Finite Group Databasebeta

← #30617879

Subgroups — #30617879

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C212yesmaximal
2#21462009C218yesmaximal, complemented
3#21461714C214yesmaximal
4C22×#14519757C211yesmaximal, characteristic
5C2×#14519856—22no—
6#16840705—22no—
7#16840704—22no—
8#16840704C2214yes—
9#16840703C2214yes—
10C4×C22⋊#6923—22no—
11C4×C22⋊#6923C2214yes—
12#16840640C22116yes—
13C23×Dic122C411yescharacteristic, complemented
14C23×C4×D61C2211yescharacteristic
15C2×#14519757—22no—
16C2×#14519757—22no—
17C2×#14519757—22no—
18C2×#14519757—22no—
19C2×#14519757C414yescomplemented
20C2×#14519757C2214yes—
21C22×C4⋊Dic61C412yescomplemented
22C22×#11758198C2212yes—
23C22×C4⋊C4×C61C411yescharacteristic, complemented
24#14519856—48no—
25C2×C4⋊#6923—24no—
26C2×C22⋊#6923—24no—
27C2×C22⋊#6923—24no—
28C2×C22⋊#6923C2318yes—
29C2×C4×#6923—24no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—24no—
32C24×D61C2311yescharacteristic
33C22×Dic122—21no—
34C22×Dic122—22no—
35C22×Dic122—21no—
36C22×Dic122—44no—
37C22×Dic122C2×C414yes—
38C23×Dic61C2×C412yes—
39C23×Dic61C2×C411yescharacteristic
40C22×C4×D61—44no—
41C22×C4×D61C2×C412yes—
42C22×C4×D61D412yes—
43C22×C4×D61D412yes—
44C22×C4×D61D412yes—
45C22×C4×D61D412yes—
46C22×C4×D61C2312yes—
47#14519757—24no—
48#14519757—24no—
49#14519757—24no—
50#14519757—24no—
51#14519757—24no—
52#14519757—24no—
53#14519757—24no—
54#14519757—24no—
55C2×C4⋊Dic61—24no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61C2×C418yes—
59C2×#11758198—24no—
60C2×#11758198—24no—
61C2×#11758198C2×C414yes—
62C2×#11758198C2314yes—
63C23×C4×C61C2×C412yes—
64C23×C4×C61C2×C411yescharacteristic
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61C2×C414yes—
69C24×C61C22×C411yescharacteristic
70C23×D61—24no—
71C23×D61—42no—
72C23×D61—42no—
73C23×D61C22×C411yescharacteristic
74C23×D61C2×D412yes—
75C23×D61C2×D412yes—
76C23×D61C2411yescharacteristic
77C22×#6923—28no—
78C22×C4×C61—24no—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—44no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C412yes—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C412yes—
86C22×C4×C61C22×C414yes—
87C22×Dic61—22no—
88C22×Dic61—22no—
89C22×Dic61—42no—
90C22×Dic61—44no—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22×C414yes—
95C22×Dic61C22×C412yes—
96C2×C4×D61—28no—
97C2×C4×D61—44no—
98C2×C4×D61—44no—
99C2×C4×D61—44no—
100C2×C4×D61—44no—
101C2×C4×D61—44no—
102C2×C4×D61C22⋊C412yes—
103C2×C4×D61C22⋊C412yes—
104C2×C4×D61C22⋊C412yes—
105C2×C4×D61C22⋊C412yes—
106C2×C4×D61C2×D412yes—
107C2×C4×D61C2×D412yes—
108C2×C4×D61C2×D412yes—
109C2×C4×D61C2×D412yes—
110C2×Dic122—24no—
111C2×Dic122—24no—
112C2×Dic122—44no—
113C2×Dic122—48no—
114C2×Dic122—44no—
115C2×Dic122—48no—
116C22⋊#6923—216no—
117C22⋊#6923—48no—
118C22⋊#6923—48no—
119C4⋊#6923—48no—
120C4×#6923—48no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—24no—
123#11758198—28no—
124#11758198—28no—
125C4⋊Dic61—28no—
126C4⋊Dic61—28no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—22no—
