Finite Group Databasebeta

← #30598725

Subgroups — #30598725

Order 15616.

910 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 100 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#21461352C212yesmaximal, complemented
2#21461300C212yesmaximal, complemented
3#21461070C212yesmaximal, complemented
4C2×#16840336C211yesmaximal, characteristic
5#16840782C2214yes—
6#16840622—24no—
7#16840618—24no—
8#16840607—22no—
9#16840605—22no—
10C22×#11758222C411yescharacteristic
11#16840336—24no—
12#16840336—24no—
13C2×#14519775C411yescharacteristic
14C2×#14519775C2211yescharacteristic
15C2×#14519766C411yescharacteristic
16C2×#14519766C2211yescharacteristic
17C2×#14519760—22no—
18C2×#14519753—22no—
19C2×#14519736C2211yescharacteristic
20C23×Q8×C61C411yescharacteristic
21#14519844—24no—
22C22⋊#92420—24no—
23#14519832—24no—
24C2×#11758222—22no—
25C2×#11758222—22no—
26C2×#11758222—42no—
27C2×#11758222—42no—
28C22×Dic122C2×C411yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C22×C4×D61—21no—
32#14519775—22no—
33#14519775—22no—
34#14519775—22no—
35#14519775D414yes—
36#14519766—22no—
37#14519766—22no—
38#14519766—22no—
39#14519766D414yes—
40#14519760—216no—
41#14519753—216no—
42#14519736—22no—
43#14519736—22no—
44C2×C4⋊Dic61D412yes—
45C2×#11758196—22no—
46C2×#11758196—22no—
47C2×C22⋊Dic61D412yes—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61C2311yescharacteristic
52C22×Q8×C61—22no—
53C22×Q8×C61—22no—
54C22×Q8×C61—21no—
55C22×Q8×C61—42no—
56C22×Q8×C61—42no—
57C22×Q8×C61C2×C411yescharacteristic
58C22×Q8×C61C2×C411yescharacteristic
59C23×C4×C61—21no—
60C23×C4×C61C2×C411yescharacteristic
61C24×C61C22⋊C411yescharacteristic
62C23×D61—21no—
63C2×Q8×C61—24no—
64C2×Q8×C61—21no—
65C2×Q8×C61—21no—
66C2×Q8×C61—48no—
67C2×Q8×C61—48no—
68C2×Q8×C61—42no—
69C2×Q8×C61—41no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C2×Q8×C61—42no—
74C2×Q8×C61C22⋊C412yes—
75C2×Q8×C61C22⋊C412yes—
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—24no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—42no—
84C22×C4×C61—42no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C2×C22⋊D61—24no—
88C2×C22⋊D61—42no—
89C2×C22⋊D61—42no—
90C22×Dic61—21no—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C2×D412yes—
93#11758222—48no—
94#11758222—48no—
95C2×D244—21no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244C22⋊C412yes—
99C2×C4×D61—24no—
100C2×C4×D61—42no—
101C2×C4×D61—42no—
102C2×Dic122—21no—
103C2×Dic122—41no—
104C2×Dic122—41no—
105C2×Dic122C22⋊C412yes—
106C2×#92420—44no—
107C22⋊Dic61—24no—
108#11758198—28no—
109#11758198—28no—
110C4⋊Dic61—24no—
111#11758196—28no—
112#11758196—28no—
113C4×Dic61—21no—
114C4×Dic61—21no—
115C4×Dic61C2×D412yes—
116C4×Dic61C2×D412yes—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—42no—
120C23×C61#10012yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—22no—
123C22×D61—44no—
124C22×D61—41no—
125C22×D61—41no—
126Q8×C61—42no—
127Q8×C61—42no—
128Q8×C61—48no—
129Q8×C61—42no—
130Q8×C61—42no—
131C2×C4×C61—24no—
132C2×C4×C61—24no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—48no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61#10012yes—
145C2×C4×C61C2×C22⋊C411yescharacteristic
146C2×C4×C61C2×C22⋊C411yescharacteristic
147C22⋊D61—48no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—42no—
154C2×Dic61#12014yes—
155D244—42no—
156D244—42no—
157C4×D61—48no—
158Dic122—42no—
159Dic122—42no—
160#92420—84no—
161#8524—611nomaximal, Sylow
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61#37911yescharacteristic
168C22×C61C2×#10011yescharacteristic
169C22×C61C2×#10011yescharacteristic
170C2×D61—44no—
171C2×D61—42no—
172C2×D61—82no—
173C2×D61—82no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—22no—
177C4×C61—42no—
178C4×C61—42no—
179Dic61—24no—
