Finite Group Databasebeta

← #29779723

Subgroups — #29779723

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454206C211yesmaximal, characteristic, complemented
2C4×#14519765C211yesmaximal, characteristic
3#21447388C212yesmaximal, complemented
4#21447061C212yesmaximal
5#21446516C212yesmaximal
6#21446509C212yesmaximal, complemented
7#21445835C211yesmaximal, characteristic, complemented
8#21445835C211yesmaximal, characteristic, complemented
9#21445830C211yesmaximal, characteristic
10#21445830C211yesmaximal, characteristic
11C61×#1767C211yesmaximal, characteristic, complemented
12C2×#14519765C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×#14519702C2212yes—
16C2×#14519702C2212yes—
17C2×#14519699C2212yes—
18C2×#14519699C2212yes—
19#16839668C2211yescharacteristic
20#16839664C2211yescharacteristic
21#16839660C2212yes—
22#16839660C2212yes—
23#16839653—22no—
24#16839646—22no—
25#16839646—22no—
26#16839628C2211yescharacteristic
27#16839628C2211yescharacteristic
28#16839628C2211yescharacteristic
29C4×#11758198—22no—
30#16839525C2212yes—
31#16839525C2212yes—
32#16839525C2212yes—
33#16839525C2212yes—
34C2×C4⋊C8×C61C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#97×C61C2212yes—
37C43×C61C2211yescharacteristic
38C61×#336C2212yes—
39C22×Dic122C2311yescharacteristic
40C22×D244C2311yescharacteristic
41C22×C4×D61—21no—
42#14519765—22no—
43#14519765—22no—
44C4×Dic122—22no—
45C4×Dic122—22no—
46C4×D244—22no—
47C4×D244—22no—
48C2×C4⋊Dic61C2311yescharacteristic
49C2×C4⋊Dic61C2311yescharacteristic
50C2×C4⋊Dic61C2312yes—
51C2×C4⋊Dic61C2312yes—
52C2×#11758196—21no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758198—22no—
56C2×#11758198—21no—
57C2×#11758198—22no—
58C2×C4×Dic61—21no—
59#14519702—24no—
60#14519702—42no—
61#14519699—24no—
62#14519699—42no—
63#14519684—22no—
64#14519684—22no—
65#14519684—21no—
66#14519684—21no—
67#14519684—21no—
68#14519684—21no—
69#14519684—22no—
70#14519684—22no—
71C22×C8×C61C2312yes—
72C22×C8×C61C2312yes—
73C2×C42×C61D412yes—
74C2×C42×C61D412yes—
75C2×C42×C61C2311yescharacteristic
76C2×C42×C61C2311yescharacteristic
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—22no—
79C4⋊C8×C61—22no—
80#97×C61—42no—
81C23×D61—21no—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—22no—
90C22×Dic61—22no—
91C22×Dic61—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244#4112yes—
98C2×C4×D61—44no—
99C2×C4×D61—42no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122—21no—
103C2×Dic122—21no—
104C2×Dic122—21no—
105C2×Dic122#4112yes—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—22no—
112C42×C61—21no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—42no—
117C42×C61—42no—
118C42×C61#4112yes—
119C42×C61#4112yes—
120#11758198—28no—
121#11758198—44no—
122#11758198—42no—
123C4⋊Dic61—22no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131#11758196—44no—
132#11758196—44no—
133#11758196—42no—
134C4×Dic61—42no—
135C23×C61C22×D411yescharacteristic
136C22×D61—22no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—42no—
140C22×D61—42no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61#12412yes—
150C2×C4×C61#12412yes—
151C2×C4×C61C2×#4111yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C2×C4×C61C2×#4111yescharacteristic
155C2×C4×C61H(5,𝔽2)11yescharacteristic
156C2×C4×C612−1+411yescharacteristic
157C2×Dic61—22no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170D244—22no—
171D244—22no—
172D244—41no—
173D244—41no—
174C4×D61—84no—
175Dic122—22no—
176Dic122—22no—
177Dic122—41no—
178Dic122—41no—
179C8×C61—42no—
180C8×C61—42no—
181#29128—611nomaximal, Sylow
182C22×C61#52511yescharacteristic
183C22×C61C2×#12411yescharacteristic
184C22×C61#53711yescharacteristic
185C22×C61#57512yes—
186C22×C61#57512yes—
187C2×D61—42no—
188C2×D61—44no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—82no—
192C4×C61—22no—
193C4×C61—22no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61#55211yescharacteristic
197C4×C61#55211yescharacteristic
198C4×C61#55311yescharacteristic
199C4×C61#55311yescharacteristic
200Dic61—42no—
201Dic61—82no—
202Dic61—81no—
203Dic61—81no—
204Dic61—82no—
205#2864—611no—
206C4×#124—611no—
207#2101—612no—
208#2042—612no—
209#1936—612no—
210#1935—612no—
211#1805—611no—
212#1805—611no—