130C23×C61C2×C22⋊C411yescharacteristic
131C23×C61C2×C22⋊C411yescharacteristic
132C23×C61C23×C411yescharacteristic
133C22×D61—22no—
134C22×D61—24no—
135C22×D61—24no—
136C22×D61—24no—
137C22×D61—24no—
138C22×D61—41no—
139C22×D61—42no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—42no—
143C22×D61—41no—
144C22×D61—416no—
145C22×D61C2×C22⋊C412yes—
146C22×D61C22×D412yes—
147C22×D612−1+414yes—
148C2×#6923—48no—
149C2×#6923—48no—
150C2×#6923—48no—
151C2×C4×C61—22no—
152C2×C4×C61—24no—
153C2×C4×C61—24no—
154C2×C4×C61—24no—
155C2×C4×C61—44no—
156C2×C4×C61—44no—
157C2×C4×C61—44no—
158C2×C4×C61#9914yes—
159C2×C4×C61C2×C22⋊C412yes—
160C2×C4×C61C2×C22⋊C412yes—
161C2×Dic61—24no—
162C2×Dic61—24no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166C2×Dic61—44no—
167C2×Dic61—44no—
168C2×Dic61—44no—
169C2×Dic61C2×C22⋊C412yes—
170C2×Dic61C2×C22⋊C412yes—
171C4×D61—24no—
172C4×D61—24no—
173C4×D61—24no—
174C4×D61—24no—
175C4×D61—42no—
176C4×D61—42no—
177C4×D61—42no—
178C4×D61—42no—
179Dic122—48no—
180Dic122—48no—
181#21825—611nomaximal, Sylow
182C22×C61—28no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61C2×#9912yes—
190C22×C61#41012yes—
191C22×C61C22×C22⋊C411yescharacteristic
192C22×C61#51912yes—
193C2×D61—22no—
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—22no—
197C2×D61—22no—
198C2×D61—44no—
199C2×D61—48no—
200C2×D61—48no—
201C2×D61—44no—
202C2×D61#53614yes—
203#6923—88no—
204C4×C61—22no—
205C4×C61—44no—
206C4×C61C2×#9914yes—
207Dic61—22no—
208Dic61—22no—
209Dic61—44no—
210C22×C22⋊Q8—611no—
211#2280—612no—
212#1996—618no—
213#1819—614no—
214C2×C61—22no—
215C2×C61—22no—
216C2×C61—22no—
217C2×C61#231011yescharacteristic
218C2×C61C22×#9911yescharacteristic
219C2×C61C2×#41011yescharacteristic
220D61—24no—
221D61—42no—
222C23×Q8—611no—
223C24×C4—611no—
224C2×C22⋊Q8—614no—
225C2×C22⋊Q8—614no—
226C2×C22⋊Q8—1222no—
227C2×C22⋊Q8—1222no—
228C2×C22⋊Q8—1222no—
229C2×C22⋊Q8—1222no—
230C2×#117—1222no—
231C22×C4⋊C4—612no—
232C22×C4⋊C4#692311yescharacteristic, complemented
233C22×C22⋊C4—612no—
234#381—1222no—
235#380—614no—
236#380—1222no—
237#379—614no—
238C4×C22⋊C4—614no—
239C4×C22⋊C4—1222no—
240#342—6116no—
241C61#2182511yescharacteristic, complemented, Sylow
242C25—611no—
243C22×Q8—614no—
244C22×Q8—1222no—
245C22×Q8—1221no—
246C22×Q8—1221no—
247C22×Q8—2444no—
248C23×C4—611no—
249C23×C4—612no—
250C23×C4—612no—
251C23×C4—612no—
252C23×C4—612no—
253C23×C4—612no—
254C23×C4—612no—
255C23×C4—612no—
256C23×C4—2444no—
257C23×C4C2×#692312yes—
258C23×C4C2×#692311yescharacteristic
259C22⋊Q8—1224no—
260C22⋊Q8—1224no—
261C22⋊Q8—1224no—
262C22⋊Q8—1224no—
263C22⋊Q8—1224no—
264C22⋊Q8—1224no—
265C22⋊Q8—1224no—
266C22⋊Q8—1224no—
267#117—2448no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—618no—
272C2×C4⋊C4—1224no—
273C2×C4⋊C4—1222no—
274C2×C4⋊C4—1222no—
275C2×C4⋊C4—1224no—
276C2×C4⋊C4C2×#692314yes—
277C2×C22⋊C4—614no—
278C2×C22⋊C4—614no—
279C2×C22⋊C4—618no—
280C2×C22⋊C4—1224no—
281C2×C22⋊C4—1224no—
282C2×C22⋊C4—1224no—