180Dic61—42no—
181Dic61—42no—
182C2×#536—611no—
183#1558—612no—
184#1533—612no—
185#1464—612no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—42no—
189C2×C61#178311yescharacteristic
190C2×C61#179612yes—
191D61—82no—
192C22×#41—611no—
193C23×Q8#692311yescharacteristic
194#536—1224no—
195#536—1224no—
196C2×C4⋊Q8—611no—
197C2×C4⋊Q8—611no—
198C2×#124—611no—
199C2×#124—611no—
200C2×#123—1222no—
201C2×C22⋊Q8—1222no—
202C2×#117—611no—
203#432—614no—
204#333—1224no—
205#329—1224no—
206#324—1222no—
207#323—1222no—
208C61#852411yescharacteristic, complemented, Sylow
209C2×#41—1222no—
210C2×#41—1222no—
211C2×#41—2442no—
212C2×#41—2442no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—21no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—611no—
219C22×Q8C2×#692311yescharacteristic
220C22×Q8C2×#692311yescharacteristic
221C22×D4—611no—
222C22×D4—1221no—
223C23×C4—21no—
224C23×C4—1221no—
225C23×C4C2×#692311yescharacteristic
226C4⋊Q8—614no—
227C4⋊Q8—1222no—
228C4⋊Q8—1222no—
229C4⋊Q8—1222no—
230#124—614no—
231#124—1222no—
232#124—1222no—
233#124—1222no—
234#123—12216no—
235C22⋊Q8—12216no—
236#117—1222no—
237#117—1222no—
238C2×C4⋊C4—612no—
239C2×C4⋊C4—1222no—
240C2×C4⋊C4—1222no—
241C2×C22⋊C4—612no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4—1222no—
244C2×C42—611no—
245C2×C42—611no—
246C4⋊C8—1224no—
247C22⋊C8—1224no—
248#97—1224no—
249C24—1221no—
250C24C22⋊#692311yescharacteristic
251#41—2448no—
252#41—2448no—
253C2×Q8—24no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—48no—
257C2×Q8—48no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—612no—
265C2×Q8—1221no—
266C2×Q8—2441no—
267C2×Q8—2441no—
268C2×Q8C22⋊#692312yes—
269C2×Q8C22⋊#692312yes—
270C2×D4—612no—
271C2×D4—1224no—
272C2×D4—1221no—
273C2×D4—2441no—
274C2×D4—2441no—
275C2×D4—2442no—
276C2×D4—2442no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—612no—
287C22×C4—612no—
288C22×C4—1224no—
289C22×C4—1221no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4C22⋊#692311yescharacteristic
293C22×C4C22×#692311yescharacteristic
294C2×C8—2444no—
295C4⋊C4—1224no—
296C4⋊C4—1228no—
297C4⋊C4—1228no—
298C22⋊C4—1228no—
299C22⋊C4—1228no—
300C22⋊C4—1224no—
301C42—612no—
302C42—612no—
303C42—1221no—
304C42—1221no—
305C23—21no—
306C23—21no—
307C23—42no—
308C23—1222no—
309C23—2444no—
310C23—2441no—
311C23—2441no—
312C23#1451983712yes—
313C23C2×C22⋊#692311yescharacteristic
314Q8—42no—
315Q8—42no—
316Q8—48no—
317Q8—42no—
318Q8—42no—
319Q8—2442no—
320Q8—2442no—
321D4—2442no—
322D4—2442no—
323D4—2448no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—48no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—614no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—2448no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2442no—
345C2×C4#1451983712yes—
346C2×C4C2×C22⋊#692311yescharacteristic
347C2×C4C2×C22⋊#692311yescharacteristic
348C8—4884no—
349C22—22no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—4882no—
355C22—4882no—
356C22—2442no—
357C22—2444no—
358C22#1684070311yescharacteristic
359C22C2×#1451983711yescharacteristic
360C22C2×#1451983711yescharacteristic
361C4—22no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—1224no—
367C4—2442no—
368C4—2442no—
369C2—21no—
370C2—21no—
371C2—42no—
372C2—4882no—
373C2#2146165111yescharacteristic
374C2#2146167112yes—
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61#379
prime radicalC22×#11758222C4
cosocleC2×C4×Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#8524
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. #21461352C2semidirect
2. #21461300C2semidirect
3. #21461070C2semidirect
208. C61#8524semidirect

Maximal subgroups

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

subgroupindex
#214613522
#214613002
#214610702
C2×#168403362
#852461