Finite Group Databasebeta

← #29598238

Subgroups — #29598238

Order 15616.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840145C211yesmaximal, characteristic
2C2×#16839567C211yesmaximal, characteristic, complemented
3C2×#16839565C211yesmaximal, characteristic, complemented
4C2×#16839563C211yesmaximal, characteristic
5C2×#16839537C211yesmaximal, characteristic, complemented
6#21445977C211yesmaximal, characteristic, complemented
7C2×C61×#357C211yesmaximal, characteristic
8C22×Dic244C2211yescharacteristic
9C22×#11758200—21no—
10C2×#14519764—21no—
11C22×#11758185C2211yescharacteristic
12#16840145—22no—
13#16840145C414yescomplemented
14C2×#14519711C2211yescharacteristic
15C2×#14519710—21no—
16C2×C8×Dic61C2211yescharacteristic
17#16839631—21no—
18#16839567—22no—
19#16839565—22no—
20#16839563—22no—
21#16839537—22no—
22C22×D8×C61C2211yescharacteristic
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C61×#357—22no—
26C22×Dic122—21no—
27C2×Dic244—21no—
28C2×Dic244—21no—
29C2×Dic244—21no—
30C2×Dic244—21no—
31C2×Dic244—21no—
32C2×Dic244C2×C412yes—
33C2×#11758200—24no—
34C2×#11758200—42no—
35C2×#11758200—42no—
36C22×#92418D411yescharacteristic
37#14519764—24no—
38#14519764—42no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—42no—
41C2×C22⋊Dic61—42no—
42C2×C4×Dic61D411yescharacteristic
43C2×#11758185—21no—
44C2×#11758185—21no—
45C2×#11758185D412yes—
46#14519711—21no—
47#14519711—21no—
48#14519710—42no—
49C8×Dic61C2×C412yes—
50C8×Dic61D412yes—
51C22×D4×C61—21no—
52C2×D8×C61—21no—
53C2×D8×C61—21no—
54C2×D8×C61—21no—
55C2×D8×C61—21no—
56C2×D8×C61—21no—
57C2×D8×C61C2×C412yes—
58C22×C8×C61C2311yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C16×C61—21no—
61C2×C16×C61—21no—
62C2×C16×C61D412yes—
63#107×C61—21no—
64#107×C61—21no—
65C24×C61—41no—
66C2×D4×C61—22no—
67C2×D4×C61—41no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—41no—
75C22×Dic61—41no—
76C22×Dic61D812yes—
77C2×Dic122—22no—
78C2×Dic122—41no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83D8×C61—22no—
84D8×C61—22no—
85D8×C61—41no—
86D8×C61—41no—
87C2×C8×C61C22×C411yescharacteristic
88C2×C8×C61C2×D411yescharacteristic
89C2×C8×C61C2×D411yescharacteristic
90C2×C8×C61C2×D411yescharacteristic
91C2×C8×C61#4111yescharacteristic
92C2×C8×C61#4111yescharacteristic
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C22⋊Dic61—48no—
96#11758200—48no—
97C4⋊Dic61—41no—
98C4⋊Dic61—41no—
99#11758196—84no—
100C4×Dic61C22⋊C412yes—
101C4×Dic61SD1612yes—
102C2×#92418C22⋊C412yes—
103C2×#92418D812yes—
104C2×#92418SD1612yes—
105Dic244—22no—
106Dic244—22no—
107Dic244—41no—
108Dic244—41no—
109C16×C61—22no—
110#11758185—22no—
111C23×C61—42no—
112C23×C61—81no—
113C23×C61—81no—
114C23×C61—82no—
115C23×C61—82no—
116C23×C61C2×D811yescharacteristic
117D4×C61—42no—
118D4×C61—42no—
119D4×C61—81no—
120D4×C61—81no—
121C2×C4×C61—81no—
122C2×C4×C61—82no—
123C2×C4×C61C2×C22⋊C411yescharacteristic
124C2×C4×C61#12411yescharacteristic
125C2×C4×C61#12711yescharacteristic
126C2×C4×C61C2×D811yescharacteristic
127C2×C4×C61C2×SD1611yescharacteristic
128C2×C4×C61C2×SD1611yescharacteristic
129C2×Dic61—22no—
130C2×Dic61—22no—
131C2×Dic61—42no—
132C2×Dic61—81no—
133C2×Dic61—81no—
134C2×Dic61—81no—
135C2×Dic61—82no—
136C2×Dic61#10214yes—
137Dic122—42no—
138Dic122—42no—
139Dic122—81no—
140Dic122—81no—
141C8×C61C4×D411yescharacteristic
142C8×C61C4×D411yescharacteristic
143C8×C61#12111yescharacteristic
144C8×C61#12111yescharacteristic
145#92418#10214yes—
146#13205—611nomaximal, Sylow
147C22×C61—84no—
148C22×C61—82no—
149C22×C61—162no—
150C22×C61—161no—
151C22×C61—161no—
152C22×C61C2×#10211yescharacteristic
153C22×C61#48611yescharacteristic
154C22×C61#49311yescharacteristic
155C22×C61#50912yes—
156C22×C61#51012yes—
157C4×C61—161no—
158C4×C61#39011yescharacteristic
159C4×C61C2×#10211yescharacteristic
160C4×C61#48611yescharacteristic
161C4×C61#49211yescharacteristic
162Dic61—24no—
163Dic61—82no—
164Dic61—161no—
165C2×#495—611no—
166C2×#358—611no—
167C2×#358—611no—
168C2×#357—611no—
