Finite Group Databasebeta

← #29834820

Subgroups — #29834820

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455396C211yesmaximal, characteristic
2C2×#107×D61C211yesmaximal, characteristic, complemented
3#21448020C212yesmaximal, complemented
4#21448015C212yesmaximal, complemented
5#21448003C212yesmaximal, complemented
6#21447998C212yesmaximal, complemented
7#21447607C211yesmaximal, characteristic, complemented
8#21447605C211yesmaximal, characteristic, complemented
9#21447531C211yesmaximal, characteristic, complemented
10#21447429C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C22×C8×D61C2211yescharacteristic
13#16840269—21no—
14C2×C4⋊C4×D61—21no—
15C2×#14519717C2211yescharacteristic
16#16840000C2211yescharacteristic
17#16840000C2211yescharacteristic
18#16839995C2211yescharacteristic
19#16839995C2211yescharacteristic
20#16839988—21no—
21#16839988—21no—
22#16839800C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519710—21no—
25#107×D61—21no—
26#107×D61—21no—
27#107×D61C2212yes—
28#107×D61C2212yes—
29#16839758—21no—
30#16839590C2212yescomplemented
31#16839589C2212yescomplemented
32#16839588C2212yescomplemented
33#16839587C2212yescomplemented
34#16839566C2212yescomplemented
35#16839563C2212yescomplemented
36#16839541C2212yes—
37#16839541C2212yes—
38C2×#135×C61C2211yescharacteristic
39C2×M32×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#358C2212yescomplemented
42C61×#357C2212yescomplemented
43C22×C4×D61—41no—
44C22×C4×D61D411yescharacteristic
45C22×#92418D411yescharacteristic
46C2×C8×D61—22no—
47C2×C8×D61—22no—
48C2×C8×D61D411yescharacteristic
49C2×C8×D61D411yescharacteristic
50C2×C8×D61C2311yescharacteristic
51C2×C8×D61C2311yescharacteristic
52#14519758—41no—
53#14519753—21no—
54#14519753—21no—
55#14519748—21no—
56#14519748—21no—
57#14519745—41no—
58C4×C22⋊D61—41no—
59C4×C22⋊D61—41no—
60#14519736—21no—
61C2×C4⋊Dic61—21no—
62C2×#11758196—41no—
63C4⋊C4×D61—22no—
64C4⋊C4×D61—22no—
65C4⋊C4×D61—41no—
66C4⋊C4×D61—41no—
67#14519717—22no—
68C2×#11758185C2312yes—
69#14519711—21no—
70#14519711C2311yescharacteristic
71#14519711C2311yescharacteristic
72#14519711C2312yes—
73#14519710—22no—
74#14519710—22no—
75#14519709—21no—
76#14519701—21no—
77#14519701—21no—
78#14519698—21no—
79#14519698—21no—
80C2×#41×C61—21no—
81#135×C61—22no—
82#135×C61—22no—
83C2×Q16×C61C2311yescharacteristic
84C2×SD16×C61—21no—
85C2×D8×C61C2311yescharacteristic
86C22×C8×C61C2311yescharacteristic
87C2×C4⋊C4×C61—21no—
88M32×C61—22no—
89C2×C16×C61C2312yes—
90#107×C61—21no—
91#107×C61C2312yes—
92C23×D61D811yescharacteristic
93#41×C61—42no—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C22×Dic61—41no—
104C22×Dic61D811yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—42no—
108C2×C4×D61—42no—
109C2×C4×D61—84no—
110C2×C4×D61D811yescharacteristic
111C2×C4×D61D811yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113C2×C4×D61C2×D411yescharacteristic
114Q16×C61—21no—
115Q16×C61—21no—
116SD16×C61—42no—
117D8×C61—21no—
118D8×C61—21no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61#4112yes—
121C2×C8×C61#4112yes—
122C2×C8×C61C2411yescharacteristic
123C4⋊C4×C61—22no—
124C4⋊C4×C61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128#11758198—41no—
129#11758198—41no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—41no—
134#11758196—42no—
135#11758196—42no—
