Finite Group Databasebeta

← #29696216

Subgroups — #29696216

Order 15616.

1062 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2#21454554C212yesmaximal
3#21446807C212yesmaximal, complemented
4#21446284C214yesmaximal, complemented
5#21445901C212yesmaximal, complemented
6C4×#14519696C211yesmaximal, characteristic, complemented
7#21445556C212yesmaximal, complemented
8C61×#1801C211yesmaximal, characteristic, complemented
9C2×M16×D61C2214yes—
10#16840485—22no—
11#16840485—24no—
12#16840485—22no—
13C2×#14519777C2211yescharacteristic
14C2×#14519772C414yes—
15C2×#14519765C412yes—
16C2×#14519753C2212yes—
17C2×C4×Dic122C412yes—
18C2×C4×D244C2211yescharacteristic
19C2×#14519735C2212yes—
20#16839839—22no—
21#16839832—24no—
22#16839827—22no—
23C2×#14519696C2212yes—
24#16839725C2214yes—
25C4×#11758194C2212yes—
26#16839531C2211yescharacteristic
27#16839531C2211yescharacteristic
28#16839530C2214yes—
29#16839530C2214yes—
30C2×C4×Q8×C61C2211yescharacteristic
31C61×#423C2212yes—
32C2×#101×C61C2212yes—
33C61×#343C2212yes—
34C22×Dic122C2×C412yes—
35C22×D244C2×C411yescharacteristic
36C22×C4×D61C2312yes—
37C2×#11758194C2312yes—
38C2×#11758194C2312yes—
39C2×#11758189—24no—
40M16×D61—28no—
41M16×D61—44no—
42C2×C8×D61—24no—
43#14519777—22no—
44#14519777—22no—
45#14519772—28no—
46#14519772—28no—
47#14519765—24no—
48#14519765—24no—
49#14519753—24no—
50#14519753—24no—
51C4×Dic122—24no—
52C4×Dic122—22no—
53C4×Dic122—22no—
54C4×D244D412yes—
55C4×D244D414yes—
56C4×D244D412yes—
57#14519735—24no—
58#14519735—24no—
59C2×C4⋊Dic61C2×C412yes—
60C2×C4⋊Dic61C2×C411yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×#11758196C2×C414yes—
63C2×#11758198C2×C414yes—
64C2×#11758198C2×C414yes—
65C2×#11758198C2×C412yes—
66C2×C4×Dic61C2×C414yes—
67#14519696—24no—
68#14519689—24no—
69#14519689—24no—
70#14519688—22no—
71C4×#92418—22no—
72C22×Q8×C61C2311yescharacteristic
73C2×M16×C61C2314yes—
74C4×Q8×C61—22no—
75C4×Q8×C61—22no—
76C2×C4⋊C4×C61C2×C412yes—
77C2×C4⋊C4×C61C2311yescharacteristic
78C2×C42×C61C2×C412yes—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61—22no—
81C4⋊C8×C61—22no—
82#101×C61—42no—
83C23×D61C22×C412yes—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C22×C4×C61C22×C412yes—
89C22×C4×C61C22×C411yescharacteristic
90C22×C4×C61C22×C412yes—
91C22×C4×C61C22×C411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C22×Dic61C22×C412yes—
94C22×Dic61C22×C414yes—
95C2×D244—42no—
96C2×D244C22⋊C412yes—
97C2×D244C22⋊C412yes—
98C2×C4×D61—24no—
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61C2×D414yes—
102C2×C4×D61C2×D414yes—
103C2×Dic122—22no—
104C2×Dic122—22no—
105C2×Dic122—44no—
106M16×C61—24no—
107M16×C61—24no—
108C2×C8×C61—24no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—22no—
111C42×C61—22no—
112C42×C61—22no—
113C42×C61C2×D412yes—
114C42×C61C2×D412yes—
115#11758198—28no—
116#11758198—28no—
117#11758198C22⋊C414yes—
118#11758198C22⋊C414yes—
119C4⋊Dic61—24no—
120C4⋊Dic61—24no—
