Finite Group Databasebeta

← #30598773

Subgroups — #30598773

Order 15616.

1222 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 104 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#21461714C212yesmaximal, complemented
2#21461714C212yesmaximal, complemented
3#21461645C212yesmaximal
4C22×#14519763C211yesmaximal, characteristic
5C2×#14519835—24no—
6#16840703C2214yes—
7#16840640—24no—
8#16840640—24no—
9C23×Dic122C411yescharacteristic, complemented
10C23×C22⋊D61C2211yescharacteristic
11C2×#14519763—24no—
12C2×#14519763—24no—
13C2×#14519763—24no—
14C22×C22⋊Dic61C411yescharacteristic, complemented
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×C4×Dic61C411yescharacteristic, complemented
18C22×C22⋊C4×C61C411yescharacteristic, complemented
19C2×C22⋊#6923—24no—
20C2×C22⋊#6923—24no—
21C2×C22⋊#6923—24no—
22#14519835—48no—
23C24×D61C2311yescharacteristic
24C22×Dic122—22no—
25C22×Dic122—22no—
26C22×Dic122—22no—
27C22×Dic122—44no—
28C22×C22⋊D61—22no—
29C22×C22⋊D61—22no—
30C22×C22⋊D61—44no—
31C22×C22⋊D61D414yes—
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35#14519763—232no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198D414yes—
44C2×#11758198D414yes—
45C2×C4×Dic61—22no—
46C2×C4×Dic61—22no—
47C2×C4×Dic61—22no—
48C25×C61C2×C411yescharacteristic
49C23×C4×C61C2×C411yescharacteristic
50C23×C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61—22no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—42no—
57C24×C61—42no—
58C24×C61—42no—
59C24×C61C22⋊C412yes—
60C24×C61C22×C411yescharacteristic
61C23×D61—42no—
62C23×D61—42no—
63C23×D61—42no—
64C23×D61C2×D412yes—
65C23×D61C2×D412yes—
66C23×D61C2×D412yes—
67C22×#6923—44no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C412yes—
75C22×C4×C61C22⋊C412yes—
76C2×C22⋊D61—28no—
77C2×C22⋊D61—28no—
78C2×C22⋊D61—44no—
79C2×C22⋊D61—48no—
80C2×C22⋊D61—48no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—24no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—42no—
89C22×Dic61—42no—
90C22×Dic61C22⋊C412yes—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C22⋊C412yes—
93C2×Dic122—28no—
94C2×Dic122—48no—
95C2×Dic122—48no—
96C2×Dic122—44no—
97C2×Dic122—44no—
98C22⋊#6923—48no—
99C22⋊#6923—48no—
100C22⋊#6923—48no—
101C22⋊C4×C61—28no—
102C22⋊Dic61—28no—
103#11758198—28no—
104#11758198—28no—
105C4×Dic61—28no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—24no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—88no—
120C23×C61C2×C22⋊C411yescharacteristic
121C23×C61C2×C22⋊C411yescharacteristic
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—24no—
124C22×D61—24no—
125C22×D61—24no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—88no—
133C22×D61#12014yes—
134C2×#6923—84no—
135C2×#6923—84no—
136C2×#6923—84no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C22⋊D61—416no—
146C2×Dic61—24no—
147C2×Dic61—24no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154C2×Dic61—42no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61#9914yes—
159Dic122—416no—
160#8544—611nomaximal, Sylow
161C22×C61—28no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—84no—
172C22×C61—84no—
173C22×C61—84no—
174C22×C61#37911yescharacteristic
175C22×C61C2×#9912yes—
176C22×C61#41012yes—
177C22×C61#41012yes—
178C2×D61—24no—
179C2×D61—24no—
180C2×D61—24no—
181C2×D61—84no—
182C2×D61—84no—
183C2×D61—84no—
184#6923—164no—
185C4×C61—42no—
186C4×C61—42no—
187Dic61—24no—
188Dic61—42no—
189Dic61—42no—
190C22×#124—611no—
191#1819—612no—
192#1819—612no—
193#1781—612no—
194C2×C61—22no—
195C2×C61—22no—
196C2×C61—22no—
197C2×C61—82no—