136#11758196—41no—
137#11758196—41no—
138C4×Dic61—41no—
139C2×#92418—22no—
140C2×#92418C2×D411yescharacteristic
141C2×#92418C2×D411yescharacteristic
142C8×D61—22no—
143C8×D61—22no—
144C8×D61—21no—
145C8×D61—21no—
146C8×D61—21no—
147C8×D61—21no—
148C16×C61—22no—
149#11758185—22no—
150C23×C61—41no—
151C23×C61C2×D811yescharacteristic
152C22×D61—22no—
153C22×D61—22no—
154C22×D61—22no—
155C22×D61C2×D811yescharacteristic
156C22×D61C2×D811yescharacteristic
157Q8×C61—41no—
158Q8×C61—41no—
159D4×C61—41no—
160D4×C61—41no—
161D4×C61—82no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
165C2×C4×C61—82no—
166C2×C4×C61#12312yes—
167C2×C4×C61C2×D811yescharacteristic
168C2×C4×C61#13712yes—
169C2×C4×C61C22×D411yescharacteristic
170C22⋊D61—82no—
171C22⋊D61—82no—
172C2×Dic61—22no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—82no—
177C2×Dic61C2×D811yescharacteristic
178C2×Dic61C2×D811yescharacteristic
179C4×D61—21no—
180C4×D61—21no—
181C4×D61—21no—
182C4×D61—21no—
183C4×D61—22no—
184C4×D61—22no—
185C4×D61—84no—
186C4×D61—84no—
187C8×C61#12312yes—
188C8×C61C2×#4111yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—21no—
191#92418—21no—
192#33646—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61#48012yes—
199C22×C61C22×D811yescharacteristic
200C2×D61—28no—
201C2×D61—21no—
202C2×D61—21no—
203C2×D61—21no—
204C2×D61—21no—
205C2×D61#50912yes—
206C2×D61#51012yes—
207C4×C61—81no—
208C4×C61—82no—
209C4×C61#48012yes—
210C4×C61C2×#12311yescharacteristic
211C4×C61C2×#13711yescharacteristic
212Dic61—21no—
213Dic61—21no—
214Dic61—82no—
215Dic61—81no—
216Dic61—81no—
217#3050—611no—
218C22×#107—611no—
219#2236—612no—
220#2235—612no—
221#2233—612no—
222#2232—612no—
223#2159—611no—
224#2159—611no—
225#2139—611no—
226#2139D6111yescharacteristic, complemented
227#2112—611no—
228C2×C61—22no—
229C2×C61—161no—
230C2×C61C2×#48011yescharacteristic
231C2×C61C2×#50911yescharacteristic
232C2×C61C2×#51011yescharacteristic
233D61—22no—
234D61—22no—
235C2×#135C2×D6111yescharacteristic
236C23×C8—611no—
237#525—1221no—
238C22×C4⋊C4—1221no—
239C2×M32—611no—
240C2×M32C2×D6111yescharacteristic
241#467—611no—
242#467—611no—
243#466—611no—
244#466—611no—
245#465—1221no—
246#465—1221no—
247#427—611no—
248C2×#107—611no—
249C2×#107—612no—
250C2×#107—612no—
251C2×#107—1221no—
252C2×#107—1221no—
253C2×#107—1221no—
254C2×#107C2×D6111yescharacteristic
255#416—1221no—
256#367—612no—
257#367—612no—
258#366—612no—
259#366—612no—
260#358—612no—
261#358C2×D6112yescomplemented
262#357—612no—
263#357C2×D6112yescomplemented
264C22⋊C16—612no—
265C22⋊C16—612no—
266C61#3364611yescharacteristic, complemented, Sylow
267C2×#41—21no—
268C23×C4—611no—
269C23×C4—2441no—
270#135—22no—
271#135—22no—
272C2×Q16C22×D6111yescharacteristic
273C2×SD16—21no—
274C2×D8C22×D6111yescharacteristic
275C22×C8—611no—
276C22×C8—611no—
277C22×C8—611no—
278C22×C8—611no—
279C22×C8—611no—
280C22×C8—1222no—
281C22×C8—1222no—
282C22×C8C22×D6111yescharacteristic
283#123—2441no—
284C22⋊Q8—1221no—
285C22⋊Q8—1221no—
286#121—1221no—
287#121—1221no—
288#120—2441no—
289C4×D4—2441no—
290C4×D4—2441no—
291#117—1221no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—1222no—
294C2×C4⋊C4—1222no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—2441no—
297C2×C4⋊C4—2441no—
298C2×C4⋊C4—2441no—
299M32—22no—
300M32—1222no—
301C2×C16—612no—
302C2×C16C22×D6112yes—