121C4⋊Dic61C22⋊C412yes—
122C4⋊Dic61C22⋊C411yescharacteristic
123C4⋊Dic61C22⋊C411yescharacteristic
124#11758196—24no—
125#11758196—24no—
126C4×Dic61—24no—
127C4×Dic61—24no—
128#11758194—22no—
129#11758194—22no—
130#11758194—22no—
131#11758194#4114yes—
132C2×#92418—22no—
133C2×#92418—22no—
134#11758189—48no—
135C8×D61—48no—
136C23×C61C23×C411yescharacteristic
137C22×D61—22no—
138C22×D61—22no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61C2×C22⋊C414yes—
142Q8×C61—42no—
143Q8×C61—42no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61C2×C22⋊C412yes—
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61C22×D412yes—
155C2×C4×C61C2×#4112yes—
156C2×C4×C61H(5,𝔽2)11yescharacteristic
157C2×C4×C612−1+411yescharacteristic
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—24no—
161C2×Dic61—44no—
162C2×Dic61C2×C22⋊C412yes—
163C2×Dic61C2×C22⋊C412yes—
164D244—24no—
165D244—42no—
166C4×D61—24no—
167C4×D61—24no—
168C4×D61—24no—
169C4×D61—24no—
170Dic122—44no—
171Dic122—44no—
172C8×C61—44no—
173#92418—24no—
174#92418—42no—
175#22484—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61C22×C22⋊C411yescharacteristic
179C22×C61C4×#4111yescharacteristic
180C22×C61#52011yescharacteristic
181C2×D61—24no—
182C2×D61—24no—
183C2×D61—24no—
184C2×D61—82no—
185C2×D61—82no—
186C4×C61—21no—
187C4×C61—22no—
188C4×C61—42no—
189C4×C61—42no—
190C4×C61#41312yes—
191C4×C61#54612yes—
192C4×C61#54712yes—
193Dic61—22no—
194Dic61—22no—
195Dic61—44no—
196C2×#556—611no—
197D4×M16—612no—
198#1997—612no—
199#1901—614no—
200#1822—612no—
201#1801D6111yescharacteristic, complemented
202C4×#101—611no—
203#1742—612no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61#233911yescharacteristic
207C2×C61C2×#41311yescharacteristic
208C2×C61#288911yescharacteristic
209D61—44no—
210C22×M16—614no—
211#556—1222no—
212#556—1224no—
213#556—1222no—
214C2×C4⋊Q8—611no—
215C2×#126—614no—
216C2×#124—612no—
217C2×C22⋊Q8—612no—
218C2×C4×Q8—612no—
219C2×C4×Q8C2×D6111yescharacteristic
220C2×C4×D4—611no—
221C2×#117—612no—
222#436—1222no—
223#435—1224no—
224C8×D4—1222no—
225#423C2×D6112yes—
226C2×#101—612no—
227C2×#101C2×D6112yes—
228#407—614no—
229C4×M16—612no—
230#343—614no—
231#343—614no—
232#343—611no—
233#343—611no—
234#343C2×D6112yes—
235C61#2248411yescharacteristic, complemented, Sylow
236C22×Q8—612no—
237C22×Q8C22×D6111yescharacteristic
238C22×D4—611no—
239C23×C4—612no—
240C2×M16—612no—
241C2×M16—612no—
242C2×M16—1224no—
243C2×M16—1228no—
244C2×M16—2444no—
245C2×M16C22×D6114yes—
246C22×C8—1224no—
247C4⋊Q8—1222no—
248C4⋊Q8—1222no—
249#126—1228no—
250#126—1228no—
251#124—1224no—
252#124—1224no—
253C22⋊Q8—1224no—
254C22⋊Q8—1224no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—1224no—
258C4×Q8—1222no—
259C4×Q8—1222no—
260C4×D4—612no—
261C4×D4—614no—
262C4×D4—612no—
263#117—1224no—
264#117—1224no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—614no—
267C2×C4⋊C4—612no—
268C2×C4⋊C4—612no—
269C2×C4⋊C4C4×D6112yes—