169C2×#357D6111yescharacteristic
170C2×#346—611no—
171#1845—611no—
172C2×C61—162no—
173C2×C61#195711yescharacteristic
174C2×C61#213912yes—
175C2×C61#224312yes—
176C2×C61#225011yescharacteristic
177C2×C61#225111yescharacteristic
178C22×Q16—611no—
179C22×SD16—1221no—
180C22×D8C2×D6111yescharacteristic
181C2×#124—1221no—
182C22×C16—611no—
183C22×C16C2×D6111yescharacteristic
184#495—614no—
185#495—1222no—
186C2×#107—611no—
187C2×#107—1221no—
188C2×#107C2×D6111yescharacteristic
189C2×C4×C8—611no—
190#384—1221no—
191#358—1222no—
192#358—1222no—
193#357—22no—
194#357—1222no—
195#346—1222no—
196C61#1320511yescharacteristic, complemented, Sylow
197C22×Q8—1221no—
198C22×D4—21no—
199C2×Q16—612no—
200C2×Q16—1221no—
201C2×Q16—1221no—
202C2×Q16—1221no—
203C2×Q16—1221no—
204C2×Q16—1221no—
205C2×SD16—1224no—
206C2×SD16—2442no—
207C2×SD16—2442no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8C4×D6112yes—
214C22×C8—611no—
215C22×C8C22×D6111yescharacteristic
216#124—1224no—
217#124—2442no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—2442no—
221C2×C22⋊C4—2442no—
222C2×C42—611no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—612no—
226C2×C16—1221no—
227C2×C16—1221no—
228C2×C16C22⋊D6112yes—
229#107—21no—
230#107—21no—
231#107—1221no—
232#107—1221no—
233#107—2442no—
234C4×C8—612no—
235C4×C8—612no—
236C24—41no—
237C2×Q8—1222no—
238C2×Q8—2441no—
239C2×Q8—2441no—
240C2×Q8—2441no—
241C2×Q8—2441no—
242C2×Q8—2441no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—612no—
251C22×C4—2441no—
252C22×C4—2441no—
253C22×C4D4×D6111yescharacteristic
254Q16—1222no—
255Q16—1222no—
256Q16—2441no—
257Q16—2441no—
258SD16—2448no—
259D8—22no—
260D8—22no—
261D8—41no—
262D8—41no—
263C2×C8—612no—
264C2×C8—612no—
265C2×C8—612no—
266C2×C8C2×C4×D6111yescharacteristic
267C2×C8#1175822211yescharacteristic
268C2×C8D4×D6111yescharacteristic
269C2×C8D4×D6111yescharacteristic
270C2×C8#1175822411yescharacteristic
271C2×C8C2×C22⋊D6111yescharacteristic
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—4884no—
275C4⋊C4—2441no—
276C4⋊C4—2441no—
277C22⋊C4—2448no—
278C42—612no—
279C42—612no—
280C16—22no—
281C16—1222no—
282C23—42no—
283C23—82no—
284C23—82no—
285C23—81no—
286C23—81no—
287C23D8×D6111yescharacteristic
288Q8—4881no—
289Q8—4881no—
290Q8—2442no—
291Q8—2442no—
292D4—42no—
293D4—42no—
294D4—81no—
295D4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—614no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—4882no—
302C2×C4—4881no—
303C2×C4—4881no—
304C2×C4—4881no—
305C2×C4—2442no—
306C2×C4C22⋊C4×D6111yescharacteristic
307C2×C4#1451976311yescharacteristic
308C2×C4#1451977311yescharacteristic
309C2×C4D8×D6111yescharacteristic
310C2×C4SD16×D6111yescharacteristic
311C2×C4SD16×D6111yescharacteristic
312C8—614no—
313C8#1451973711yescharacteristic
314C8C4×C22⋊D6111yescharacteristic
315C8#1451974911yescharacteristic
316C8#1451975011yescharacteristic
317C22—84no—
318C22—82no—
319C22—162no—
320C22—161no—
321C22—161no—
322C22#102×D6111yescharacteristic
323C22#1684010711yescharacteristic
324C22#1684013711yescharacteristic
325C22#1684019612yes—
326C22#1684020112yes—
327C4—161no—
328C4—1224no—
329C4—9761no—
330C4—4882no—
331C4#1683966211yescharacteristic
332C4#102×D6111yescharacteristic
333C4#1684010711yescharacteristic
334C4#1684013311yescharacteristic
335C2—162no—
336C2#2144662311yescharacteristic
337C2#2144752812yes—
338C2#2144805412yes—
339C2#2144808211yescharacteristic
340C2#2144808911yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#357C2
socleC23×C61C2×D8
prime radicalC22×D8×C61C22
cosocleC22×C8×C61C23
hypercenterC2×#357D61
hypocenterC61#13205
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16839567C2semidirect
3. C2×#16839565C2semidirect
5. C2×#16839537C2semidirect
6. #21445977C2semidirect
13. #16840145C4semidirect
196. C61#13205semidirect

Maximal subgroups

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

subgroupindex
C2×#168401452
C2×#168395672
C2×#168395652
C2×#168395632
C2×#168395372
#214459772
C2×C61×#3572
#1320561