Finite Group Databasebeta

← #30598229

Subgroups — #30598229

Order 15616.

876 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 130 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#21461711C211yesmaximal, characteristic
2C2×#16840615C214yesmaximal
3#21460995C211yesmaximal, characteristic
4C2×#16840317C211yesmaximal, characteristic
5C2×#14519834C2212yes—
6C2×(C2×#6923)⋊C4—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4C2212yes—
9#16840628—21no—
10#16840628—21no—
11#16840615—28no—
12#16840317—28no—
13C2×#14519777C411yescharacteristic, complemented
14C2×#14519768—21no—
15C2×#14519757C2212yes—
16C2×#14519753—21no—
17C2×Q8×Dic61—21no—
18C2×#14519742C412yescomplemented
19C2×#14519735—21no—
20C2×C42×D61C2211yescharacteristic
21C2×C4⋊Q8×C61C411yescharacteristic, complemented
22C23×#6923—22no—
23C22×#92420—22no—
24#14519834—24no—
25(C2×#6923)⋊C4—24no—
26(C2×#6923)⋊C4—24no—
27(C2×#6923)⋊C4—42no—
28C22×Dic122C2×C412yes—
29C22×C4×D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61C2311yescharacteristic
32#14519777—24no—
33#14519768—28no—
34#14519757—24no—
35#14519757D418yes—
36#14519753—28no—
37Q8×Dic61—28no—
38#14519742—24no—
39#14519742—24no—
40#14519735—28no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—22no—
44C2×#11758196—22no—
45C2×#11758198—22no—
46C2×C4×Dic61—22no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C42×D61—24no—
51C22×Q8×C61—21no—
52C4⋊Q8×C61—24no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C412yes—
55C2×C42×C61C2×C411yescharacteristic
56C23×D61C2×D411yescharacteristic
57C22×#6923—24no—
58C22×#6923—42no—
59C22×#6923—42no—
60C22×#6923—42no—
61C22×#6923—42no—
62C22×#6923—42no—
63C22×#6923—42no—
64C2×Q8×C61—24no—
65C2×Q8×C61—42no—
66C2×Q8×C61—42no—
67C22×C4×C61—22no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—22no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61SD1612yes—
83C2×C4×D61Q1612yes—
84C2×C4×D61C2×D414yes—
85C2×Dic122—22no—
86C2×Dic122—42no—
87C2×Dic122—42no—
88C2×Dic122C22⋊C414yes—
89C2×#92420—42no—
90C2×#92420—42no—
91C2×#92420—42no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61C22⋊C414yes—
95C42×C61—22no—
96#11758198—28no—
97C4⋊Dic61—24no—
98C4⋊Dic61—24no—
99C4⋊Dic61—28no—
100#11758196—28no—
101C4×Dic61—28no—
102C4×Dic61—22no—
103C4×Dic61—22no—
104C4×Dic61—22no—
105C23×C61C2×C22⋊C411yescharacteristic
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—21no—
110C22×D61#12012yes—
111C22×D61C2×SD1611yescharacteristic
112C22×D61C2×Q1611yescharacteristic
113C22×D61#13712yes—
114C2×#6923—44no—
115C2×#6923—44no—
116C2×#6923—44no—
117C2×#6923—82no—
118C2×#6923—82no—
119C2×#6923—82no—
120C2×#6923—82no—
121Q8×C61—48no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—24no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61#10312yes—
132C2×C4×C61#10412yes—
133C2×C4×C61C2×C22⋊C412yes—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—24no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61#10312yes—
145C2×Dic61#12012yes—
146C4×D61—24no—
147C4×D61—24no—
148C4×D61—44no—
149C4×D61—44no—
150Dic122—44no—
151Dic122—44no—
152#92420—82no—
153#8333—611nomaximal, Sylow
154C22×C61#37911yescharacteristic
155C22×C61#41011yescharacteristic
156C22×C61#41011yescharacteristic
157C22×C61C2×#10311yescharacteristic
158C22×C61#41911yescharacteristic
159C22×C61C2×#10411yescharacteristic
160C22×C61#42111yescharacteristic
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—22no—
164C2×D61—22no—
165C2×D61—21no—
166C2×D61—21no—
167C2×D61—22no—
168C2×D61—22no—
169C2×D61#44812yes—
170C2×D61#45112yes—
171#6923—84no—
172C4×C61—22no—
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—44no—
176Dic61—22no—
177Dic61—22no—
178Dic61—44no—
179Dic61—41no—
180C2×#533—611no—
181#1817—611no—
182C2×#328—614no—
183#1434—611no—
184C2×C61#150311yescharacteristic
185C2×C61#150611yescharacteristic
186C2×C61#151511yescharacteristic