303#107—21no—
304#107—612no—
305#107—611no—
306#107—611no—
307#107—1222no—
308#107—1222no—
309#107—1221no—
310#107C22×D6112yes—
311#106—1221no—
312#103—1221no—
313#103—1221no—
314#102—1221no—
315#102—1221no—
316C24—611no—
317#41—42no—
318#41—42no—
319C2×Q8—21no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4—2441no—
323C2×D4—2441no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—611no—
331C22×C4—1222no—
332C22×C4—1222no—
333C22×C4—4884no—
334C22×C4—2442no—
335C22×C4—2442no—
336C22×C4—2441no—
337C22×C4D4×D6111yescharacteristic
338Q16—21no—
339Q16—21no—
340SD16—42no—
341D8—21no—
342D8—21no—
343C2×C8—611no—
344C2×C8—611no—
345C2×C8—1222no—
346C2×C8—1221no—
347C2×C8—1221no—
348C2×C8—1221no—
349C2×C8—1221no—
350C2×C8—1222no—
351C2×C8—1222no—
352C2×C8#1175822212yes—
353C2×C8D4×D6111yescharacteristic
354C2×C8#1175822412yes—
355C2×C8C23×D6111yescharacteristic
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1222no—
361C4⋊C4—2442no—
362C4⋊C4—2442no—
363C4⋊C4—2441no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—2441no—
367C22⋊C4—2441no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C42—2441no—
372C16—22no—
373C16—1222no—
374C23—41no—
375C23—611no—
376C23—611no—
377C23—1222no—
378C23—1222no—
379C23—1222no—
380C23D8×D6111yescharacteristic
381Q8—41no—
382Q8—41no—
383D4—41no—
384D4—41no—
385D4—82no—
386D4—4882no—
387D4—4882no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—1222no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1222no—
400C2×C4—1222no—
401C2×C4—4884no—
402C2×C4—4884no—
403C2×C4—4882no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4#1451975912yes—
408C2×C4D8×D6111yescharacteristic
409C2×C4#1451980612yes—
410C2×C4C2×D4×D6111yescharacteristic
411C8—1221no—
412C8—1221no—
413C8#1451975912yes—
414C8C2×#1175822411yescharacteristic
415C8C2×#1175822211yescharacteristic
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—612no—
422C22—612no—
423C22—1228no—
424C22—1221no—
425C22—1221no—
426C22—1221no—
427C22—1221no—
428C22#1684007412yes—
429C22C2×D8×D6111yescharacteristic
430C4—81no—
431C4—82no—
432C4—1221no—
433C4—1221no—
434C4—4881no—
435C4—4882no—
436C4—4881no—
437C4#1684007412yes—
438C4C2×#1451975911yescharacteristic
439C4C2×#1451980611yescharacteristic
440C2—22no—
441C2—161no—
442C2—1222no—
443C2—1222no—
444C2C2×#1684007411yescharacteristic
445C2#509×D6111yescharacteristic
446C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radical#21455396C2
hypercenter#2139D61
hypocenterC61#33646
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. C2×#107×D61C2semidirect
3. #21448020C2semidirect
4. #21448015C2semidirect
5. #21448003C2semidirect
6. #21447998C2semidirect
7. #21447607C2semidirect
8. #21447605C2semidirect
9. #21447531C2semidirect
10. #21447429C2semidirect
11. C61×#2139C2semidirect
30. #16839590C22semidirect
31. #16839589C22semidirect
32. #16839588C22semidirect
33. #16839587C22semidirect
34. #16839566C22semidirect
35. #16839563C22semidirect
41. C61×#358C22semidirect
42. C61×#357C22semidirect
226. #2139D61semidirect
261. #358C2×D61semidirect
263. #357C2×D61semidirect
266. C61#33646semidirect

Maximal subgroups

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

subgroupindex
#214553962
C2×#107×D612
#214480202
#214480152
#214480032
#214479982
#214476072
#214476052
#214475312
#214474292
C61×#21392
#3364661