Finite Group Databasebeta

← #30602885

Subgroups — #30602885

Order 15616.

762 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 170 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#21462257C211yesmaximal, characteristic
2#21461522C211yesmaximal, characteristic
3#21461498C212yesmaximal
4C2×#16840629C212yesmaximal
5#21448440C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8C4⋊C4×#6923—24no—
9C4×C22⋊#6923—22no—
10#16840642—22no—
11#16840632—24no—
12#16840629C418yescomplemented
13C2×#14519735C2211yescharacteristic
14C2×#14519734C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16#16839657—22no—
17#16839633—22no—
18#16839631C411yescharacteristic, complemented
19#16839628C411yescharacteristic, complemented
20#16839627C411yescharacteristic, complemented
21C61×#384C411yescharacteristic, complemented
22C2×#11758213—22no—
23D61⋊M16—22no—
24D61⋊M16—22no—
25D61⋊M16—22no—
26D61⋊M16C2×C414yes—
27C2×#11758217—22no—
28C2×D61⋊C8—22no—
29#14519856—22no—
30#14519856—22no—
31#14519856C2×C414yes—
32#14519856Q814yes—
33C2×C4⋊#6923—22no—
34C2×C22⋊#6923—22no—
35C2×C4×#6923—24no—
36C2×C4×#6923D414yes—
37(C2×#6923)⋊C4—24no—
38C22×C4×D61D412yes—
39C22×C4×D61C2311yescharacteristic
40#14519735—22no—
41#14519735—22no—
42#14519734—22no—
43#14519734C2×C414yes—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—22no—
48C2×#11758196D412yes—
49C4⋊C4×D61—22no—
50C4⋊C4×D61—22no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×C4×Dic61—21no—
54C2×C4×Dic61C2×C411yescharacteristic
55#14519684—22no—
56#14519684—22no—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61C23×D61C2×D411yescharacteristic
62C22×#6923—24no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C412yes—
70C22×Dic61C2×D411yescharacteristic
71C2×C4×D61—22no—
72C2×C4×D61—22no—
73C2×C4×D61—82no—
74C2×C4×D61C22×C412yes—
75C2×C4×D61C2×D412yes—
76C2×C4×D61C2×Q812yes—
77C2×C4×D61#4112yes—
78C2×C4×D61#4112yes—
79C2×C4×D61#4112yes—
80C22⋊#6923—24no—
81C22⋊#6923—24no—
82#11758217—24no—
83#11758217—42no—
84C2×#92420—42no—
85C4⋊#6923—24no—
86C4⋊#6923—24no—
87C4×#6923—24no—
88C4×#6923—84no—
89C4×#6923C4⋊C418yescomplemented
90#11758213—24no—
91#11758213—42no—
92D61⋊C8—24no—
93D61⋊C8—42no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—82no—
97C42×C61—21no—
98C42×C61C4212yescomplemented
99#11758198—24no—
100#11758198—41no—
101#11758198—41no—
102C4⋊Dic61—22no—
103C4⋊Dic61—22no—
104#11758196—24no—
105#11758196—24no—
106#11758196—82no—
107C4×Dic61—21no—
108C4×Dic61—42no—
109C4×Dic61C4212yescomplemented
110C23×C61C2×C22⋊C411yescharacteristic
111C22×D61—41no—
112C22×D61—41no—
113C22×D61C4×D412yes—
114C22×D61#12011yescharacteristic
115C22×D61C22⋊Q812yes—
116C22×D61#12311yescharacteristic
117C2×#6923—28no—
118C2×#6923—44no—
119C2×#6923—44no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61C2×C4211yescharacteristic
128C2×C4×C61#11711yescharacteristic
129C2×C4×C61C4×D412yes—
130C2×C4×C61C4×Q812yes—
131C2×Dic61—22no—
132C2×Dic61—22no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—41no—
138C2×Dic61C4×D412yes—
139C2×Dic61#12112yes—
140C2×Dic61#12411yescharacteristic
141C2×Dic61C4⋊Q811yescharacteristic
142C4×D61—44no—
143C4×D61—82no—
144C4×D61C2×C4⋊C412yes—
145C4×D61#11712yes—
146C4×D61C4×D412yes—
147C4×D61C4×Q812yes—
148C4×D61C22⋊Q812yes—
149C4×D61#12312yes—
150C4×D61#12512yes—
151C4×D61#12612yes—
152#92420—44no—
153#10888—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61C4×C22⋊C411yescharacteristic
157C22×C61#38611yescharacteristic
158C22×C61#39111yescharacteristic
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—42no—
164C2×D61—42no—
165C2×D61#38512yes—
166C2×D61#38712yes—
167#6923—48no—
168C4×C61—42no—
169C4×C61—42no—
170C4×C61—81no—
171C4×C61C4×C4⋊C412yes—
172C4×C61#38712yes—
173Dic61—42no—
174Dic61—42no—
175Dic61—81no—
176Dic61#38812yes—
177Dic61#38912yes—
178#2317—611no—
179#2274—611no—
180C2×#343—611no—
181#1733—612no—
182C2×H(3,ℤ/4)—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61#143611yescharacteristic
