Finite Group Databasebeta

← #30598208

Subgroups — #30598208

Order 15616.

684 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461123C212yesmaximal, complemented
2#21461121C212yesmaximal, complemented
3#21461050C211yesmaximal, characteristic
4#21461050C211yesmaximal, characteristic
5C2×#16840372C211yesmaximal, characteristic
6#16840778C2212yes—
7#16840778C2212yes—
8#16840612—24no—
9Q8⋊#92420—24no—
10#16840607—21no—
11#16840607—21no—
12#16840605—21no—
13#16840605—21no—
14#16840372—24no—
15#16840372—24no—
16C2×#14519775C411yescharacteristic
17C2×#14519770—21no—
18C2×#14519770—21no—
19C2×#14519768—21no—
20C2×#14519768C411yescharacteristic
21C2×#14519767—21no—
22C2×#14519756C411yescharacteristic
23C2×Q8×Dic61C2211yescharacteristic
24C2×#14519742C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26C2×C22⋊Q8×C61C411yescharacteristic
27C4⋊#92420—22no—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C4×#92420—22no—
32C4×#92420—22no—
33C22×Dic122C2×C411yescharacteristic
34#14519775—22no—
35#14519775—22no—
36#14519770—28no—
37#14519770—28no—
38#14519768—28no—
39#14519768—22no—
40#14519768—22no—
41#14519767—28no—
42#14519756—22no—
43#14519756—22no—
44Q8×Dic61—22no—
45Q8×Dic61D414yes—
46#14519742—22no—
47#14519742D414yes—
48#14519736—22no—
49#14519736—22no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×C4⋊Dic61D411yescharacteristic
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61D411yescharacteristic
62C2×C22⋊Dic61D411yescharacteristic
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C22×Q8×C61C2×C411yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C24×C61C22⋊C411yescharacteristic
75C2×Q8×C61—21no—
76C2×Q8×C61—24no—
77C2×Q8×C61C22⋊C412yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—24no—
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C22×Dic61C2×D411yescharacteristic
91C2×Dic122—24no—
92C2×Dic122—21no—
93C2×Dic122C22⋊C412yes—
94C2×#92420—42no—
95C2×#92420—42no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61C22⋊C412yes—
99C22⋊C4×C61—24no—
100C22⋊Dic61—24no—
101C22⋊Dic61—22no—
102C22⋊Dic61—22no—
103C4⋊Dic61—24no—
104C4⋊Dic61—21no—
105C4⋊Dic61—24no—
106C4⋊Dic61—22no—
107C4⋊Dic61—22no—
108C4⋊Dic61C22⋊C412yes—
109#11758196—24no—
110#11758196—24no—
111#11758196—24no—
112#11758196—24no—
113#11758196—24no—
114C4×Dic61—21no—
115C4×Dic61—21no—
116C4×Dic61—24no—
117C4×Dic61—24no—
118C4×Dic61C2×D412yes—
119C4×Dic61C2×D412yes—
120C23×C61—21no—
121C23×C61—22no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61C2×C22⋊C411yescharacteristic
125Q8×C61—24no—
126Q8×C61—42no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61C2×C22⋊C411yescharacteristic
137C2×C4×C61#13114yes—
138C2×Dic61—21no—
139C2×Dic61—22no—
140C2×Dic61—21no—
141C2×Dic61—22no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61#12012yes—
149C2×Dic61#12012yes—
150Dic122—24no—
151Dic122—42no—
152#92420—82no—
153#92420—82no—
154#8327—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—41no—
158C22×C61—41no—
159C22×C61—41no—
160C22×C61—82no—
161C22×C61#37911yescharacteristic
162C22×C61#40812yes—
163C4×C61—42no—
164C4×C61—41no—
165C4×C61—42no—
166C4×C61#40812yes—
167C4×C61#40812yes—
168Dic61—22no—
169Dic61—22no—
170Dic61—41no—
171Dic61—41no—
172Dic61—42no—
173Dic61—42no—
174C2×#541—611no—
175#1479—612no—
176#1479—612no—
177#1458—611no—
