Finite Group Databasebeta

← #30601648

Subgroups — #30601648

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461673C211yesmaximal, characteristic
2C4×C22⋊#92420C211yesmaximal, characteristic
3C2×#16840607C214yesmaximal
4C2×#16839661C211yesmaximal, characteristic
5C2×C4⋊#92420—21no—
6C2×C22⋊#92420C2212yes—
7C2×C22⋊#92420C2212yes—
8C2×C4×#92420—21no—
9#16840628—21no—
10#16840628—21no—
11#16840607—28no—
12C22×C4⋊Dic61C411yescharacteristic
13C22×C4⋊Dic61C2212yes—
14C22×C4×Dic61C2211yescharacteristic
15#16839661—21no—
16#16839661—21no—
17#16839661—21no—
18#16839661—21no—
19C2×#14519684C412yes—
20C22×C4⋊C4×C61C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23C4⋊#92420—42no—
24C22⋊#92420—22no—
25C22⋊#92420—24no—
26C22⋊#92420D414yes—
27C4×#92420—42no—
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2×C412yes—
30C23×Dic61C2311yescharacteristic
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—24no—
34C2×C4⋊Dic61—22no—
35C2×C4⋊Dic61—22no—
36C2×C4⋊Dic61—24no—
37C2×C4⋊Dic61—42no—
38C2×C4×Dic61—42no—
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—22no—
44#14519684—22no—
45C23×C4×C61C2×C412yes—
46C23×C4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61C814yescomplemented
51C24×C61C22×C411yescharacteristic
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—24no—
56C22×C4×C61—22no—
57C22×C4×C61—24no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C2×C814yes—
61C22×C4×C61C2×C812yes—
62C22×C4×C61M1612yes—
63C22×Dic61—22no—
64C22×Dic61—22no—
65C22×Dic61—22no—
66C22×Dic61—24no—
67C22×Dic61—41no—
68C22×Dic61—41no—
69C22×Dic61—41no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C22×Dic61C2×D412yes—
77C22×Dic61#4112yes—
78C2×#92420—24no—
79C2×#92420—42no—
80C2×#92420—42no—
81C2×#92420—42no—
82C2×#92420—42no—
83C2×#92420—42no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61—24no—
86C4⋊Dic61—42no—
87C4⋊Dic61—42no—
88C4⋊Dic61—44no—
89C4⋊Dic61—42no—
90C4⋊Dic61—42no—
91C4×Dic61—22no—
92C4×Dic61—22no—
93C4×Dic61—41no—
94C4×Dic61—41no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61C2×C22⋊C411yescharacteristic
100C23×C61C4×D412yes—
101C23×C61C22×C811yescharacteristic
102C23×C61C2×M1611yescharacteristic
103C23×C61#13112yes—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—24no—
107C2×C4×C61—24no—
108C2×C4×C61—24no—
109C2×C4×C61—28no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—44no—
113C2×C4×C61C22⋊C812yes—
114C2×C4×C61C22⋊C812yes—
115C2×C4×C61#9912yes—
116C2×C4×C61#10112yes—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—44no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—84no—
133C2×Dic61#10112yes—
134C2×Dic61#12112yes—
135C2×Dic61#12411yescharacteristic
136C2×Dic61#12711yescharacteristic
137#92420—44no—
138#92420—82no—
139#9650—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—22no—
148C22×C61#39011yescharacteristic
149C22×C61C2×C22⋊C811yescharacteristic
150C22×C61#40811yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61#41011yescharacteristic
153C22×C61C2×#10111yescharacteristic
154C22×C61C2×#10111yescharacteristic
155C22×C61C8×D412yes—
156C22×C61#43612yes—
157C4×C61—22no—
158C4×C61—44no—
159C4×C61#32414yescomplemented
160Dic61—22no—
161Dic61—21no—
162Dic61—21no—
163Dic61—22no—
164Dic61—41no—
165Dic61—82no—
166C2×#389—611no—
167#1797—611no—
168C4×C22⋊C8—611no—
169C2×#324—614no—
170C2×C61—22no—
171C2×C61—22no—
172C2×C61C2×#32411yescharacteristic
173C2×C61#146111yescharacteristic
174C2×C61#147011yescharacteristic
175C2×C61#147211yescharacteristic
