Finite Group Databasebeta

← #30601957

Subgroups — #30601957

Order 15616.

522 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462008C212yesmaximal
2#21461696C211yesmaximal, characteristic
3C8×C22⋊#6923C211yesmaximal, characteristic
4#21461259C212yesmaximal
5C2×#16840000C211yesmaximal, characteristic
6C2×#14519846—21no—
7D61⋊(C22⋊C8)C2211yescharacteristic
8C2×C8×#6923—21no—
9#16840704C2212yes—
10C4×C22⋊#6923C2211yescharacteristic
11#16840640—24no—
12#16840636—21no—
13(C2×#6923)⋊C8—21no—
14#16840616—24no—
15C22×C8×D61C2211yescharacteristic
16C2×#14519753C2212yes—
17#16840000—21no—
18#16840000—21no—
19#16840000—21no—
20#16840000—21no—
21C2×#14519711C411yescharacteristic, complemented
22C2×#14519701C412yescomplemented
23C22×Q16×C61C411yescharacteristic, complemented
24C2×D61⋊C8—21no—
25C2×C4⋊#6923—22no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923D412yes—
28C2×C4×#6923—21no—
29#14519846—42no—
30#14519834D412yes—
31C22⋊#92420—21no—
32C8×#6923—42no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—21no—
35C22×C4×D61C2311yescharacteristic
36C22×#92418C2×C411yescharacteristic
37C2×C8×D61—42no—
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41C2×C8×D61D411yescharacteristic
42#14519753—22no—
43#14519753—22no—
44#14519753—22no—
45#14519753—22no—
46C2×C4⋊Dic61C2×C412yes—
47C2×#11758196—22no—
48C2×#11758198—22no—
49#14519711—21no—
50#14519711—21no—
51#14519701—22no—
52#14519701—22no—
53C22×Q8×C61C2×C412yes—
54C2×Q16×C61—21no—
55C2×Q16×C61—21no—
56C2×Q16×C61—42no—
57C22×C8×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—22no—
60C22×#6923—42no—
61C2×Q8×C61—22no—
62C2×Q8×C61—22no—
63C2×Q8×C61—42no—
64C2×Q8×C61—42no—
65C22×C4×C61—22no—
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61#4111yescharacteristic
69C2×C4×D61—41no—
70C2×C4×D61—41no—
71C2×C4×D61C2×D411yescharacteristic
72C2×C4×D61C2×D411yescharacteristic
73C2×C4×D61C2×D411yescharacteristic
74C2×C4×D61#4111yescharacteristic
75C22⋊#6923—22no—
76C22⋊#6923—44no—
77C22⋊#6923Q1614yescomplemented
78C2×#92420—22no—
79C4⋊#6923—42no—
80C4⋊#6923—42no—
81C4×#6923—41no—
82C4×#6923—41no—
83D61⋊C8—41no—
84D61⋊C8—41no—
85Q16×C61—42no—
86Q16×C61—42no—
87C2×C8×C61—22no—
88C2×C8×C61C22⋊C411yescharacteristic
89C2×C8×C61C22⋊C411yescharacteristic
90#11758198—42no—
91#11758198—42no—
92C4⋊Dic61—22no—
93C4⋊Dic61—22no—
94#11758196—42no—
95#11758196—42no—
96C2×#92418—41no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C22⋊C411yescharacteristic
99C8×D61—22no—
100C8×D61—22no—
101C8×D61—41no—
102C8×D61—41no—
103C23×C61C4×D411yescharacteristic
104C22×D61—41no—
105C22×D61—41no—
106C22×D61#12111yescharacteristic
107C22×D61#12711yescharacteristic
108C22×D61C2×Q1612yes—
109C22×D61#13512yes—
110C2×#6923—24no—
111C2×#6923—42no—
112C2×#6923—42no—
113C2×#6923—82no—
114C2×#6923—82no—
115C2×#6923—82no—
116Q8×C61—42no—
117Q8×C61—42no—
118Q8×C61—82no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—42no—
122C2×C4×C61—82no—
123C2×C4×C61C2×C22⋊C411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×Dic61—41no—
126C2×Dic61—42no—
127C2×Dic61—82no—
128C2×Dic61#12111yescharacteristic
129C2×Dic61#12411yescharacteristic
130C4×D61—21no—
131C4×D61—21no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—41no—
135C4×D61—41no—
136#92420—42no—
137C8×C61—21no—
138C8×C61#9912yes—
139#92418—21no—
140#92418—21no—
141#9782—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#39011yescharacteristic
145C22×C61C4×Q1612yes—
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—42no—
151C2×D61—42no—
152C2×D61#46712yes—
153C2×D61#49411yescharacteristic
154C2×D61#49511yescharacteristic
155#6923—44no—
156#6923—162no—
157C4×C61—21no—
158C4×C61—82no—
159C4×C61C2×#9911yescharacteristic
160C4×C61#41011yescharacteristic