198C2×C61#178111yescharacteristic
199C2×C61#178211yescharacteristic
200C2×C61#178211yescharacteristic
201D61—44no—
202C23×Q8—611no—
203C23×D4—611no—
204C2×#124—1224no—
205C2×#124—1224no—
206C2×#124—1224no—
207C22×C22⋊C4—611no—
208C22×C22⋊C4—611no—
209C22×C22⋊C4—611no—
210C22×C22⋊C4#692311yescharacteristic, complemented
211C22×C42—611no—
212C2×#99—1224no—
213#379—614no—
214#342—1224no—
215#342—1224no—
216C61#854411yescharacteristic, complemented, Sylow
217C25—611no—
218C25C2×#692311yescharacteristic
219C22×Q8—1222no—
220C22×Q8—1222no—
221C22×Q8—1222no—
222C22×Q8—2444no—
223C22×D4—614no—
224C22×D4—1222no—
225C22×D4—1222no—
226C22×D4—2444no—
227C23×C4—611no—
228C23×C4—611no—
229C23×C4—611no—
230C23×C4C2×#692311yescharacteristic
231C23×C4C2×#692311yescharacteristic
232#124—12232no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—614no—
237C2×C22⋊C4—614no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1224no—
246C2×C22⋊C4—1224no—
247C2×C22⋊C4—1224no—
248C2×C42—1222no—
249C2×C42—1222no—
250C2×C42—1222no—
251#99—2448no—
252C24—21no—
253C24—21no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—612no—
258C24—612no—
259C24—612no—
260C24—2442no—
261C24—2442no—
262C24—2442no—
263C24C22⋊#692312yes—
264C24C22×#692311yescharacteristic
265C2×Q8—1228no—
266C2×Q8—2444no—
267C2×Q8—2448no—
268C2×Q8—2444no—
269C2×Q8—2448no—
270C2×D4—1228no—
271C2×D4—1228no—
272C2×D4—2444no—
273C2×D4—2448no—
274C2×D4—2448no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—612no—
282C22×C4—612no—
283C22×C4—612no—
284C22×C4—1224no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—2444no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4C22⋊#692312yes—
295C22×C4C22⋊#692312yes—
296C22⋊C4—28no—
297C22⋊C4—1228no—
298C22⋊C4—1228no—
299C22⋊C4—1228no—
300C22⋊C4—2448no—
301C22⋊C4—2448no—
302C22⋊C4—2448no—
303C42—1228no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—88no—
318C23—614no—
319C23—1224no—
320C23—1224no—
321C23—1224no—
322C23—4888no—
323C23—2442no—
324C23—2442no—
325C23—2442no—
326C23—2442no—
327C23—2442no—
328C23—2442no—
329C23C2×C22⋊#692311yescharacteristic
330C23C2×C22⋊#692311yescharacteristic
331C23C2×C22⋊#692311yescharacteristic
332Q8—24416no—
333D4—24416no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—614no—
343C2×C4—1224no—
344C2×C4—1224no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—1222no—
348C2×C4—1222no—
349C2×C4—4884no—
350C2×C4—4884no—
351C2×C4—4884no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C22—28no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—84no—
369C22—84no—
370C22—84no—
371C22—1224no—
372C22—1224no—
373C22—1224no—
374C22—4884no—
375C22—4884no—
376C22—4884no—
377C22#1684070311yescharacteristic
378C22D61⋊#9912yes—
379C22#1684074112yes—
380C22#1684074212yes—
381C4—42no—
382C4—42no—
383C4—1224no—
384C4—9764no—
385C4—2442no—
386C4—2442no—
387C2—22no—
388C2—22no—
389C2—22no—
390C2—82no—
391C2—2444no—
392C2#2146164611yescharacteristic
393C2#2146164811yescharacteristic
394C2#2146164911yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC22×C61#379
prime radicalC23×C22⋊D61C22
cosocleC24×D61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#8544
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21461714C2semidirect
2. #21461714C2semidirect
9. C23×Dic122C4semidirect
14. C22×C22⋊Dic61C4semidirect
17. C22×C4×Dic61C4semidirect
18. C22×C22⋊C4×C61C4semidirect
210. C22×C22⋊C4#6923semidirect
216. C61#8544semidirect

Maximal subgroups

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

subgroupindex
#214617142
#214616452
C22×#145197632
#854461