Finite Group Databasebeta

← #29838549

Subgroups — #29838549

Order 15616.

2550 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic
2#21448386C214yesmaximal, complemented
3#21448374C214yesmaximal, complemented
4#21448347C218yesmaximal
5#21448301C212yesmaximal
6#21448293C214yesmaximal, complemented
7C4×#14519742C214yesmaximal, complemented
8#117×Dic61C212yesmaximal
9C2×C42×Dic61C211yesmaximal, characteristic
10C61×#2285C211yesmaximal, characteristic
11#16840319C4116yescomplemented
12C2×#14519777C2212yes—
13C2×#14519769C2212yes—
14C2×#14519756C2214yes—
15C2×C4×Dic122C2214yes—
16C2×#14519742C2218yes—
17C2×#14519736C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19C22×C4×Dic61C2214yes—
20C22×C4×Dic61C2211yescharacteristic
21#16839667C2218yes—
22#16839658C2218yes—
23#16839658C2218yes—
24#16839644C22116yes—
25#16839634C2218yes—
26C4⋊C4×Dic61C2218yes—
27C4⋊C4×Dic61C2218yes—
28C4×#11758196C2218yes—
29#16839613C22116yes—
30C22⋊C4×Dic61C2218yes—
31#16839608C2214yes—
32C42×Dic61C2212yes—
33C42×Dic61C2212yes—
34C42×Dic61C2212yes—
35C42×Dic61C2212yes—
36C42×Dic61C2214yes—
37C2×#117×C61C2212yes—
38C22×C42×C61C2211yescharacteristic
39C61×#384C2212yes—
40C61×#382C2212yes—
41C61×#380C2214yes—
42C4×C4⋊C4×C61C2214yes—
43C22×Dic122C2314yes—
44C23×Dic61C2312yes—
45C23×Dic61C2312yes—
46#14519777C2×C4116yescomplemented
47#14519769C2×C4116yescomplemented
48#14519756C2×C4132yes—
49C4×Dic122C2×C4132yescomplemented
50#14519742—232no—
51#14519736C2×C4116yes—
52C2×C4⋊Dic61C2314yes—
53C2×C4⋊Dic61C2314yes—
54C2×#11758196C23116yes—
55C2×#11758196C2318yes—
56C2×C22⋊Dic61C2314yes—
57C2×C4×Dic61—24no—
58C2×C4×Dic61—216no—
59C2×C4×Dic61Q818yescomplemented
60C2×C4×Dic61C2314yes—
61C2×C4×Dic61C2312yes—
62C2×C4×Dic61C2314yes—
63C2×C4×Dic61C2312yes—
64C2×C4×Dic61C2314yes—
65C2×C4×Dic61C23116yes—
66C2×C4×Dic61C2314yes—
67C2×C4×Dic61C2311yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C2×C4×Dic61C2312yes—
70C2×C4×Dic61C2314yes—
71#14519684C23116yes—
72#14519684C23116yes—
73C23×C4×C61C2312yes—
74C23×C4×C61C2311yescharacteristic
75#117×C61—28no—
76C2×C4⋊C4×C61C2318yes—
77C2×C4⋊C4×C61C2314yes—
78C2×C22⋊C4×C61C2314yes—
79C2×C42×C61C2×C418yes—
80C2×C42×C61C2314yes—
81C2×C42×C61C2312yes—
82C2×C42×C61C2311yescharacteristic
83C2×C42×C61C2311yescharacteristic
84#95×C61C2318yes—
85C24×C61C2411yescharacteristic
86C22×C4×C61—28no—
87C22×C4×C61C22×C418yes—
88C22×C4×C61C22×C414yes—
89C22×C4×C61C2×Q814yes—
90C22×C4×C61#4114yes—
91C22×C4×C61C2418yes—
92C22×C4×C61C2412yes—
93C22×C4×C61C2411yescharacteristic
94C22×C4×C61C2411yescharacteristic
95C22×C4×C61C2412yes—
96C22×Dic61—24no—
97C22×Dic61—28no—
98C22×Dic61—28no—
99C22×Dic61C2×Q818yes—
100C22×Dic61C2418yes—
101C22×Dic61C2412yes—
102C22×Dic61C2414yes—
103C22×Dic61C2412yes—
104C2×Dic122—216no—
105C2×Dic122C22×C4116yes—
106C4⋊C4×C61—28no—
107C4⋊C4×C61—216no—
108C22⋊C4×C61—28no—
109C42×C61—28no—
110C42×C61C22×C418yes—
111C42×C61C22×C414yes—
112C42×C61C22×C414yes—
113C22⋊Dic61C22×C4116yes—
