Finite Group Databasebeta

← #29779566

Subgroups — #29779566

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454124C211yesmaximal, characteristic
2#391×D61C211yesmaximal, characteristic
3#21447239C211yesmaximal, characteristic, complemented
4#21447237C211yesmaximal, characteristic, complemented
5#21446967C211yesmaximal, characteristic, complemented
6#21446507C212yesmaximal, complemented
7#21446506C212yesmaximal, complemented
8#21445908C212yesmaximal
9#21445752C212yesmaximal
10#21445580C211yesmaximal, characteristic
11C61×#2026C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×C4⋊C4×D61C2211yescharacteristic
14C2×C42×D61C2211yescharacteristic
15C2×#14519702C2212yes—
16C2×#14519701C2212yes—
17C2×#14519700C2212yes—
18C2×#14519689C2212yes—
19C2×#14519687C2212yes—
20#16839669C2211yescharacteristic
21#16839668C2211yescharacteristic
22#16839667C2211yescharacteristic
23#16839666—22no—
24#16839658C2212yes—
25#16839643—22no—
26#16839633—22no—
27#16839628C2211yescharacteristic
28#16839619C2212yes—
29#95×D61—22no—
30#16839525C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2211yescharacteristic
34C2×#125×C61C2211yescharacteristic
35C2×#103×C61C2212yes—
36C61×#391C2211yescharacteristic
37C61×#340C2212yes—
38C61×#336C2211yescharacteristic
39C22×Dic122—21no—
40C22×Dic122C2311yescharacteristic
41C2×Q8×D61—22no—
42C2×Q8×D61—22no—
43C2×Q8×D61—24no—
44C2×Q8×D61—42no—
45C22×C4×D61—21no—
46C22×C4×D61—21no—
47C22×C4×D61D412yes—
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2312yes—
52C2×#11758196—21no—
53C2×#11758196—22no—
54C4⋊C4×D61—24no—
55C2×#11758198—22no—
56C2×C4×Dic61D412yes—
57C2×C4×Dic61C2311yescharacteristic
58C42×D61—42no—
59#14519702—24no—
60#14519701—24no—
61#14519700—42no—
62#14519689—24no—
63#14519687—42no—
64#14519684—22no—
65#14519684—22no—
66#14519684—22no—
67#14519684—22no—
68#14519684—22no—
69#14519684—22no—
70C22×Q8×C61C2311yescharacteristic
71C22×C8×C61C2312yes—
72#125×C61—42no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61C2311yescharacteristic
75C2×C4⋊C4×C61C2312yes—
76C2×C42×C61C2311yescharacteristic
77#103×C61—42no—
78#95×C61—22no—
79C23×D61C2×D411yescharacteristic
80C2×Q8×C61—21no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61#4112yes—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2411yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—22no—
92C22×Dic61C2×D412yes—
93C22×Dic61C2×D411yescharacteristic
94Q8×D61—48no—
95Q8×D61—48no—
96C2×C4×D61—21no—
97C2×C4×D61—21no—
98C2×C4×D61—22no—
99C2×C4×D61—24no—
100C2×C4×D61—24no—
101C2×C4×D61—42no—
102C2×C4×D61—42no—
103C2×C4×D61—42no—
104C2×C4×D61—42no—
105C2×C4×D61—44no—
106C2×C4×D61—42no—
107C2×C4×D61#4114yes—
108C2×Dic122—22no—
109C2×Dic122—22no—
110C2×Dic122—24no—
111C2×Dic122—21no—
112C2×Dic122—42no—
113C2×Dic122—42no—
114C2×Dic122#4112yes—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—44no—
122C42×C61—22no—
123#11758198—44no—
124C4⋊Dic61—22no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127#11758196—44no—
128#11758196—42no—
129C4×Dic61—22no—
130C4×Dic61—42no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C23×C61C22×D411yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—22no—
138C22×D61#12412yes—
139C22×D61#13714yes—
140Q8×C61—24no—
141Q8×C61—42no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61C2×#4111yescharacteristic
154C2×C4×C61C2×#4112yes—
155C2×C4×C61C2×#4111yescharacteristic
156C2×C4×C61H(5,𝔽2)12yes—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61#12412yes—
171C4×D61—24no—
172C4×D61—24no—
173C4×D61—48no—
174C4×D61—48no—
175C4×D61—84no—
176Dic122—24no—
177Dic122—48no—
178Dic122—42no—
179C8×C61—42no—
180#92418—42no—
181#29112—611nomaximal, Sylow
182C22×C61#52511yescharacteristic
183C22×C61C2×#12411yescharacteristic
184C22×C61#53411yescharacteristic
185C22×C61C2×#13712yes—
186C22×C61#57512yes—
187C2×D61—22no—
188C2×D61—24no—
189C2×D61—22no—
190C2×D61—24no—
191C2×D61#49114yes—
192C4×C61—42no—
193C4×C61—42no—
194C4×C61—81no—
195C4×C61—82no—
196C4×C61#55112yes—
197C4×C61#55112yes—
198Dic61—22no—
199Dic61—42no—
200Dic61—42no—
201Dic61—82no—
202Dic61—81no—
203#2851—611no—
204C2×#391—611no—
205#2074—611no—
206#2074—611no—
207#2026—611no—
208#2026D6111yescharacteristic, complemented
209#1935—612no—
210#1935—612no—
211#1824—612no—
212#1787—612no—
213#1749—611no—
