Finite Group Databasebeta

← #30598784

Subgroups — #30598784

Order 15616.

1198 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 1198 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#21461651C212yesmaximal
2#21461069C212yesmaximal, complemented
3#21461067C212yesmaximal, complemented
4C22×#14519759C211yesmaximal, characteristic
5C2×#14519837—24no—
6#16840732C2214yes—
7#16840608—24no—
8#16840607—24no—
9C23×C22⋊D61C2211yescharacteristic
10C23×C4×D61C411yescharacteristic
11C2×#14519759—24no—
12C2×#14519759—24no—
13C2×#14519759—24no—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758196C2211yescharacteristic
16C22×C22⋊Dic61C411yescharacteristic
17C22×#11758198C411yescharacteristic
18C22×C22⋊C4×C61C411yescharacteristic
19C2×#11758217—24no—
20#14519837—48no—
21C22⋊#92420—24no—
22C22⋊#92420—24no—
23C24×D61C2×C411yescharacteristic
24C22×C22⋊D61—22no—
25C22×C22⋊D61—22no—
26C22×C22⋊D61—44no—
27C22×C22⋊D61D414yes—
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2311yescharacteristic
31C22×C4×D61—22no—
32C22×C4×D61—22no—
33C22×C4×D61—22no—
34C22×C4×D61—44no—
35#14519759—232no—
36C2×C4⋊Dic61—22no—
37C2×C4⋊Dic61—22no—
38C2×C4⋊Dic61D414yes—
39C2×#11758196—22no—
40C2×#11758196—22no—
41C2×#11758196D414yes—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C2×#11758198—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—24no—
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—28no—
77C2×C22⋊D61—44no—
78C2×C22⋊D61—48no—
79C2×C22⋊D61—48no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—42no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C22×Dic61C22⋊C412yes—
88C22×Dic61C22⋊C412yes—
89C22×Dic61C2×D412yes—
90C22×Dic61C2×D412yes—
91C22×Dic61C2×D412yes—
92C2×C4×D61—28no—
93C2×C4×D61—48no—
94C2×C4×D61—44no—
95C2×C4×D61—48no—
96C2×C4×D61—44no—
97#11758217—48no—
98C2×#92420—44no—
99C22⋊C4×C61—28no—
100C22⋊Dic61—28no—
101#11758198—28no—
102C4⋊Dic61—28no—
103#11758196—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—24no—
122C22×D61—24no—
123C22×D61—22no—
124C22×D61—22no—
125C22×D61—416no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61#9914yes—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—24no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C22⋊D61—416no—
142C2×Dic61—24no—
143C2×Dic61—22no—
144C2×Dic61—22no—
145C2×Dic61—24no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61#12014yes—
154C4×D61—416no—
155#92420—84no—
156#8548—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—24no—
175C2×D61—44no—
176C2×D61—48no—
177C2×D61—48no—
178C2×D61—42no—
179C2×D61—42no—
180C2×D61—44no—
181C4×C61—42no—
182C4×C61—42no—
183Dic61—24no—
184Dic61—42no—
185Dic61—42no—
186C22×#123—611no—
187#1783—612no—
188#1463—612no—
189#1463—612no—
190C2×C61—22no—
191C2×C61—22no—
192C2×C61—22no—
193C2×C61—82no—
194C2×C61#178111yescharacteristic
195C2×C61#178511yescharacteristic
196C2×C61#178511yescharacteristic
197D61—44no—
198C23×D4—611no—
199C24×C4—611no—
200C2×#123—1224no—
201C2×#123—1224no—
202C2×#123—1224no—
203C22×C4⋊C4—611no—
204C22×C4⋊C4—611no—
205C22×C22⋊C4—611no—
206C22×C22⋊C4—611no—
207C22×C22⋊C4#692311yescharacteristic
208C2×#100—1224no—
209#407—614no—
210#324—1224no—
211#324—1224no—
212C61#854811yescharacteristic, complemented, Sylow
213C25—611no—
214C25C2×#692311yescharacteristic
215C22×D4—614no—
216C22×D4—1222no—
217C22×D4—1222no—
218C22×D4—2444no—
219C23×C4—611no—
220C23×C4—611no—
221C23×C4—611no—
222C23×C4—1222no—
223C23×C4—1222no—
224C23×C4—1222no—
225C23×C4—2444no—
226C23×C4C2×#692311yescharacteristic
227C23×C4C2×#692311yescharacteristic
228C2×M16—1224no—
229#123—12232no—
230C2×C4⋊C4—614no—
231C2×C4⋊C4—614no—
232C2×C4⋊C4—1222no—
233C2×C4⋊C4—1222no—
234C2×C4⋊C4—1222no—
235C2×C4⋊C4—1222no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245#100—2448no—
246C22⋊C8—1224no—
247C22⋊C8—1224no—
248C24—21no—
249C24—21no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—612no—
254C24—1224no—
255C24—1221no—
256C24—1221no—
257C24—2442no—
258C24—2442no—
259C24C22⋊#692312yes—
260C24C22×#692311yescharacteristic
261C2×D4—1228no—
262C2×D4—1228no—
263C2×D4—2444no—
264C2×D4—2448no—
265C2×D4—2448no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—612no—
273C22×C4—612no—
274C22×C4—612no—
275C22×C4—612no—
276C22×C4—612no—
277C22×C4—1228no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—2444no—
283C22×C4—2448no—
284C22×C4—2444no—
285C22×C4—2448no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4C22⋊#692312yes—
290C22×C4C22⋊#692312yes—
291M16—2448no—
292C2×C8—2444no—
293C4⋊C4—1228no—
294C4⋊C4—1228no—
295C22⋊C4—28no—
296C22⋊C4—1228no—
297C22⋊C4—1228no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—24no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—88no—
312C23—614no—
313C23—1224no—
314C23—1224no—
315C23—1222no—
316C23—1222no—
317C23—2442no—
318C23—2442no—
319C23—24416no—
320C23—2442no—
321C23—2442no—
322C23—2442no—
323C23—2442no—
324C23C2×C22⋊#692311yescharacteristic
325C23C2×C22⋊#692311yescharacteristic
326C23C2×C22⋊#692311yescharacteristic
327D4—24416no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—614no—
337C2×C4—1224no—
338C2×C4—1224no—
339C2×C4—1222no—
340C2×C4—1222no—
341C2×C4—24416no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2442no—
349C8—4884no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—84no—
361C22—84no—
362C22—84no—
363C22—1224no—
364C22—2444no—
365C22—2444no—
366C22—2448no—
367C22—2448no—
368C22—2442no—
369C22—2442no—
370C22#1684070311yescharacteristic
371C22D61⋊#9912yes—
372C22#1684074812yes—
373C22#1684074912yes—
374C4—42no—
375C4—42no—
376C4—1224no—
377C4—2442no—
378C4—2442no—
379C2—22no—
380C2—22no—
381C2—22no—
382C2—82no—
383C2—2444no—
384C2#2146164611yescharacteristic
385C2#2146165511yescharacteristic
386C2#2146165611yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC22×C61#379
prime radicalC23×C22⋊D61C22
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#8548
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
2. #21461069C2semidirect
3. #21461067C2semidirect
212. C61#8548semidirect

Maximal subgroups

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

subgroupindex
#214616512
#214610692
#214610672
C22×#145197592
#854861