Finite Group Databasebeta

← #30601154

Subgroups — #30601154

Order 15616.

528 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 78 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#21462296C211yesmaximal, characteristic
2#21461719C211yesmaximal, characteristic
3#21461227C212yesmaximal
4#21460667C212yesmaximal
5#21450294C211yesmaximal, characteristic
6#16840734C2212yes—
7#16840711—24no—
8#16840705C2212yes—
9#16840704—21no—
10C4×C22⋊#6923—21no—
11#16840642—21no—
12#16840642—21no—
13#16840615—24no—
14C2×#14519757C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16#16839695—21no—
17#16839695—21no—
18#16839668C411yescharacteristic, complemented
19#16839667C411yescharacteristic, complemented
20#16839633—21no—
21#16839633—21no—
22#16839631C411yescharacteristic, complemented
23C4×#11758198C2211yescharacteristic
24C61×#384C411yescharacteristic, complemented
25C2×#11758217—22no—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28C2×C4×#6923—22no—
29#14519834—22no—
30#14519834—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4D414yes—
35C22×Dic122C2×C411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519757—22no—
40#14519757—22no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196D411yescharacteristic
46C2×#11758196D411yescharacteristic
47C4⋊C4×D61—22no—
48C4⋊C4×D61—22no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×C4×Dic61C2×C411yescharacteristic
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C42×C61C2×C411yescharacteristic
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—41no—
79C2×C4×D61—41no—
80C2×C4×D61—81no—
81C2×C4×D61—81no—
82C2×C4×D61C2×D412yes—
83C2×C4×D61#4112yes—
84C2×Dic122—22no—
85C2×Dic122—41no—
86C2×Dic122—41no—
87C22⋊#6923—42no—
88C22⋊#6923—42no—
89#11758217—42no—
90#11758217—42no—
91C2×#92420—42no—
92C4⋊#6923—82no—
93C4×#6923—82no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—81no—
97C4⋊C4×C61—81no—
98C42×C61—41no—
99#11758198—24no—
100#11758198—41no—
101#11758198—41no—
102C4⋊Dic61—22no—
103C4⋊Dic61—42no—
104#11758196—22no—
105#11758196—22no—
106#11758196—81no—
107#11758196—81no—
108C4×Dic61—41no—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—22no—
113C22×D61—41no—
114C22×D61#12011yescharacteristic
115C22×D61#12311yescharacteristic
116C2×#6923—44no—
117C2×#6923—44no—
118C2×#6923—44no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61C4×D412yes—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—22no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61#12112yes—
145C4×D61—42no—
146C4×D61—42no—
147C4×D61—42no—
148C4×D61—42no—
149C4×D61—81no—
150C4×D61—81no—
151Dic122—82no—
152#92420—82no—
153#9461—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61#38611yescharacteristic
157C22×C61#41011yescharacteristic
158C22×C61#41011yescharacteristic
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—42no—
165C2×D61—42no—
166C2×D61#45612yes—
167#6923—84no—
168C4×C61—22no—
169C4×C61—42no—
170C4×C61—41no—
171C4×C61—41no—
172C4×C61—81no—
173Dic61—22no—
174Dic61—42no—
175Dic61—41no—
176Dic61—41no—
177Dic61—81no—
178#2502—611no—
179#2313—611no—
180#1821—611no—
181#1511—612no—
182#1284—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61#188611yescharacteristic
186C2×C61#189911yescharacteristic
187C2×C61#191111yescharacteristic
188D61—22no—
189D61—41no—
190C2×C22⋊Q8—611no—
191C22×C4⋊C4—611no—
192#407—612no—
193#398—1221no—
194#398—1221no—
195#391—611no—
196#391—611no—
197#387—1224no—
198#385—1221no—
199#385—1221no—
200#384—611no—
201#384#692311yescharacteristic, complemented
202#381—612no—
203#380—1221no—
204C4×C22⋊C4—611no—
205C4×C22⋊C4—1221no—
206#343—1221no—
207#343—1221no—
208#328—1224no—
209C61#946111yescharacteristic, complemented, Sylow
210C22×Q8—611no—
211C23×C4—611no—
212C23×C4—611no—
213C23×C4—611no—
214C2×M16—1222no—
215C22⋊Q8—1222no—
216C22⋊Q8—1222no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—611no—
222C2×C4⋊C4—1222no—
223C2×C4⋊C4—1222no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4C2×#692311yescharacteristic
229C2×C4⋊C4C2×#692311yescharacteristic
230C2×C22⋊C4—1221no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1222no—
233C2×C42—611no—
234C2×C42—1222no—
235C2×C42C2×#692311yescharacteristic
236C22⋊C8—1222no—
237C22⋊C8—1222no—
238#95—614no—
239#95—1221no—
240#95—1221no—
241#95—1221no—
242#95—1222no—
243#95—1222no—
244#95—1222no—
245#95—1221no—
246#95—1221no—
247C24—611no—
248C2×Q8—1222no—
249C2×Q8—2441no—
250C2×Q8—2441no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—611no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—1224no—
264C22×C4—1221no—
265C22×C4—4881no—
266C22×C4—4881no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4C22⋊#692311yescharacteristic
270C22×C4C22⋊#692311yescharacteristic
271C22×C4C22×#692311yescharacteristic
272M16—2442no—
273M16—2442no—
274C2×C8—2442no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—1222no—
280C4⋊C4—1222no—
281C4⋊C4—1222no—
282C4⋊C4—4881no—
283C4⋊C4—4881no—
284C4⋊C4—4882no—
285C4⋊C4—2442no—
286C22⋊C4—1224no—
287C22⋊C4—2441no—
288C22⋊C4—2441no—
289C22⋊C4—2442no—
290C22⋊C4—2442no—
291C42—41no—
292C42—4882no—
293C42—2441no—
294C23—611no—
295C23—611no—
296C23—1222no—
297C23—1221no—
298C23—1221no—
299C23—2441no—
300C23C2×C22⋊#692311yescharacteristic
301Q8—4882no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—612no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1222no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2444no—
327C2×C4—2444no—
328C2×C4—2444no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4D4×#692312yes—
337C8—4882no—
338C22—21no—
339C22—21no—
340C22—612no—
341C22—1221no—
342C22—1221no—
343C22—1221no—
344C22—1221no—
345C22—1221no—
346C22—2442no—
347C22—2442no—
348C22#1684071011yescharacteristic
349C22#1684074111yescharacteristic
350C22#1684074311yescharacteristic
351C4—22no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—1222no—
357C4—4884no—
358C4—4881no—
359C4—2442no—
360C4—2441no—
361C4—2441no—
362C2—21no—
363C2—21no—
364C2—1222no—
365C2—2441no—
366C2#2146182511yescharacteristic
367C2#2146184311yescharacteristic
368C2#2146186511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#9461
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
18. #16839668C4semidirect
19. #16839667C4semidirect
22. #16839631C4semidirect
24. C61×#384C4semidirect
201. #384#6923semidirect
209. C61#9461semidirect

Maximal subgroups

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

subgroupindex
#214622962
#214617192
#214612272
#214606672
#214502942
#946161