270C2×C4⋊C4C22×D6111yescharacteristic
271C2×C22⋊C4—612no—
272C2×C22⋊C4—614no—
273C2×C22⋊C4—614no—
274C2×C42—614no—
275C2×C42C4×D6112yes—
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—22no—
278C4⋊C8—22no—
279#101—42no—
280#101—1224no—
281C22⋊C8—1224no—
282C22⋊C8—1224no—
283#97—1222no—
284C4×C8—1222no—
285C24—612no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—1222no—
291C2×Q8—1222no—
292C2×Q8—2444no—
293C2×D4—612no—
294C2×D4—612no—
295C2×D4—2442no—
296C22×C4—612no—
297C22×C4—614no—
298C22×C4—614no—
299C22×C4—614no—
300C22×C4—1224no—
301C22×C4—1222no—
302C22×C4—1222no—
303C22×C4C2×C4×D6112yes—
304C22×C4C2×C4×D6111yescharacteristic
305C22×C4C2×C4×D6112yes—
306C22×C4C2×C4×D6111yescharacteristic
307C22×C4C23×D6111yescharacteristic
308M16—24no—
309M16—24no—
310M16—614no—
311M16—1222no—
312M16—1222no—
313M16—1222no—
314M16—2448no—
315C2×C8—24no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—2448no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—612no—
322C4⋊C4—611no—
323C4⋊C4—611no—
324C4⋊C4—1224no—
325C4⋊C4—1224no—
326C4⋊C4—1224no—
327C4⋊C4—1224no—
328C22⋊C4—614no—
329C22⋊C4—614no—
330C22⋊C4—1228no—
331C22⋊C4—1228no—
332C42—22no—
333C42—22no—
334C42—1224no—
335C42—1224no—
336C42D4×D6112yes—
337C42D4×D6112yes—
338C23—614no—
339C23—1222no—
340C23—1222no—
341C23—2442no—
342C23—2442no—
343C23C22×C4×D6111yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—2444no—
347Q8—2444no—
348D4—1224no—
349D4—2442no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—612no—
358C2×C4—612no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—1224no—
362C2×C4—1224no—
363C2×C4—1224no—
364C2×C4—1224no—
365C2×C4—1224no—
366C2×C4—2444no—
367C2×C4C22⋊C4×D6111yescharacteristic
368C2×C4C22⋊C4×D6112yes—
369C2×C4C22⋊C4×D6111yescharacteristic
370C2×C4C2×D4×D6112yes—
371C2×C4#41×D6112yes—
372C2×C4#1451982211yescharacteristic
373C2×C4#1451982511yescharacteristic
374C8—44no—
375C8—1224no—
376C8—2442no—
377C22—21no—
378C22—21no—
379C22—1224no—
380C22—1224no—
381C22—1224no—
382C22—4882no—
383C22—4882no—
384C22C2×C22⋊C4×D6111yescharacteristic
385C22#1684023011yescharacteristic
386C22#1684024311yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—1222no—
392C4—1222no—
393C4—2444no—
394C4#1683974512yes—
395C4#1684040912yes—
396C4#1684041412yes—
397C2—21no—
398C2—21no—
399C2—2444no—
400C2#2144864711yescharacteristic
401C2C2×#1683974511yescharacteristic
402C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1801C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#1801D61
hypocenterC61#22484
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
3. #21446807C2semidirect
4. #21446284C2semidirect
5. #21445901C2semidirect
6. C4×#14519696C2semidirect
7. #21445556C2semidirect
8. C61×#1801C2semidirect
201. #1801D61semidirect
235. C61#22484semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
#214545542
#214468072
#214462842
#214459012
C4×#145196962
#214455562
C61×#18012
#2248461