114C4⋊Dic61C22×C4116yes—
115C4⋊Dic61C22×C4116yes—
116#11758196—232no—
117#11758196C22×C4132yes—
118C4×Dic61—24no—
119C4×Dic61—24no—
120C4×Dic61—28no—
121C4×Dic61—28no—
122C4×Dic61—28no—
123C4×Dic61—232no—
124C4×Dic61C22×C4116yes—
125C4×Dic61C2×Q814yes—
126C4×Dic61C2×Q818yes—
127C4×Dic61C2×Q814yes—
128C4×Dic61#41116yes—
129C23×C61—24no—
130C23×C61C23×C412yes—
131C23×C61C22×Q812yes—
132C23×C61C2×#4112yes—
133C23×C61C2511yescharacteristic
134C2×C4×C61—24no—
135C2×C4×C61—216no—
136C2×C4×C61—216no—
137C2×C4×C61—28no—
138C2×C4×C61C4×Q8116yescomplemented
139C2×C4×C61C23×C414yes—
140C2×C4×C61C23×C412yes—
141C2×C4×C61C23×C412yes—
142C2×C4×C61C23×C414yes—
143C2×C4×C61C22×Q812yes—
144C2×C4×C61C22×Q812yes—
145C2×C4×C61C2×#4112yes—
146C2×C4×C61C2×#4112yes—
147C2×C4×C61C2×#4118yes—
148C2×Dic61—216no—
149C2×Dic61—24no—
150C2×Dic61—24no—
151C2×Dic61—24no—
152C2×Dic61—216no—
153C2×Dic61—216no—
154C2×Dic61C23×C4116yes—
155C2×Dic61C22×Q814yes—
156C2×Dic61C22×Q814yes—
157C2×Dic61C2×#4118yes—
158C2×Dic61C2×#4118yes—
159Dic122—232no—
160C4×#533—611nomaximal, Sylow
161C22×C61—24no—
162C22×C61—24no—
163C22×C61—24no—
164C22×C61C2×C4×Q814yes—
165C22×C61C24×C411yescharacteristic
166C22×C61C23×Q811yescharacteristic
167C22×C61C22×#4114yes—
168C22×C61C22×#4111yescharacteristic
169C4×C61—24no—
170C4×C61—48no—
171C4×C61C2×C4×Q814yes—
172C4×C61C2×C4×Q814yes—
173C4×C61C4×#4118yes—
174Dic61—216no—
175Dic61—24no—
176Dic61—28no—
177Dic61#53318yescomplemented
178C2×#533—611no—
179C4×C4⋊Q8—614no—
180C4×#125—614no—
181C4×C22⋊Q8—618no—
182#2296—612no—
183#2295—614no—
184#2285D6111yescharacteristic
185C42×Q8—614no—
186C4×#117—612no—
187C2×C43—611no—
188C2×C61—24no—
189C2×C61C22×C4×Q811yescharacteristic
190C2×C61C2×C4×#4112yes—
191C2×C61C2×#53312yes—
192C2×C61#344412yes—
193#533—6116no—
194C2×C4⋊Q8—612no—
195C2×#125—612no—
196C2×C22⋊Q8—614no—
197C2×C4×Q8—618no—
198C2×C4×Q8—614no—
199C2×#117—612no—
200C2×#117C2×D6112yes—
201C22×C42—611no—
202C22×C42—611no—
203C22×C42—614no—
204C22×C42C2×D6111yescharacteristic
205#391—618no—
206#389—618no—
207#389—618no—
208#387—6116no—
209#385—618no—
210#384C2×D6112yes—
211#382C2×D6112yes—
212#380C2×D6114yes—
213C4×C4⋊C4—618no—
214C4×C4⋊C4—618no—
215C4×C4⋊C4—618no—
216C4×C4⋊C4—6116no—
217C4×C4⋊C4C2×D6114yes—
218C4×C22⋊C4—618no—
219#376—614no—
220C43—612no—
221C43—612no—
222C43—612no—
223C43—612no—
224C43—614no—
225C61C4×#53311yescharacteristic, complemented, Sylow
226C22×Q8—614no—
227C23×C4—612no—
228C23×C4—612no—
229C23×C4C22×D6112yes—
230C23×C4C22×D6111yescharacteristic
231C4⋊Q8—6116no—
232#125—6116no—
233C22⋊Q8—6132no—
234C4×Q8—6132no—
235C4×Q8—12232no—
236#117—28no—
237#117—6116no—
238C2×C4⋊C4—614no—
239C2×C4⋊C4—618no—
240C2×C4⋊C4—614no—
241C2×C4⋊C4—6116no—
242C2×C4⋊C4C22×D6114yes—
243C2×C4⋊C4C22×D6118yes—
244C2×C22⋊C4—614no—
245C2×C22⋊C4C22×D6114yes—
246C2×C42—618no—
247C2×C42—614no—
248C2×C42—612no—
249C2×C42—614no—
250C2×C42—611no—
251C2×C42—611no—
252C2×C42—6116no—
253C2×C42—614no—
254C2×C42—614no—
255C2×C42—612no—