213#1804—611no—
214#1804—611no—
215#1767D6111yescharacteristic, complemented
216C2×C61#256211yescharacteristic
217C2×C61#311112yes—
218C2×C61#313511yescharacteristic
219C2×C61#314011yescharacteristic
220C2×C61#325612yes—
221D61—82no—
222C2×#124—611no—
223C2×C4×Q8—611no—
224C2×C4×D4—611no—
225C2×C4⋊C8C2×D6111yescharacteristic
226C2×C4⋊C8C2×D6111yescharacteristic
227C2×#103—612no—
228C2×#103—612no—
229C2×#102—612no—
230C2×#102—612no—
231C2×#97C2×D6112yes—
232#391—611no—
233#390—611no—
234#389—612no—
235#389—612no—
236#388—1222no—
237#387—1222no—
238#387—1222no—
239#382—611no—
240#382—611no—
241#382—611no—
242C4×C22⋊C4—1222no—
243C43C2×D6111yescharacteristic
244#340—612no—
245#340—612no—
246#340—612no—
247#340—612no—
248#336C2×D6112yes—
249C61#2912811yescharacteristic, complemented, Sylow
250C22×Q8—611no—
251C22×D4—611no—
252C23×C4—1221no—
253C22×C8C22×D6112yes—
254C22×C8C22×D6112yes—
255#124—1222no—
256#124—1222no—
257C4×Q8—1222no—
258C4×Q8—1222no—
259C4×D4—1222no—
260C4×D4—1222no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—612no—
264C2×C4⋊C4—612no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1222no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1222no—
271C2×C42—1221no—
272C2×C42D24412yes—
273C2×C42D24412yes—
274C2×C42C22×D6111yescharacteristic
275C2×C42C22×D6111yescharacteristic
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—22no—
278C4⋊C8—22no—
279#103—1224no—
280#103—2442no—
281#102—1224no—
282#102—2442no—
283#97—42no—
284#95—1222no—
285#95—1222no—
286#95—1221no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290#95—1222no—
291#95—1222no—
292C24—1221no—
293C2×Q8—612no—
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—1221no—
299C2×D4—612no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—1221no—
310C22×C4—2444no—
311C22×C4—2442no—
312C22×C4C2×D24412yes—
313C22×C4C2×D24412yes—
314C22×C4C2×D24411yescharacteristic
315C22×C4C2×D24411yescharacteristic
316C22×C4C23×D6111yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C4⋊C4—1222no—
324C4⋊C4—1222no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4—1222no—
330C4⋊C4—1222no—
331C4⋊C4—2444no—
332C4⋊C4—2444no—
333C4⋊C4—2442no—
334C22⋊C4—1228no—
335C22⋊C4—2444no—
336C22⋊C4—2442no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—42no—
343C42—2442no—
344C42#1175822212yes—
345C42#1175822212yes—
346C23—1222no—
347C23—2441no—
348C23—2442no—
349C23—2442no—
350C23—2441no—
351C23C22×D24411yescharacteristic
352Q8—1222no—
353Q8—1222no—
354Q8—2441no—
355Q8—2441no—
356D4—1222no—
357D4—1222no—
358D4—2441no—
359D4—2441no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—1222no—
369C2×C4—4884no—
370C2×C4—2442no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4#1451976512yes—
383C2×C4#1451976512yes—
384C2×C4#41×D6111yescharacteristic
385C2×C4#41×D6111yescharacteristic
386C2×C4C2×#1175822211yescharacteristic
387C2×C4C2×#1175822211yescharacteristic
388C2×C4#1451982211yescharacteristic
389C2×C4#1451982511yescharacteristic
390C8—42no—
391C8—42no—
392C22—4881no—
393C22—4882no—
394C22—4881no—
395C22—2444no—
396C22—2442no—
397C22#1684027411yescharacteristic
398C22C2×#1451976511yescharacteristic
399C22#1684034311yescharacteristic
400C22#1684056212yes—
401C22#1684056212yes—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—42no—
406C4—4881no—
407C4—4882no—
408C4—4882no—
409C4—4881no—
410C4—2442no—
411C4#1684045311yescharacteristic
412C4#1684045311yescharacteristic
413C4#1684046211yescharacteristic
414C4#1684046211yescharacteristic
415C2—4882no—
416C2#2145098011yescharacteristic
417C2#2145593012yes—
418C2#2145616511yescharacteristic
419C2#2145620611yescharacteristic
420C2#2145722012yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1767C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1767D61
hypocenterC61#29128
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
1. #21454206C2semidirect
3. #21447388C2semidirect
6. #21446509C2semidirect
7. #21445835C2semidirect
8. #21445835C2semidirect
11. C61×#1767C2semidirect
215. #1767D61semidirect
249. C61#29128semidirect

Maximal subgroups

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

subgroupindex
#214542062
C4×#145197652
#214473882
#214470612
#214465162
#214465092
#214458352
#214458302
C61×#17672
#2912861