Finite Group Databasebeta

← #30601759

Subgroups — #30601759

Order 15616.

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

normal subgroups only

This group has 630 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#21461546C211yesmaximal, characteristic
2C2×#16840616C214yesmaximal
3#21460995C211yesmaximal, characteristic
4#21454166C211yesmaximal, characteristic
5C2×#14519834C2212yes—
6C2×#14519832—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840616—28no—
12C2×#14519753C2212yes—
13C2×C42×D61C2211yescharacteristic
14#16839669C412yescomplemented
15#16839658—21no—
16#16839658—21no—
17#16839632C411yescharacteristic, complemented
18#16839619—21no—
19#16839619—21no—
20C2×C4⋊Q8×C61C411yescharacteristic, complemented
21C23×#6923—22no—
22C22×#92420—22no—
23#14519834—22no—
24#14519834D414yes—
25#14519832—42no—
26(C2×#6923)⋊C4—24no—
27(C2×#6923)⋊C4—24no—
28(C2×#6923)⋊C4—42no—
29C22×C4×D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61C2311yescharacteristic
32#14519753—24no—
33#14519753—24no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61C2×C412yes—
37C2×#11758196—22no—
38C2×#11758198—22no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C42×D61—42no—
43#14519684—22no—
44#14519684—22no—
45C22×Q8×C61C2×C412yes—
46C4⋊Q8×C61—42no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C42×C61C2×C411yescharacteristic
50C23×D61C2×D411yescharacteristic
51C22×#6923—24no—
52C22×#6923—42no—
53C22×#6923—42no—
54C22×#6923—42no—
55C22×#6923—42no—
56C22×#6923—42no—
57C22×#6923—42no—
58C2×Q8×C61—24no—
59C2×Q8×C61—42no—
60C2×Q8×C61—42no—
61C22×C4×C61—22no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—22no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C2×C4×D61—22no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—22no—
73C2×C4×D61—41no—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×C4×D61C2×D412yes—
78C2×C4×D61#4112yes—
79C2×#92420—24no—
80C2×#92420—42no—
81C2×#92420—42no—
82C4⋊C4×C61—42no—
83C4⋊C4×C61—42no—
84C42×C61—22no—
85#11758198—42no—
86#11758198—42no—
87C4⋊Dic61—24no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90#11758196—44no—
91C4×Dic61—22no—
92C4×Dic61—41no—
93C4×Dic61—41no—
94C23×C61C2×C22⋊C411yescharacteristic
95C22×D61—21no—
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—21no—
99C22×D61#12411yescharacteristic
100C22×D61#12711yescharacteristic
101C22×D61#13712yes—
102C22×D61#13712yes—
103C2×#6923—44no—
104C2×#6923—44no—
105C2×#6923—44no—
106C2×#6923—82no—
107C2×#6923—82no—
108C2×#6923—82no—
109C2×#6923—82no—
110Q8×C61—84no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61#9912yes—
121C2×C4×C61#10412yes—
122C2×C4×C61C4×D412yes—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61#12112yes—
135C4×D61—42no—
136C4×D61—42no—
137C4×D61—44no—
138C4×D61—44no—
139#92420—44no—
140#9683—611nomaximal, Sylow
141C22×C61#39011yescharacteristic
142C22×C61C2×#9911yescharacteristic
143C22×C61#41011yescharacteristic
144C22×C61#41911yescharacteristic
145C22×C61#41911yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C22×C61#42111yescharacteristic
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—22no—
153C2×D61—22no—
154C2×D61—22no—
155C2×D61—22no—
156C2×D61#49112yes—
157C2×D61#49712yes—
158#6923—84no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—82no—
163Dic61—22no—
164Dic61—41no—
165Dic61—41no—
166Dic61—82no—
167#2857—611no—
168#1749—611no—
169C2×#328—614no—
170#1434—611no—
171C2×C61#150611yescharacteristic
172C2×C61#150811yescharacteristic
173C2×C61#151111yescharacteristic
174C2×C61#151511yescharacteristic