214C2×C61#256111yescharacteristic
215C2×C61#311712yes—
216C2×C61C2×#49111yescharacteristic
217C2×C61#314011yescharacteristic
218C2×C61#319612yes—
219D61—24no—
220C23×Q8—611no—
221C2×#125C2×D6111yescharacteristic
222C22×C4⋊C4—611no—
223C22×C42—611no—
224C2×#103—612no—
225C2×#103—612no—
226C2×#103—612no—
227C2×#103C2×D6112yes—
228C2×C22⋊C8—612no—
229C2×#97—612no—
230#391—611no—
231#391—611no—
232#391—611no—
233#391—1222no—
234#391C2×D6111yescharacteristic
235#389—612no—
236#387—1222no—
237#385—1222no—
238#382—611no—
239#380—612no—
240C2×#95—1222no—
241#340—612no—
242#340—612no—
243#340—612no—
244#340C2×D6112yes—
245#336—611no—
246#336C2×D6111yescharacteristic
247C61#2911211yescharacteristic, complemented, Sylow
248C22×Q8—611no—
249C22×Q8—1221no—
250C22×Q8—1224no—
251C22×Q8—1222no—
252C22×Q8—1222no—
253C22×Q8—2442no—
254C22×Q8C22×D6111yescharacteristic
255C23×C4—611no—
256C23×C4—612no—
257C23×C4—1221no—
258C23×C4—1221no—
259C22×C8—612no—
260C22×C8C22×D6112yes—
261#125—42no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—612no—
265C2×C4⋊C4—1224no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1222no—
268C2×C4⋊C4C22×D6111yescharacteristic
269C2×C4⋊C4C22×D6112yes—
270C2×C22⋊C4—1222no—
271C2×C42—611no—
272C2×C42—612no—
273C2×C42—2442no—
274C2×C42C22×D6111yescharacteristic
275#103—42no—
276#103—1224no—
277#103—1224no—
278#103—2442no—
279C22⋊C8—1224no—
280#97—2442no—
281#95—22no—
282#95—1222no—
283#95—1222no—
284#95—1222no—
285#95—1222no—
286#95—1222no—
287#95—1222no—
288C24—611no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—612no—
293C2×Q8—1224no—
294C2×Q8—1221no—
295C2×Q8—1222no—
296C2×Q8—1222no—
297C2×Q8—2442no—
298C2×Q8—2442no—
299C2×Q8—2448no—
300C2×Q8—2448no—
301C2×Q8#1175822612yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—611no—
306C22×C4—614no—
307C22×C4—612no—
308C22×C4—1224no—
309C22×C4—1224no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1222no—
313C22×C4—1221no—
314C22×C4—1222no—
315C22×C4—1221no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2444no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4D4×D6112yes—
323C22×C4C23×D6111yescharacteristic
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—1222no—
328C2×C8—1222no—
329C2×C8—1222no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—1222no—
335C4⋊C4—1222no—
336C4⋊C4—1222no—
337C4⋊C4—2442no—
338C4⋊C4—2444no—
339C22⋊C4—2444no—
340C42—22no—
341C42—1222no—
342C42—2442no—
343C23—612no—
344C23—614no—
345C23—1221no—
346C23—1222no—
347C23—1221no—
348C23C2×D4×D6111yescharacteristic
349Q8—24no—
350Q8—42no—
351Q8—1224no—
352Q8—2448no—
353Q8—2442no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—612no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—1224no—
369C2×C4—1224no—
370C2×C4—1222no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—4884no—
375C2×C4—2448no—
376C2×C4—2448no—
377C2×C4—2442no—
378C2×C4—2441no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4#41×D6112yes—
385C2×C4#41×D6111yescharacteristic
386C2×C4C2×#1175822611yescharacteristic
387C2×C4#1451982212yes—
388C8—42no—
389C8—2442no—
390C22—614no—
391C22—1224no—
392C22—1222no—
393C22—1224no—
394C22—1222no—
395C22#1684027511yescharacteristic
396C22#124×D6111yescharacteristic
397C22#1684032211yescharacteristic
398C22#137×D6112yes—
399C22#1684055912yes—
400C4—42no—
401C4—42no—
402C4—82no—
403C4—81no—
404C4—1222no—
405C4—4881no—
406C4—4882no—
407C4—2442no—
408C4—2442no—
409C4#1684043912yes—
410C4#1684044212yes—
411C2—1224no—
412C2#2145097311yescharacteristic
413C2#2145599512yes—
414C2#491×D6111yescharacteristic
415C2#2145620211yescharacteristic
416C2#2145664712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2026C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2026D61
hypocenterC61#29112
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2026

Complements

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

NH ≅ G/Ntype
3. #21447239C2semidirect
4. #21447237C2semidirect
5. #21446967C2semidirect
6. #21446507C2semidirect
7. #21446506C2semidirect
11. C61×#2026C2semidirect
208. #2026D61semidirect
247. C61#29112semidirect

Maximal subgroups

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

subgroupindex
#214541242
#391×D612
#214472392
#214472372
#214469672
#214465072
#214465062
#214459082
#214457522
#214455802
C61×#20262
#2911261