Finite Group Databasebeta

← #29834821

Subgroups — #29834821

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455398C211yesmaximal, characteristic
2C2×#107×D61C211yesmaximal, characteristic, complemented
3#21448022C211yesmaximal, characteristic, complemented
4#21448021C211yesmaximal, characteristic, complemented
5#21448018C212yesmaximal, complemented
6#21448002C211yesmaximal, characteristic, complemented
7#21447998C212yesmaximal, complemented
8#21447606C212yesmaximal, complemented
9#21447528C212yesmaximal, complemented
10#21447428C211yesmaximal, characteristic, complemented
11C61×#2233C211yesmaximal, characteristic, complemented
12C2×#14519796C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840272—21no—
15C2×C4⋊C4×D61—21no—
16C2×#14519716C2212yes—
17#16840004C2211yescharacteristic
18#16840003C2211yescharacteristic
19#16839995C2211yescharacteristic
20#16839990—21no—
21#16839989—21no—
22#16839799C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519710—21no—
25#107×D61—21no—
26#107×D61—21no—
27#107×D61C2212yes—
28#107×D61C2212yes—
29#16839755—21no—
30#16839589C2212yescomplemented
31#16839588C2212yescomplemented
32#16839587C2212yescomplemented
33#16839567C2212yescomplemented
34#16839565C2212yescomplemented
35#16839563C2212yescomplemented
36C22⋊#11758185C2211yescharacteristic
37#16839541C2212yes—
38C61×#466C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#367C2212yescomplemented
41C61×#357C2212yescomplemented
42C61×C22⋊C16C2211yescharacteristic
43C2×#11758224—21no—
44C22×C4×D61—41no—
45C22×C4×D61D411yescharacteristic
46#14519796—24no—
47C2×Dic244C2311yescharacteristic
48C2×#11758200—21no—
49C22×#92418D411yescharacteristic
50C2×C8×D61—24no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61D411yescharacteristic
53C2×C8×D61C2311yescharacteristic
54C2×C8×D61C2311yescharacteristic
55#14519759—41no—
56#14519757—21no—
57#14519756—21no—
58#14519748—21no—
59#14519745—41no—
60D4×Dic61—41no—
61#14519737—41no—
62#14519735—21no—
63C2×C4⋊Dic61—21no—
64C2×#11758196—41no—
65C4⋊C4×D61—22no—
66C4⋊C4×D61—22no—
67C4⋊C4×D61—41no—
68C4⋊C4×D61—41no—
69#14519716—24no—
70C2×#11758185C2312yes—
71#14519711—21no—
72#14519711C2311yescharacteristic
73#14519711C2312yes—
74#14519710—22no—
75#14519710—22no—
76#14519708—21no—
77#14519702—21no—
78#14519700—21no—
79#14519698—21no—
80C2×D8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82#121×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2311yescharacteristic
87#107×C61C2312yes—
88#102×C61—21no—
89C23×D61D811yescharacteristic
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98C22×Dic61D811yescharacteristic
99#11758224—42no—
100#11758224—42no—
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61—42no—
104C2×C4×D61—42no—
105C2×C4×D61—84no—
106C2×C4×D61D811yescharacteristic
107C2×C4×D61D811yescharacteristic
108C2×C4×D61C2×D411yescharacteristic
109C2×C4×D61C2×D411yescharacteristic
110C2×Dic122—21no—
111D8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2411yescharacteristic
115C4⋊C4×C61—21no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊Dic61—41no—
120C22⋊Dic61—41no—
121#11758200—42no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128#11758196—41no—
129#11758196—42no—
130#11758196—42no—
131C4×Dic61—41no—
132C2×#92418—22no—
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135Dic244—22no—
136C8×D61—22no—
137C8×D61—22no—
138C8×D61—22no—
139C8×D61#4114yes—
140C16×C61—22no—
141#11758185—22no—
142C23×C61—41no—
143C23×C61C2×D811yescharacteristic
144C22×D61—22no—
145C22×D61—22no—
146C22×D61—22no—
147C22×D61C2×D811yescharacteristic
148C22×D61C2×D811yescharacteristic
149D4×C61—41no—
150D4×C61—41no—
151D4×C61—82no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—82no—
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—82no—
160C22⋊D61—82no—
161C2×Dic61—22no—
162C2×Dic61—41no—
163C2×Dic61—42no—
164C2×Dic61—41no—
165C2×Dic61—82no—
166C2×Dic61—82no—
167C2×Dic61C2×D811yescharacteristic
168C2×Dic61C2×D811yescharacteristic
169C4×D61—22no—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—22no—
173C4×D61—84no—
174C4×D61—84no—
175C4×D61#12312yes—
176C4×D61#13712yes—
177Dic122—41no—
178Dic122—41no—
179C8×C61—21no—
180C8×C61C2×#4112yes—
181#92418—21no—
182#92418#12312yes—
183#33646—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61#51012yes—
190C22×C61C22×D811yescharacteristic
191C2×D61—28no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61#48012yes—
197C2×D61#50912yes—
198C4×C61—21no—
199C4×C61—82no—
200C4×C61—81no—
201C4×C61C2×#12311yescharacteristic
202C4×C61C2×#13711yescharacteristic
203Dic61—21no—
204Dic61—81no—
205Dic61—82no—
206Dic61—81no—
207Dic61#48012yes—
208#3050—611no—
209C22×#107—611no—
210#2236—611no—
211#2236—611no—
212#2235—612no—
213#2233—611no—
214#2233D6111yescharacteristic, complemented
215#2232—612no—