283C2×C22⋊C4—1224no—
284C2×C42—1224no—
285#95—1224no—
286#95—1224no—
287C24—611no—
288C24—612no—
289C24—612no—
290C24—611no—
291C24—1224no—
292C24—2442no—
293C24—2442no—
294C24C22×#692311yescharacteristic
295C2×Q8—1224no—
296C2×Q8—1224no—
297C2×Q8—2448no—
298C2×Q8—2444no—
299C2×Q8—2448no—
300C2×Q8—2444no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—44no—
305C22×C4—612no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—612no—
309C22×C4—612no—
310C22×C4—612no—
311C22×C4—612no—
312C22×C4—612no—
313C22×C4—612no—
314C22×C4—612no—
315C22×C4—612no—
316C22×C4—612no—
317C22×C4—614no—
318C22×C4—1228no—
319C22×C4—1222no—
320C22×C4—1222no—
321C22×C4—1228no—
322C22×C4—2444no—
323C22×C4—2444no—
324C22×C4—2444no—
325C22×C4—2444no—
326C22×C4—2444no—
327C22×C4—2442no—
328C22×C4—2444no—
329C22×C4C22⋊#692311yescharacteristic
330C22×C4C22⋊#692312yes—
331C22×C4C22⋊#692311yescharacteristic
332C22×C4C22×#692312yes—
333C22×C4C22×#692314yes—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—1228no—
337C4⋊C4—1228no—
338C4⋊C4—2448no—
339C22⋊C4—1228no—
340C22⋊C4—1228no—
341C22⋊C4—12216no—
342C22⋊C4—2448no—
343C22⋊C4—2448no—
344C42—2448no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—612no—
349C23—614no—
350C23—612no—
351C23—1224no—
352C23—1224no—
353C23—1224no—
354C23—1224no—
355C23—1222no—
356C23—2442no—
357C23—2441no—
358C23—2441no—
359C23—2442no—
360C23—2441no—
361C23—2441no—
362C23—24416no—
363C23C2×C22⋊#692311yescharacteristic
364C23C2×C22⋊#692311yescharacteristic
365C23C23×#692311yescharacteristic
366Q8—2448no—
367Q8—2448no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—612no—
376C2×C4—612no—
377C2×C4—1224no—
378C2×C4—1224no—
379C2×C4—1224no—
380C2×C4—1224no—
381C2×C4—1224no—
382C2×C4—1224no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2448no—
391C2×C4—2448no—
392C2×C4—2448no—
393C2×C4—2444no—
394C2×C4—2444no—
395C2×C4—2444no—
396C2×C4#1451983614yes—
397C2×C4C2×C22⋊#692312yes—
398C2×C4C2×C22⋊#692312yes—
399C22—28no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—614no—
407C22—1222no—
408C22—1222no—
409C22—1222no—
410C22—1222no—
411C22—1222no—
412C22—2444no—
413C22—2448no—
414C22—2448no—
415C22—2444no—
416C22C2×#1451983612yes—
417C22#1684074212yes—
418C22C22×C22⋊#692311yescharacteristic
419C22#1684083012yes—
420C4—22no—
421C4—44no—
422C4—1222no—
423C4—1222no—
424C4—4888no—
425C4—2444no—
426C4C2×#1451983614yes—
427C2—22no—
428C2—22no—
429C2—22no—
430C2—1224no—
431C2—2442no—
432C2#2146229311yescharacteristic
433C2C22×#1451983611yescharacteristic
434C2C2×#1684074211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61#21825
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21462009C2semidirect
13. C23×Dic122C4semidirect
19. C2×#14519757C4semidirect
21. C22×C4⋊Dic61C4semidirect
23. C22×C4⋊C4×C61C4semidirect
232. C22×C4⋊C4#6923semidirect
241. C61#21825semidirect

Maximal subgroups

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

subgroupindex
#214622632
#214620092
#214617142
C22×#145197572
#2182561