186C2×C61#191111yescharacteristic
187C2×C61#191211yescharacteristic
188D61—22no—
189D61—22no—
190C22×M16—612no—
191C2×#117—612no—
192C2×#117—611no—
193C2×#117—611no—
194C22×C4⋊C4—611no—
195#389—1222no—
196#385—1222no—
197#384—611no—
198#384#692311yescharacteristic, complemented
199#382—611no—
200#382—611no—
201C4×C4⋊C4—1224no—
202C4×C22⋊C4—1222no—
203#343—1222no—
204#339—1224no—
205H(3,ℤ/4)—618no—
206C61#1088811yescharacteristic, complemented, Sylow
207C23×C4—611no—
208C23×C4—612no—
209C2×M16—614no—
210C2×M16—1222no—
211C2×M16—1222no—
212C2×M16—1222no—
213C2×M16—1222no—
214C2×M16—1222no—
215C22×C8—1222no—
216#117—614no—
217#117—614no—
218#117—614no—
219#117—1222no—
220#117—1222no—
221#117—1222no—
222#117—1222no—
223#117—1222no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—612no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—611no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1222no—
232C2×C4⋊C4—1222no—
233C2×C4⋊C4C2×#692311yescharacteristic
234C2×C4⋊C4C2×#692311yescharacteristic
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—1222no—
237C2×C22⋊C4—1221no—
238C2×C42—614no—
239C2×C42—611no—
240C2×C42—1224no—
241C2×C42—1221no—
242C2×C42C2×#692311yescharacteristic
243#95—1222no—
244#95—1224no—
245#95—1222no—
246C24—611no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—611no—
250C22×C4—612no—
251C22×C4—612no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—1222no—
258C22×C4—1222no—
259C22×C4—1224no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—4882no—
263C22×C4C22⋊#692312yes—
264C22×C4C22×#692311yescharacteristic
265M16—1224no—
266M16—1224no—
267M16—2442no—
268M16—2442no—
269C2×C8—1224no—
270C2×C8—2442no—
271C2×C8—2442no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—82no—
275C4⋊C4—1224no—
276C4⋊C4—1224no—
277C4⋊C4—1224no—
278C4⋊C4—1224no—
279C4⋊C4—1222no—
280C4⋊C4—1222no—
281C4⋊C4—4882no—
282C22⋊C4—1224no—
283C22⋊C4—1224no—
284C22⋊C4—1224no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—21no—
288C42—618no—
289C42—612no—
290C42—1224no—
291C42—1221no—
292C42—4884no—
293C42—2442no—
294C42C4×#692312yescomplemented
295C23—611no—
296C23—611no—
297C23—612no—
298C23—612no—
299C23—2441no—
300C23—2441no—
301C23C2×C22⋊#692311yescharacteristic
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—612no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—612no—
313C2×C4—612no—
314C2×C4—612no—
315C2×C4—612no—
316C2×C4—612no—
317C2×C4—612no—
318C2×C4—612no—
319C2×C4—612no—
320C2×C4—612no—
321C2×C4—1228no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—4882no—
325C2×C4—2444no—
326C2×C4—2442no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2444no—
331C2×C4—2444no—
332C2×C4—2442no—
333C2×C4C2×C4×#692311yescharacteristic
334C2×C4#1451985611yescharacteristic
335C2×C4D4×#692312yes—
336C2×C4Q8×#692312yes—
337C8—2444no—
338C22—21no—
339C22—21no—
340C22—612no—
341C22—612no—
342C22—1221no—
343C22—1221no—
344C22—1221no—
345C22—1221no—
346C22—2442no—
347C22—2442no—
348C22C4×C22⋊#692311yescharacteristic
349C22#1684071011yescharacteristic
350C22(C2×#6923)⋊Q811yescharacteristic
351C4—42no—
352C4—42no—
353C4—81no—
354C4—612no—
355C4—612no—
356C4—4881no—
357C4—2442no—
358C4—2448no—
359C4—2442no—
360C4C4⋊C4×#692312yes—
361C4#1684071112yes—
362C2—21no—
363C2—21no—
364C2—1222no—
365C2—1222no—
366C2#2146099811yescharacteristic
367C2#2146186511yescharacteristic
368C2#2146186611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#10888
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
12. #16840629C4semidirect
18. #16839631C4semidirect
19. #16839628C4semidirect
20. #16839627C4semidirect
21. C61×#384C4semidirect
89. C4×#6923C4⋊C4semidirect
98. C42×C61C42semidirect
109. C4×Dic61C42semidirect
198. #384#6923semidirect
206. C61#10888semidirect
294. C42C4×#6923semidirect

Maximal subgroups

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

subgroupindex
#214622572
#214615222
#214614982
C2×#168406292
#214484402
#1088861