216#2159—612no—
217#2139—612no—
218#2112—611no—
219C2×C61—22no—
220C2×C61—161no—
221C2×C61C2×#48011yescharacteristic
222C2×C61C2×#50911yescharacteristic
223C2×C61C2×#51011yescharacteristic
224D61—22no—
225D61—22no—
226C2×#135—611no—
227C23×C8—611no—
228#525—1221no—
229C22×C4⋊C4—1221no—
230C2×M32—612no—
231#467—611no—
232#467—611no—
233#466—611no—
234#466C2×D6111yescharacteristic
235#465—1221no—
236#465—1221no—
237#427—611no—
238C2×#107—611no—
239C2×#107—612no—
240C2×#107—612no—
241C2×#107—1221no—
242C2×#107—1221no—
243C2×#107—1221no—
244C2×#107C2×D6111yescharacteristic
245#416—1221no—
246#367—612no—
247#367C2×D6112yescomplemented
248#366—612no—
249#366—612no—
250#358—612no—
251#358—612no—
252#357—612no—
253#357C2×D6112yescomplemented
254C22⋊C16—611no—
255C22⋊C16—612no—
256C22⋊C16C2×D6111yescharacteristic
257C61#3364611yescharacteristic, complemented, Sylow
258C2×#41—1221no—
259C23×C4—611no—
260C23×C4—2441no—
261#135—1224no—
262C2×Q16—611no—
263C2×SD16—1221no—
264C2×D8C22×D6111yescharacteristic
265C22×C8—611no—
266C22×C8—611no—
267C22×C8—611no—
268C22×C8—611no—
269C22×C8—611no—
270C22×C8—1224no—
271C22×C8C22×D6111yescharacteristic
272#123—2441no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275#121—21no—
276#121—1221no—
277#120—2441no—
278C4×D4—2441no—
279C4×D4—2441no—
280#117—1221no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4—2441no—
286C2×C4⋊C4—2441no—
287C2×C4⋊C4—2441no—
288M32—1224no—
289C2×C16—612no—
290C2×C16C22×D6112yes—
291#107—21no—
292#107—612no—
293#107—611no—
294#107—1222no—
295#107—1222no—
296#107—1221no—
297#107C22×D6112yes—
298#107C22×D6111yescharacteristic
299#106—1221no—
300#103—1221no—
301#103—1221no—
302#102—21no—
303#102—1221no—
304C24—611no—
305#41—2442no—
306#41—2442no—
307C2×Q8—1221no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—2441no—
311C2×D4—2441no—
312C22×C4—41no—
313C22×C4—611no—
314C22×C4—611no—
315C22×C4—611no—
316C22×C4—611no—
317C22×C4—611no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—4884no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4D4×D6111yescharacteristic
326Q16—1222no—
327SD16—2442no—
328D8—22no—
329C2×C8—22no—
330C2×C8—611no—
331C2×C8—614no—
332C2×C8—611no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1222no—
336C2×C8—1222no—
337C2×C8D4×D6111yescharacteristic
338C2×C8C23×D6111yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—1221no—
343C4⋊C4—1222no—
344C4⋊C4—2442no—
345C4⋊C4—2442no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C22⋊C4—41no—
350C22⋊C4—2441no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C22⋊C4—2441no—
354C42—2441no—
355C16—22no—
356C16—1222no—
357C23—41no—
358C23—611no—
359C23—611no—
360C23—1222no—
361C23—1222no—
362C23—1222no—
363C23D8×D6111yescharacteristic
364Q8—2441no—
365Q8—2441no—
366D4—41no—
367D4—41no—
368D4—82no—
369D4—4882no—
370D4—4882no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—611no—
377C2×C4—612no—
378C2×C4—612no—
379C2×C4—611no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—1222no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—4884no—
386C2×C4—4884no—
387C2×C4—4882no—
388C2×C4—4882no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4D8×D6111yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C8—21no—
395C8—612no—
396C8—1221no—
397C8#41×D6112yes—
398C22—21no—
399C22—21no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—612no—
404C22—612no—
405C22—1228no—
406C22—1221no—
407C22—1221no—
408C22—1221no—
409C22—1221no—
410C22#1684020112yes—
411C22C2×D8×D6111yescharacteristic
412C4—21no—
413C4—81no—
414C4—82no—
415C4—612no—
416C4—1221no—
417C4—4881no—
418C4—4882no—
419C4—4881no—
420C4#123×D6111yescharacteristic
421C4#137×D6111yescharacteristic
422C2—22no—
423C2—161no—
424C2—1222no—
425C2—1222no—
426C2#480×D6111yescharacteristic
427C2#509×D6111yescharacteristic
428C2C2×#1684020111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2233C2
socleC22×C61C22×D8
prime radical#21455398C2
hypercenter#2233D61
hypocenterC61#33646
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
2. C2×#107×D61C2semidirect
3. #21448022C2semidirect
4. #21448021C2semidirect
5. #21448018C2semidirect
6. #21448002C2semidirect
7. #21447998C2semidirect
8. #21447606C2semidirect
9. #21447528C2semidirect
10. #21447428C2semidirect
11. C61×#2233C2semidirect
30. #16839589C22semidirect
31. #16839588C22semidirect
32. #16839587C22semidirect
33. #16839567C22semidirect
34. #16839565C22semidirect
35. #16839563C22semidirect
40. C61×#367C22semidirect
41. C61×#357C22semidirect
214. #2233D61semidirect
247. #367C2×D61semidirect
253. #357C2×D61semidirect
257. C61#33646semidirect

Maximal subgroups

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

subgroupindex
#214553982
C2×#107×D612
#214480222
#214480212
#214480182
#214480022
#214479982
#214476062
#214475282
#214474282
C61×#22332
#3364661