256C2×C42—614no—
257C2×C42—612no—
258C2×C42—12216no—
259C2×C42—1224no—
260C2×C42C4×D6118yes—
261C2×C42C22×D6114yes—
262C2×C42C22×D6111yescharacteristic
263C2×C42C22×D6111yescharacteristic
264C2×C42C22×D6112yes—
265#95—6116no—
266#95—6116no—
267#95C22×D6118yes—
268C24C23×D6111yescharacteristic
269C2×Q8—6116no—
270C2×Q8—12216no—
271C22×C4—28no—
272C22×C4—612no—
273C22×C4—618no—
274C22×C4—612no—
275C22×C4—614no—
276C22×C4—618no—
277C22×C4—1228no—
278C22×C4—1228no—
279C22×C4—1224no—
280C22×C4C2×C4×D6118yes—
281C22×C4C2×C4×D6114yes—
282C22×C4#1175822414yes—
283C22×C4Q8×D6114yes—
284C22×C4C23×D6118yes—
285C22×C4C23×D6112yes—
286C22×C4C23×D6111yescharacteristic
287C22×C4C23×D6112yes—
288C22×C4C23×D6111yescharacteristic
289C4⋊C4—216no—
290C4⋊C4—28no—
291C4⋊C4—6132no—
292C4⋊C4—6116no—
293C4⋊C4—6116no—
294C4⋊C4—12232no—
295C22⋊C4—28no—
296C22⋊C4—6116no—
297C42—28no—
298C42—6116no—
299C42—614no—
300C42—618no—
301C42—614no—
302C42—6116no—
303C42—1228no—
304C42—1228no—
305C42—12232no—
306C42—1228no—
307C42—1224no—
308C42—1224no—
309C42C2×C4×D6118yes—
310C42C2×C4×D6114yes—
311C42C2×C4×D6114yes—
312C23—24no—
313C23C22×C4×D6112yes—
314C23C2×Q8×D6112yes—
315C23C2×#1175822412yes—
316C23C24×D6111yescharacteristic
317Q8—12232no—
318C2×C4—24no—
319C2×C4—216no—
320C2×C4—216no—
321C2×C4—28no—
322C2×C4—614no—
323C2×C4—618no—
324C2×C4—6116no—
325C2×C4—614no—
326C2×C4—618no—
327C2×C4—12216no—
328C2×C4—12216no—
329C2×C4—1224no—
330C2×C4—1224no—
331C2×C4—12216no—
332C2×C4—1224no—
333C2×C4#14519742116yescomplemented
334C2×C4C22×C4×D6114yes—
335C2×C4C22×C4×D6112yes—
336C2×C4C22×C4×D6112yes—
337C2×C4C22×C4×D6114yes—
338C2×C4C2×Q8×D6112yes—
339C2×C4C2×Q8×D6112yes—
340C2×C4#41×D6118yes—
341C2×C4C2×#1175822412yes—
342C2×C4C2×#1175822412yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22C2×#1451974214yes—
347C22C23×C4×D6111yescharacteristic
348C22C22×Q8×D6111yescharacteristic
349C22C2×#41×D6114yes—
350C22C22×#1175822411yescharacteristic
351C4—24no—
352C4—48no—
353C4—618no—
354C4—12216no—
355C4—1224no—
356C4—1228no—
357C4C2×#1451974214yes—
358C4C2×#1451974214yes—
359C4#1684023018yes—
360C2—24no—
361C2C22×#1451974211yescharacteristic
362C2C2×#1684023012yes—
363C2#533×D6112yes—
364C2#2145911412yes—
subgroupG/H
centerC22×C4C23×D61
commutatorC22×C61C24×C4
FrattiniC23C24×D61
FittingC61×#2285C2
socle, cosocleC23×C61C25
prime radicalC24×C61C24
hypercenter#2285D61
hypocenterC61C4×#533
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #21448386C2semidirect
3. #21448374C2semidirect
6. #21448293C2semidirect
7. C4×#14519742C2semidirect
11. #16840319C4semidirect
46. #14519777C2×C4semidirect
47. #14519769C2×C4semidirect
49. C4×Dic122C2×C4semidirect
59. C2×C4×Dic61Q8semidirect
138. C2×C4×C61C4×Q8semidirect
177. Dic61#533semidirect
225. C61C4×#533semidirect
333. C2×C4#14519742semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
#214483862
#214483742
#214483472
#214483012
#214482932
C4×#145197422
#117×Dic612
C2×C42×Dic612
C61×#22852
C4×#53361