175C2×C61#196011yescharacteristic
176C2×C61#196411yescharacteristic
177C2×C61#198011yescharacteristic
178D61—22no—
179D61—22no—
180C2×C4⋊Q8#692311yescharacteristic, complemented
181C2×C22⋊Q8—612no—
182C22×C42—611no—
183C2×C22⋊C8—612no—
184C2×#97—1221no—
185#391—612no—
186#389—1221no—
187#389—1221no—
188#384—611no—
189#380—1221no—
190#380—1221no—
191C2×#95—612no—
192C2×#95—1221no—
193C2×#95—1221no—
194#336—1221no—
195#328—1228no—
196C61#968311yescharacteristic, complemented, Sylow
197C22×Q8C2×#692312yes—
198C23×C4—611no—
199C23×C4—611no—
200C23×C4—611no—
201C23×C4—1222no—
202C22×C8—1222no—
203C4⋊Q8—42no—
204C22⋊Q8—1224no—
205C22⋊Q8—1224no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—612no—
209C2×C4⋊C4—1222no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C22⋊C4—1222no—
213C2×C42—611no—
214C2×C42—611no—
215C2×C42—611no—
216C2×C42—2442no—
217C2×C42C2×#692311yescharacteristic
218C22⋊C8—614no—
219C22⋊C8—1222no—
220#97—2442no—
221#95—1222no—
222#95—1222no—
223#95—1224no—
224#95—1224no—
225#95—2442no—
226C24—611no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C22×C4—22no—
231C22×C4—611no—
232C22×C4—612no—
233C22×C4—612no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—1222no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—1222no—
240C22×C4—1224no—
241C22×C4—1222no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4—2441no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4—2442no—
249C22×C4—2442no—
250C22×C4—2442no—
251C22×C4—2442no—
252C22×C4C22⋊#692311yescharacteristic
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255C2×C8—1224no—
256C2×C8—2442no—
257C2×C8—2442no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—1224no—
261C4⋊C4—2442no—
262C4⋊C4—2444no—
263C4⋊C4—2442no—
264C22⋊C4—2442no—
265C22⋊C4—2442no—
266C42—22no—
267C42—1222no—
268C42—2441no—
269C42—2441no—
270C23—611no—
271C23—611no—
272C23—612no—
273C23—612no—
274C23—1221no—
275C23—1221no—
276C23—1221no—
277C23—1221no—
278C23C2×C22⋊#692311yescharacteristic
279Q8—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—612no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—4882no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2444no—
305C2×C4—2444no—
306C2×C4—2444no—
307C2×C4—2444no—
308C2×C4—2444no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4—2442no—
312C2×C4#1451983512yes—
313C2×C4#1451984212yes—
314C2×C4D4×#692312yes—
315C8—2444no—
316C22—612no—
317C22—612no—
318C22—1221no—
319C22—1221no—
320C22—1222no—
321C22—1221no—
322C22—1221no—
323C22—1222no—
324C22—1222no—
325C22—1222no—
326C22#1684071411yescharacteristic
327C22C2×#1451983511yescharacteristic
328C22#1684074211yescharacteristic
329C22#1684075511yescharacteristic
330C22#1684075511yescharacteristic
331C22C2×#1451984211yescharacteristic
332C22#1684076111yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—82no—
337C4—1222no—
338C4—4884no—
339C4—4882no—
340C4—2441no—
341C4—2441no—
342C2—1222no—
343C2—1222no—
344C2#2146121311yescharacteristic
345C2#2146121911yescharacteristic
346C2#2146122911yescharacteristic
347C2#2146124311yescharacteristic
348C2#2146195011yescharacteristic
349C2#2146195811yescharacteristic
350C2#2146198211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#9683
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #16839669C4semidirect
17. #16839632C4semidirect
20. C2×C4⋊Q8×C61C4semidirect
180. C2×C4⋊Q8#6923semidirect
196. C61#9683semidirect

Maximal subgroups

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

subgroupindex
#214615462
C2×#168406162
#214609952
#214541662
#968361