C2#2146178611yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×D4⋊C8C4
socleC22×C61#385
prime radicalC2×#11758224D4
cosocleC42×D61C23
hypercenterD4⋊C8#6923
hypocenterC61#8904
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
3. #21461387C2semidirect
4. #21461375C2semidirect
5. #21460716C2semidirect
6. #21460654C2semidirect
25. #16839494C4semidirect
26. #16839492C4semidirect
27. D4⋊#92418C4semidirect
28. C61×D4⋊C8C4semidirect
212. D4⋊C8#6923semidirect
213. C61#8904semidirect

Maximal subgroups

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

subgroupindex
#214624642
#214616912
#214613872
#214613752
#214607162
#214606542
#214441982
#890461