176C2×C61#195411yescharacteristic
177C2×C61#196411yescharacteristic
178C2×C61#196611yescharacteristic
179C22×C4⋊C4—612no—
180C22×C4⋊C4—611no—
181C22×C4⋊C4#692311yescharacteristic
182C22×C42—611no—
183C2×C4⋊C8—1221no—
184C2×C22⋊C8—612no—
185C2×C22⋊C8—612no—
186C2×C4×C8—1221no—
187#389—1221no—
188#389—1221no—
189#389—1221no—
190#389—1221no—
191C2×#95—612no—
192#336—1221no—
193#336—1221no—
194#324—1228no—
195C61#965011yescharacteristic, complemented, Sylow
196C23×C4—611no—
197C23×C4—611no—
198C23×C4—612no—
199C23×C4C2×#692312yes—
200C23×C4C2×#692311yescharacteristic
201C22×C8—1222no—
202C22×C8—1222no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—1224no—
207C2×C4⋊C4—1222no—
208C2×C4⋊C4—1222no—
209C2×C4⋊C4—1224no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—2442no—
213C2×C4⋊C4#9242014yescomplemented
214C2×C42—611no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—2442no—
219C4⋊C8—2442no—
220C22⋊C8—614no—
221C22⋊C8—1222no—
222C22⋊C8—1224no—
223C4×C8—2442no—
224#95—1222no—
225#95—1222no—
226C24C22×#692311yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—612no—
237C22×C4—612no—
238C22×C4—611no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—1224no—
242C22×C4—1222no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4—2441no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4C2×#9242014yes—
249C22×C4C2×#9242012yes—
250C22×C4#1175821712yes—
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22⋊#692311yescharacteristic
253C2×C8—1224no—
254C2×C8—2442no—
255C2×C8—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8—2442no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—2442no—
262C4⋊C4—2444no—
263C4⋊C4—2442no—
264C4⋊C4—2442no—
265C4⋊C4—2442no—
266C42—1222no—
267C42—1222no—
268C42—2441no—
269C42—2441no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23C2×C22⋊#692311yescharacteristic
275C23D4×#692312yes—
276C23C22×#9242011yescharacteristic
277C23C2×#1175821711yescharacteristic
278C23#1451986412yes—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—612no—
289C2×C4—611no—
290C2×C4—611no—
291C2×C4—612no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—4884no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2444no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4C22⋊#9242012yes—
309C2×C4C22⋊#9242012yes—
310C2×C4#1451983612yes—
311C2×C4#1451983812yes—
312C8—4882no—
313C8—2444no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22#1684071411yescharacteristic
323C22C2×C22⋊#9242011yescharacteristic
324C22#1684073611yescharacteristic
325C22C2×#1451983611yescharacteristic
326C22#1684074311yescharacteristic
327C22C2×#1451983811yescharacteristic
328C22D61⋊#10111yescharacteristic
329C22D4×#9242012yes—
330C22#1684079712yes—
331C4—22no—
332C4—44no—
333C4—1222no—
334C4—1221no—
335C4—1222no—
336C4—1221no—
337C4—4882no—
338C4—2441no—
339C4#1684060714yescomplemented
340C2—22no—
341C2—22no—
342C2C2×#1684060711yescharacteristic
343C2#2146106211yescharacteristic
344C2#2146109211yescharacteristic
345C2#2146109711yescharacteristic
346C2#2146194011yescharacteristic
347C2#2146195911yescharacteristic
348C2#2146196211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#9650
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
50. C2×C4⋊C4×C61C8semidirect
159. C4×C61#324semidirect
195. C61#9650semidirect
213. C2×C4⋊C4#92420semidirect
339. C4#16840607semidirect

Maximal subgroups

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

subgroupindex
#214616732
C4×C22⋊#924202
C2×#168406072
C2×#168396612
#965061