Finite Group Databasebeta

← #30598827

Subgroups — #30598827

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461675C212yesmaximal
2C2×#16840607C212yesmaximal
3#21460657C212yesmaximal
4C2×C4×C4⋊Dic61C211yesmaximal, characteristic
5C2×#14519844—22no—
6C2×C22⋊#92420C2214yes—
7#16840628—22no—
8#16840607—24no—
9C22×C4⋊Dic61C411yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×C4×Dic61C411yescharacteristic
12C22×C4×Dic61C2211yescharacteristic
13C4×C4⋊Dic61—22no—
14C4×C4⋊Dic61—22no—
15C2×#14519684C411yescharacteristic
16C2×#14519684C2211yescharacteristic
17C22×C42×C61C411yescharacteristic
18C22×#92420—24no—
19#14519844—44no—
20C22⋊#92420—24no—
21C22⋊#92420—24no—
22C23×Dic61C2×C411yescharacteristic
23C23×Dic61C2×C411yescharacteristic
24C23×Dic61C2×C411yescharacteristic
25C23×Dic61C2311yescharacteristic
26C2×C4⋊Dic61—22no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61—22no—
30C2×C4⋊Dic61C814yescomplemented
31C2×C4⋊Dic61D412yes—
32C2×C4⋊Dic61Q812yes—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—42no—
36C2×C4×Dic61—42no—
37C2×C4×Dic61D412yes—
38C2×C4×Dic61D412yes—
39#14519684—22no—
40#14519684—22no—
41C23×C4×C61C2×C411yescharacteristic
42C23×C4×C61C2×C411yescharacteristic
43C23×C4×C61C2×C411yescharacteristic
44C2×C42×C61—21no—
45C2×C42×C61—21no—
46C2×C42×C61—22no—
47C2×C42×C61—22no—
48C24×C61C22×C411yescharacteristic
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C4⋊C412yes—
64C22×C4×C61C2×C812yes—
65C22×C4×C61M1612yes—
66C22×C4×C61M1612yes—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—41no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—41no—
76C22×Dic61—41no—
77C22×Dic61—41no—
78C22×Dic61—41no—
79C22×Dic61C22⋊C412yes—
80C22×Dic61C4⋊C412yes—
81C22×Dic61C2×C812yes—
82C22×Dic61C2×C812yes—
83C22×Dic61M1612yes—
84C22×Dic61C2×D412yes—
85C22×Dic61C2×D411yescharacteristic
86C22×Dic61C2×Q811yescharacteristic
87C22×Dic61#4112yes—
88C2×#92420—44no—
89C2×#92420—44no—
90C2×#92420—44no—
91C42×C61—24no—
92C42×C61—24no—
93C4⋊Dic61—28no—
94C4⋊Dic61—24no—
95C4⋊Dic61—24no—
96C4×Dic61—24no—
97C4×Dic61—42no—
98C4×Dic61—42no—
99C4×Dic61—42no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C2×C4⋊C411yescharacteristic
106C23×C61#11711yescharacteristic
107C23×C61C22×C811yescharacteristic
108C23×C61C2×M1611yescharacteristic
109C23×C61C2×M1611yescharacteristic
110C23×C61C2×M1611yescharacteristic
111C2×C4×C61—24no—
112C2×C4×C61—24no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61#9912yes—
129C2×C4×C61#10112yes—
130C2×C4×C61C4⋊C814yescomplemented
131C2×Dic61—22no—
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—22no—
135C2×Dic61—22no—
136C2×Dic61—22no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61C22⋊C814yes—
151C2×Dic61#12112yes—
152C2×Dic61C22⋊Q812yes—
153#92420—84no—
154#8560—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61#38011yescharacteristic
168C22×C61C2×C22⋊C811yescharacteristic
169C22×C61#40711yescharacteristic
170C22×C61C2×#9911yescharacteristic
171C22×C61#41011yescharacteristic
172C22×C61C2×#10111yescharacteristic
173C22×C61#41311yescharacteristic
174C22×C61C2×C4⋊C811yescharacteristic
175C22×C61#42311yescharacteristic
176C22×C61#43111yescharacteristic
177C22×C61#43211yescharacteristic
178C4×C61—24no—
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—42no—
182Dic61—22no—
183Dic61—22no—
184Dic61—24no—
185Dic61—42no—
186Dic61—42no—
187C2×C4×C4⋊C4—611no—
188#1797—612no—
189C2×#324—612no—
190#1280—612no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61#145911yescharacteristic
196C2×C61#146111yescharacteristic
197C2×C61#146311yescharacteristic
198C2×C61#146411yescharacteristic
199C2×C61#179711yescharacteristic
200C2×C61#179911yescharacteristic
201C2×C61#180111yescharacteristic