187C2×C61#151811yescharacteristic
188C2×C61#178211yescharacteristic
189C2×C61#178711yescharacteristic
190C2×C61#178911yescharacteristic
191D61—22no—
192D61—22no—
193#533—1228no—
194C2×C4⋊Q8—611no—
195C2×C4⋊Q8#692311yescharacteristic, complemented
196C2×#125—1221no—
197C2×C22⋊Q8—612no—
198C2×C22⋊Q8—1221no—
199C2×C4×Q8—612no—
200C2×C4×Q8—1221no—
201C2×#117—1221no—
202C22×C42—611no—
203C2×C22⋊C8—612no—
204C2×#95—612no—
205C2×#95—1221no—
206C2×#95—1221no—
207#336—1221no—
208#336—1221no—
209#328—1228no—
210C61#833311yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C22×Q8—612no—
213C23×C4—611no—
214C23×C4—611no—
215C23×C4—611no—
216C23×C4—1222no—
217C22×C8—1222no—
218C4⋊Q8—24no—
219C4⋊Q8—1224no—
220#125—1228no—
221C22⋊Q8—618no—
222C22⋊Q8—1224no—
223C22⋊Q8—1228no—
224C4×Q8—1224no—
225C4×Q8—1228no—
226C4×Q8—1224no—
227#117—1228no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1222no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4C2×#692312yes—
234C2×C22⋊C4—1222no—
235C2×C42—611no—
236C2×C42—611no—
237C2×C42—611no—
238C2×C42—1224no—
239C2×C42—1222no—
240C2×C42C2×#692311yescharacteristic
241C22⋊C8—1224no—
242#95—1224no—
243#95—1224no—
244#95—2442no—
245C24—611no—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—614no—
250C2×Q8—1222no—
251C2×Q8—2442no—
252C2×Q8—2442no—
253C22×C4—22no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—614no—
259C22×C4—611no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—1224no—
266C22×C4—2441no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4—2442no—
270C22×C4—2442no—
271C22×C4—2442no—
272C22×C4—2442no—
273C22×C4—2442no—
274C22×C4—2442no—
275C22×C4C22⋊#692311yescharacteristic
276C22×C4C22⋊#692311yescharacteristic
277C22×C4C22×#692311yescharacteristic
278C2×C8—2442no—
279C2×C8—2442no—
280C2×C8—2442no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—1228no—
284C4⋊C4—1228no—
285C4⋊C4—1224no—
286C4⋊C4—1224no—
287C4⋊C4C22⋊#692314yes—
288C22⋊C4—1228no—
289C42—22no—
290C42—1222no—
291C42—1222no—
292C42—1228no—
293C42—1222no—
294C23—611no—
295C23—611no—
296C23—612no—
297C23—612no—
298C23—1221no—
299C23—1221no—
300C23—1221no—
301C23—1221no—
302C23C2×C22⋊#692311yescharacteristic
303Q8—48no—
304Q8—2444no—
305Q8—2444no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—612no—
316C2×C4—612no—
317C2×C4—1224no—
318C2×C4—1224no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—1224no—
327C2×C4—4882no—
328C2×C4—4882no—
329C2×C4—4882no—
330C2×C4—4882no—
331C2×C4—2444no—
332C2×C4—2444no—
333C2×C4—2444no—
334C2×C4—2444no—
335C2×C4—2444no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4#1451984012yes—
339C2×C4#1451984112yes—
340C2×C4C2×C22⋊#692312yes—
341C8—4882no—
342C22—612no—
343C22—612no—
344C22—1221no—
345C22—1221no—
346C22—1222no—
347C22—1222no—
348C22—1221no—
349C22—1221no—
350C22—1222no—
351C22—1222no—
352C22#1684070311yescharacteristic
353C22#1684074311yescharacteristic
354C22#1684074311yescharacteristic
355C22C2×#1451984011yescharacteristic
356C22#1684075511yescharacteristic
357C22C2×#1451984111yescharacteristic
358C22#1684076011yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—44no—
363C4—1222no—
364C4—1222no—
365C4—4884no—
366C4—2444no—
367C4—2441no—
368C2—1222no—
369C2—1222no—
370C2#2146120211yescharacteristic
371C2#2146121111yescharacteristic
372C2#2146124211yescharacteristic
373C2#2146125111yescharacteristic
374C2#2146165011yescharacteristic
375C2#2146165811yescharacteristic
376C2#2146166011yescharacteristic
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#8333
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
13. C2×#14519777C4semidirect
18. C2×#14519742C4semidirect
21. C2×C4⋊Q8×C61C4semidirect
195. C2×C4⋊Q8#6923semidirect
210. C61#8333semidirect

Maximal subgroups

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

subgroupindex
#214617112
C2×#168406152
#214609952
C2×#168403172
#833361