Finite Group Databasebeta

← #30598798

Subgroups — #30598798

Order 15616.

1154 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 1154 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#21461653C212yesmaximal, complemented
2#21461066C212yesmaximal, complemented
3#21461064C212yesmaximal, complemented
4C22×#14519763C211yesmaximal, characteristic
5C2×#14519838—24no—
6#16840732C2214yes—
7#16840606—24no—
8#16840605—24no—
9C23×Dic122C2211yescharacteristic
10C23×C22⋊D61C411yescharacteristic
11C2×#14519763—24no—
12C2×#14519763—24no—
13C2×#14519763—24no—
14C22×C22⋊Dic61C2211yescharacteristic
15C22×#11758198C411yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×C4×Dic61C411yescharacteristic
18C22×C22⋊C4×C61C411yescharacteristic
19C2×#11758217—24no—
20#14519838—48no—
21C22⋊#92420—24no—
22C22⋊#92420—24no—
23C24×D61C2×C411yescharacteristic
24C22×Dic122—22no—
25C22×Dic122—22no—
26C22×Dic122—44no—
27C22×Dic122D414yes—
28C22×C22⋊D61—22no—
29C22×C22⋊D61—22no—
30C22×C22⋊D61—22no—
31C22×C22⋊D61—44no—
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2311yescharacteristic
35#14519763—232no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61D414yes—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198—22no—
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—21no—
62C23×D61—21no—
63C23×D61—42no—
64C23×D61—42no—
65C23×D61—42no—
66C23×D61C22⋊C412yes—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C412yes—
74C22×C4×C61C22⋊C412yes—
75C2×C22⋊D61—28no—
76C2×C22⋊D61—48no—
77C2×C22⋊D61—48no—
78C2×C22⋊D61—44no—
79C2×C22⋊D61—44no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—24no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C22×Dic61C22⋊C412yes—
88C22×Dic61C22⋊C412yes—
89C22×Dic61C2×D412yes—
90C22×Dic61C2×D412yes—
91C22×Dic61C2×D412yes—
92C2×Dic122—28no—
93C2×Dic122—28no—
94C2×Dic122—44no—
95C2×Dic122—48no—
96C2×Dic122—48no—
97#11758217—48no—
98C2×#92420—44no—
99C22⋊C4×C61—28no—
100C22⋊Dic61—28no—
101#11758198—28no—
102#11758198—28no—
103C4×Dic61—28no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—24no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—88no—
118C23×C61C2×C22⋊C411yescharacteristic
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61C2×C22⋊C411yescharacteristic
121C22×D61—22no—
122C22×D61—24no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—42no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—88no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—24no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C22⋊D61—416no—
141C2×Dic61—24no—
142C2×Dic61—24no—
143C2×Dic61—22no—
144C2×Dic61—24no—
145C2×Dic61—22no—
146C2×Dic61—22no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61#9914yes—
153C2×Dic61#12014yes—
154Dic122—416no—
155#92420—84no—
156#8552—611nomaximal, Sylow
157C22×C61—28no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—84no—
168C22×C61—84no—
169C22×C61—84no—
170C22×C61#37911yescharacteristic
171C22×C61C2×#9912yes—
172C22×C61#41312yes—
173C22×C61#41312yes—
174C2×D61—42no—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—84no—
178C2×D61—84no—
179C2×D61—84no—
180C4×C61—42no—
181C4×C61—42no—
182Dic61—24no—
183Dic61—24no—
184Dic61—42no—
185C22×#124—611no—
186#1784—612no—
187#1462—612no—
188#1462—612no—
189C2×C61—22no—
190C2×C61—22no—
191C2×C61—22no—
192C2×C61—82no—
193C2×C61#178111yescharacteristic
194C2×C61#178511yescharacteristic
195C2×C61#178511yescharacteristic
196D61—82no—
197C23×Q8—611no—
198C23×D4—611no—
199C2×#124—1224no—
200C2×#124—1224no—
201C2×#124—1224no—
202C22×C22⋊C4—611no—
203C22×C22⋊C4—611no—
204C22×C22⋊C4—611no—
205C22×C22⋊C4#692311yescharacteristic
206C22×C42—611no—
207C2×#101—1224no—
208#407—614no—
209#323—1224no—
210#323—1224no—
211C61#855211yescharacteristic, complemented, Sylow
212C25—611no—
213C25C2×#692311yescharacteristic
214C22×Q8—614no—
215C22×Q8—1222no—
216C22×Q8—1222no—
217C22×Q8—2444no—
218C22×D4—1222no—
219C22×D4—1222no—
220C22×D4—1222no—
221C22×D4—2444no—
222C23×C4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C23×C4C2×#692311yescharacteristic
226C23×C4C2×#692311yescharacteristic
227C2×M16—1224no—
228#124—12232no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—614no—
233C2×C22⋊C4—614no—
234C2×C22⋊C4—1222no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1222no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C42—1222no—
242C2×C42—1222no—
243C2×C42—1222no—
244#101—2448no—
245C22⋊C8—1224no—
246C22⋊C8—1224no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—612no—
253C24—1221no—
254C24—1221no—
255C24—2442no—
256C24—2442no—
257C24—2442no—
258C24C22⋊#692312yes—
259C24C22×#692311yescharacteristic
260C2×Q8—1228no—
261C2×Q8—1228no—
262C2×Q8—2444no—
263C2×Q8—2448no—
264C2×Q8—2448no—
265C2×D4—1228no—
266C2×D4—2444no—
267C2×D4—2448no—
268C2×D4—2448no—
269C2×D4—2444no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—612no—
277C22×C4—612no—
278C22×C4—612no—
279C22×C4—612no—
280C22×C4—612no—
281C22×C4—1224no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4C22⋊#692312yes—
289C22×C4C22⋊#692312yes—
290M16—2448no—
291C2×C8—2444no—
292C22⋊C4—28no—
293C22⋊C4—1228no—
294C22⋊C4—1228no—
295C22⋊C4—1228no—
296C42—1228no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—88no—
311C23—1224no—
312C23—1222no—
313C23—4888no—
314C23—2442no—
315C23—2442no—
316C23—2442no—
317C23—2442no—
318C23—2442no—
319C23—2442no—
320C23—2442no—
321C23—2442no—
322C23C2×C22⋊#692311yescharacteristic
323C23C2×C22⋊#692311yescharacteristic
324C23C2×C22⋊#692311yescharacteristic
325Q8—24416no—
326D4—24416no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—614no—
336C2×C4—614no—
337C2×C4—1224no—
338C2×C4—1224no—
339C2×C4—1224no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2442no—
348C8—4884no—
349C22—28no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—84no—
360C22—84no—
361C22—84no—
362C22—4884no—
363C22—4884no—
364C22—4884no—
365C22—2442no—
366C22—2442no—
367C22—2442no—
368C22#1684070311yescharacteristic
369C22D61⋊#9912yes—
370C22#1684074812yes—
371C22#1684074912yes—
372C4—42no—
373C4—42no—
374C4—1224no—
375C4—1224no—
376C4—2442no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—82no—
381C2—4882no—
382C2#2146164611yescharacteristic
383C2#2146165511yescharacteristic
384C2#2146165611yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC22×C61#379
prime radicalC23×C22⋊D61C4
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#8552
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. #21461653C2semidirect
2. #21461066C2semidirect
3. #21461064C2semidirect
211. C61#8552semidirect

Maximal subgroups

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

subgroupindex
#214616532
#214610662
#214610642
C22×#145197632
#855261