202C22×C4⋊C4—611no—
203C22×C4⋊C4—611no—
204C22×C42—611no—
205C22×C42—611no—
206C22×C42#692311yescharacteristic
207C2×C4⋊C8—1222no—
208C2×C22⋊C8—614no—
209C4×C4⋊C4—1222no—
210C4×C4⋊C4—1222no—
211C2×#95—611no—
212C2×#95—611no—
213#336—1222no—
214#324—1224no—
215C61#856011yescharacteristic, complemented, Sylow
216C23×C4—611no—
217C23×C4—611no—
218C23×C4—611no—
219C23×C4—611no—
220C23×C4C2×#692311yescharacteristic
221C23×C4C2×#692311yescharacteristic
222C23×C4C2×#692311yescharacteristic
223C22×C8—1224no—
224C2×C4⋊C4—614no—
225C2×C4⋊C4—612no—
226C2×C4⋊C4—612no—
227C2×C4⋊C4—1222no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1222no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—612no—
236C2×C42—612no—
237C2×C42—1221no—
238C2×C42—1221no—
239C2×C42—2442no—
240C2×C42—2442no—
241C4⋊C8—2444no—
242C22⋊C8—1224no—
243C22⋊C8—1224no—
244#95—1222no—
245#95—1222no—
246C24C22×#692311yescharacteristic
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—612no—
260C22×C4—612no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—612no—
264C22×C4—612no—
265C22×C4—612no—
266C22×C4—612no—
267C22×C4—612no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—2441no—
273C22×C4—2441no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4—2441no—
277C22×C4—2441no—
278C22×C4—2441no—
279C22×C4—2441no—
280C22×C4D61⋊C812yes—
281C22×C4C4⋊#692312yes—
282C22×C4#1175821712yes—
283C22×C4#1175821712yes—
284C22×C4C22⋊#692311yescharacteristic
285C22×C4C22⋊#692311yescharacteristic
286C2×C8—2444no—
287C2×C8—2444no—
288C2×C8—2444no—
289C4⋊C4—1224no—
290C4⋊C4—1224no—
291C4⋊C4—1228no—
292C42—24no—
293C42—24no—
294C42—1224no—
295C42—2442no—
296C42—2442no—
297C42—2442no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×C22⋊#692311yescharacteristic
303C23C2×C4⋊#692311yescharacteristic
304C23#1451985611yescharacteristic
305C23C2×D61⋊C811yescharacteristic
306C23C2×#1175821711yescharacteristic
307C23C2×#1175821711yescharacteristic
308C23D61⋊M1611yescharacteristic
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—612no—
327C2×C4—612no—
328C2×C4—614no—
329C2×C4—1222no—
330C2×C4—1222no—
331C2×C4—1222no—
332C2×C4—1222no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2441no—
338C2×C4—2442no—
339C2×C4—2441no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4#1451983612yes—
349C2×C4#1451983812yes—
350C2×C4#1451984414yescomplemented
351C8—4884no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22#1684070411yescharacteristic
365C22D61⋊(C22⋊C8)11yescharacteristic
366C22#1684073211yescharacteristic
367C22C2×#1451983611yescharacteristic
368C22#1684074111yescharacteristic
369C22C2×#1451983811yescharacteristic
370C22#1684074811yescharacteristic
371C22C2×#1451984411yescharacteristic
372C22#1684076511yescharacteristic
373C22#1684077911yescharacteristic
374C22#1684078211yescharacteristic
375C4—22no—
376C4—22no—
377C4—24no—
378C4—42no—
379C4—1224no—
380C4—1222no—
381C4—1222no—
382C4—2442no—
383C4—2442no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2#2146105611yescharacteristic
389C2#2146106111yescharacteristic
390C2#2146106711yescharacteristic
391C2#2146107011yescharacteristic
392C2#2146167511yescharacteristic
393C2#2146168011yescharacteristic
394C2#2146168511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC22×C42×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC23×Dic61C23
hypercenterC22×C42#6923
hypocenterC61#8560
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
30. C2×C4⋊Dic61C8semidirect
130. C2×C4×C61C4⋊C8semidirect
215. C61#8560semidirect
350. C2×C4#14519844semidirect

Maximal subgroups

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

subgroupindex
#214616752
C2×#168406072
#214606572
C2×C4×C4⋊Dic612
#856061