161Dic61—21no—
162Dic61—21no—
163Dic61—82no—
164C2×#467—611no—
165#1996—612no—
166#1808—611no—
167C8×C22⋊C4—611no—
168#1520—612no—
169C2×C61—22no—
170C2×C61#196411yescharacteristic
171C2×C61#196911yescharacteristic
172C2×C61#197611yescharacteristic
173D61—22no—
174D61—21no—
175D61—21no—
176C22×Q16#692311yescharacteristic, complemented
177C23×C8—611no—
178C2×C22⋊Q8—612no—
179#467—1221no—
180#467—1221no—
181#467—1221no—
182#467—1221no—
183C2×#107—611no—
184C2×#107—1221no—
185C2×#103—612no—
186C2×C22⋊C8—611no—
187C2×C4×C8—1221no—
188#380—612no—
189C4×C22⋊C4—611no—
190#342—1224no—
191#340—1221no—
192#336—1221no—
193#328—1224no—
194C61#978211yescharacteristic, complemented, Sylow
195C22×Q8C2×#692312yes—
196C23×C4—611no—
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—42no—
200C22×C8—611no—
201C22×C8—611no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—611no—
205C22×C8—1221no—
206C22×C8—2442no—
207C22×C8C2×#692311yescharacteristic
208C22⋊Q8—1222no—
209C22⋊Q8—1222no—
210C22⋊Q8—1222no—
211C22⋊Q8—1222no—
212C2×C4⋊C4—612no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4—1222no—
215C2×C22⋊C4—612no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1222no—
218C2×C42—1221no—
219#107—1221no—
220#107—1221no—
221#107—2442no—
222#103—1222no—
223#103—1222no—
224C22⋊C8—612no—
225C22⋊C8—1221no—
226C4×C8—2442no—
227#95—1221no—
228#95—1222no—
229C24—611no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—42no—
233C2×Q8—42no—
234C22×C4—22no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2442no—
245C22×C4C22×#692311yescharacteristic
246Q16—42no—
247Q16—42no—
248C2×C8—22no—
249C2×C8—611no—
250C2×C8—611no—
251C2×C8—1222no—
252C2×C8—1222no—
253C2×C8—1222no—
254C2×C8—2441no—
255C2×C8—2441no—
256C2×C8—2441no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8C22⋊#692311yescharacteristic
260C2×C8C22⋊#692311yescharacteristic
261C4⋊C4—1222no—
262C4⋊C4—1222no—
263C4⋊C4—2442no—
264C4⋊C4—2442no—
265C4⋊C4—2442no—
266C4⋊C4—2442no—
267C22⋊C4—614no—
268C22⋊C4—1222no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C22⋊C4—2444no—
272C42—2441no—
273C42—2441no—
274C23—611no—
275C23—611no—
276C23—612no—
277C23—612no—
278C23—2441no—
279C23—2441no—
280C23D4×#692311yescharacteristic
281Q8—42no—
282Q8—42no—
283Q8—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—611no—
289C2×C4—611no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1224no—
295C2×C4—4882no—
296C2×C4—4882no—
297C2×C4—4882no—
298C2×C4—4882no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4C2×C22⋊#692311yescharacteristic
306C2×C4D4×#692311yescharacteristic
307C8—21no—
308C8—1221no—
309C8—1221no—
310C8—2442no—
311C8#1451983512yes—
312C22—21no—
313C22—21no—
314C22—612no—
315C22—611no—
316C22—611no—
317C22—1221no—
318C22—1221no—
319C22—1221no—
320C22—1221no—
321C22—2442no—
322C22—2442no—
323C22#1684071411yescharacteristic
324C22Q16×#692312yes—
325C4—21no—
326C4—82no—
327C4—1221no—
328C4—1221no—
329C4—9762no—
330C4—4882no—
331C4—2444no—
332C4C2×#1451983511yescharacteristic
333C4#1684074211yescharacteristic
334C2—22no—
335C2—1221no—
336C2—1222no—
337C2—1221no—
338C2#2146195811yescharacteristic
339C2#2146196511yescharacteristic
340C2#2146197811yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×Q16×C61C4
socleC22×C61#390
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC22×Q16#6923
hypocenterC61#9782
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
21. C2×#14519711C4semidirect
22. C2×#14519701C4semidirect
23. C22×Q16×C61C4semidirect
77. C22⋊#6923Q16semidirect
176. C22×Q16#6923semidirect
194. C61#9782semidirect

Maximal subgroups

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

subgroupindex
#214620082
#214616962
C8×C22⋊#69232
#214612592
C2×#168400002
#978261