Finite Group Databasebeta

← #30598801

Subgroups — #30598801

Order 15616.

1166 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1166 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#21461718C212yesmaximal, complemented
2#21461064C212yesmaximal, complemented
3#21461064C212yesmaximal, complemented
4C22×#14519758C211yesmaximal, characteristic
5#16840732C2214yes—
6#16840642—24no—
7#16840605—24no—
8#16840605—24no—
9C24×Dic61C411yescharacteristic
10C2×#14519758—24no—
11C2×#14519758—24no—
12C2×#14519758—24no—
13C22×#11758196C411yescharacteristic
14C22×#11758196C2211yescharacteristic
15C22×C22⋊Dic61C411yescharacteristic
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2211yescharacteristic
18C23×D4×C61C411yescharacteristic
19C2×#11758217—24no—
20C22⋊#92420—24no—
21C22⋊#92420—24no—
22C23×Dic61—22no—
23C23×Dic61—22no—
24C23×Dic61—22no—
25C23×Dic61—44no—
26C23×Dic61C2×C411yescharacteristic
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2311yescharacteristic
30#14519758—232no—
31C2×#11758196—22no—
32C2×#11758196—22no—
33C2×#11758196—22no—
34C2×#11758196—22no—
35C2×#11758196—22no—
36C2×#11758196D414yes—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61D414yes—
45C2×C22⋊Dic61D414yes—
46C25×C61C2×C411yescharacteristic
47C25×C61C2×C411yescharacteristic
48C22×D4×C61—22no—
49C22×D4×C61—22no—
50C22×D4×C61—22no—
51C22×D4×C61—44no—
52C23×C4×C61C2×C411yescharacteristic
53C24×C61—24no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—21no—
57C24×C61—21no—
58C24×C61—42no—
59C24×C61—42no—
60C24×C61—42no—
61C24×C61—42no—
62C24×C61—42no—
63C24×C61C22⋊C412yes—
64C24×C61C22⋊C412yes—
65C24×C61C22×C411yescharacteristic
66C2×D4×C61—28no—
67C2×D4×C61—48no—
68C2×D4×C61—44no—
69C2×D4×C61—48no—
70C2×D4×C61—44no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C412yes—
75C22×Dic61—28no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—42no—
83C22×Dic61—48no—
84C22×Dic61—48no—
85C22×Dic61—44no—
86C22×Dic61—44no—
87C22×Dic61—42no—
88C22×Dic61—42no—
89C22×Dic61—42no—
90C22×Dic61C22⋊C412yes—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C22⋊C412yes—
93C22×Dic61C2×D412yes—
94C22×Dic61C2×D412yes—
95C22×Dic61C2×D412yes—
96#11758217—48no—
97C2×#92420—44no—
98C22⋊Dic61—28no—
99C22⋊Dic61—28no—
100C22⋊Dic61—28no—
101#11758196—28no—
102#11758196—28no—
103C23×C61—22no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—24no—
107C23×C61—24no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—24no—
112C23×C61—416no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61—88no—
128C23×C61#10014yes—
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61C2×C22⋊C411yescharacteristic
131C23×C61C2×C22⋊C411yescharacteristic
132D4×C61—416no—
133C2×C4×C61—22no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×Dic61—24no—
138C2×Dic61—22no—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—416no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61#12014yes—
154#92420—84no—
155#8553—611nomaximal, Sylow
156C22×C61—28no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—24no—
164C22×C61—44no—
165C22×C61—48no—
166C22×C61—44no—
167C22×C61—48no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—42no—
173C22×C61—84no—
174C22×C61—84no—
175C22×C61—84no—
176C22×C61#37911yescharacteristic
177C22×C61#41012yes—
178C22×C61C2×#10012yes—
179C22×C61#41312yes—
180C4×C61—42no—
181Dic61—24no—
182Dic61—42no—
183Dic61—42no—
184Dic61—42no—
185C22×#123—611no—
186#1821—612no—
187#1462—612no—
188#1462—612no—
189C2×C61—22no—
190C2×C61—22no—
191C2×C61—22no—
192C2×C61—44no—
193C2×C61—82no—
194C2×C61#178211yescharacteristic
195C2×C61#178311yescharacteristic
196C2×C61#178511yescharacteristic
197C23×D4#692311yescharacteristic
198C24×C4—611no—
199C2×#123—1224no—
200C2×#123—1224no—
201C2×#123—1224no—
202C22×C4⋊C4—611no—
203C22×C4⋊C4—611no—
204C22×C22⋊C4—611no—
205C22×C22⋊C4—611no—
206C22×C22⋊C4—611no—
207#407—614no—
208#343—1224no—
209#323—1224no—
210#323—1224no—
211C61#855311yescharacteristic, complemented, Sylow
212C25C2×#692311yescharacteristic
213C25C2×#692311yescharacteristic
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C23×C4—611no—
219C23×C4—611no—
220C23×C4—611no—
221C23×C4—611no—
222C23×C4—1222no—
223C23×C4—1222no—
224C23×C4—1222no—
225C23×C4—2444no—
226C23×C4C2×#692311yescharacteristic
227C2×M16—1224no—
228#123—12232no—
229C2×C4⋊C4—614no—
230C2×C4⋊C4—1222no—
231C2×C4⋊C4—1222no—
232C2×C4⋊C4—1222no—
233C2×C4⋊C4—1222no—
234C2×C4⋊C4—1222no—
235C2×C22⋊C4—614no—
236C2×C22⋊C4—614no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4—1222no—
244C22⋊C8—1224no—
245C22⋊C8—1224no—
246C24—24no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C22⋊#692312yes—
257C24C22⋊#692312yes—
258C24C22×#692311yescharacteristic
259C2×D4—28no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—612no—
268C22×C4—612no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—612no—
272C22×C4—612no—
273C22×C4—1228no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—2442no—
281C22×C4—2448no—
282C22×C4—2448no—
283C22×C4—2444no—
284C22×C4—2444no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4C22⋊#692312yes—
289M16—2448no—
290C2×C8—2444no—
291C4⋊C4—1228no—
292C4⋊C4—1228no—
293C22⋊C4—1228no—
294C22⋊C4—1228no—
295C22⋊C4—1228no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23#1451983714yes—
322C23C2×C22⋊#692311yescharacteristic
323C23C2×C22⋊#692311yescharacteristic
324C23C2×C22⋊#692311yescharacteristic
325D4—416no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—614no—
331C2×C4—1224no—
332C2×C4—1222no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—24416no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C8—4884no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—24no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22#1684070311yescharacteristic
369C22#1684074212yes—
370C22C2×#1451983712yes—
371C22#1684074912yes—
372C4—42no—
373C4—1224no—
374C4—2442no—
375C4—2442no—
376C4—2442no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—44no—
381C2—82no—
382C2#2146164911yescharacteristic
383C2#2146165111yescharacteristic
384C2#2146165611yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
Fitting, prime radicalC23×D4×C61C4
socleC22×C61#379
cosocleC23×Dic61C23
hypercenterC23×D4#6923
hypocenterC61#8553
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461718C2semidirect
2. #21461064C2semidirect
3. #21461064C2semidirect
211. C61#8553semidirect

Maximal subgroups

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

subgroupindex
#214617182
#214610642
C22×#145197582
#855361