178#1458—611no—
179C2×C61—22no—
180C2×C61—81no—
181C2×C61#178011yescharacteristic
182C2×C61#179611yescharacteristic
183C2×C61#179611yescharacteristic
184#541—1224no—
185#541—1224no—
186C2×C4⋊Q8—611no—
187C2×#126—1221no—
188C2×#126—1221no—
189C2×#125—611no—
190C2×#125—1221no—
191C2×#125—1221no—
192C2×C22⋊Q8—611no—
193C2×C22⋊Q8#692311yescharacteristic
194C2×C4×Q8—611no—
195C2×C4×Q8—611no—
196C2×#117—611no—
197#431—612no—
198#431—612no—
199Q8⋊C8—1224no—
200Q8⋊C8—1224no—
201#324—1221no—
202#324—1221no—
203#323—1221no—
204#323—1221no—
205C61#832711yescharacteristic, complemented, Sylow
206C22×Q8—611no—
207C22×Q8C2×#692311yescharacteristic
208C23×C4C2×#692311yescharacteristic
209C4⋊Q8—1222no—
210C4⋊Q8—1222no—
211#126—1228no—
212#126—1228no—
213#125—1222no—
214#125—1228no—
215#125—1228no—
216#125—1222no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—1222no—
220C22⋊Q8—1222no—
221C4×Q8—614no—
222C4×Q8—614no—
223C4×Q8—1222no—
224C4×Q8—1222no—
225#117—1222no—
226#117—1222no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—611no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4C2×#692311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—611no—
242C2×C22⋊C4—1221no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—1221no—
246C2×C42—1221no—
247C4⋊C8—1222no—
248C4⋊C8—1222no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251C4×C8—1222no—
252C4×C8—1222no—
253C24C22⋊#692311yescharacteristic
254C2×Q8—21no—
255C2×Q8—24no—
256C2×Q8—612no—
257C2×Q8—1221no—
258C2×Q8—1224no—
259C2×Q8C22⋊#692312yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4C22⋊#692311yescharacteristic
272C22×C4C22×#692311yescharacteristic
273C2×C8—2442no—
274C2×C8—2442no—
275C4⋊C4—24no—
276C4⋊C4—21no—
277C4⋊C4—612no—
278C4⋊C4—1222no—
279C4⋊C4—1222no—
280C4⋊C4—1224no—
281C4⋊C4—1224no—
282C4⋊C4—1224no—
283C4⋊C4—1224no—
284C4⋊C4—1224no—
285C4⋊C4—1224no—
286C4⋊C4—1224no—
287C4⋊C4—1221no—
288C4⋊C4C22⋊#692312yes—
289C22⋊C4—24no—
290C22⋊C4—1222no—
291C22⋊C4—1222no—
292C22⋊C4—1224no—
293C42—612no—
294C42—612no—
295C42—1224no—
296C42—1224no—
297C42—1221no—
298C42—1221no—
299C23—21no—
300C23—22no—
301C23—41no—
302C23—41no—
303C23C2×C22⋊#692311yescharacteristic
304Q8—24no—
305Q8—42no—
306Q8—1224no—
307Q8—2442no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—612no—
317C2×C4—612no—
318C2×C4—1222no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1222no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4C2×C22⋊#692311yescharacteristic
329C2×C4C2×C22⋊#692311yescharacteristic
330C2×C4#1451986414yes—
331C8—4882no—
332C8—4882no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—82no—
339C22#1684070311yescharacteristic
340C22#1684073512yes—
341C4—41no—
342C4—42no—
343C4—42no—
344C4—1222no—
345C4—1222no—
346C4—2442no—
347C4—2441no—
348C4—2442no—
349C4—2441no—
350C4#1684073512yes—
351C4#1684073612yes—
352C2—22no—
353C2—81no—
354C2#2146164311yescharacteristic
355C2#2146167111yescharacteristic
356C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61#379
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#8327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21461123C2semidirect
2. #21461121C2semidirect
205. C61#8327semidirect

Maximal subgroups

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

subgroupindex
#214611232
#214611212
#214610502
C2×#168403722
#832761